Research Article Nonlinear Dynamic Analysis of Plates Stiffened … · 2019. 7. 31. · Research...

Research ArticleNonlinear Dynamic Analysis of Plates Stiffened by ParallelBeams with Deformable Connection

J A Dourakopoulos and E J Sapountzakis

School of Civil Engineering National Technical University of Athens Zografou Campus GR-157 80 Athens Greece

Correspondence should be addressed to E J Sapountzakis cvsapouncentralntuagr

Received 30 May 2014 Accepted 6 August 2014 Published 14 September 2014

Academic Editor Bassam A Izzuddin

Copyright copy 2014 J A Dourakopoulos and E J Sapountzakis This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

In this paper a general solution to the geometrically nonlinear dynamic analysis of plates stiffened by arbitrarily placed parallelbeams of arbitrary doubly symmetric cross-section subjected to dynamic loading is presentedTheplate-beam structure is assumedto undergo moderate large deflections and the nonlinear analysis is carried out by retaining nonlinear terms in the kinematicalrelations According to the proposedmodel the arbitrarily placed parallel stiffening beams are isolated from the plate by sections inthe lower outer surface of the plate making the hypothesis that the plate and the beams can slip in all directions of the connectionwithout separation and taking into account the arising tractions in all directions at the fictitious interfaces These tractions areintegrated with respect to each half of the interface width resulting in two interface lines along which the loading of the beams andthe additional loading of the plate are defined Six boundary value problems are formulated and solved using the analog equationmethod (AEM) a BEM-based method Both free and forced transverse vibrations are considered and numerical examples withgreat practical interest are presented demonstrating the effectiveness wherever possible the accuracy and the range of applicationsof the proposed method

1 Introduction

Stiffened plate panels are structural elements of practicalimportance in applications such as ship superstructuresbridge decks and aircraft structures Stiffening of the plateprovides the benefit of added load-carrying capability with arelatively small additional weight penalty while buckling isprevented especially in case of inplane loading The uniquepeculiarities of the aforementioned structures are obtaineddue to the behavior of the bond between the plate and thebeams however this bond is the usual reason why thesestructures are prone to failureMoreover since these stiffenedplates are frequently subjected to dynamic loading such asair blasts or underwater explosions a clear understanding ofthe dynamic response requires development of an efficientdynamic analysis capability

When the deflections of the structure are small a widerange of linear analysis tools such as modal analysis canbe used and some analytical results are possible As the

deflections become larger the induced geometric nonlin-earities result in effects that are not observed in linearsystems making therefore the determination of an analyticalsolution extremely difficult and in most cases impossibleMoreover having in mind the importance of weight savingin engineering structures the study of nonlinear effects inthe analysis (large deflection analysis) of stiffened platesbecomes essential These nonlinearities result from retainingthe squares of the slopes in the strain-displacement relations(intermediate nonlinear theory) avoiding in this way theinaccuracies arising from a linear or a linearized second-order analysis

The linear dynamic behavior of stiffened plates has beenwidely studied employing the Rayleigh-Ritz method [1ndash4]the transfer matrix method [5] the finite difference method[6ndash9] the finite element method [10ndash12] the finite stripmethod [13] and the boundary element method (BEM)[14ndash16] To the authorsrsquo knowledge a limited amount oftechnical literature is also available on the nonlinear dynamic

Hindawi Publishing CorporationAdvances in Civil EngineeringVolume 2014 Article ID 942763 22 pageshttpdxdoiorg1011552014942763

2 Advances in Civil Engineering

analysis of stiffened plate systems According to this theintegral equation method has been applied for the largedeflection analysis of clamped laterally loaded skew plateswith stiffener parallel to the skew directions [17] Moreoverthe finite element method has been widely used for thefree large-amplitude flexural vibration of stiffened [18] andlaminated plates [19 20] and for forced vibrations underinstantaneous loading taking into account both geomet-ric and material nonlinearities [21] The spline finite stripmethod has also been applied for the examination of non-linear transient vibration of stiffened plates [22] and theGalerkin method has been employed for the solution ofdiscrete equations derived after the application of Fourierseries for the deflection of the plate [23 24] Neverthelessin all of the aforementioned research efforts restrictions areencountered arising from the ignorance of the tractionsin the fictitious plate-beams interfaces of the nonuniformdistribution of the interface transverse shear force or of thenonuniform torsional response of the beams

The behavior of composite slab-and-beam structure isaffected significantly by the deformability of the shear con-nection Therefore much research effort has been done con-cerning the linear dynamic analysis of composite beams withdeformable connectionsWu et al examined the free vibrationof partial interaction composite beams with axial force andproposed an approximate simple expression to predict thefundamental frequency of the partial-interaction compositemembers with axial force [26] Xu and Wu investigated thestatic dynamic and buckling behavior of partial-interactioncomposite members by taking into account the influences ofrotary inertia and shear deformations [27] Moreover theyalso employed the state-space method in order to developan exact two-dimensional plane stress model of compositebeams with interlayer slips [28 29]

On the other hand less research effort has been made onthe forced vibration of partial-interaction composite beamsGirhammar and Pan [30] investigated the dynamic behaviorof partial-interaction composite beams without axial forceand presented the orthogonality relation of vibration modesThey also obtained the analytical expressions for the transientresponse of composite beams under impulsive and step loadsAdam et al [31] decomposed the dynamic response intotwo parts the quasi-static component solved by integratingthe influence function and the complementary componentsolved by themode superposition Recently Girhammar et al[32] employed Hamiltonrsquos principle to derive the governingequations for dynamic problems of composite beams undergeneralized boundary conditions At the same time theystudied the forced vibration of composite beams by virtueof mode superposition However in the literature largedeflection analysis of partial interaction has received limitedattention and focused especially on the static behavior of thestructure [33ndash38]

In this paper a general solution to the geometricallynonlinear dynamic analysis of plates stiffened by arbitrarilyplaced parallel beams of arbitrary doubly symmetric cross-section with deformable connections subjected to arbitrarydynamic loading is presented The solution is based on thestructural model proposed by Sapountzakis and Mokos in

[39] andDourakopoulos and Sapountzakis in [40] accordingto which the stiffening beams are isolated from the plateby sections in the lower outer surface of the plate makingthe hypothesis that the plate and the beams can slip inall directions of the connection without separation (ieuplift is neglected) and taking into account the arisingtractions in all directions at the fictitious interfaces Due tothe slip relative inplane displacements between beam andplate arise assuming however that they are continuouslyin contact and taking into account the arising tractions inall directions at the fictitious interfaces These tractions areintegrated with respect to each half of the interface widthresulting in two interface lines along which the loading ofthe beams and the additional loading of the plate are definedThe unknown distribution of the aforementioned integratedtractions is established by applying continuity conditions inall directions at the two interface lines taking into accounttheir relationship with the interface slip through the shearconnector stiffness Any distribution of connectors in eachdirection of the interfaces can be handled The utilization oftwo interface lines for each beam enables the nonuniformtorsional response of the beams to be taken into account asthe angle of twist is indirectly equatedwith the correspondingplate slope Six boundary value problems are formulated andsolved using the analog equationmethod (AEM) [41] a BEM-based methodThe essential features and novel aspects of thepresent formulation are summarized as follows

(i) The adopted model permits the evaluation of theinplane shear forces (longitudinal and transverse) atthe interfaces in both directions taking into accountthe influence of interface slip the knowledge of whichis very important in the design of shear connectors inplate-beam structures

(ii) Utilization of two interface lines permits the nonuni-form distribution of the inplane forces along theinterface width to be taken into account

(iii) Both free and forced transverse vibrations are consid-ered taking into account geometric nonlinearities

(iv) The stiffened plate is of arbitrary shape and is sub-jected to arbitrary dynamic loading while both thenumber and the placement of the parallel beams arealso arbitrary (eccentric beams are included)

(v) The cross-section of the stiffening beams is an arbi-trary doubly symmetric thin or thick-walled oneTheformulation does not stand on the assumption of athin-walled structure according to which torsionaland warping rigidities can be evaluated employingclosed-form expressions without significant error aslong as the cross-section thickness is small [42]Therefore in this paper the cross-sectionrsquos torsionaland warping rigidities are evaluated ldquoexactlyrdquo in anumerical sense employing the BEM (requiring onlyboundary discretization for the cross-sectional anal-ysis)

(vi) The plate and the beams are supported by the mostgeneral boundary conditions including elastic sup-port or restraint

Advances in Civil Engineering 3

s

Corner 1t

n

(Γ1)

(Γ0)

(ΓK)li

lI Beam I

Beam i

Beam 1 l1

Corner ky p

z wp x up

bIf

bif

b1f

xi

zi

yi

Cihib

(Ω)

fij=1

fij=2

fij interface line (j = 1 2)

Γ =K

⋃j=1

Γj

Figure 1 Two-dimensional region Ω occupied by the plate

(vii) The nonuniform torsion in which the stiffeningbeams are subjected is taken into account by solvingthe corresponding problem and by comprehendingthe arising twisting andwarping in the correspondingdisplacement continuity conditions The distributedwarpingmoment arising from the nonuniform distri-bution of longitudinal inplane forces is also taken intoaccount

(viii) Contrary to previous research efforts where thenumerical analysis is based on BEM using a lumpedmass assumptionmodel after evaluating the flexibilitymatrix at the mass nodal points in this work adistributed mass model is employed

Based on the numerical solution developed several exampleswith great practical interest are presented demonstrating theeffectiveness wherever possible the accuracy and the rangeof applications of the proposed method

2 Statement of the Problem

Let us consider a thin plate of homogeneous isotropicand linearly elastic material with modulus of elasticity 119864119901volume mass density 120588119901 shear modulus 119866119901 and Poissonrsquosratio ]119901 having constant thickness ℎ119901 and occupying thetwo-dimensional multiple connected region Ω of the 119909119910

plane bounded by the piecewise Γ119895 (119895 = 0 1 2 119870)

boundary curves The plate is stiffened by a set of 119894 =

1 2 119868 arbitrarily placed parallel beams of arbitrary doublysymmetric cross-section of area119860119894

119887and length 119897

119894Thematerialof the beams is considered to be homogeneous isotropic andlinearly elastic with modulus of elasticity 119864

119894

119887 mass density 120588

119894

119887

shear modulus 119866119894119887 and Poissonrsquos ratio ]119894

119887 Therefore material

nonlinearities are not considered in the following analysisFor the sake of convenience the 119909 axis is taken parallel

to the beams of length 119897119894 which may have either internal or

boundary point supports The stiffened plate is subjected tothe arbitrary lateral dynamic load119892 = 119892(x 119905) x 119909 119910 119905 ge 0Owing to this loading the plate and the beams can slip in alldirections of the connection without separation (ie uplift isneglected) For the analysis of the aforementioned problem aglobal coordinate system119874119909119910 for the analysis of the plate andlocal coordinate ones 119862

119894119909119894119910119894 corresponding to the centroid

axes of each beam are employed (Figure 1)The solution of the problem at hand is approached

employing the improved model proposed by Sapountzakisand Mokos in [39] and Dourakopoulos and Sapountzakisin [40] According to this model the stiffening beams areisolated from the plate by sections in its lower outer surfacetaking into account the arising tractions at the fictitiousinterfaces (Figure 2) Integration of these tractions along eachhalf of the width of the 119894th beam results in line forces perunit length in all directions in two interface lines which aredenoted by 119902

119894

119909119895 119902119894119910119895 and 119902

119894

119911119895(119895 = 1 2) encountering in this

way the nonuniform distribution of the interface transverseshear forces 119902119894

119910 which in previous models [43] was ignored

The aforementioned integrated tractions result in the loadingof the 119894th beam and the additional loading of the plate Theirdistribution is unknown and can be established by imposingdisplacement continuity conditions in all directions alongthe two interface lines enabling in this way the nonuniformtorsional response of the beams to be taken into accountThus the arising additional loading (Figure 3) at the middlesurface of the plate and the loading along the centroid axis


g

aa

(a)

O

g

Ep p 120588p

hp

fij=1 fi

j=2

Ω

(Γ)

qizjqiyjx up

y p

z wp

qixj

qixj

qiyjqizj

xi uib

zi wib

hib

bif

Eib ib 120588ib

yi ib

bif4

Ci centroidSi shear center

Oi equiv Ci equiv Si

(b)

Figure 2 Thin elastic plate stiffened by beams (a) and isolation ofthe beams from the plate (b)

(coinciding with the shear center axis) of each beam can besummarized as follows

(a) In the Plate (at the Traces of the Two Interface Lines 119895 = 1 2

of the 119894th Plate-Beam Interface)

(i) An inplane line body force 119902119894

119909119895at the middle surface

of the plate

(ii) An inplane line body force 119902119894


of the plate

(iii) A lateral line load 119902119894

119911119895

O

x up

z wp

y p

zi wib

yi ibOi

mibxj

mibzj mi

byj

qiyj

qizj

qixj

xi uib

mipy2

mipy1

mipx1 mi

px2

fij=2

fij=1

qiz1qiy1

qiy2

qiz2

qix2qix1

mibwj

Figure 3 Structural model and directions of the additional loadingof the plate and the 119894th beam

(iv) A lateral line load 120597119898119894

119901119910119895120597119909 due to the eccentricity

of the component 119902119894119909119895from the middle surface of the

plate 119898119894119901119910119895

= 119902119894

119909119895ℎ1199012 is the bending moment

(v) A lateral line load 120597119898119894



plate 119898119894119901119909119895

= 119902119894


(b) In Each (119894th) Beam (119862119894119909119894119910119894119911119894 System of Axes)

(i) An axially distributed line load 119902119894

119909119895along the beam

centroid axis 119862119894119909119894(ii) A transversely distributed line load 119902

119894

119910119895along the

beam centroid axis 119862119894119909119894(iii) A perpendicularly distributed line load 119902

119894

119911119895along the

beam centroid axis 119862119894119909119894(iv) A distributed bending moment 119898119894

119887119910119895= 119902119894

119909119895119890119894

119911119895along

119862119894119910119894 local beam centroid axis due to the eccentricities

119890119894

119911119895of the components 119902

119894

119909119895from the beam centroid

axis 1198901198941199111

= 119890119894

1199112= minusℎ119894

1198872 are the eccentricities

(v) A distributed bending moment 119898119894119887119911119895

= minus119902119894

119909119895119890119894

119910119895along


119890119894


119894


axis 1198901198941199101

= minus119887119894

1198914 and 119890

119894

1199102= 119887119894


(vi) A distributed twisting moment119898119894119887119909119895

= 119902119894

119911119895119890119894

119910119895minus 119902119894

119910119895119890119894

119911119895

along 119862119894119909119894 local beam shear center axis due to the

eccentricities 119890119894

119911119895and 119890119894


119894

119910119895and

119902119894

119911119895from the beam shear center axis respectively 119890119894

1199111=

119890119894

1199112= minusℎ119894

1198872 and 119890

119894

1199101= minus119887119894

1198914 and 119890

119894

1199102= 119887119894

1198914 are the

eccentricities(vii) A distributed warping moment 119898119894

119887119908119895= minus119902119894

119909119895(120593119875119894

119878)119891119895

along 119862119894119909119894 local beam shear center axis was ignored


in previous models [39 43] (120593119875119894119878)119891119895

is the value ofthe primary warping function 120593

119875119894

119878with respect to the

shear center of the beam cross-section (coincidingwith its centroid) at the point of the 119895th interface lineof the 119894th plate-beam interface

On the basis of the above considerations the response ofthe plate and the beams may be described by the followingboundary value problems

(a) For the Plate The analysis of the plate is based on theVon Karman plate theory according to which the deflectionof the plate cannot be regarded as small as compared to theplate thickness while it remains small in comparison withthe rest dimensions of the plate Due to this assumptiongeometrical nonlinearities should be taken into account andthe displacement field of an arbitrary point of the plate asimplied by the Kirchhoff hypothesis is given as

119906119901 (119909 119910 119911 119905) = 119906119901 (119909 119910 119905) minus 119911

120597119908119901 (119909 119910 119905)

120597119909 (1a)

V119901 (119909 119910 119911 119905) = V119901 (119909 119910 119905) minus 119911

120597119908119901 (119909 119910 119905)

120597119910 (1b)

119908119901 (119909 119910 119911 119905) = 119908119901 (119909 119910 119905) (1c)

where 119906119901 V119901 119908119901 x 119909 119910 119905 ge 0 are the timedependent inplane and transverse displacement componentsof an arbitrary point of the plate and 119906119901 = 119906119901(x 119905) V119901 =

V119901(x 119905) and 119908119901 = 119908119901(x 119905) x 119909 119910 119905 ge 0 are thecorresponding components of a point at its middle surfaceEmploying the strain-displacement relations of the three-dimensional elasticity for moderate large displacements [4445] the strain components can be written as

120576119909119909 =

120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

(2a)

120576119910119910 =

120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

(2b)

120574119909119910 =

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119909

120597119908119901

120597119910 (2c)

120576119911119911 = 120574119909119911 = 120574119910119911 = 0 (2d)

Substituting (1a)ndash(1c) and (2a)ndash(2d) to the stress-strain rela-tions defined by the Hookersquos law

119878119909119909

119878119910119910

119878119909119910

=

[[[[[[[[[[

[

119864119901

(1 minus ]119901)2

119864119901]119901

(1 minus ]119901)2

0

119864119901]119901

(1 minus ]119901)2

119864119901

(1 minus ]119901)2

0

0 0

119864119901

2 (1 + ]119901)

]]]]]]]]]]

]

120576119909119909

120576119910119910

120574119909119910

(3)

the nonvanishing components of the second Piola-Kirchhoffstress tensor are obtained as

119878119909119909 =

119864119901

(1 minus ]2119901)

times [

120597119906119901

120597119909minus 119911

1205972119908119901

1205971199092+

1

2(

120597119908119901

120597119909)

2

]

+]119901 [120597V119901120597119910

minus 119911

1205972119908119901

1205971199102+

1

2(

120597119908119901

120597119910)

2

]

(4a)

119878119910119910 =

119864119901

(1 minus ]2119901)

]119901 [120597119906119901

120597119909minus 119911

1205972119908119901

1205971199092+

1

2(

120597119908119901

120597119909)

2

]

+[

120597V119901120597119910

minus 119911

1205972119908119901

1205971199102+

1

2(

120597119908119901

120597119910)

2

]

(4b)

119878119909119910 =

119864119901

2 (1 + ]119901)(

120597119906119901

120597119910+

120597V119901120597119909

minus 2119911

1205972119908119901

120597119909120597119910+

120597119908119901

120597119909

120597119908119901

120597119910)

(4c)

Subsequently integrating the stress components over theplate thickness the stress resultants acting on the plate arewritten as

119873119901119909 = 119862[

120597119906119901

120597119909+ ]119901

120597V119901120597119909

+1

2(

120597119908119901

120597119909)

2

+1

2]119901(

120597119908119901

120597119910)

2

]

(5a)

119873119901119910 = 119862[

120597V119901120597119909

+ ]119901120597119906119901

120597119909+

1

2(

120597119908119901

120597119910)

2

+1

2]119901(

120597119908119901

120597119909)

2

]

(5b)

119873119901119909119910 = 119862

1 minus ]1199012

(

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119910

120597119908119901

120597119909) (5c)

119872119901119909 = minus119863(

1205972119908119901

1205971199092+ ]119901

1205972119908119901

1205971199102) (5d)

119872119901119910 = minus119863(

1205972119908119901

1205971199102+ ]119901

1205972119908119901

1205971199092) (5e)

119872119901119909119910 = minus119863(1 minus ]119901)1205972119908119901

120597119909120597119910 (5f)

where 119862 = 119864119901ℎ119901(1 minus ]2119901) and 119863 = 119864119901ℎ

3

11990112(1 minus ]2

119901) are the

membrane and bending rigidities of the plate respectively


On the basis of Hamiltonrsquos principle the system of partialdifferential equations of motion of the plate in terms of thestress resultants is obtained as

120597119873119901119909

120597119909+

120597119873119901119909119910

120597119910minus 120588119901ℎ119901119901 =

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119909119895120575 (119910 minus 119910119895)) (6a)

120597119873119901119910

120597119910+

120597119873119901119909119910

120597119909minus 120588119901ℎ119901V119901 =

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119910119895120575 (119910 minus 119910119895)) (6b)

minus

1205972119872119901119909

1205971199092minus 2

1205972119872119901119909119910

120597119909120597119910minus

1205972119872119901119910

1205971199102minus 119873119901119909

1205972119908119901

1205971199092

minus 2119873119901119909119910

1205972119908119901

120597119909120597119910minus 119873119901119910

1205972119908119901

1205971199102

+ 120588119901ℎ119901119901 minus 120588119901ℎ119901119901

120597119908119901

120597119909minus 120588119901ℎ119901V119901

120597119908119901

120597119910

minus

120588119901ℎ3

119901

12

1205972119901

1205971199092minus

120588119901ℎ3

119901

12

1205972119901

1205971199102

= 119892 minus

119868

sum

119894=1

[

[

2

sum

119895=1

(119902119894

119911119895+

120597119898119894

119901119909119895

120597119910+

120597119898119894

119901119910119895

120597119909minus 119902119894

119909119895

120597119908119894

119901119895

120597119909

minus119902119894

119910119895

120597119908119894

119901119895

120597119910)120575 (119910 minus 119910119895)

]

]

(6c)

where 120575(119910minus119910119894) is the Diracrsquos delta function in the 119910 directionEmploying relations (5a)ndash(5f) the governing differentialequations (6a)ndash(6c) in the domain Ω can be expressed interms of the displacement components as

119866119901ℎ119901 [nabla2119906119901 +

1 + ]1199011 minus ]119901

120597

120597119909(

120597119906119901

120597119909+

120597V119901120597119910

)

+(2

1 minus ]119901

1205972119908119901

1205971199092+

1205972119908119901

1205971199102)

120597119908119901

120597119909

+

1 + ]1199011 minus ]119901

1205972119908119901

120597119909120597119910

120597119908119901

120597119910] minus 120588119901ℎ119901119901

=

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119909119895120575 (119910 minus 119910119895))

(7a)

119866119901ℎ119901 [nabla2V119901 +

1 + ]1199011 minus ]119901

120597

120597119910(

120597119906119901

120597119909+

120597V119901120597119910

)

+ (2

1 minus ]119901

1205972119908119901

1205971199102+

1205972119908119901

1205971199092)

120597119908119901

120597119910

+

1 + ]1199011 minus ]119901

1205972119908119901

120597119909120597119910

120597119908119901

120597119909] minus 120588119901ℎ119901V119901

=

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119910119895120575 (119910 minus 119910119895))

(7b)

119863nabla4119908119901 minus 119862

times [(

120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

)

+]119901(120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

)]

times

1205972119908119901

1205971199092+ (1 minus ]119901)

sdot (

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119909

120597119908119901

120597119910)

1205972119908119901

120597119909120597119910

+ [(

120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

)

+]119901(120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

)] sdot

1205972119908119901

1205971199102

+ 120588119901ℎ119901119901 minus 120588119901ℎ119901119901

120597119908119901

120597119909minus 120588119901ℎ119901V119901

120597119908119901

120597119910

minus

120588119901ℎ3

119901

12

1205972119901

1205971199092minus

120588119901ℎ3

119901

12

1205972119901

1205971199102

= 119892 minus

119868

sum

119894=1

[

[

2

sum

119895=1

(119902119894

119911119895+

120597119898119894

119901119909119895

120597119910+

120597119898119894

119901119910119895

120597119909minus 119902119894

119909119895

120597119908119894

119901119895

120597119909

minus119902119894

119910119895

120597119908119894

119901119895

120597119910)120575 (119910 minus 119910119895)

]

]

(7c)

The governing differential equations (7a)ndash(7c) are also sub-jected to the pertinent boundary conditions of the problemat hand

1198861199011119906119901119899 + 1198861199012119873119901119899 = 1198861199013 (8a)

1205731199011119906119901119905 + 1205731199012119873119901119905 = 1205731199013 (8b)

1205741199011119908119901 + 1205741199012119877119901119899 = 1205741199013 (8c)

1205751199011

120597119908119901

120597119899+ 1205751199012119872119901119899 = 1205751199013

(8d)

1205761119896119908119901 + 1205762119896

10038171003817100381710038171003817119879119908119901

10038171003817100381710038171003817119896= 1205763119896 1205762119896 = 0 (8e)

and to the initial conditions

119908119901 (x 0) = 1199081199010 (x) (9a)

119901 (x 0) = 1199081199010 (x) (9b)


where 119886119901119897 120573119901119897 120574119901119897 and 120575119901119897 (119897 = 1 2 3) are functions specifiedat the boundary Γ 120576119897119896 (119897 = 1 2 3) are functions specifiedat the 119896 corners of the plate 1199081199010(x) 1199081199010(x) and x 119909 119910

are the initial deflection and velocity of the points of themiddle surface of the plate 119906119901119899 119906119901119905 and 119873119901119899 119873119901119905 are theboundary membrane displacements and forces in the normaland tangential directions to the boundary respectively 119877119901119899and 119872119901119899 are the effective reaction along the boundary andthe bending moment normal to it respectively which byemploying intrinsic coordinates (ie the distance along theoutward normal 119899 to the boundary and the arc length 119904) arewritten as

119877119901119899 = minus119863[120597

120597119899nabla2119908119901 minus (]119901 minus 1)

120597

120597119904(

1205972119908119901

120597119904120597119899minus 120581

120597119908119901

120597119904)]

+ 119873119901119899

120597119908119901

120597119899+ 119873119901119905

120597119908119901

120597119904

(10a)

119872119901119899 = minus119863[nabla2119908119901 + (]119901 minus 1)(

1205972119908119901

1205971199042+ 120581

120597119908119901

120597119899)] (10b)

in which 120581(119904) is the curvature of the boundary Finally119879119908119901119896

is the discontinuity jump of the twisting moment119879119908119901 at the corner 119896 of the plate while 119879119908119901 along theboundary is given by the following relation

119879119908119901 = 119863(]119901 minus 1)(

1205972119908119901

120597119904120597119899minus 120581

120597119908119901

120597119904) (11)

The boundary conditions (8a)ndash(8d) are the most generalboundary conditions for the plate problem including alsoelastic support while the corner condition (8e) holds for freeor transversely elastically restrained corners k It is apparentthat all types of the conventional boundary conditions can bederived from these equations by specifying appropriately thefunctions 119886119901119897 120573119901119897 120574119901119897 and 120575119901119897 (119897 = 1 2 3) (eg for a clampededge it is 1198861199011 = 1205731199011 = 1205741199011 = 1205751199011 = 1 1198861199012 = 1198861199013 = 1205731199012 = 1205731199013 =

1205741199012 = 1205741199013 = 1205751199012 = 1205751199013 = 0)

(b) For Each (119894th) Beam Each beam undergoes transversedeflectionwith respect to 119911

119894 and119910119894 axes and axial deformation

and nonuniform angle of twist along 119909119894 axis Based on

the Bernoulli theory the displacement field of an arbitrarypoint of a cross-section (taking into account moderate largedisplacements and considering the angle of rotation of twistto have relatively small values) can be derived with respect tothose of its centroid as

119906119894

119887(119909119894 119910119894 119911119894 119905) = 119906

119894

119887(119909119894 119905) minus 119910

119894120579119894

119887119911(119909119894 119905)

+ 119911119894120579119894

119887119910(119909119894 119905) +

120597120579119894

119887119909

120597119909119894120593119875119894

119878(119910119894 119911119894 119905)

(12a)

V119894119887(119909119894 119910119894 119911119894 119905) = V119894

119887(119909119894 119905) minus 119911

119894120579119894

119887119909(119909119894 119905) (12b)

119908119894

119887(119909119894 119910119894 119911119894 119905) = 119908

119894

119887(119909119894 119905) + 119910

119894120579119894

119887119909(119909119894 119905) (12c)

120579119894

119887119910(119909119894 119905) = minus

120597119908119894

119887(119909119894 119905)

120597119909119894 (12d)

120579119894

119887119911(119909119894 119905) =

120597V119894119887(119909119894 119905)

120597119909119894 (12e)

where 119906119894119887 V119894119887 and119908

119894

119887are the axial and transverse displacement

components with respect to the 119862119894119909119894119910119894119911119894 system of axes 119906119894

119887=

119906119894

119887(119909119894) V119894119887= V119894119887(119909119894) and 119908

119894

119887= 119908119894

119887(119909119894) are the corresponding

components of the centroid 119862119894 120579119894119887119910

= 120579119894

119887119910(119909119894) and 120579

119894

119887119911=

120579119894

119887119911(119909119894) are the angles of rotation of the cross-section due to

bending with respect to its centroid 120597120579119894119887119909119889119909119894 denotes the

rate of change of the angle of twist 120579119894

119887119909(119909119894) regarded as the

torsional curvature and 120593119875119894

119878is the primary warping function

with respect to the cross-sectionrsquos shear center (coincidingwith its centroid) Employing again the strain-displacementrelations of the three-dimensional elasticity for moderatedisplacements [44 45] the strain components are given as

120576119909119909 =120597119906119894

119887

120597119909119894+

1

2

[

[

(120597V119894119887

120597119909119894)

2

+ (120597119908119894

119887

120597119909119894)

2

]

]

(13a)

120574119909119911 =120597119908119894

119887

120597119909119894+

120597119906119894

119887

120597119911119894+ (

120597V119894119887

120597119909119894

120597V119894119887

120597119911119894+

120597119908119894

119887

120597119909119894

120597119908119894

119887

120597119911119894) (13b)

120574119909119910 =120597V119894119887

120597119909119894+

120597119906119894

119887

120597119910119894+ (

120597V119894119887

120597119909119894

120597V119894119887

120597119910119894+

120597119908119894

119887

120597119909119894

120597119908119894

119887

120597119910119894) (13c)

120576119910119910 = 120576119911119911 = 120574119910119911 = 0 (13d)

Employing the Hookersquos stress-strain law and integrating thearising stress components over the beamrsquos cross-section afterignoring the nonlinear terms with respect to the angle oftwist and its derivatives the stress resultants of the beam arederived as

119873119894

119887= 119864119894

119887119860119894

119887[120597119906119894

119887

120597119909119894+

1

2((

120597V119894119887

120597119909119894)

2

+ (120597119908119894

119887

120597119909119894)

2

)] (14a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (14b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (14c)

119872119875119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894 (14d)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (14e)

where 119872119875119894

119887119905is the primary twisting moment [15] resulting

from the primary shear stress distribution 119872119894

119887119908is the


warping moment due to torsional curvature Furthermore119868119894

119910and 119868

119894

119911are the principal moments of inertia 119868

119894

119878is the

polar moment of inertia while 119868119894

119905and 119862

119894

119878are the torsion

and warping constants of the 119894th beam with respect to thecross-sectionrsquos shear center (coinciding with its centroid)respectively given as [46]

119868119894

119905= intΩ

(1199101198942+ 1199111198942+ 119910119894 120597120593119875119894

119878

120597119911119894minus 119911119894 120597120593119875119894

119878

120597119910119894)119889Ω (15a)

119862119894

119878= intΩ

(120593119875119894

119878)2

119889Ω (15b)

On the basis ofHamiltonrsquos principle the differential equationsof motion in terms of displacements are obtained as

minus120597119873119894

119887

120597119909119894+ 120588119894

119887Α119894

119887119894

119887=

2

sum

119895=1

119902119894

119909119895 (16a)

minus 119873119894

119887

1205972V119894119887

1205971199091198942+

1205972119872119894

119887119911

1205971199091198942

+ 120588119894

119887Α119894

119887V119894119887minus 120588119894

119887119868119894

119911

1205972V119894119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(16b)

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942minus

1205972119872119894

119887119910

1205971199091198942

+ 120588119894

119887Α119894

119887119894

119887minus 120588119894

119887119868119894

119910

1205972119894

119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(16c)

minus120597119872119894

119887119905

120597119909119894minus

1205972119872119894

119887119908

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(16d)

Substituting the expressions of the stress resultants of (14a)ndash(14e) in (16a)ndash(16d) the differential equations of motion areobtained as

minus 119864119894

119887119860119894

119887(1205972119906119894

119887

1205971199091198942+

120597119908119894

119887

120597119909119894

1205972119908119894

119887

1205971199091198942

+120597V119894119887

120597119909119894

1205972V119894119887

1205971199091198942) + 120588

119894

119887Α119894

119887119894

119887

=

2

sum

119895=1

119902119894

119909119895

(17a)

119864119894

119887119868119894

119911

1205974V119894119887

1205971199091198944minus 119873119894

119887

1205972V119894119887

1205971199091198942minus 120588119894

119887119868119894

119911

1205972V1198871205971199092

+ 120588119894

119887Α119894

119887V119887 minus 120588

119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(17b)

119864119894

119887119868119894

119910

1205974119908119894

119887

1205971199091198944

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942

minus 120588119894

119887119868119894

119910

1205972119887

1205971199092+ 120588119894

119887Α119894

119887119887 minus 120588

119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(17c)

119864119894

119887119862119894

119878

1205974120579119894

119887119909

1205971199091198944

minus 119866119894

119887119868119894

119905

1205972120579119894

119887119909

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(17d)

Moreover the corresponding boundary conditions of the 119894thbeam at its ends 119909119894 = 0 119897

119894 are given as

119886119894

1198871119906119894

119887+ 120572119894

1198872119873119894

119887= 120572119894

1198873 (18)

120573119894

1198871V119894119887+ 120573119894

1198872119877119894

119887119910= 120573119894

1198873 (19a)

120573119894

1198871120579119894

119887119911+ 120573119894

1198872119872119894

119887119911= 120573119894

1198873 (19b)

120574119894

1198871119908119894

119887+ 120574119894

1198872119877119894

119887119911= 120574119894

1198873 (20a)

120574119894

1198871120579119894

119887119910+ 120574119894

1198872119872119894

119887119910= 120574119894

1198873 (20b)

120575119894

1198871120579119894

119887119909+ 120575119894

1198872119872119894

119887119905= 120575119894

1198873 (21a)

120575119894

1198871

119889120579119894

119887119909

119889119909119894+ 120575119894

1198872119872119894

119887119908= 120575119894

1198873(21b)

and the initial conditions as

119908119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22a)

119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22b)

where the angles of rotation of the cross-section due tobending 120579

119894

119887119910 120579119894

119887119911are given from (12d) and (12e) 119877

119894

119887119910 119877119894

119887119911

and 119872119894

119887119911 119872119894119887119910

are the reactions and bending moments withrespect to 119910

119894 119911119894 axes respectively which after applying theaforementioned simplifications are given as

119877119894

119887119910= 119873119894

119887

120597V119894119887

120597119909119894minus 119864119894

119887119868119894

119911

1205973V119894119887

1205971199091198943 (23a)

119877119894

119887119911= 119873119894

119887

120597119908119894

119887

120597119909119894minus 119864119894

119887119868119894

119910

1205973119908119894

119887

1205971199091198943 (23b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (24a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (24b)


and 119872119894

119887119905 119872119894119887119908

are the torsional and warping moments at theboundaries of the beam respectively given as

119872119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894minus 119864119894

119887119862119894

119878

1205973120579119894

119887119909

1205971199091198943 (25a)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (25b)

Finally 120572119894119887119896 120573119894

119887119896 120573119894

119887119896 120574119894

119887119896 120574119894

119887119896 120575119894

119887119896 120575119894

119887119896(119896 = 1 2 3) are func-

tions specified at the 119894th beam ends (119909119894 = 0 119897119894)The boundary

conditions (18)ndash(21b) are the most general boundary condi-tions for the beam problem including also the elastic supportIt is apparent that all types of the conventional boundary con-ditions (clamped simply supported free or guided edge) canbe derived from these equations by specifying appropriatelythe aforementioned coefficients

Equations (7a)ndash(17d) constitute a set of seven coupled andnonlinear partial differential equations including thirteenunknowns namely 119906119901 V119901119908119901 119906

119894

119887 V119894119887119908119894119887 120579119894119887119909 1199021198941199091 1199021198941199101 1199021198941199111 119902119894

1199092

119902119894

1199102 and 119902

119894

1199112 Six additional equations are required which

result from the displacement continuity conditions in thedirections of 119909119894 119910119894 and 119911

119894 local axes along the two interfacelines of each (119894th) plate-beam interface Taking into accountthe displacement fields expressed by (1a)ndash(1c) and (12a)ndash(12e)the displacement continuity conditions [47] can be expressedas follows

In the direction of 119909119894 local axis

119906119894

1199011minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199011

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894+

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198911

+119902119894

1199091

1198961198941199091

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]

along interface line 1 (119891119894

119895=1)

(26a)

119906119894

1199012minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199012

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894minus

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198912

+119902119894

1199092

1198961198941199092

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=2)

(26b)


V1198941199011

minus V119894119887=

ℎ119901

2

120597119908119894

1199011

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199101

1198961198941199101

[1 minus1

2(120579119894

119887119909)2

]


119895=1)

(27a)

V1198941199012

minus V119894119887=

ℎ119901

2

120597119908119894

1199012

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199102

1198961198941199102

[1 minus1

2(120579119894

119887119909)2

]


119895=2)

(27b)


119908119894

1199011minus 119908119894

119887= minus

119887119894

119891

4120579119894

119887119909along interface line 1 (119891

119894

119895=1) (28a)

119908119894

1199012minus 119908119894

119887=

119887119894

119891

4120579119894


119894

119895=2) (28b)

where (120593119875119894

119878)119891119895

is the value of the primary warping functionwith respect to the shear center of the beam cross-section(coinciding with its centroid) at the point of the 119895th interfaceline of the 119894th plate-beam interface 119891

119894

119895and 119896

119894

119909119895 119896119894119910119895

are thestiffness of the arbitrarily distributed shear connectors alongthe 119909119894 and 119910

119894 directions respectively It is noted that 119896119894119909119895

=

119896119894

119909119895(119904119894

119909119895) and 119896

119894

119910119895= 119896119894

119910119895(119904119894

119910119895) can represent any linear or

nonlinear relationship between the inplane interface forcesand the interface slip 119904

119894

119895in the corresponding direction

In all of the aforementioned equations the values of theprimary warping function 120593

119875119894

119878(119910119894 119911119894) should be set having the

appropriate algebraic sign corresponding to the local beamaxes

3 Integral Representations-NumericalSolution

The solution of the presented dynamic problem requires theintegration of the set of (7a)ndash(7c) and (17a)ndash(17d) subjected tothe prescribed boundary and initial conditionsMoreover thedisplacement continuity conditions should also be fulfilledDue to the nonlinear and coupling character of the equationsof motion an analytical solution is out of question There-fore a numerical solution is derived employing the analogequation method [41] a BEM-based method Contrary toprevious research efforts where the numerical analysis isbased on BEM using a lumped mass assumption model afterevaluating the flexibility matrix at the mass nodal points [15]in this work a distributed mass model is employed

In the following sections the plate and beam problemsrepresented by (7a)ndash(7c) and (17a)ndash(17d) respectively areexamined independently and the connection between theseproblems is achieved employing the displacement continuityconditions

31 For the Plate Displacement Components 119906119901 V119901 119908119901According to the precedent analysis the large deflectionanalysis of the plate becomes equivalent to establishingthe inplane displacement components 119906119901 and V119901 havingcontinuous partial derivatives up to the second order withrespect to 119909 119910 and the deflection 119908119901 having continuouspartial derivatives up to the fourth order with respect to119909 119910 and all displacement components having derivativesup to the second order with respect to 119905 Moreover thesedisplacement components must satisfy the boundary valueproblem described by the nonlinear and coupled governingdifferential equations of equilibrium (equations (7a)ndash(7c))inside the domain the conditions (equations (8a)ndash(8e)) at theboundary Γ and the initial conditions (equations (9a)-(9b))Equations (7a)ndash(7c) and (8a)ndash(8e) are solved using the analog


equation method [41] More specifically setting as 1199061199011 = 1199061199011199061199012 = V119901 1199061199013 = 119908119901 and applying the Laplacian operator to1199061199011 1199061199012 and the biharmonic operator to 1199061199013 yields

nabla2119906119901119894 = 119901119901119894 (119909 119910 119905) (119894 = 1 2) (29a)

nabla41199061199013 = 1199011199013 (119909 119910 119905) (29b)

Equations (29a) and (29b) are called analog equationsand they indicate that the solution of (7a)ndash(7c) and(8a)ndash(8e) can be established by solving (29a) and (29b)under the same boundary conditions (equations (8a)ndash(8e)) provided that the fictitious load distributions119901119901119894(119909 119910) (119894 = 1 2 3) are first established Thesedistributions can be determined using BEM Followingthe procedure presented in [47] the unknown boundaryquantities 119906119901119894(119875 119905) 119906119901119894119909(119875 119905) 119906119901119894119909119909(119875 119905) and 119906119901119894119909119909119909(119875 119905)

(119875 isin boundary 119894 = 1 2 3) can be expressed in terms of119901119901119894 (119894 = 1 2 3) after applying the integral representationsof the displacement components 119906119894 (119894 = 1 2 3) and theirderivatives with respect to 119909 119910 to the boundary of the plate

By discretizing the boundary of the plate into 119873 bound-ary elements and the domain Ω into 119872 domain cellsand employing the constant element assumption (as thenumerical implementation becomes very simple and theobtained results are of high accuracy) the application ofthe integral representations and the corresponding one ofthe Laplacian nabla

21199061199013 and of the boundary conditions (8a)ndash

(8e) to the 119873 boundary nodal points results in a set of8 times 119873 nonlinear algebraic equations relating the unknownboundary quantities with the fictitious load distributions 119901119901119894(119894 = 1 2 3) that can be written as

[

[

E11990111 0 00 E11990122 00 0 E11990133

]

]

d1199011d1199012d1199013

+

0D1198991198971199011

0D1198991198971199012

00

D1198991198971199013

=

01205721199013

01205731199013

001205741199013

1205751199013

(30)

where

E11990111 = [A1 H1 H20 D11990122 D11990123

] (31a)

E11990122 = [A1 H1 H20 D11990144 D11990145

] (31b)

E11990133 =[[[

[

A2 H1 H2 G1 G2A1 0 0 H1 H20 D11990178 D11990179 0 D1199017110 D11990188 D11990189 D119901810 0

]]]

]

(31c)

D11990122 to D119901810 are 119873 times 119873 rectangular known matricesincluding the values of the functions 119886119901119895 120573119901119895 120574119901119895 120575119901119895 (119895 = 1 2)of (8a)ndash(8d) 1205721199013 1205731199013 1205741199013 and 1205751199013 are119873times 1 known columnmatrices including the boundary values of the functions

1198861199013 1205731199013 1205741199013 1205751199013 of (8a)ndash(8d) H119894 (119894 = 1 2) H2 and G119894(119894 = 1 2) are rectangular 119873 times 119873 known coefficient matricesresulting from the values of kernels at the boundary elementsof the plate A1 A1 and A2 are 119873 times 119872 rectangular knownmatrices originating from the integration of kernels on thedomain cells of the plate D119899119897

119901119894(119894 = 1 2) are 119873 times 1 and

D1198991198971199013

is 2119873 times 1 column matrices containing the nonlinearterms included in the expressions of the boundary conditions(equations (8a)ndash(8d)) It is noted that the derivatives ofthe unknown boundary quantities with respect to the arclength 119904 appearing in (8a)ndash(8d) are approximated employingappropriate central backward or forward finite differenceschemes Finally

d119901119894 = p119901119894 u119901119894 u119901119894119899119879 (119894 = 1 2) (32a)

d1199013 = p1199013 u1199013 u1199013119899 nabla2u1199013 (nabla

2u1199013)119899119879 (32b)

are generalized unknown vectors where

u119901119894 = (119906119901119894)1(119906119901119894)2

sdot sdot sdot (119906119901119894)119873119879

(119894 = 1 2 3)

(33a)

u119901119894119899 = (

120597119906119901119894

120597119899)

1

(

120597119906119901119894

120597119899)

2

sdot sdot sdot (

120597119906119901119894

120597119899)

119873

119879

(119894 = 1 2 3)

(33b)

nabla2u1199013 = (nabla

21199061199013)1

(nabla21199061199013)2

sdot sdot sdot (nabla21199061199013)119873

119879

(33c)

(nabla2u1199013)119899

= (

120597nabla21199061199013

120597119899)

1

(

120597nabla21199061199013

120597119899)

2

sdot sdot sdot (

120597nabla21199061199013

120597119899)

119873

119879

(33d)

are vectors including the unknown boundary valuesof the respective boundary quantities and p119901119894 =

(119901119901119894)1(119901119901119894)2

sdot sdot sdot (119901119901119894)119872119879

(119894 = 1 2 3) are vectorscontaining the 119872 unknown nodal values of the fictitiousloads at the domain cells of the plate In the case that theboundary Γ has 119896 free or transversely elastically restrainedcorners 119896 additional equations must be satisfied togetherwith (30) These additional equations result from theapplication of the corner condition (8e) on the 119896 cornersfollowing the procedure presented in [48]

Discretization of the integral representations of the dis-placement components 119906119894 and their derivatives with respectto 119909 119910 [49] and application to the 119872 domain nodal pointsyields

u119901119894 = B119901119894d119901119894 (119894 = 1 2 3) (34a)u119901119894119909 = B119901119894119909d119901119894 (119894 = 1 2 3) (34b)

u119901119894119910 = B119901119894119910d119901119894 (119894 = 1 2 3) (34c)


u119901119894119909119909 = B119901119894119909119909d119901119894 (119894 = 1 2 3) (34d)

u119901119894119910119910 = B119901119894119910119910d119901119894 (119894 = 1 2 3) (34e)

u119901119894119909119910 = B119901119894119909119910d119901119894 (119894 = 1 2 3) (34f)

where B119901119894 B119901119894119909 B119901119894119910 B119901119894119909119909 B119901119894119910119910 B119901119894119909119910 (119894 = 1 2) are119872 times (2119873 + 119872) and B1199013 B1199013119909 B1199013119910 B1199013119909119909 B1199013119910119910 B1199013119909119910are119872 times (4119873 + 119872) known coefficient matrices In evaluatingthe domain integrals of the kernels over the domain cellssingular and hypersingular integrals ariseThey are computedby transforming them to line integrals on the boundary ofthe cell [50 51] The final step of the AEM is to apply (7a)ndash(7c) to the 119872 nodal points inside Ω yielding the followingequations

119866119901ℎ119901 [p1199011 +1 + ]1199011 minus ]119901

(B1199011119909119909d1199011 + B1199012119909119910d1199012)

+ 2

1 minus ]119901(B1199013119909119909d1199013)119889119892B1199013119909d1199013

+(B1199013119910119910d1199013)119889119892 sdot B1199013119909d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119910d1199013]

minus 120588119901ℎ119901B1199011d1199011 = Zq119909(35a)

119866119901ℎ119901 [p1199012 +1 + ]1199011 minus ]119901

(B1199011119909119910d1199011 + B1199012119910119910d1199012)

+ 2

1 minus ]119901(B1199013119910119910d1199013)119889119892B1199013119910d1199013

+ (B1199013119909119909d1199013)119889119892 sdot B1199013119910d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119909d1199013]

minus 120588119901ℎ119901B1199012d1199012 = Zq119910

(35b)

119863p1199013 minus 119862

times [(B1199011119909d1199011)119889119892B1199013119909119909d1199013

+1

2(B1199013119909d1199013)119889119892(B1199013119909d1199013)119889119892B1199013119909119909d1199013

+ ]119901(B1199012119910d1199012)119889119892B1199013119909119909d1199013 +1

2]119901(B1199013119910d1199013)119889119892

times (B1199013119910d1199013)119889119892B1199013119909119909d1199013]

+ (1 minus ]119901)

times [(B1199011119910d1199011)119889119892B1199013119909119910d1199013

+ (B1199012119909d1199012)119889119892B1199013119909119910d1199013

+(B1199013119909d1199013)119889119892 sdot (B1199013119910d1199013)119889119892B1199013119909119910d1199013]

+ [(B1199012119910d1199012)119889119892B1199013119910119910d1199013 +1

2(B1199013119910d1199013)119889119892

sdot (B1199013119910d1199013)119889119892B1199013119910119910d1199013

+ ]119901(B1199011119909d1199011)119889119892B1199013119910119910d1199013

+1

2]119901(B1199013119909d1199013)119889119892

sdot (B1199013119909d1199013)119889119892B1199013119910119910d1199013]

+ 120588119901ℎ119901B1199013d1199013 minus 120588119901ℎ119901(B1199011d1199011)119889119892B1199013119909d1199013

minus 120588119901ℎ119901(B1199012d1199012)119889119892B1199013119910d1199013 minus120588119901ℎ3

119901

12B1199013119909119909d1199013

minus

120588119901ℎ3

119901

12B1199013119910119910d1199013

= g minus Zq119911 minus ZX119910q119910 minus ZX119909q119909 + (Zq119909)119889119892B1199013119909d1199013

+ (Zq119910)119889119892B1199013119910d1199013

(35c)

where q119909 = q1199091 q1199092119879 q119910 = q1199101 q1199102

119879 andq119911 = q1199111 q1199112

119879 are vectors of dimension 2119871 including theunknown 119902

119894

119909119895 119902119894119910119895 119902119894119911119895(119895 = 1 2) interface forces 2119871 is the total

number of the nodal points at each plate-beam interface Z isa position 119872 times 2119871 matrix which converts the vectors q119909 q119910q119911 into corresponding ones with length 119872 the symbol (sdot)119889119892indicates a diagonal 119872 times 119872 matrix with the elements of theincluded column matrix Matrices X119909 and X119910 of dimension2119871 times 2119871 result after approximating the bending momentderivatives of 119898119894

119901119910119895= 119902119894

119909119895ℎ1199012 119898

119894

119901119909119895= 119902119894

119910119895ℎ1199012 respectively

using appropriately central backward or forward differences

32 For the Beam Displacement Components 119906119894

119887 V119894119887 and 119908

119894

119887

and for the Angle of Twist 120579119894

119887119909 According to the prece-

dent analysis the nonlinear dynamic analysis of the 119894thbeam reduces in establishing the displacement components119906119894

119887(119909119894 119905) and V119894

119887(119909119894 119905) 119908119894

119887(119909119894 119905) 120579119894119887119909(119909119894 119905) having continuous

derivatives up to the second order and up to the fourthorder with respect to 119909 respectively and also having deriva-tives up to the second order with respect to 119905 Moreoverthese displacement components must satisfy the coupledgoverning differential equations (17a)ndash(17d) inside the beam


the boundary conditions at the beam ends (18)ndash(21b) and theinitial conditions (22a) and (22b)

Let 119906119894

1198871= 119906119894

119887(119909119894 119905) 119906

119894

1198872= V119894119887(119909119894 119905) 119906

119894

1198873= 119908119894

119887(119909119894 119905)

and 119906119894

1198874= 120579119894

119887119909(119909119894 119905) be the sought solutions of the problem

represented by (17a)ndash(17d) Differentiating with respect to 119909

these functions two and four times respectively yields

1205972119906119894

1198871

1205971199091198942

= 119901119894

1198871(119909119894 119905) (36a)

1205974119906119894

119887119895

1205971199091198944= 119901119894

119887119895(119909119894 119905) (119895 = 2 3 4) (36b)

Equations (36a) and (36b) are quasi-static and indicate thatthe solution of (17a)ndash(17d) can be established by solving (36a)and (36b) under the same boundary conditions (equations(18)ndash(21b)) provided that the fictitious load distributions119901119894

119887119895(119909119894 119905) (119895 = 1 2 3 4) are first established These distribu-

tions can be determined following the procedure presentedin [49] and employing the constant element assumptionfor the load distributions 119901

119894

119887119895along the 119871 internal beam

elements (as the numerical implementation becomes verysimple and the obtained results are of high accuracy) Thusthe integral representations of the displacement components119906119894

119887119895(119895 = 1 2 3 4) and their derivativeswith respect to119909

119894whenapplied to the beam ends (0 119897119894) together with the boundaryconditions (18)ndash(21b) are employed to express the unknownboundary quantities 119906119894

119887119895(120577119894) 119906119894119887119895119909

(120577119894) 119906119894119887119895119909119909

(120577119894) and 119906

119894

119887119895119909119909119909(120577119894)

(120577119894 = 0 119897119894) in terms of 119901119894

119887119895(119895 = 1 2 3 4) Thus the following

set of 28 nonlinear algebraic equations for the 119894th beam isobtained as

[[[

[

E11989411988711

0 0 00 E119894

119887220 0

0 0 E11989411988733

00 0 0 E119894

11988744

]]]

]

d1198941198871

d1198941198872

d1198941198873

d1198941198874

+

0D119894 1198991198971198871

00

D119894 1198991198971198872

00

D119894 1198991198971198873

000

=

0120572119894

1198873

00120573119894

1198873

00120574119894

1198873

00120575119894

1198873

(37)

where E11989411988711

and E11989411988722

E11989411988733

E11989411988744

are knownmatrices of dimen-sion 4 times (119873 + 4) and 8 times (119873 + 8) respectively D119894 119899119897

1198871is

a 2 times 1 and D119894 119899119897119887119895

(119895 = 2 3) are 4 times 1 column matricescontaining the nonlinear terms included in the expressionsof the boundary conditions (equations (18)ndash(20b)) 120572119894

1198873and

120573119894

1198873 1205741198941198873 1205751198941198873

are 2 times 1 and 4 times 1 known column matrices

respectively including the boundary values of the functions119886119894

1198873and 120573

119894

1198873 120573119894

1198873 120574119894

1198873 120574119894

1198873 120575119894

1198873 120575119894

1198873of (18)ndash(21b) Finally

d1198941198871

= p1198941198871

u1198941198871

u1198941198871119909

119879

(38a)

d119894119887119895

= p119894119887119895 u119894119887119895

u119894119887119895119909

u119894119887119895119909119909

u119894119887119895119909119909119909

119879

(119895 = 2 3 4)

(38b)


u119894119887119895

= 119906119894

119887119895(0 119905) 119906

119894

119887119895(119897119894 119905)119879

(119895 = 1 2 3 4) (39a)

u119894119887119895119909

= 120597119906119894

119887119895(0 119905)

120597119909119894

120597119906119894

119887119895(119897119894 119905)

120597119909119894

119879

(119895 = 1 2 3 4) (39b)

u119894119887119895119909119909

= 1205972119906119894

119887119895(0 119905)

1205971199091198942

1205972119906119894

119887119895(119897119894 119905)

1205971199091198942

119879

(119895 = 2 3 4)

(39c)

u119894119887119895119909119909119909

= 1205973119906119894

119887119895(0 119905)

1205971199091198943

1205973119906119894

119887119895(119897119894 119905)

1205971199091198943

119879

(119895 = 2 3 4)

(39d)

are vectors including the two unknown boundary val-ues of the respective boundary quantities and p119894

119887119895=

(119901119894

119887119895)1

(119901119894

119887119895)2

sdot sdot sdot (119901119894

119887119895)119871119879

(119895 = 1 2 3 4) are vectorsincluding the 119871 unknown nodal values of the fictitious loads

Discretization of the integral representations of theunknown quantities 119906

119894

119887119895(119895 = 1 2 3 4) and those of their

derivatives with respect to 119909119894 inside the beam 119909

119894isin (0 119897

119894) and

application to the 119871 collocation nodal points yields

u1198941198871

= B1198941198871d1198941198871 (40a)

u1198941198871119909

= B1198941198871119909

d1198941198871 (40b)

u119894119887119895

= B119894119887119895d119894119887119895 (119895 = 2 3 4) (40c)

u119894119887119895119909

= B119894119887119895119909

d119894119887119895 (119895 = 2 3 4) (40d)

u119894119887119895119909119909

= B119894119887119895119909119909

d119894119887119895 (119895 = 2 3 4) (40e)

u119894119887119895119909119909119909

= B119894119887119895119909119909119909

d119894119887119895 (119895 = 2 3 4) (40f)

where B1198941198871B1198941198871119909

are 119871 times (119871 + 4) and B119894119887119895B119894119887119895119909

B119894119887119895119909119909

B119894119887119895119909119909119909

(119895 = 2 3 4) are 119871 times (119871 + 8) known coefficient matricesrespectively


x

y

a

a

Fr

Fr

ClCl

Lx = 18 cm

Ly=9

cm A

(a)

g

02 cmhb = 05 cm

1 cm 5 cm3 cm

Ly = 9 cm

(b)

Figure 4 Plan view (a) and section a-a (b) of the stiffened plate of Example 1

Time (ms)

060

040

020

000

minus020

000 050 100 150 200 250

Disp

lace

men

tw (c

m)

AEM-nonlinear analysisAEM-linear analysis

FEM-nonlinear analysisFEM-linear analysis

Figure 5 Time history of deflection 119908 (cm) at the middle point Aof the free edge of the plate of Example 1

Applying (17a)ndash(17d) to the 119871 collocation points andemploying (40a)ndash(40f) 4 times 119871 nonlinear algebraic equationsfor each (119894th) beam are formulated as

minus 119864119894

119887119860119894

119887[p1198941198871

+ (B1198941198872119909

d1198941198872)119889119892B1198941198872119909119909

d1198941198872

+ (B1198941198873119909

d1198941198873)119889119892B1198941198873119909119909

d1198941198873]

+ 120588119894

119887Α119894

119887T1q1 = q119894

1199091+ q1198941199092

(41a)

119864119894

119887119868119894

119911p1198941198872

minus (N119894119887)119889119892B1198941198872119909119909

d1198941198872

minus 120588119894

119887119868119894

119911B1198941198872119909119909

d1198941198872

+ 120588119894

119887Α119894

119887B1198941198872d1198941198872

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198872119909

d1198941198872

= q1198941199101

+ q1198941199102

minus (B1198941198872119909

d1198941198872)119889119892

(q1198941199091

+ q1198941199092) minus X119894119887119910

(q1198941199091

+ q1198941199092)

(41b)

119864119894

119887119868119894

119910p1198941198873

minus (N119894119887)119889119892B1198941198873119909119909

d1198941198873

minus 120588119894

119887119868119894

119910B1198941198873119909119909

d1198941198873

+ 120588119894

119887Α119894

119887B1198941198873d1198941198873

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198873119909

d1198941198873

= q1198941199111

+ q1198941199112

minus (B1198941198873119909

d1198941198873)119889119892

(q1198941199091

+ q1198941199092) + X119894119887119911

(q1198941199091

+ q1198941199092)

(41c)

119864119894

119887119862119894

119878p1198941198874

minus 119866119894

119887119868119894

119905B1198941198874119909119909

d1198941198874

+ 120588119894

119887119868119894

119901B1198941198874d1198941198874

minus 120588119894

119887119862119894

119878B1198941198874119909119909

d1198941198874

= e1198941199101q1198941199111

+ e1198941199102q1198941199112

minus e1198941199111q1198941199101

minus e1198941199112q1198941199102

+ Χ119894

119887119908(q1198941199091

+ q1198941199092)

(41d)

where (N119894119887)119889119892

is a diagonal 119871 times 119871 matrix including thevalues of the axial forces of the 119894th beam the symbol (sdot)119889119892indicates a diagonal 119871 times 119871 matrix with the elements ofthe included column matrix The matrices X119894

119887119911 X119894119887119910 X119894119887119908

result after approximating the derivatives of 119898119894119887119910119895

119898119894119887119911119895 119898119894119887119908119895

using appropriately central backward or forward differencesTheir dimensions are also 119871 times 119871 Moreover e119894

1199101 e1198941199102 e1198941199111

e1198941199112

are diagonal 119871 times 119871 matrices including the values ofthe eccentricities 119890

119894

119910119895 119890119894


119894

119911119895 119902119894

119910119895with

respect to the 119894th beam shear center axis (coinciding with itscentroid) respectively

Employing (34a)ndash(34f) and (40a)ndash(40f) the discretizedcounterpart of (26a)-(26b) (27a)-(27b) and (28a)-(28b) atthe 119871 nodal points of each interface is written as

Y1B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y1B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

+

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198911

B1198941198874119909

d1198874

+ K1198941199091

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42a)


008

008

018

018

028

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036040044048

(a) 119908119901max = 0484 cm at 119905 = 119msec of linear AEM analysis

000067300301006080091501220153018402140245027603070337036803990430460491

(b) 119908119901max = 0491 cm at 119905 = 120msec of linear FEM [25] analysis

008

008

008

00801

8018

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036

(c) 119908119901max = 0394 cmat 119905 = 108msec of nonlinearAEManalysis

000064600236004780072009620120145016901930217024102660290314033803620387

(d) 119908119901max = 0387 cm at 119905 = 106msec of nonlinear FEM [25]analysis

Figure 6 Contour lines of 119908119901 of the stiffened plate of Example 1 employing the present study (a c) and a FEM [25] solution using solidelements (b d) at the time of maximum transverse displacement

Y2B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y2B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

minus

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198912

B1198941198874119909

d1198874

+ K1198941199092

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42b)

Y1B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y1B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199101

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43a)

Y2B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y2B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199102

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43b)

Y1B1199013d1199013 minus B1198941198873d1198941198873

= minus

119887119894

119891

4B1198941198874d1198941198874 (44a)

Y2B1199013d1199013 minus B1198941198873d1198941198873

=

119887119894

119891

4B1198941198874d1198941198874 (44b)

000 050 100 150 200 250

060

040

020

000

Disp

lace

men

tw (c

m)

Time (ms)

K = 01 linearK = 01 nonlinearK = 100 linear

K = 100 nonlinearFull con linearFull con nonlinear

minus020

Figure 7 Time history of the deflection 119908 (cm) at the middle pointA of the free edge of the stiffened plate of Example 1 for variousvalues of connectorsrsquo stiffness

Equations (30) (35a)ndash(35c) (37) and (41a)ndash(41d)together with continuity conditions (42a) (42b) (43a)(43b) (44a) and (44b) constitute a nonlinear system ofalgebraic equations with respect to q1199091 q1199092 q1199101 q1199102 q1199111 andq1199112 (interface forces) and d119901119894 (119894 = 1 2 3) and d119894

119887119895(119894 = 1 sdot sdot sdot 119868)

(119895 = 1 2 3 4) (generalized unknown vectors of the plate andthe beams) This system is solved using iterative numerical


000015030045060075

9

6

3

0

1815

12

9

6

3

0

(a) 119908119901max = 0725 cm at 119905 = 147msec of linear AEM analysis for119870 = 0005

000010020030040050

9

6

3

0

1815

12

9

6

3

0

(b) 119908119901max = 0501 cm at 119905 = 123msec of nonlinear AEM analysisfor119870 = 0005

000010020030040050060

9

6

3

0

1815

12

9

6

3

0

(c) 119908119901max = 0521 cm at 119905 = 122msec of linear AEM analysis for119870 = 10

00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0

(d) 119908119901max = 0419 cm at 119905 = 111msec of nonlinear AEManalysis for119870 = 10

00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000

(e) 119908119901max = 0495 cm at 119905 = 120msec of linear AEM analysis for119870 =100

00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000

(f) 119908119901max = 0401 cm at 119905 = 108msec of nonlinear AEM analysisfor119870 = 100

Figure 8 Contour lines of 119908119901 of the stiffened plate of Example 1 at the time of maximum transverse deflection ignoring (a c e) or takinginto account geometrical nonlinearities (b d f)


0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001

(a) 119906119901max = 45 times 10minus3 cm at 119905 = 123msec for119870 = 0005

00005

00005

0 3 6 9 12 15 180

3

6

900005minus00005minus00015minus00025minus00035minus00045minus00055minus00065minus00075minus00085minus00095

minus0002

minus00045

minus000

2

(b) V119901max = 93 times 10minus3 cm at 119905 = 123msec for119870 = 0005

00005

00005

0 3 6 9 12 15 180

3

6

9

500E minus 004150E minus 003250E minus 003350E minus 003450E minus 003

minus450E minus 003minus350E minus 003minus250E minus 003minus150E minus 003minus500E minus 004

minus0002

(c) 119906119901max = 43 times 10minus3 cm at 119905 = 111msec for119870 = 10

00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005

minus00065minus00055minus00045minus00035minus00025minus00015minus00005

minus00015

(d) V119901max = 64 times 10minus3 cm at 119905 = 111msec for119870 = 10

0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015

(e) 119906119901max = 40 times 10minus3 cm at 119905 = 108msec for119870 = 100

00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050

minus00060minus00050minus00040minus00030minus00020minus00010

minus0001

(f) V119901max = 60 times 10minus3 cm at 119905 = 108msec for119870 = 100

Figure 9 Contour lines of 119906119901 V119901 of the stiffened plate of Example 1 at the time of maximum transverse deflection 119908119901 taking into accountgeometrical nonlinearities

x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)


methods [52] It is worth noting here that Y1 Y2 are position119871 times 119872 matrices which convert the matrices B119901119894 (119894 = 1 2 3)B1199013119909 and B1199013119910 into corresponding ones with dimensions119871 times 119872 appropriately referring to the nodal points of the twointerface lines 119891

119894

119895=1 119891119894119895=2

respectively Moreover K1198941199091 K1198941199092

K1198941199101 and K119894

1199102are diagonal 119871 times 119871matrices including the value

of the flexibilities 1119896119894

119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary

distributed shear connectors along the 119909119894 and 119910

119894 directionsrespectively

Finally it is worth noting that beams placed along theboundary of the plate are treated as every other stiffeningbeam since the lines of action of the integrated interface force


3000

2000

1000

000

000 040 080 120 160 200

Super finite elementFinite strip methodSpline finite strip method

Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)

Figure 11 Time history of linear (a) and nonlinear (b) response of the deflection 119908 (mm) of the half panel center A of the stiffened plate ofExample 2

x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)


components 119902119894

119909119895 119902119894119910119895 and 119902

119894

119911119895(119895 = 1 2) will also be internal

ones taking special care during the numerical evaluationof the line integrals in order to avoid their ldquonear singularintegral behaviourrdquo According to this boundary elementsthat are very close to each other (distance smaller than theirlength) are divided in subelements in each of which Gaussintegration is applied [51]

4 Numerical Examples

On the basis of the analytical and numerical procedurespresented in the previous sections a FORTRAN programhas been written and representative examples have beenstudied to demonstrate the effectiveness wherever possiblethe accuracy and the range of applications of the proposedmethod It is noted that the term ldquolinear analysisrdquo appearingin all of the following sections refers to the solution of thepreviously obtained system of equations neglecting all of thenonlinear terms

010m

001m

001m

0006m

010m

Figure 13 Cross-section of the stiffener of Example 3

Example 1 A rectangular plate (ℎ119901 = 02 cm 119864119901 = 119864119887 =

3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)

with dimensions 119871119909 times 119871119910 = 18 cm times 9 cm subjected to asuddenly applied uniform load 119892 = 50 kNm2 and stiffenedby a rectangular beam of 05 cm height and 10 cm widtheccentrically placed with respect to the plate center line(Figure 4) has been studied The plate is clamped along itssmall edges while the rest of the edges are free according tothe transverse and inplane boundary conditions


Table 1 Torsion warping constants and primary warping function at the interface lines of the stiffening beam of Example 1

ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912

(cm2)05 2859119864 minus 02 3175119864 minus 04 minus4979119864 minus 02 4979119864 minus 02

3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)

FEM-nonlinear analysis FEM-linear analysis

analysis analysisProposed method-nonlinear Proposed method-linear

Figure 14 Time history of the maximum displacement 119908 (mm) of the stiffened plate of Example 3

000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024

(a) 119908119901max = 245mm at 119905 = 259msec of AEM linear analysis

000000013400029700046000624000787000950011100128001440016001770019300209002260024200258

(b) 119908119901max = 258mmat 119905 = 265msec of FEM linear analysis

001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022

(c) 119908119901max = 230mm at 119905 = 254msec of AEM nonlinear analysis

00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226

(d) 119908119901max = 226mm at 119905 = 240msec of FEM nonlinearanalysis

Figure 15 Contour lines of deflection119908119901 of the stiffened plate of Example 3 employing the present study (a c) and a FEM [25] solution usingsolid elements (b d) at the time of maximum transverse deflection


2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2

(a) 119872119909 at 119905 = 259msec of AEM linear analysis

2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2

(b) 119872119909 at 119905 = 254msec of AEM linear analysis

5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060

(c) 119872119910 at 119905 = 259msec of AEM nonlinear analysis

5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500

350025001500500minus500minus1500minus2500minus3500minus4500

minus5 minus5 minus5 minus5minus5

(d) 119872119910 at 119905 = 254msec of AEM nonlinear analysis

Figure 16 Contour lines of moments 119872119909 and 119872119910 (kNmm) at the time of maximum transverse displacement

Table 2 Maximum deflection 119908119901 (cm) at point A of the first cycleof motion of the stiffened plate for various values of119870 of Example 1

119870 Linear analysis Nonlinear analysisFull connection 0484 03911000 0488 0395100 0495 040110 0521 041901 0575 0449001 0696 04940005 0725 0501

In Table 1 the torsion 119868119905 and warping 119862119878 constants of thebeam cross-section and the values of the primary warpingfunction (120593

119875

119878)119895(119895 = 1 2) at the nodes of the two interface

lines are presentedThe connection between the slab and the beam is

accomplished using a linear distribution of shear connectorsalong each interface The adopted relationship for the shearconnectorsrsquo stiffness is given as

119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)

where 119870 is a dimensionless magnification factor In Table 2the obtained maximum deflections 119908119901 of the first cycle ofmotion of the stiffened plate at the middle of the free edgeA are shown for various values of the factor 119870 performing

Table 3 Torsion warping constants and primary warping functionat the interface lines of the stiffening beam

119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912

(m2)6984119864 minus 08 3374119864 minus 09 minus994119864 minus 04 994119864 minus 04

either a linear or a nonlinear analysis In Figure 5 the timehistories of the deflection 119908119901(119905) at the middle point Aof the free edge of the examined stiffened plate for thecase of full connection between the plate and the beamtaking into account or ignoring geometrical nonlinearitiesare presented as compared with those obtained from FEMsolutions employing 8-nodedhexahedral solid finite elements[25]

Moreover in Figure 6 the contour lines of the deflection119908119901 of the stiffened plate (full connection) at the time ofmaximum transverse displacement are presented as com-pared with those obtained from the aforementioned FEMsolutions ignoring (Figures 6(a) and 6(b)) or taking intoaccount (Figures 6(c) and 6(d)) geometrical nonlinearitiesIn order to demonstrate the influence of the shear connectorsin the dynamic behaviour of the stiffened plate in Figure 7the time histories of the deflection 119908119901(119905) at the same point Aare presented for various values of the factor 119870 performingeither a linear or a nonlinear analysis In Figure 8 the contourlines of the deflections 119908119901 of the stiffened plate at the timeof maximum displacement are presented for various cases ofconnectorsrsquo stiffness ignoring (Figures 8(a) 8(c) and 8(e)) ortaking into account (Figures 8(b) 8(d) and 8(f)) geometricalnonlinearities Moreover in Figure 9 the contour lines of


the displacements 119906119901 V119901 of the stiffened plate at the time ofmaximum deflection 119908119901 are presented employing nonlineardynamic analysis

Example 2 The linear and nonlinear response of a rectan-gular plate having a central stiffener as shown in Figure 10has been studied (119864 = 689GPa V = 030 120588 = 267 tnm3)The plate is subjected to a suddenly applied uniform load of119892 = 300 kPa while its boundaries are simply supported withrestraint against inplane motion In Figure 11 the linear andnonlinear time history response of the deflection of the halfpanel center A is depicted In this figure the obtained resultsare compared with those presented by Jiang and Olson [53]employing conventional finite strip method by Koko [54]employing super finite element method and by Sheikh andMukhopadhyay [22] employing the spline finite stripmethod

As it can easily be observed there is significant agree-ment between the results concerning the linear dynamicanalysis while deviation between the different types ofanalysis is observed in nonlinear dynamic analysis whichhas been already mentioned by the previous investigators[22 53 54]

Example 3 A rectangular plate with dimensions 119871119909 times 119871119910 =

18m times 09m (Figure 12) stiffened by an 119868 cross-sectionbeam (Figure 13) eccentrically placedwith respect to the platecentre line clamped along all its edges and subjected to asuddenly applied uniform load 119892 = 750 kNm2 has beenstudied (ℎ119901 = 002m 119864119901 = 119864119887 = 689 times 10

6 kNm2 120588119901 =

120588119887 = 267 tnm3 ]119901 = ]119887 = 03) In Table 3 the torsion 119868119905

and warping 119862119878 constants of the beam cross-section and thevalues of the primary warping function (120593

119875

119878)119895(119895 = 1 2) at the

nodes of the two interface lines are presented In Figure 14the time histories of the maximum deflection 119908119901(119905) of theplate for the cases of linear and nonlinear dynamic analysisare presented as compared with those obtained from FEMsolutions using 8-noded hexahedral solid finite elements [25]Moreover in Figure 15 the contour lines of the displacement119908119901 of the stiffened plate at the time of maximum transversedisplacement are presented as compared with those obtainedfrom the aforementioned FEM solutions ignoring (Figures15(a) and 15(b)) or taking into account (Figures 15(c) and15(d)) geometrical nonlinearities The convergence of theresults between the two methods is noteworthy Finally inFigure 16 the contour lines of the corresponded moments119872119909119872119910 at the time of maximum transverse displacement arepresented

Taking into account the aforementioned convergenceof the results between the two methods (AEM FEM)the importance of the reduction of calculation time whenemploying the proposedmethod should be highlightedMorespecifically for the determination of the beamrsquos response tothe dynamic loading a personal computer of 8Gb RAM andIntel I7 processor of 4 cores withmaximum clock speed equalto 38GHz was used The time step used for the nonlinearanalysis is 1 microsecond (120583sec) and for a full circle responsethe calculation time employing the proposed method was

20 minutes as compared with FEM analysis which lasted 50minutes

5 Concluding Remarks

A general solution for the geometrically nonlinear dynamicanalysis of plates stiffened by arbitrarily placed parallel beamsof arbitrary doubly symmetric cross-section subjected toarbitrary loading is presented The proposed model takesinto account the nonuniform distribution of the interfaceshear forces and the nonuniform torsional response of thebeams The main conclusions that can be drawn from thisinvestigation are as follows

(a) The proposed model permits the dynamic responseof stiffened plates subjected to arbitrary loadingwhile both the number and the placement of theparallel beams are also arbitrary (eccentric beams areincluded) The plate and the beams are supportedby the most general boundary conditions includingelastic support or restraint

(b) The adopted model permits the evaluation of thelongitudinal and transverse inplane shear forces atthe interfaces between the plate and the beams inthe geometrically nonlinear analysis of the stiffenedplate the knowledge of which is very important in thedesign of shear connectors in stiffened structures

(c) The nonuniform torsion in which the stiffeningbeams are subjected is taken into account by solvingthe corresponding problem and by comprehendingthe arising twisting andwarping in the correspondingdisplacement continuity conditions The distributedwarpingmoment arising from the nonuniform distri-bution of longitudinal inplane forces is also taken intoaccount

(d) The accuracy of the results and the validity of theproposed model are noteworthy

(e) The influence of geometrical nonlinearity on thedeformation of the examined stiffened plates isremarkable In the case of immovable inplane bound-ary conditions the displacements decrement can beverified

(f) The increment of the deflection with the decrementof the connectorsrsquo stiffness is easily verified while thisdecrement results in a more pronounced influence ofthe geometrical nonlinearity on the response of thestiffened plate

(g) The developed procedure retains most of the advan-tages of a BEM solution over a FEM approachalthough it requires domain discretization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Acknowledgment

This work has been funded by the State Scholarships Founda-tion of Greece as a part of postdoctoral research project

References

[1] T Mizusawa T Kajita and M Naruoka ldquoVibration of stiffenedskew plates by using B-spline functionsrdquo Computers and Struc-tures vol 10 no 5 pp 821ndash826 1979

[2] R B Bhat ldquoVibrations of panels with non-uniformly spacedstiffenersrdquo Journal of Sound and Vibration vol 84 no 3 pp449ndash452 1982

[3] D W Fox and V G Sigillito ldquoBounds for frequencies of ribreinforced platesrdquo Journal of Sound and Vibration vol 69 no4 pp 497ndash507 1980

[4] D W Fox and V G Sigillito ldquoBounds for eigenfrequencies of aplate with an elastically attached reinforcing ribrdquo InternationalJournal of Solids and Structures vol 18 no 3 pp 235ndash247 1982

[5] Y K Lin and B K Donaldson ldquoA brief survey of transfermatrixtechniques with special reference to the analysis of aircraftpanelsrdquo Journal of Sound and Vibration vol 10 no 1 pp 103ndash143 1969

[6] G Aksu and R Ali ldquoFree vibration analysis of stiffened platesusing finite difference methodrdquo Journal of Sound and Vibrationvol 48 no 1 pp 15ndash25 1976

[7] G Aksu ldquoFree vibration analysis of stiffened plates by includingthe effect of inplane inertiardquo Journal of Applied Mechanics vol49 no 1 pp 206ndash212 1982

[8] MMukhopadhyay ldquoVibration and stability analysis of stiffenedplates by semi-analytic finite difference method part I consid-eration of bending displacements onlyrdquo Journal of Sound andVibration vol 130 no 1 pp 27ndash39 1989

[9] MMukhopadhyay ldquoVibration and stability analysis of stiffenedplates by semi-analytic finite difference method part II consid-eration of bending and axial displacementsrdquo Journal of Soundand Vibration vol 130 no 1 pp 41ndash53 1989

[10] M D Olson and C R Hazell ldquoVibration studies on someintegral rib-stiffened platesrdquo Journal of Sound andVibration vol50 no 1 pp 43ndash61 1977

[11] A Mukherjee and M Mukhopadhyay ldquoFinite element freevibration of eccentrically stiffened platesrdquo Computers amp Struc-tures vol 30 no 6 pp 1303ndash1317 1988

[12] A Mukherjee and M Mukopadhyay ldquoRecent advances in thedynamic behavior of stiffened platesrdquo The Shock and VibrationDigest vol 27 article 6 1989

[13] R S Srinivasan and K Munaswamy ldquoDynamic responseanalysis of stiffened slab bridgesrdquo Computers amp Structures vol9 no 6 pp 559ndash566 1978

[14] E J Sapountzakis and J T Katsikadelis ldquoDynamic analysisof elastic plates reinforced with beams of doubly-symmetricalcross sectionrdquoComputational Mechanics vol 23 no 5 pp 430ndash439 1999

[15] E J Sapountzakis and V G Mokos ldquoAn improved model forthe dynamic analysis of plates stiffened by parallel beamsrdquoEngineering Structures vol 30 no 6 pp 1720ndash1733 2008

[16] E J Sapountzakis andVGMokos ldquoShear deformation effect inthe dynamic analysis of plates stiffened by parallel beamsrdquo ActaMechanica vol 204 no 3-4 pp 249ndash272 2009

[17] R S Srinivasan and S V Ramachandran ldquoLinear and nonlinearanalysis of stiffened platesrdquo International Journal of Solids andStructures vol 13 no 10 pp 897ndash912 1977

[18] S R Rao A H Sheikh and M Mukhopadhyay ldquoLarge-amplitude finite element flexural vibration of platesstiffenedplatesrdquo Journal of the Acoustical Society of America vol 93 no6 pp 3250ndash3257 1993

[19] M M Hegaze ldquoNonlinear dynamic analysis of stiffened andunstiffened laminated composite plates using a high orderelementrdquo in Proceedings of the 13th International Conference onAerospace SciencesampAviation Technology (ASAT rsquo09) vol 26 282009

[20] M Kolli and K Chandrashekhara ldquoNon-linear static anddynamic analysis of stiffened laminated platesrdquo InternationalJournal of Non-Linear Mechanics vol 32 no 1 pp 89ndash101 1997

[21] Z Qingjle L Shiqi and Z Jijia ldquoNonlinear dynamic behaviorof stiffened plate under instantaneous loadingrdquo Computers ampStructures vol 40 no 6 pp 1351ndash1356 1991

[22] A H Sheikh and M Mukhopadhyay ldquoLinear and nonlineartransient vibration analysis of stiffened plate structuresrdquo FiniteElements in Analysis and Design vol 38 no 6 pp 477ndash5022002

[23] T Zhang T-G Liu Y Zhao and J-Z Luo ldquoNonlinear dynamicbuckling of stiffened plates under in-plane impact loadrdquo Journalof Zhejiang University Science vol 5 no 5 pp 609ndash617 2004

[24] A Karimin and M Belhaq ldquoEffect of stiffener on nonlinearcharacteristic behavior of a rectangular plate a single modeapproachrdquo Mechanics Research Communications vol 36 no 6pp 699ndash706 2009

[25] Siemens PLM Software Inc ldquoNX Nastran Userrsquos Guiderdquo 2008[26] Y F Wu R Xu and W Chen ldquoFree vibrations of the partial-

interaction composite members with axial forcerdquo Journal ofSound and Vibration vol 299 no 4-5 pp 1074ndash1093 2007

[27] R Xu and Y Wu ldquoStatic dynamic and buckling analysis ofpartial interaction composite members using Timoshenkosbeam theoryrdquo International Journal of Mechanical Sciences vol49 no 10 pp 1139ndash1155 2007

[28] R Xu and Y F Wu ldquoTwo-dimensional analytical solutionsof simply supported composite beams with interlayer slipsrdquoInternational Journal of Solids and Structures vol 44 no 1 pp165ndash175 2007

[29] R Q Xu and Y-F Wu ldquoFree vibration and buckling of com-posite beams with interlayer slip by two-dimensional theoryrdquoJournal of Sound and Vibration vol 313 no 3ndash5 pp 875ndash8902008

[30] U A Girhammar and D Pan ldquoDynamic analysis of compositememberswith interlayer sliprdquo International Journal of Solids andStructures vol 30 no 6 pp 797ndash823 1993

[31] C Adam R Heuer and A Jeschko ldquoFlexural vibrations ofelastic composite beams with interlayer sliprdquo Acta Mechanicavol 125 no 1ndash4 pp 17ndash30 1997

[32] U A Girhammar D H Pan and A Gustafsson ldquoExactdynamic analysis of composite beams with partial interactionrdquoInternational Journal of Mechanical Sciences vol 51 no 8 pp565ndash582 2009

[33] U A Girhammar and V K A Gopu ldquoComposite beam-columns with interlayer slipmdashexact analysisrdquo Journal of Struc-tural Engineering vol 119 no 4 pp 1265ndash1282 1993

[34] U A Girhammar and D H Pan ldquoExact static analysis ofpartially composite beams and beam-columnsrdquo InternationalJournal of Mechanical Sciences vol 49 no 2 pp 239ndash255 2007

[35] V A Oven I W Burgess R J Plank and A A Abdul WalildquoAn analytical model for the analysis of composite beams withpartial interactionrdquo Computers and Structures vol 62 no 3 pp493ndash504 1997


[36] S Schnabl I Planinc M Saje B Cas and G Turk ldquoAnanalytical model of layered continuous beams with partialinteractionrdquo Structural Engineering and Mechanics vol 22 no3 pp 263ndash278 2006

[37] J B M Sousa Jr C E M Oliveira and A R da SilvaldquoDisplacement-based nonlinear finite element analysis of com-posite beam-columns with partial interactionrdquo Journal of Con-structional Steel Research vol 66 no 6 pp 772ndash779 2010

[38] G Ranzi A DallrsquoAsta L Ragni and A Zona ldquoA geometricnonlinear model for composite beams with partial interactionrdquoEngineering Structures vol 32 no 5 pp 1384ndash1396 2010

[39] E J Sapountzakis andV GMokos ldquoAn improvedmodel for theanalysis of plates stiffened by parallel beams with deformableconnectionrdquo Computers and Structures vol 86 no 23-24 pp2166ndash2181 2008

[40] J A Dourakopoulos and E J Sapountzakis ldquoNonlineardynamic analysis of plates stiffened by parallel beams withdeformable connectionrdquo in Proceedings of the 4th ECCOMASThematic Conference on Computational Methods in StructuralDynamics and Earthquake Engineering (COMPDYN rsquo13) KosIsland Greece June 2013

[41] J T Katsikadelis ldquoThe analog equation method A boundary-only integral equationmethod for nonlinear static and dynamicproblems in general bodiesrdquoTheoretical and AppliedMechanicsvol 27 pp 13ndash38 2002

[42] V Z Vlasov Thin-Walled Elastic Beams Israel Program forScientific Translations 1961

[43] E J Sapountzakis and V G Mokos ldquoAnalysis of plates stiffenedby parallel beamsrdquo International Journal for Numerical Methodsin Engineering vol 70 no 10 pp 1209ndash1240 2007

[44] E Ramm and T J Hofmann ldquoStabtragwerke Der Ingenieur-baurdquo in Band BaustatikBaudynamik G Mehlhorn Ed Ernstamp Sohn Berlin Germany 1995

[45] H Rothert and V Gensichen Nichtlineare Stabstatik SpringerBerlin Germany 1987

[46] E J Sapountzakis and V G Mokos ldquoWarping shear stresses innonuniform torsion by BEMrdquo Computational Mechanics vol30 no 2 pp 131ndash142 2003

[47] E J Sapountzakis and I CDikaros ldquoLarge deflection analysis ofplates stiffened by parallel beams with deformable connectionrdquoJournal of Engineering Mechanics vol 138 no 8 pp 1021ndash10412012

[48] J T Katsikadelis and A E Armenakas ldquoA new boundaryequation solution to the plate problemrdquo Journal of AppliedMechanics vol 56 no 2 pp 364ndash374 1989

[49] E J Sapountzakis and I C Dikaros ldquoLarge deflection analysisof plates stiffened by parallel beamsrdquoEngineering Structures vol35 pp 254ndash271 2012

[50] J T Katsikadelis and A E Armenakas ldquoNumerical evaluationof double integrals with a logarithmic or Cauchy-type singular-ityrdquo Journal of Applied Mechanics vol 50 no 3 pp 682ndash6841983

[51] J T Katsikadelis Boundary Elements Theory and ApplicationsElsevier Amsterdam The Netherlands 2002

[52] K E Brenan S L Campbell and L R Petzold NumericalSolution of Initial-Value Problems in Differential-Algebraic Equa-tions Classics in AppliedMathematics Society for Industrial andApplied Mathematics Philadelphia Pa USA 1996

[53] J Jiang and M D Olson ldquoNonlinear dynamic analysis of blastloaded cylindrical shell structuresrdquo Computers and Structuresvol 41 no 1 pp 41ndash52 1991

[54] T S Koko Super finite elements for nonlinear static and dynamicanalysis of stiffened plate structures [PhD thesis] Department ofCivil Engineering University of British Columbia VancouverCanada 1990

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Navigation and Observation



DistributedSensor Networks



analysis of stiffened plate systems According to this theintegral equation method has been applied for the largedeflection analysis of clamped laterally loaded skew plateswith stiffener parallel to the skew directions [17] Moreoverthe finite element method has been widely used for thefree large-amplitude flexural vibration of stiffened [18] andlaminated plates [19 20] and for forced vibrations underinstantaneous loading taking into account both geomet-ric and material nonlinearities [21] The spline finite stripmethod has also been applied for the examination of non-linear transient vibration of stiffened plates [22] and theGalerkin method has been employed for the solution ofdiscrete equations derived after the application of Fourierseries for the deflection of the plate [23 24] Neverthelessin all of the aforementioned research efforts restrictions areencountered arising from the ignorance of the tractionsin the fictitious plate-beams interfaces of the nonuniformdistribution of the interface transverse shear force or of thenonuniform torsional response of the beams

The behavior of composite slab-and-beam structure isaffected significantly by the deformability of the shear con-nection Therefore much research effort has been done con-cerning the linear dynamic analysis of composite beams withdeformable connectionsWu et al examined the free vibrationof partial interaction composite beams with axial force andproposed an approximate simple expression to predict thefundamental frequency of the partial-interaction compositemembers with axial force [26] Xu and Wu investigated thestatic dynamic and buckling behavior of partial-interactioncomposite members by taking into account the influences ofrotary inertia and shear deformations [27] Moreover theyalso employed the state-space method in order to developan exact two-dimensional plane stress model of compositebeams with interlayer slips [28 29]

On the other hand less research effort has been made onthe forced vibration of partial-interaction composite beamsGirhammar and Pan [30] investigated the dynamic behaviorof partial-interaction composite beams without axial forceand presented the orthogonality relation of vibration modesThey also obtained the analytical expressions for the transientresponse of composite beams under impulsive and step loadsAdam et al [31] decomposed the dynamic response intotwo parts the quasi-static component solved by integratingthe influence function and the complementary componentsolved by themode superposition Recently Girhammar et al[32] employed Hamiltonrsquos principle to derive the governingequations for dynamic problems of composite beams undergeneralized boundary conditions At the same time theystudied the forced vibration of composite beams by virtueof mode superposition However in the literature largedeflection analysis of partial interaction has received limitedattention and focused especially on the static behavior of thestructure [33ndash38]

In this paper a general solution to the geometricallynonlinear dynamic analysis of plates stiffened by arbitrarilyplaced parallel beams of arbitrary doubly symmetric cross-section with deformable connections subjected to arbitrarydynamic loading is presented The solution is based on thestructural model proposed by Sapountzakis and Mokos in

[39] andDourakopoulos and Sapountzakis in [40] accordingto which the stiffening beams are isolated from the plateby sections in the lower outer surface of the plate makingthe hypothesis that the plate and the beams can slip inall directions of the connection without separation (ieuplift is neglected) and taking into account the arisingtractions in all directions at the fictitious interfaces Due tothe slip relative inplane displacements between beam andplate arise assuming however that they are continuouslyin contact and taking into account the arising tractions inall directions at the fictitious interfaces These tractions areintegrated with respect to each half of the interface widthresulting in two interface lines along which the loading ofthe beams and the additional loading of the plate are definedThe unknown distribution of the aforementioned integratedtractions is established by applying continuity conditions inall directions at the two interface lines taking into accounttheir relationship with the interface slip through the shearconnector stiffness Any distribution of connectors in eachdirection of the interfaces can be handled The utilization oftwo interface lines for each beam enables the nonuniformtorsional response of the beams to be taken into account asthe angle of twist is indirectly equatedwith the correspondingplate slope Six boundary value problems are formulated andsolved using the analog equationmethod (AEM) [41] a BEM-based methodThe essential features and novel aspects of thepresent formulation are summarized as follows

(i) The adopted model permits the evaluation of theinplane shear forces (longitudinal and transverse) atthe interfaces in both directions taking into accountthe influence of interface slip the knowledge of whichis very important in the design of shear connectors inplate-beam structures

(ii) Utilization of two interface lines permits the nonuni-form distribution of the inplane forces along theinterface width to be taken into account

(iii) Both free and forced transverse vibrations are consid-ered taking into account geometric nonlinearities

(iv) The stiffened plate is of arbitrary shape and is sub-jected to arbitrary dynamic loading while both thenumber and the placement of the parallel beams arealso arbitrary (eccentric beams are included)

(v) The cross-section of the stiffening beams is an arbi-trary doubly symmetric thin or thick-walled oneTheformulation does not stand on the assumption of athin-walled structure according to which torsionaland warping rigidities can be evaluated employingclosed-form expressions without significant error aslong as the cross-section thickness is small [42]Therefore in this paper the cross-sectionrsquos torsionaland warping rigidities are evaluated ldquoexactlyrdquo in anumerical sense employing the BEM (requiring onlyboundary discretization for the cross-sectional anal-ysis)

(vi) The plate and the beams are supported by the mostgeneral boundary conditions including elastic sup-port or restraint


s

Corner 1t

n

(Γ1)

(Γ0)

(ΓK)li

lI Beam I

Beam i

Beam 1 l1

Corner ky p

z wp x up

bIf

bif

b1f

xi

zi

yi

Cihib

(Ω)

fij=1

fij=2


Γ =K

⋃j=1

Γj










119887and length 119897


119894


119894

119887









119894

119909119895 119902119894119910119895 and 119902

119894






g

aa

(a)

O

g

Ep p 120588p

hp

fij=1 fi

j=2

Ω

(Γ)

qizjqiyjx up

y p

z wp

qixj

qixj

qiyjqizj

xi uib

zi wib

hib

bif

Eib ib 120588ib

yi ib

bif4



(b)







of the plate



of the plate


119911119895

O

x up

z wp

y p

zi wib

yi ibOi

mibxj

mibzj mi

byj

qiyj

qizj

qixj

xi uib

mipy2

mipy1

mipx1 mi

px2

fij=2

fij=1

qiz1qiy1

qiy2

qiz2

qix2qix1

mibwj





plate 119898119894119901119910119895

= 119902119894





plate 119898119894119901119909119895

= 119902119894






119894

119910119895along the


119894

119911119895along the


119887119910119895= 119902119894

119909119895119890119894

119911119895along


119890119894


119894


axis 1198901198941199111

= 119890119894

1199112= minusℎ119894



= minus119902119894

119909119895119890119894

119910119895along


119890119894


119894


axis 1198901198941199101

= minus119887119894

1198914 and 119890

119894

1199102= 119887119894



= 119902119894

119911119895119890119894

119910119895minus 119902119894

119910119895119890119894

119911119895



119911119895and 119890119894


119894

119910119895and

119902119894


1199111=

119890119894

1199112= minusℎ119894

1198872 and 119890

119894

1199101= minus119887119894

1198914 and 119890

119894

1199102= 119887119894

1198914 are the


119887119908119895= minus119902119894

119909119895(120593119875119894

119878)119891119895





119875119894





119906119901 (119909 119910 119911 119905) = 119906119901 (119909 119910 119905) minus 119911

120597119908119901 (119909 119910 119905)

120597119909 (1a)

V119901 (119909 119910 119911 119905) = V119901 (119909 119910 119905) minus 119911

120597119908119901 (119909 119910 119905)

120597119910 (1b)

119908119901 (119909 119910 119911 119905) = 119908119901 (119909 119910 119905) (1c)



120576119909119909 =

120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

(2a)

120576119910119910 =

120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

(2b)

120574119909119910 =

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119909

120597119908119901

120597119910 (2c)

120576119911119911 = 120574119909119911 = 120574119910119911 = 0 (2d)


119878119909119909

119878119910119910

119878119909119910

=

[[[[[[[[[[

[

119864119901

(1 minus ]119901)2

119864119901]119901

(1 minus ]119901)2

0

119864119901]119901

(1 minus ]119901)2

119864119901

(1 minus ]119901)2

0

0 0

119864119901

2 (1 + ]119901)

]]]]]]]]]]

]

120576119909119909

120576119910119910

120574119909119910

(3)


119878119909119909 =

119864119901

(1 minus ]2119901)

times [

120597119906119901

120597119909minus 119911

1205972119908119901

1205971199092+

1

2(

120597119908119901

120597119909)

2

]

+]119901 [120597V119901120597119910

minus 119911

1205972119908119901

1205971199102+

1

2(

120597119908119901

120597119910)

2

]

(4a)

119878119910119910 =

119864119901

(1 minus ]2119901)

]119901 [120597119906119901

120597119909minus 119911

1205972119908119901

1205971199092+

1

2(

120597119908119901

120597119909)

2

]

+[

120597V119901120597119910

minus 119911

1205972119908119901

1205971199102+

1

2(

120597119908119901

120597119910)

2

]

(4b)

119878119909119910 =

119864119901

2 (1 + ]119901)(

120597119906119901

120597119910+

120597V119901120597119909

minus 2119911

1205972119908119901

120597119909120597119910+

120597119908119901

120597119909

120597119908119901

120597119910)

(4c)


119873119901119909 = 119862[

120597119906119901

120597119909+ ]119901

120597V119901120597119909

+1

2(

120597119908119901

120597119909)

2

+1

2]119901(

120597119908119901

120597119910)

2

]

(5a)

119873119901119910 = 119862[

120597V119901120597119909

+ ]119901120597119906119901

120597119909+

1

2(

120597119908119901

120597119910)

2

+1

2]119901(

120597119908119901

120597119909)

2

]

(5b)

119873119901119909119910 = 119862

1 minus ]1199012

(

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119910

120597119908119901

120597119909) (5c)

119872119901119909 = minus119863(

1205972119908119901

1205971199092+ ]119901

1205972119908119901

1205971199102) (5d)

119872119901119910 = minus119863(

1205972119908119901

1205971199102+ ]119901

1205972119908119901

1205971199092) (5e)

119872119901119909119910 = minus119863(1 minus ]119901)1205972119908119901

120597119909120597119910 (5f)


3

11990112(1 minus ]2

119901) are the




120597119873119901119909

120597119909+

120597119873119901119909119910

120597119910minus 120588119901ℎ119901119901 =

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119909119895120575 (119910 minus 119910119895)) (6a)

120597119873119901119910

120597119910+

120597119873119901119909119910

120597119909minus 120588119901ℎ119901V119901 =

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119910119895120575 (119910 minus 119910119895)) (6b)

minus

1205972119872119901119909

1205971199092minus 2

1205972119872119901119909119910

120597119909120597119910minus

1205972119872119901119910

1205971199102minus 119873119901119909

1205972119908119901

1205971199092

minus 2119873119901119909119910

1205972119908119901

120597119909120597119910minus 119873119901119910

1205972119908119901

1205971199102

+ 120588119901ℎ119901119901 minus 120588119901ℎ119901119901

120597119908119901

120597119909minus 120588119901ℎ119901V119901

120597119908119901

120597119910

minus

120588119901ℎ3

119901

12

1205972119901

1205971199092minus

120588119901ℎ3

119901

12

1205972119901

1205971199102

= 119892 minus

119868

sum

119894=1

[

[

2

sum

119895=1

(119902119894

119911119895+

120597119898119894

119901119909119895

120597119910+

120597119898119894

119901119910119895

120597119909minus 119902119894

119909119895

120597119908119894

119901119895

120597119909

minus119902119894

119910119895

120597119908119894

119901119895

120597119910)120575 (119910 minus 119910119895)

]

]

(6c)


119866119901ℎ119901 [nabla2119906119901 +

1 + ]1199011 minus ]119901

120597

120597119909(

120597119906119901

120597119909+

120597V119901120597119910

)

+(2

1 minus ]119901

1205972119908119901

1205971199092+

1205972119908119901

1205971199102)

120597119908119901

120597119909

+

1 + ]1199011 minus ]119901

1205972119908119901

120597119909120597119910

120597119908119901

120597119910] minus 120588119901ℎ119901119901

=

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119909119895120575 (119910 minus 119910119895))

(7a)

119866119901ℎ119901 [nabla2V119901 +

1 + ]1199011 minus ]119901

120597

120597119910(

120597119906119901

120597119909+

120597V119901120597119910

)

+ (2

1 minus ]119901

1205972119908119901

1205971199102+

1205972119908119901

1205971199092)

120597119908119901

120597119910

+

1 + ]1199011 minus ]119901

1205972119908119901

120597119909120597119910

120597119908119901

120597119909] minus 120588119901ℎ119901V119901

=

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119910119895120575 (119910 minus 119910119895))

(7b)

119863nabla4119908119901 minus 119862

times [(

120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

)

+]119901(120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

)]

times

1205972119908119901

1205971199092+ (1 minus ]119901)

sdot (

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119909

120597119908119901

120597119910)

1205972119908119901

120597119909120597119910

+ [(

120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

)

+]119901(120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

)] sdot

1205972119908119901

1205971199102

+ 120588119901ℎ119901119901 minus 120588119901ℎ119901119901

120597119908119901

120597119909minus 120588119901ℎ119901V119901

120597119908119901

120597119910

minus

120588119901ℎ3

119901

12

1205972119901

1205971199092minus

120588119901ℎ3

119901

12

1205972119901

1205971199102

= 119892 minus

119868

sum

119894=1

[

[

2

sum

119895=1

(119902119894

119911119895+

120597119898119894

119901119909119895

120597119910+

120597119898119894

119901119910119895

120597119909minus 119902119894

119909119895

120597119908119894

119901119895

120597119909

minus119902119894

119910119895

120597119908119894

119901119895

120597119910)120575 (119910 minus 119910119895)

]

]

(7c)


1198861199011119906119901119899 + 1198861199012119873119901119899 = 1198861199013 (8a)

1205731199011119906119901119905 + 1205731199012119873119901119905 = 1205731199013 (8b)

1205741199011119908119901 + 1205741199012119877119901119899 = 1205741199013 (8c)

1205751199011

120597119908119901

120597119899+ 1205751199012119872119901119899 = 1205751199013

(8d)

1205761119896119908119901 + 1205762119896

10038171003817100381710038171003817119879119908119901

10038171003817100381710038171003817119896= 1205763119896 1205762119896 = 0 (8e)


119908119901 (x 0) = 1199081199010 (x) (9a)

119901 (x 0) = 1199081199010 (x) (9b)




119877119901119899 = minus119863[120597


120597

120597119904(

1205972119908119901

120597119904120597119899minus 120581

120597119908119901

120597119904)]

+ 119873119901119899

120597119908119901

120597119899+ 119873119901119905

120597119908119901

120597119904

(10a)


1205972119908119901

1205971199042+ 120581

120597119908119901

120597119899)] (10b)



119879119908119901 = 119863(]119901 minus 1)(

1205972119908119901

120597119904120597119899minus 120581

120597119908119901

120597119904) (11)


1205741199012 = 1205741199013 = 1205751199012 = 1205751199013 = 0)





119906119894

119887(119909119894 119910119894 119911119894 119905) = 119906

119894

119887(119909119894 119905) minus 119910

119894120579119894

119887119911(119909119894 119905)

+ 119911119894120579119894

119887119910(119909119894 119905) +

120597120579119894

119887119909

120597119909119894120593119875119894

119878(119910119894 119911119894 119905)

(12a)

V119894119887(119909119894 119910119894 119911119894 119905) = V119894

119887(119909119894 119905) minus 119911

119894120579119894

119887119909(119909119894 119905) (12b)

119908119894

119887(119909119894 119910119894 119911119894 119905) = 119908

119894

119887(119909119894 119905) + 119910

119894120579119894

119887119909(119909119894 119905) (12c)

120579119894

119887119910(119909119894 119905) = minus

120597119908119894

119887(119909119894 119905)

120597119909119894 (12d)

120579119894

119887119911(119909119894 119905) =

120597V119894119887(119909119894 119905)

120597119909119894 (12e)

where 119906119894119887 V119894119887 and119908

119894



119887=

119906119894

119887(119909119894) V119894119887= V119894119887(119909119894) and 119908

119894

119887= 119908119894



= 120579119894

119887119910(119909119894) and 120579

119894

119887119911=

120579119894




119887119909(119909119894) regarded as the




120576119909119909 =120597119906119894

119887

120597119909119894+

1

2

[

[

(120597V119894119887

120597119909119894)

2

+ (120597119908119894

119887

120597119909119894)

2

]

]

(13a)

120574119909119911 =120597119908119894

119887

120597119909119894+

120597119906119894

119887

120597119911119894+ (

120597V119894119887

120597119909119894

120597V119894119887

120597119911119894+

120597119908119894

119887

120597119909119894

120597119908119894

119887

120597119911119894) (13b)

120574119909119910 =120597V119894119887

120597119909119894+

120597119906119894

119887

120597119910119894+ (

120597V119894119887

120597119909119894

120597V119894119887

120597119910119894+

120597119908119894

119887

120597119909119894

120597119908119894

119887

120597119910119894) (13c)

120576119910119910 = 120576119911119911 = 120574119910119911 = 0 (13d)


119873119894

119887= 119864119894

119887119860119894

119887[120597119906119894

119887

120597119909119894+

1

2((

120597V119894119887

120597119909119894)

2

+ (120597119908119894

119887

120597119909119894)

2

)] (14a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (14b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (14c)

119872119875119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894 (14d)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (14e)

where 119872119875119894



119887119908is the



119910and 119868

119894


119894

119878is the


119905and 119862

119894



119868119894

119905= intΩ

(1199101198942+ 1199111198942+ 119910119894 120597120593119875119894

119878

120597119911119894minus 119911119894 120597120593119875119894

119878

120597119910119894)119889Ω (15a)

119862119894

119878= intΩ

(120593119875119894

119878)2

119889Ω (15b)


minus120597119873119894

119887

120597119909119894+ 120588119894

119887Α119894

119887119894

119887=

2

sum

119895=1

119902119894

119909119895 (16a)

minus 119873119894

119887

1205972V119894119887

1205971199091198942+

1205972119872119894

119887119911

1205971199091198942

+ 120588119894

119887Α119894

119887V119894119887minus 120588119894

119887119868119894

119911

1205972V119894119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(16b)

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942minus

1205972119872119894

119887119910

1205971199091198942

+ 120588119894

119887Α119894

119887119894

119887minus 120588119894

119887119868119894

119910

1205972119894

119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(16c)

minus120597119872119894

119887119905

120597119909119894minus

1205972119872119894

119887119908

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(16d)


minus 119864119894

119887119860119894

119887(1205972119906119894

119887

1205971199091198942+

120597119908119894

119887

120597119909119894

1205972119908119894

119887

1205971199091198942

+120597V119894119887

120597119909119894

1205972V119894119887

1205971199091198942) + 120588

119894

119887Α119894

119887119894

119887

=

2

sum

119895=1

119902119894

119909119895

(17a)

119864119894

119887119868119894

119911

1205974V119894119887

1205971199091198944minus 119873119894

119887

1205972V119894119887

1205971199091198942minus 120588119894

119887119868119894

119911

1205972V1198871205971199092

+ 120588119894

119887Α119894

119887V119887 minus 120588

119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(17b)

119864119894

119887119868119894

119910

1205974119908119894

119887

1205971199091198944

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942

minus 120588119894

119887119868119894

119910

1205972119887

1205971199092+ 120588119894

119887Α119894

119887119887 minus 120588

119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(17c)

119864119894

119887119862119894

119878

1205974120579119894

119887119909

1205971199091198944

minus 119866119894

119887119868119894

119905

1205972120579119894

119887119909

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(17d)


119894 are given as

119886119894

1198871119906119894

119887+ 120572119894

1198872119873119894

119887= 120572119894

1198873 (18)

120573119894

1198871V119894119887+ 120573119894

1198872119877119894

119887119910= 120573119894

1198873 (19a)

120573119894

1198871120579119894

119887119911+ 120573119894

1198872119872119894

119887119911= 120573119894

1198873 (19b)

120574119894

1198871119908119894

119887+ 120574119894

1198872119877119894

119887119911= 120574119894

1198873 (20a)

120574119894

1198871120579119894

119887119910+ 120574119894

1198872119872119894

119887119910= 120574119894

1198873 (20b)

120575119894

1198871120579119894

119887119909+ 120575119894

1198872119872119894

119887119905= 120575119894

1198873 (21a)

120575119894

1198871

119889120579119894

119887119909

119889119909119894+ 120575119894

1198872119872119894

119887119908= 120575119894

1198873(21b)


119908119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22a)

119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22b)


119894

119887119910 120579119894


119894

119887119910 119877119894

119887119911

and 119872119894

119887119911 119872119894119887119910



119877119894

119887119910= 119873119894

119887

120597V119894119887

120597119909119894minus 119864119894

119887119868119894

119911

1205973V119894119887

1205971199091198943 (23a)

119877119894

119887119911= 119873119894

119887

120597119908119894

119887

120597119909119894minus 119864119894

119887119868119894

119910

1205973119908119894

119887

1205971199091198943 (23b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (24a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (24b)


and 119872119894

119887119905 119872119894119887119908


119872119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894minus 119864119894

119887119862119894

119878

1205973120579119894

119887119909

1205971199091198943 (25a)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (25b)

Finally 120572119894119887119896 120573119894

119887119896 120573119894

119887119896 120574119894

119887119896 120574119894

119887119896 120575119894

119887119896 120575119894

119887119896(119896 = 1 2 3) are func-




119894

119887 V119894119887119908119894119887 120579119894119887119909 1199021198941199091 1199021198941199101 1199021198941199111 119902119894

1199092

119902119894

1199102 and 119902

119894





119906119894

1199011minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199011

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894+

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198911

+119902119894

1199091

1198961198941199091

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=1)

(26a)

119906119894

1199012minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199012

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894minus

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198912

+119902119894

1199092

1198961198941199092

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=2)

(26b)


V1198941199011

minus V119894119887=

ℎ119901

2

120597119908119894

1199011

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199101

1198961198941199101

[1 minus1

2(120579119894

119887119909)2

]


119895=1)

(27a)

V1198941199012

minus V119894119887=

ℎ119901

2

120597119908119894

1199012

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199102

1198961198941199102

[1 minus1

2(120579119894

119887119909)2

]


119895=2)

(27b)


119908119894

1199011minus 119908119894

119887= minus

119887119894

119891

4120579119894


119894

119895=1) (28a)

119908119894

1199012minus 119908119894

119887=

119887119894

119891

4120579119894


119894

119895=2) (28b)

where (120593119875119894

119878)119891119895


119894

119895and 119896

119894

119909119895 119896119894119910119895



=

119896119894

119909119895(119904119894

119909119895) and 119896

119894

119910119895= 119896119894

119910119895(119904119894



119894



119875119894









nabla2119906119901119894 = 119901119901119894 (119909 119910 119905) (119894 = 1 2) (29a)

nabla41199061199013 = 1199011199013 (119909 119910 119905) (29b)






[

[

E11990111 0 00 E11990122 00 0 E11990133

]

]

d1199011d1199012d1199013

+

0D1198991198971199011

0D1198991198971199012

00

D1198991198971199013

=

01205721199013

01205731199013

001205741199013

1205751199013

(30)

where

E11990111 = [A1 H1 H20 D11990122 D11990123

] (31a)

E11990122 = [A1 H1 H20 D11990144 D11990145

] (31b)

E11990133 =[[[

[

A2 H1 H2 G1 G2A1 0 0 H1 H20 D11990178 D11990179 0 D1199017110 D11990188 D11990189 D119901810 0

]]]

]

(31c)



119901119894(119894 = 1 2) are 119873 times 1 and

D1198991198971199013


d119901119894 = p119901119894 u119901119894 u119901119894119899119879 (119894 = 1 2) (32a)


2u1199013)119899119879 (32b)


u119901119894 = (119906119901119894)1(119906119901119894)2

sdot sdot sdot (119906119901119894)119873119879

(119894 = 1 2 3)

(33a)

u119901119894119899 = (

120597119906119901119894

120597119899)

1

(

120597119906119901119894

120597119899)

2

sdot sdot sdot (

120597119906119901119894

120597119899)

119873

119879

(119894 = 1 2 3)

(33b)


21199061199013)1

(nabla21199061199013)2


119879

(33c)

(nabla2u1199013)119899

= (

120597nabla21199061199013

120597119899)

1

(

120597nabla21199061199013

120597119899)

2

sdot sdot sdot (

120597nabla21199061199013

120597119899)

119873

119879

(33d)


(119901119901119894)1(119901119901119894)2

sdot sdot sdot (119901119901119894)119872119879



u119901119894 = B119901119894d119901119894 (119894 = 1 2 3) (34a)u119901119894119909 = B119901119894119909d119901119894 (119894 = 1 2 3) (34b)

u119901119894119910 = B119901119894119910d119901119894 (119894 = 1 2 3) (34c)


u119901119894119909119909 = B119901119894119909119909d119901119894 (119894 = 1 2 3) (34d)

u119901119894119910119910 = B119901119894119910119910d119901119894 (119894 = 1 2 3) (34e)

u119901119894119909119910 = B119901119894119909119910d119901119894 (119894 = 1 2 3) (34f)


119866119901ℎ119901 [p1199011 +1 + ]1199011 minus ]119901

(B1199011119909119909d1199011 + B1199012119909119910d1199012)

+ 2

1 minus ]119901(B1199013119909119909d1199013)119889119892B1199013119909d1199013

+(B1199013119910119910d1199013)119889119892 sdot B1199013119909d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119910d1199013]

minus 120588119901ℎ119901B1199011d1199011 = Zq119909(35a)

119866119901ℎ119901 [p1199012 +1 + ]1199011 minus ]119901

(B1199011119909119910d1199011 + B1199012119910119910d1199012)

+ 2

1 minus ]119901(B1199013119910119910d1199013)119889119892B1199013119910d1199013

+ (B1199013119909119909d1199013)119889119892 sdot B1199013119910d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119909d1199013]

minus 120588119901ℎ119901B1199012d1199012 = Zq119910

(35b)

119863p1199013 minus 119862

times [(B1199011119909d1199011)119889119892B1199013119909119909d1199013

+1

2(B1199013119909d1199013)119889119892(B1199013119909d1199013)119889119892B1199013119909119909d1199013

+ ]119901(B1199012119910d1199012)119889119892B1199013119909119909d1199013 +1

2]119901(B1199013119910d1199013)119889119892

times (B1199013119910d1199013)119889119892B1199013119909119909d1199013]

+ (1 minus ]119901)

times [(B1199011119910d1199011)119889119892B1199013119909119910d1199013

+ (B1199012119909d1199012)119889119892B1199013119909119910d1199013

+(B1199013119909d1199013)119889119892 sdot (B1199013119910d1199013)119889119892B1199013119909119910d1199013]

+ [(B1199012119910d1199012)119889119892B1199013119910119910d1199013 +1

2(B1199013119910d1199013)119889119892

sdot (B1199013119910d1199013)119889119892B1199013119910119910d1199013

+ ]119901(B1199011119909d1199011)119889119892B1199013119910119910d1199013

+1

2]119901(B1199013119909d1199013)119889119892

sdot (B1199013119909d1199013)119889119892B1199013119910119910d1199013]

+ 120588119901ℎ119901B1199013d1199013 minus 120588119901ℎ119901(B1199011d1199011)119889119892B1199013119909d1199013


119901

12B1199013119909119909d1199013

minus

120588119901ℎ3

119901

12B1199013119910119910d1199013


+ (Zq119910)119889119892B1199013119910d1199013

(35c)

where q119909 = q1199091 q1199092119879 q119910 = q1199101 q1199102

119879 andq119911 = q1199111 q1199112


119894



119901119910119895= 119902119894

119909119895ℎ1199012 119898

119894

119901119909119895= 119902119894

119910119895ℎ1199012 respectively



119887 V119894119887 and 119908

119894

119887




119887(119909119894 119905) and V119894

119887(119909119894 119905) 119908119894





Let 119906119894

1198871= 119906119894

119887(119909119894 119905) 119906

119894

1198872= V119894119887(119909119894 119905) 119906

119894

1198873= 119908119894

119887(119909119894 119905)

and 119906119894

1198874= 120579119894




1205972119906119894

1198871

1205971199091198942

= 119901119894

1198871(119909119894 119905) (36a)

1205974119906119894

119887119895

1205971199091198944= 119901119894

119887119895(119909119894 119905) (119895 = 2 3 4) (36b)




119894





119887119895(120577119894) 119906119894119887119895119909

(120577119894) 119906119894119887119895119909119909

(120577119894) and 119906

119894

119887119895119909119909119909(120577119894)

(120577119894 = 0 119897119894) in terms of 119901119894



[[[

[

E11989411988711

0 0 00 E119894

119887220 0

0 0 E11989411988733

00 0 0 E119894

11988744

]]]

]

d1198941198871

d1198941198872

d1198941198873

d1198941198874

+

0D119894 1198991198971198871

00

D119894 1198991198971198872

00

D119894 1198991198971198873

000

=

0120572119894

1198873

00120573119894

1198873

00120574119894

1198873

00120575119894

1198873

(37)

where E11989411988711

and E11989411988722

E11989411988733

E11989411988744


1198871is

a 2 times 1 and D119894 119899119897119887119895


1198873and

120573119894

1198873 1205741198941198873 1205751198941198873



1198873and 120573

119894

1198873 120573119894

1198873 120574119894

1198873 120574119894

1198873 120575119894

1198873 120575119894


d1198941198871

= p1198941198871

u1198941198871

u1198941198871119909

119879

(38a)

d119894119887119895

= p119894119887119895 u119894119887119895

u119894119887119895119909

u119894119887119895119909119909

u119894119887119895119909119909119909

119879

(119895 = 2 3 4)

(38b)


u119894119887119895

= 119906119894

119887119895(0 119905) 119906

119894

119887119895(119897119894 119905)119879

(119895 = 1 2 3 4) (39a)

u119894119887119895119909

= 120597119906119894

119887119895(0 119905)

120597119909119894

120597119906119894

119887119895(119897119894 119905)

120597119909119894

119879

(119895 = 1 2 3 4) (39b)

u119894119887119895119909119909

= 1205972119906119894

119887119895(0 119905)

1205971199091198942

1205972119906119894

119887119895(119897119894 119905)

1205971199091198942

119879

(119895 = 2 3 4)

(39c)

u119894119887119895119909119909119909

= 1205973119906119894

119887119895(0 119905)

1205971199091198943

1205973119906119894

119887119895(119897119894 119905)

1205971199091198943

119879

(119895 = 2 3 4)

(39d)


119887119895=

(119901119894

119887119895)1

(119901119894

119887119895)2


119887119895)119871119879



119894



119894isin (0 119897

119894) and


u1198941198871

= B1198941198871d1198941198871 (40a)

u1198941198871119909

= B1198941198871119909

d1198941198871 (40b)

u119894119887119895

= B119894119887119895d119894119887119895 (119895 = 2 3 4) (40c)

u119894119887119895119909

= B119894119887119895119909

d119894119887119895 (119895 = 2 3 4) (40d)

u119894119887119895119909119909

= B119894119887119895119909119909

d119894119887119895 (119895 = 2 3 4) (40e)

u119894119887119895119909119909119909

= B119894119887119895119909119909119909

d119894119887119895 (119895 = 2 3 4) (40f)

where B1198941198871B1198941198871119909

are 119871 times (119871 + 4) and B119894119887119895B119894119887119895119909

B119894119887119895119909119909

B119894119887119895119909119909119909



x

y

a

a

Fr

Fr

ClCl

Lx = 18 cm

Ly=9

cm A

(a)

g

02 cmhb = 05 cm

1 cm 5 cm3 cm

Ly = 9 cm

(b)


Time (ms)

060

040

020

000

minus020

000 050 100 150 200 250

Disp

lace

men

tw (c

m)





minus 119864119894

119887119860119894

119887[p1198941198871

+ (B1198941198872119909

d1198941198872)119889119892B1198941198872119909119909

d1198941198872

+ (B1198941198873119909

d1198941198873)119889119892B1198941198873119909119909

d1198941198873]

+ 120588119894

119887Α119894

119887T1q1 = q119894

1199091+ q1198941199092

(41a)

119864119894

119887119868119894

119911p1198941198872

minus (N119894119887)119889119892B1198941198872119909119909

d1198941198872

minus 120588119894

119887119868119894

119911B1198941198872119909119909

d1198941198872

+ 120588119894

119887Α119894

119887B1198941198872d1198941198872

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198872119909

d1198941198872

= q1198941199101

+ q1198941199102

minus (B1198941198872119909

d1198941198872)119889119892

(q1198941199091

+ q1198941199092) minus X119894119887119910

(q1198941199091

+ q1198941199092)

(41b)

119864119894

119887119868119894

119910p1198941198873

minus (N119894119887)119889119892B1198941198873119909119909

d1198941198873

minus 120588119894

119887119868119894

119910B1198941198873119909119909

d1198941198873

+ 120588119894

119887Α119894

119887B1198941198873d1198941198873

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198873119909

d1198941198873

= q1198941199111

+ q1198941199112

minus (B1198941198873119909

d1198941198873)119889119892

(q1198941199091

+ q1198941199092) + X119894119887119911

(q1198941199091

+ q1198941199092)

(41c)

119864119894

119887119862119894

119878p1198941198874

minus 119866119894

119887119868119894

119905B1198941198874119909119909

d1198941198874

+ 120588119894

119887119868119894

119901B1198941198874d1198941198874

minus 120588119894

119887119862119894

119878B1198941198874119909119909

d1198941198874

= e1198941199101q1198941199111

+ e1198941199102q1198941199112

minus e1198941199111q1198941199101

minus e1198941199112q1198941199102

+ Χ119894

119887119908(q1198941199091

+ q1198941199092)

(41d)

where (N119894119887)119889119892


119887119911 X119894119887119910 X119894119887119908


119898119894119887119911119895 119898119894119887119908119895


1199101 e1198941199102 e1198941199111

e1198941199112


119894

119910119895 119890119894


119894

119911119895 119902119894

119910119895with



Y1B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y1B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

+

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198911

B1198941198874119909

d1198874

+ K1198941199091

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42a)


008

008

018

018

028

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036040044048


000067300301006080091501220153018402140245027603070337036803990430460491


008

008

008

00801

8018

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036


000064600236004780072009620120145016901930217024102660290314033803620387



Y2B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y2B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

minus

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198912

B1198941198874119909

d1198874

+ K1198941199092

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42b)

Y1B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y1B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199101

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43a)

Y2B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y2B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199102

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43b)

Y1B1199013d1199013 minus B1198941198873d1198941198873

= minus

119887119894

119891

4B1198941198874d1198941198874 (44a)

Y2B1199013d1199013 minus B1198941198873d1198941198873

=

119887119894

119891

4B1198941198874d1198941198874 (44b)

000 050 100 150 200 250

060

040

020

000

Disp

lace

men

tw (c

m)

Time (ms)



minus020



119887119895(119894 = 1 sdot sdot sdot 119868)



000015030045060075

9

6

3

0

1815

12

9

6

3

0


000010020030040050

9

6

3

0

1815

12

9

6

3

0


000010020030040050060

9

6

3

0

1815

12

9

6

3

0


00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0


00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000


00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000




0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










s

Corner 1t

n

(Γ1)

(Γ0)

(ΓK)li

lI Beam I

Beam i

Beam 1 l1

Corner ky p

z wp x up

bIf

bif

b1f

xi

zi

yi

Cihib

(Ω)

fij=1

fij=2


Γ =K

⋃j=1

Γj










119887and length 119897


119894


119894

119887









119894

119909119895 119902119894119910119895 and 119902

119894






g

aa

(a)

O

g

Ep p 120588p

hp

fij=1 fi

j=2

Ω

(Γ)

qizjqiyjx up

y p

z wp

qixj

qixj

qiyjqizj

xi uib

zi wib

hib

bif

Eib ib 120588ib

yi ib

bif4



(b)







of the plate



of the plate


119911119895

O

x up

z wp

y p

zi wib

yi ibOi

mibxj

mibzj mi

byj

qiyj

qizj

qixj

xi uib

mipy2

mipy1

mipx1 mi

px2

fij=2

fij=1

qiz1qiy1

qiy2

qiz2

qix2qix1

mibwj





plate 119898119894119901119910119895

= 119902119894





plate 119898119894119901119909119895

= 119902119894






119894

119910119895along the


119894

119911119895along the


119887119910119895= 119902119894

119909119895119890119894

119911119895along


119890119894


119894


axis 1198901198941199111

= 119890119894

1199112= minusℎ119894



= minus119902119894

119909119895119890119894

119910119895along


119890119894


119894


axis 1198901198941199101

= minus119887119894

1198914 and 119890

119894

1199102= 119887119894



= 119902119894

119911119895119890119894

119910119895minus 119902119894

119910119895119890119894

119911119895



119911119895and 119890119894


119894

119910119895and

119902119894


1199111=

119890119894

1199112= minusℎ119894

1198872 and 119890

119894

1199101= minus119887119894

1198914 and 119890

119894

1199102= 119887119894

1198914 are the


119887119908119895= minus119902119894

119909119895(120593119875119894

119878)119891119895





119875119894





119906119901 (119909 119910 119911 119905) = 119906119901 (119909 119910 119905) minus 119911

120597119908119901 (119909 119910 119905)

120597119909 (1a)

V119901 (119909 119910 119911 119905) = V119901 (119909 119910 119905) minus 119911

120597119908119901 (119909 119910 119905)

120597119910 (1b)

119908119901 (119909 119910 119911 119905) = 119908119901 (119909 119910 119905) (1c)



120576119909119909 =

120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

(2a)

120576119910119910 =

120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

(2b)

120574119909119910 =

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119909

120597119908119901

120597119910 (2c)

120576119911119911 = 120574119909119911 = 120574119910119911 = 0 (2d)


119878119909119909

119878119910119910

119878119909119910

=

[[[[[[[[[[

[

119864119901

(1 minus ]119901)2

119864119901]119901

(1 minus ]119901)2

0

119864119901]119901

(1 minus ]119901)2

119864119901

(1 minus ]119901)2

0

0 0

119864119901

2 (1 + ]119901)

]]]]]]]]]]

]

120576119909119909

120576119910119910

120574119909119910

(3)


119878119909119909 =

119864119901

(1 minus ]2119901)

times [

120597119906119901

120597119909minus 119911

1205972119908119901

1205971199092+

1

2(

120597119908119901

120597119909)

2

]

+]119901 [120597V119901120597119910

minus 119911

1205972119908119901

1205971199102+

1

2(

120597119908119901

120597119910)

2

]

(4a)

119878119910119910 =

119864119901

(1 minus ]2119901)

]119901 [120597119906119901

120597119909minus 119911

1205972119908119901

1205971199092+

1

2(

120597119908119901

120597119909)

2

]

+[

120597V119901120597119910

minus 119911

1205972119908119901

1205971199102+

1

2(

120597119908119901

120597119910)

2

]

(4b)

119878119909119910 =

119864119901

2 (1 + ]119901)(

120597119906119901

120597119910+

120597V119901120597119909

minus 2119911

1205972119908119901

120597119909120597119910+

120597119908119901

120597119909

120597119908119901

120597119910)

(4c)


119873119901119909 = 119862[

120597119906119901

120597119909+ ]119901

120597V119901120597119909

+1

2(

120597119908119901

120597119909)

2

+1

2]119901(

120597119908119901

120597119910)

2

]

(5a)

119873119901119910 = 119862[

120597V119901120597119909

+ ]119901120597119906119901

120597119909+

1

2(

120597119908119901

120597119910)

2

+1

2]119901(

120597119908119901

120597119909)

2

]

(5b)

119873119901119909119910 = 119862

1 minus ]1199012

(

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119910

120597119908119901

120597119909) (5c)

119872119901119909 = minus119863(

1205972119908119901

1205971199092+ ]119901

1205972119908119901

1205971199102) (5d)

119872119901119910 = minus119863(

1205972119908119901

1205971199102+ ]119901

1205972119908119901

1205971199092) (5e)

119872119901119909119910 = minus119863(1 minus ]119901)1205972119908119901

120597119909120597119910 (5f)


3

11990112(1 minus ]2

119901) are the




120597119873119901119909

120597119909+

120597119873119901119909119910

120597119910minus 120588119901ℎ119901119901 =

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119909119895120575 (119910 minus 119910119895)) (6a)

120597119873119901119910

120597119910+

120597119873119901119909119910

120597119909minus 120588119901ℎ119901V119901 =

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119910119895120575 (119910 minus 119910119895)) (6b)

minus

1205972119872119901119909

1205971199092minus 2

1205972119872119901119909119910

120597119909120597119910minus

1205972119872119901119910

1205971199102minus 119873119901119909

1205972119908119901

1205971199092

minus 2119873119901119909119910

1205972119908119901

120597119909120597119910minus 119873119901119910

1205972119908119901

1205971199102

+ 120588119901ℎ119901119901 minus 120588119901ℎ119901119901

120597119908119901

120597119909minus 120588119901ℎ119901V119901

120597119908119901

120597119910

minus

120588119901ℎ3

119901

12

1205972119901

1205971199092minus

120588119901ℎ3

119901

12

1205972119901

1205971199102

= 119892 minus

119868

sum

119894=1

[

[

2

sum

119895=1

(119902119894

119911119895+

120597119898119894

119901119909119895

120597119910+

120597119898119894

119901119910119895

120597119909minus 119902119894

119909119895

120597119908119894

119901119895

120597119909

minus119902119894

119910119895

120597119908119894

119901119895

120597119910)120575 (119910 minus 119910119895)

]

]

(6c)


119866119901ℎ119901 [nabla2119906119901 +

1 + ]1199011 minus ]119901

120597

120597119909(

120597119906119901

120597119909+

120597V119901120597119910

)

+(2

1 minus ]119901

1205972119908119901

1205971199092+

1205972119908119901

1205971199102)

120597119908119901

120597119909

+

1 + ]1199011 minus ]119901

1205972119908119901

120597119909120597119910

120597119908119901

120597119910] minus 120588119901ℎ119901119901

=

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119909119895120575 (119910 minus 119910119895))

(7a)

119866119901ℎ119901 [nabla2V119901 +

1 + ]1199011 minus ]119901

120597

120597119910(

120597119906119901

120597119909+

120597V119901120597119910

)

+ (2

1 minus ]119901

1205972119908119901

1205971199102+

1205972119908119901

1205971199092)

120597119908119901

120597119910

+

1 + ]1199011 minus ]119901

1205972119908119901

120597119909120597119910

120597119908119901

120597119909] minus 120588119901ℎ119901V119901

=

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119910119895120575 (119910 minus 119910119895))

(7b)

119863nabla4119908119901 minus 119862

times [(

120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

)

+]119901(120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

)]

times

1205972119908119901

1205971199092+ (1 minus ]119901)

sdot (

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119909

120597119908119901

120597119910)

1205972119908119901

120597119909120597119910

+ [(

120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

)

+]119901(120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

)] sdot

1205972119908119901

1205971199102

+ 120588119901ℎ119901119901 minus 120588119901ℎ119901119901

120597119908119901

120597119909minus 120588119901ℎ119901V119901

120597119908119901

120597119910

minus

120588119901ℎ3

119901

12

1205972119901

1205971199092minus

120588119901ℎ3

119901

12

1205972119901

1205971199102

= 119892 minus

119868

sum

119894=1

[

[

2

sum

119895=1

(119902119894

119911119895+

120597119898119894

119901119909119895

120597119910+

120597119898119894

119901119910119895

120597119909minus 119902119894

119909119895

120597119908119894

119901119895

120597119909

minus119902119894

119910119895

120597119908119894

119901119895

120597119910)120575 (119910 minus 119910119895)

]

]

(7c)


1198861199011119906119901119899 + 1198861199012119873119901119899 = 1198861199013 (8a)

1205731199011119906119901119905 + 1205731199012119873119901119905 = 1205731199013 (8b)

1205741199011119908119901 + 1205741199012119877119901119899 = 1205741199013 (8c)

1205751199011

120597119908119901

120597119899+ 1205751199012119872119901119899 = 1205751199013

(8d)

1205761119896119908119901 + 1205762119896

10038171003817100381710038171003817119879119908119901

10038171003817100381710038171003817119896= 1205763119896 1205762119896 = 0 (8e)


119908119901 (x 0) = 1199081199010 (x) (9a)

119901 (x 0) = 1199081199010 (x) (9b)




119877119901119899 = minus119863[120597


120597

120597119904(

1205972119908119901

120597119904120597119899minus 120581

120597119908119901

120597119904)]

+ 119873119901119899

120597119908119901

120597119899+ 119873119901119905

120597119908119901

120597119904

(10a)


1205972119908119901

1205971199042+ 120581

120597119908119901

120597119899)] (10b)



119879119908119901 = 119863(]119901 minus 1)(

1205972119908119901

120597119904120597119899minus 120581

120597119908119901

120597119904) (11)


1205741199012 = 1205741199013 = 1205751199012 = 1205751199013 = 0)





119906119894

119887(119909119894 119910119894 119911119894 119905) = 119906

119894

119887(119909119894 119905) minus 119910

119894120579119894

119887119911(119909119894 119905)

+ 119911119894120579119894

119887119910(119909119894 119905) +

120597120579119894

119887119909

120597119909119894120593119875119894

119878(119910119894 119911119894 119905)

(12a)

V119894119887(119909119894 119910119894 119911119894 119905) = V119894

119887(119909119894 119905) minus 119911

119894120579119894

119887119909(119909119894 119905) (12b)

119908119894

119887(119909119894 119910119894 119911119894 119905) = 119908

119894

119887(119909119894 119905) + 119910

119894120579119894

119887119909(119909119894 119905) (12c)

120579119894

119887119910(119909119894 119905) = minus

120597119908119894

119887(119909119894 119905)

120597119909119894 (12d)

120579119894

119887119911(119909119894 119905) =

120597V119894119887(119909119894 119905)

120597119909119894 (12e)

where 119906119894119887 V119894119887 and119908

119894



119887=

119906119894

119887(119909119894) V119894119887= V119894119887(119909119894) and 119908

119894

119887= 119908119894



= 120579119894

119887119910(119909119894) and 120579

119894

119887119911=

120579119894




119887119909(119909119894) regarded as the




120576119909119909 =120597119906119894

119887

120597119909119894+

1

2

[

[

(120597V119894119887

120597119909119894)

2

+ (120597119908119894

119887

120597119909119894)

2

]

]

(13a)

120574119909119911 =120597119908119894

119887

120597119909119894+

120597119906119894

119887

120597119911119894+ (

120597V119894119887

120597119909119894

120597V119894119887

120597119911119894+

120597119908119894

119887

120597119909119894

120597119908119894

119887

120597119911119894) (13b)

120574119909119910 =120597V119894119887

120597119909119894+

120597119906119894

119887

120597119910119894+ (

120597V119894119887

120597119909119894

120597V119894119887

120597119910119894+

120597119908119894

119887

120597119909119894

120597119908119894

119887

120597119910119894) (13c)

120576119910119910 = 120576119911119911 = 120574119910119911 = 0 (13d)


119873119894

119887= 119864119894

119887119860119894

119887[120597119906119894

119887

120597119909119894+

1

2((

120597V119894119887

120597119909119894)

2

+ (120597119908119894

119887

120597119909119894)

2

)] (14a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (14b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (14c)

119872119875119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894 (14d)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (14e)

where 119872119875119894



119887119908is the



119910and 119868

119894


119894

119878is the


119905and 119862

119894



119868119894

119905= intΩ

(1199101198942+ 1199111198942+ 119910119894 120597120593119875119894

119878

120597119911119894minus 119911119894 120597120593119875119894

119878

120597119910119894)119889Ω (15a)

119862119894

119878= intΩ

(120593119875119894

119878)2

119889Ω (15b)


minus120597119873119894

119887

120597119909119894+ 120588119894

119887Α119894

119887119894

119887=

2

sum

119895=1

119902119894

119909119895 (16a)

minus 119873119894

119887

1205972V119894119887

1205971199091198942+

1205972119872119894

119887119911

1205971199091198942

+ 120588119894

119887Α119894

119887V119894119887minus 120588119894

119887119868119894

119911

1205972V119894119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(16b)

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942minus

1205972119872119894

119887119910

1205971199091198942

+ 120588119894

119887Α119894

119887119894

119887minus 120588119894

119887119868119894

119910

1205972119894

119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(16c)

minus120597119872119894

119887119905

120597119909119894minus

1205972119872119894

119887119908

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(16d)


minus 119864119894

119887119860119894

119887(1205972119906119894

119887

1205971199091198942+

120597119908119894

119887

120597119909119894

1205972119908119894

119887

1205971199091198942

+120597V119894119887

120597119909119894

1205972V119894119887

1205971199091198942) + 120588

119894

119887Α119894

119887119894

119887

=

2

sum

119895=1

119902119894

119909119895

(17a)

119864119894

119887119868119894

119911

1205974V119894119887

1205971199091198944minus 119873119894

119887

1205972V119894119887

1205971199091198942minus 120588119894

119887119868119894

119911

1205972V1198871205971199092

+ 120588119894

119887Α119894

119887V119887 minus 120588

119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(17b)

119864119894

119887119868119894

119910

1205974119908119894

119887

1205971199091198944

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942

minus 120588119894

119887119868119894

119910

1205972119887

1205971199092+ 120588119894

119887Α119894

119887119887 minus 120588

119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(17c)

119864119894

119887119862119894

119878

1205974120579119894

119887119909

1205971199091198944

minus 119866119894

119887119868119894

119905

1205972120579119894

119887119909

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(17d)


119894 are given as

119886119894

1198871119906119894

119887+ 120572119894

1198872119873119894

119887= 120572119894

1198873 (18)

120573119894

1198871V119894119887+ 120573119894

1198872119877119894

119887119910= 120573119894

1198873 (19a)

120573119894

1198871120579119894

119887119911+ 120573119894

1198872119872119894

119887119911= 120573119894

1198873 (19b)

120574119894

1198871119908119894

119887+ 120574119894

1198872119877119894

119887119911= 120574119894

1198873 (20a)

120574119894

1198871120579119894

119887119910+ 120574119894

1198872119872119894

119887119910= 120574119894

1198873 (20b)

120575119894

1198871120579119894

119887119909+ 120575119894

1198872119872119894

119887119905= 120575119894

1198873 (21a)

120575119894

1198871

119889120579119894

119887119909

119889119909119894+ 120575119894

1198872119872119894

119887119908= 120575119894

1198873(21b)


119908119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22a)

119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22b)


119894

119887119910 120579119894


119894

119887119910 119877119894

119887119911

and 119872119894

119887119911 119872119894119887119910



119877119894

119887119910= 119873119894

119887

120597V119894119887

120597119909119894minus 119864119894

119887119868119894

119911

1205973V119894119887

1205971199091198943 (23a)

119877119894

119887119911= 119873119894

119887

120597119908119894

119887

120597119909119894minus 119864119894

119887119868119894

119910

1205973119908119894

119887

1205971199091198943 (23b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (24a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (24b)


and 119872119894

119887119905 119872119894119887119908


119872119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894minus 119864119894

119887119862119894

119878

1205973120579119894

119887119909

1205971199091198943 (25a)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (25b)

Finally 120572119894119887119896 120573119894

119887119896 120573119894

119887119896 120574119894

119887119896 120574119894

119887119896 120575119894

119887119896 120575119894

119887119896(119896 = 1 2 3) are func-




119894

119887 V119894119887119908119894119887 120579119894119887119909 1199021198941199091 1199021198941199101 1199021198941199111 119902119894

1199092

119902119894

1199102 and 119902

119894





119906119894

1199011minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199011

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894+

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198911

+119902119894

1199091

1198961198941199091

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=1)

(26a)

119906119894

1199012minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199012

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894minus

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198912

+119902119894

1199092

1198961198941199092

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=2)

(26b)


V1198941199011

minus V119894119887=

ℎ119901

2

120597119908119894

1199011

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199101

1198961198941199101

[1 minus1

2(120579119894

119887119909)2

]


119895=1)

(27a)

V1198941199012

minus V119894119887=

ℎ119901

2

120597119908119894

1199012

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199102

1198961198941199102

[1 minus1

2(120579119894

119887119909)2

]


119895=2)

(27b)


119908119894

1199011minus 119908119894

119887= minus

119887119894

119891

4120579119894


119894

119895=1) (28a)

119908119894

1199012minus 119908119894

119887=

119887119894

119891

4120579119894


119894

119895=2) (28b)

where (120593119875119894

119878)119891119895


119894

119895and 119896

119894

119909119895 119896119894119910119895



=

119896119894

119909119895(119904119894

119909119895) and 119896

119894

119910119895= 119896119894

119910119895(119904119894



119894



119875119894









nabla2119906119901119894 = 119901119901119894 (119909 119910 119905) (119894 = 1 2) (29a)

nabla41199061199013 = 1199011199013 (119909 119910 119905) (29b)






[

[

E11990111 0 00 E11990122 00 0 E11990133

]

]

d1199011d1199012d1199013

+

0D1198991198971199011

0D1198991198971199012

00

D1198991198971199013

=

01205721199013

01205731199013

001205741199013

1205751199013

(30)

where

E11990111 = [A1 H1 H20 D11990122 D11990123

] (31a)

E11990122 = [A1 H1 H20 D11990144 D11990145

] (31b)

E11990133 =[[[

[

A2 H1 H2 G1 G2A1 0 0 H1 H20 D11990178 D11990179 0 D1199017110 D11990188 D11990189 D119901810 0

]]]

]

(31c)



119901119894(119894 = 1 2) are 119873 times 1 and

D1198991198971199013


d119901119894 = p119901119894 u119901119894 u119901119894119899119879 (119894 = 1 2) (32a)


2u1199013)119899119879 (32b)


u119901119894 = (119906119901119894)1(119906119901119894)2

sdot sdot sdot (119906119901119894)119873119879

(119894 = 1 2 3)

(33a)

u119901119894119899 = (

120597119906119901119894

120597119899)

1

(

120597119906119901119894

120597119899)

2

sdot sdot sdot (

120597119906119901119894

120597119899)

119873

119879

(119894 = 1 2 3)

(33b)


21199061199013)1

(nabla21199061199013)2


119879

(33c)

(nabla2u1199013)119899

= (

120597nabla21199061199013

120597119899)

1

(

120597nabla21199061199013

120597119899)

2

sdot sdot sdot (

120597nabla21199061199013

120597119899)

119873

119879

(33d)


(119901119901119894)1(119901119901119894)2

sdot sdot sdot (119901119901119894)119872119879



u119901119894 = B119901119894d119901119894 (119894 = 1 2 3) (34a)u119901119894119909 = B119901119894119909d119901119894 (119894 = 1 2 3) (34b)

u119901119894119910 = B119901119894119910d119901119894 (119894 = 1 2 3) (34c)


u119901119894119909119909 = B119901119894119909119909d119901119894 (119894 = 1 2 3) (34d)

u119901119894119910119910 = B119901119894119910119910d119901119894 (119894 = 1 2 3) (34e)

u119901119894119909119910 = B119901119894119909119910d119901119894 (119894 = 1 2 3) (34f)


119866119901ℎ119901 [p1199011 +1 + ]1199011 minus ]119901

(B1199011119909119909d1199011 + B1199012119909119910d1199012)

+ 2

1 minus ]119901(B1199013119909119909d1199013)119889119892B1199013119909d1199013

+(B1199013119910119910d1199013)119889119892 sdot B1199013119909d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119910d1199013]

minus 120588119901ℎ119901B1199011d1199011 = Zq119909(35a)

119866119901ℎ119901 [p1199012 +1 + ]1199011 minus ]119901

(B1199011119909119910d1199011 + B1199012119910119910d1199012)

+ 2

1 minus ]119901(B1199013119910119910d1199013)119889119892B1199013119910d1199013

+ (B1199013119909119909d1199013)119889119892 sdot B1199013119910d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119909d1199013]

minus 120588119901ℎ119901B1199012d1199012 = Zq119910

(35b)

119863p1199013 minus 119862

times [(B1199011119909d1199011)119889119892B1199013119909119909d1199013

+1

2(B1199013119909d1199013)119889119892(B1199013119909d1199013)119889119892B1199013119909119909d1199013

+ ]119901(B1199012119910d1199012)119889119892B1199013119909119909d1199013 +1

2]119901(B1199013119910d1199013)119889119892

times (B1199013119910d1199013)119889119892B1199013119909119909d1199013]

+ (1 minus ]119901)

times [(B1199011119910d1199011)119889119892B1199013119909119910d1199013

+ (B1199012119909d1199012)119889119892B1199013119909119910d1199013

+(B1199013119909d1199013)119889119892 sdot (B1199013119910d1199013)119889119892B1199013119909119910d1199013]

+ [(B1199012119910d1199012)119889119892B1199013119910119910d1199013 +1

2(B1199013119910d1199013)119889119892

sdot (B1199013119910d1199013)119889119892B1199013119910119910d1199013

+ ]119901(B1199011119909d1199011)119889119892B1199013119910119910d1199013

+1

2]119901(B1199013119909d1199013)119889119892

sdot (B1199013119909d1199013)119889119892B1199013119910119910d1199013]

+ 120588119901ℎ119901B1199013d1199013 minus 120588119901ℎ119901(B1199011d1199011)119889119892B1199013119909d1199013


119901

12B1199013119909119909d1199013

minus

120588119901ℎ3

119901

12B1199013119910119910d1199013


+ (Zq119910)119889119892B1199013119910d1199013

(35c)

where q119909 = q1199091 q1199092119879 q119910 = q1199101 q1199102

119879 andq119911 = q1199111 q1199112


119894



119901119910119895= 119902119894

119909119895ℎ1199012 119898

119894

119901119909119895= 119902119894

119910119895ℎ1199012 respectively



119887 V119894119887 and 119908

119894

119887




119887(119909119894 119905) and V119894

119887(119909119894 119905) 119908119894





Let 119906119894

1198871= 119906119894

119887(119909119894 119905) 119906

119894

1198872= V119894119887(119909119894 119905) 119906

119894

1198873= 119908119894

119887(119909119894 119905)

and 119906119894

1198874= 120579119894




1205972119906119894

1198871

1205971199091198942

= 119901119894

1198871(119909119894 119905) (36a)

1205974119906119894

119887119895

1205971199091198944= 119901119894

119887119895(119909119894 119905) (119895 = 2 3 4) (36b)




119894





119887119895(120577119894) 119906119894119887119895119909

(120577119894) 119906119894119887119895119909119909

(120577119894) and 119906

119894

119887119895119909119909119909(120577119894)

(120577119894 = 0 119897119894) in terms of 119901119894



[[[

[

E11989411988711

0 0 00 E119894

119887220 0

0 0 E11989411988733

00 0 0 E119894

11988744

]]]

]

d1198941198871

d1198941198872

d1198941198873

d1198941198874

+

0D119894 1198991198971198871

00

D119894 1198991198971198872

00

D119894 1198991198971198873

000

=

0120572119894

1198873

00120573119894

1198873

00120574119894

1198873

00120575119894

1198873

(37)

where E11989411988711

and E11989411988722

E11989411988733

E11989411988744


1198871is

a 2 times 1 and D119894 119899119897119887119895


1198873and

120573119894

1198873 1205741198941198873 1205751198941198873



1198873and 120573

119894

1198873 120573119894

1198873 120574119894

1198873 120574119894

1198873 120575119894

1198873 120575119894


d1198941198871

= p1198941198871

u1198941198871

u1198941198871119909

119879

(38a)

d119894119887119895

= p119894119887119895 u119894119887119895

u119894119887119895119909

u119894119887119895119909119909

u119894119887119895119909119909119909

119879

(119895 = 2 3 4)

(38b)


u119894119887119895

= 119906119894

119887119895(0 119905) 119906

119894

119887119895(119897119894 119905)119879

(119895 = 1 2 3 4) (39a)

u119894119887119895119909

= 120597119906119894

119887119895(0 119905)

120597119909119894

120597119906119894

119887119895(119897119894 119905)

120597119909119894

119879

(119895 = 1 2 3 4) (39b)

u119894119887119895119909119909

= 1205972119906119894

119887119895(0 119905)

1205971199091198942

1205972119906119894

119887119895(119897119894 119905)

1205971199091198942

119879

(119895 = 2 3 4)

(39c)

u119894119887119895119909119909119909

= 1205973119906119894

119887119895(0 119905)

1205971199091198943

1205973119906119894

119887119895(119897119894 119905)

1205971199091198943

119879

(119895 = 2 3 4)

(39d)


119887119895=

(119901119894

119887119895)1

(119901119894

119887119895)2


119887119895)119871119879



119894



119894isin (0 119897

119894) and


u1198941198871

= B1198941198871d1198941198871 (40a)

u1198941198871119909

= B1198941198871119909

d1198941198871 (40b)

u119894119887119895

= B119894119887119895d119894119887119895 (119895 = 2 3 4) (40c)

u119894119887119895119909

= B119894119887119895119909

d119894119887119895 (119895 = 2 3 4) (40d)

u119894119887119895119909119909

= B119894119887119895119909119909

d119894119887119895 (119895 = 2 3 4) (40e)

u119894119887119895119909119909119909

= B119894119887119895119909119909119909

d119894119887119895 (119895 = 2 3 4) (40f)

where B1198941198871B1198941198871119909

are 119871 times (119871 + 4) and B119894119887119895B119894119887119895119909

B119894119887119895119909119909

B119894119887119895119909119909119909



x

y

a

a

Fr

Fr

ClCl

Lx = 18 cm

Ly=9

cm A

(a)

g

02 cmhb = 05 cm

1 cm 5 cm3 cm

Ly = 9 cm

(b)


Time (ms)

060

040

020

000

minus020

000 050 100 150 200 250

Disp

lace

men

tw (c

m)





minus 119864119894

119887119860119894

119887[p1198941198871

+ (B1198941198872119909

d1198941198872)119889119892B1198941198872119909119909

d1198941198872

+ (B1198941198873119909

d1198941198873)119889119892B1198941198873119909119909

d1198941198873]

+ 120588119894

119887Α119894

119887T1q1 = q119894

1199091+ q1198941199092

(41a)

119864119894

119887119868119894

119911p1198941198872

minus (N119894119887)119889119892B1198941198872119909119909

d1198941198872

minus 120588119894

119887119868119894

119911B1198941198872119909119909

d1198941198872

+ 120588119894

119887Α119894

119887B1198941198872d1198941198872

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198872119909

d1198941198872

= q1198941199101

+ q1198941199102

minus (B1198941198872119909

d1198941198872)119889119892

(q1198941199091

+ q1198941199092) minus X119894119887119910

(q1198941199091

+ q1198941199092)

(41b)

119864119894

119887119868119894

119910p1198941198873

minus (N119894119887)119889119892B1198941198873119909119909

d1198941198873

minus 120588119894

119887119868119894

119910B1198941198873119909119909

d1198941198873

+ 120588119894

119887Α119894

119887B1198941198873d1198941198873

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198873119909

d1198941198873

= q1198941199111

+ q1198941199112

minus (B1198941198873119909

d1198941198873)119889119892

(q1198941199091

+ q1198941199092) + X119894119887119911

(q1198941199091

+ q1198941199092)

(41c)

119864119894

119887119862119894

119878p1198941198874

minus 119866119894

119887119868119894

119905B1198941198874119909119909

d1198941198874

+ 120588119894

119887119868119894

119901B1198941198874d1198941198874

minus 120588119894

119887119862119894

119878B1198941198874119909119909

d1198941198874

= e1198941199101q1198941199111

+ e1198941199102q1198941199112

minus e1198941199111q1198941199101

minus e1198941199112q1198941199102

+ Χ119894

119887119908(q1198941199091

+ q1198941199092)

(41d)

where (N119894119887)119889119892


119887119911 X119894119887119910 X119894119887119908


119898119894119887119911119895 119898119894119887119908119895


1199101 e1198941199102 e1198941199111

e1198941199112


119894

119910119895 119890119894


119894

119911119895 119902119894

119910119895with



Y1B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y1B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

+

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198911

B1198941198874119909

d1198874

+ K1198941199091

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42a)


008

008

018

018

028

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036040044048


000067300301006080091501220153018402140245027603070337036803990430460491


008

008

008

00801

8018

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036


000064600236004780072009620120145016901930217024102660290314033803620387



Y2B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y2B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

minus

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198912

B1198941198874119909

d1198874

+ K1198941199092

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42b)

Y1B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y1B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199101

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43a)

Y2B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y2B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199102

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43b)

Y1B1199013d1199013 minus B1198941198873d1198941198873

= minus

119887119894

119891

4B1198941198874d1198941198874 (44a)

Y2B1199013d1199013 minus B1198941198873d1198941198873

=

119887119894

119891

4B1198941198874d1198941198874 (44b)

000 050 100 150 200 250

060

040

020

000

Disp

lace

men

tw (c

m)

Time (ms)



minus020



119887119895(119894 = 1 sdot sdot sdot 119868)



000015030045060075

9

6

3

0

1815

12

9

6

3

0


000010020030040050

9

6

3

0

1815

12

9

6

3

0


000010020030040050060

9

6

3

0

1815

12

9

6

3

0


00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0


00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000


00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000




0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










g

aa

(a)

O

g

Ep p 120588p

hp

fij=1 fi

j=2

Ω

(Γ)

qizjqiyjx up

y p

z wp

qixj

qixj

qiyjqizj

xi uib

zi wib

hib

bif

Eib ib 120588ib

yi ib

bif4



(b)







of the plate



of the plate


119911119895

O

x up

z wp

y p

zi wib

yi ibOi

mibxj

mibzj mi

byj

qiyj

qizj

qixj

xi uib

mipy2

mipy1

mipx1 mi

px2

fij=2

fij=1

qiz1qiy1

qiy2

qiz2

qix2qix1

mibwj





plate 119898119894119901119910119895

= 119902119894





plate 119898119894119901119909119895

= 119902119894






119894

119910119895along the


119894

119911119895along the


119887119910119895= 119902119894

119909119895119890119894

119911119895along


119890119894


119894


axis 1198901198941199111

= 119890119894

1199112= minusℎ119894



= minus119902119894

119909119895119890119894

119910119895along


119890119894


119894


axis 1198901198941199101

= minus119887119894

1198914 and 119890

119894

1199102= 119887119894



= 119902119894

119911119895119890119894

119910119895minus 119902119894

119910119895119890119894

119911119895



119911119895and 119890119894


119894

119910119895and

119902119894


1199111=

119890119894

1199112= minusℎ119894

1198872 and 119890

119894

1199101= minus119887119894

1198914 and 119890

119894

1199102= 119887119894

1198914 are the


119887119908119895= minus119902119894

119909119895(120593119875119894

119878)119891119895





119875119894





119906119901 (119909 119910 119911 119905) = 119906119901 (119909 119910 119905) minus 119911

120597119908119901 (119909 119910 119905)

120597119909 (1a)

V119901 (119909 119910 119911 119905) = V119901 (119909 119910 119905) minus 119911

120597119908119901 (119909 119910 119905)

120597119910 (1b)

119908119901 (119909 119910 119911 119905) = 119908119901 (119909 119910 119905) (1c)



120576119909119909 =

120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

(2a)

120576119910119910 =

120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

(2b)

120574119909119910 =

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119909

120597119908119901

120597119910 (2c)

120576119911119911 = 120574119909119911 = 120574119910119911 = 0 (2d)


119878119909119909

119878119910119910

119878119909119910

=

[[[[[[[[[[

[

119864119901

(1 minus ]119901)2

119864119901]119901

(1 minus ]119901)2

0

119864119901]119901

(1 minus ]119901)2

119864119901

(1 minus ]119901)2

0

0 0

119864119901

2 (1 + ]119901)

]]]]]]]]]]

]

120576119909119909

120576119910119910

120574119909119910

(3)


119878119909119909 =

119864119901

(1 minus ]2119901)

times [

120597119906119901

120597119909minus 119911

1205972119908119901

1205971199092+

1

2(

120597119908119901

120597119909)

2

]

+]119901 [120597V119901120597119910

minus 119911

1205972119908119901

1205971199102+

1

2(

120597119908119901

120597119910)

2

]

(4a)

119878119910119910 =

119864119901

(1 minus ]2119901)

]119901 [120597119906119901

120597119909minus 119911

1205972119908119901

1205971199092+

1

2(

120597119908119901

120597119909)

2

]

+[

120597V119901120597119910

minus 119911

1205972119908119901

1205971199102+

1

2(

120597119908119901

120597119910)

2

]

(4b)

119878119909119910 =

119864119901

2 (1 + ]119901)(

120597119906119901

120597119910+

120597V119901120597119909

minus 2119911

1205972119908119901

120597119909120597119910+

120597119908119901

120597119909

120597119908119901

120597119910)

(4c)


119873119901119909 = 119862[

120597119906119901

120597119909+ ]119901

120597V119901120597119909

+1

2(

120597119908119901

120597119909)

2

+1

2]119901(

120597119908119901

120597119910)

2

]

(5a)

119873119901119910 = 119862[

120597V119901120597119909

+ ]119901120597119906119901

120597119909+

1

2(

120597119908119901

120597119910)

2

+1

2]119901(

120597119908119901

120597119909)

2

]

(5b)

119873119901119909119910 = 119862

1 minus ]1199012

(

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119910

120597119908119901

120597119909) (5c)

119872119901119909 = minus119863(

1205972119908119901

1205971199092+ ]119901

1205972119908119901

1205971199102) (5d)

119872119901119910 = minus119863(

1205972119908119901

1205971199102+ ]119901

1205972119908119901

1205971199092) (5e)

119872119901119909119910 = minus119863(1 minus ]119901)1205972119908119901

120597119909120597119910 (5f)


3

11990112(1 minus ]2

119901) are the




120597119873119901119909

120597119909+

120597119873119901119909119910

120597119910minus 120588119901ℎ119901119901 =

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119909119895120575 (119910 minus 119910119895)) (6a)

120597119873119901119910

120597119910+

120597119873119901119909119910

120597119909minus 120588119901ℎ119901V119901 =

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119910119895120575 (119910 minus 119910119895)) (6b)

minus

1205972119872119901119909

1205971199092minus 2

1205972119872119901119909119910

120597119909120597119910minus

1205972119872119901119910

1205971199102minus 119873119901119909

1205972119908119901

1205971199092

minus 2119873119901119909119910

1205972119908119901

120597119909120597119910minus 119873119901119910

1205972119908119901

1205971199102

+ 120588119901ℎ119901119901 minus 120588119901ℎ119901119901

120597119908119901

120597119909minus 120588119901ℎ119901V119901

120597119908119901

120597119910

minus

120588119901ℎ3

119901

12

1205972119901

1205971199092minus

120588119901ℎ3

119901

12

1205972119901

1205971199102

= 119892 minus

119868

sum

119894=1

[

[

2

sum

119895=1

(119902119894

119911119895+

120597119898119894

119901119909119895

120597119910+

120597119898119894

119901119910119895

120597119909minus 119902119894

119909119895

120597119908119894

119901119895

120597119909

minus119902119894

119910119895

120597119908119894

119901119895

120597119910)120575 (119910 minus 119910119895)

]

]

(6c)


119866119901ℎ119901 [nabla2119906119901 +

1 + ]1199011 minus ]119901

120597

120597119909(

120597119906119901

120597119909+

120597V119901120597119910

)

+(2

1 minus ]119901

1205972119908119901

1205971199092+

1205972119908119901

1205971199102)

120597119908119901

120597119909

+

1 + ]1199011 minus ]119901

1205972119908119901

120597119909120597119910

120597119908119901

120597119910] minus 120588119901ℎ119901119901

=

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119909119895120575 (119910 minus 119910119895))

(7a)

119866119901ℎ119901 [nabla2V119901 +

1 + ]1199011 minus ]119901

120597

120597119910(

120597119906119901

120597119909+

120597V119901120597119910

)

+ (2

1 minus ]119901

1205972119908119901

1205971199102+

1205972119908119901

1205971199092)

120597119908119901

120597119910

+

1 + ]1199011 minus ]119901

1205972119908119901

120597119909120597119910

120597119908119901

120597119909] minus 120588119901ℎ119901V119901

=

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119910119895120575 (119910 minus 119910119895))

(7b)

119863nabla4119908119901 minus 119862

times [(

120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

)

+]119901(120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

)]

times

1205972119908119901

1205971199092+ (1 minus ]119901)

sdot (

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119909

120597119908119901

120597119910)

1205972119908119901

120597119909120597119910

+ [(

120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

)

+]119901(120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

)] sdot

1205972119908119901

1205971199102

+ 120588119901ℎ119901119901 minus 120588119901ℎ119901119901

120597119908119901

120597119909minus 120588119901ℎ119901V119901

120597119908119901

120597119910

minus

120588119901ℎ3

119901

12

1205972119901

1205971199092minus

120588119901ℎ3

119901

12

1205972119901

1205971199102

= 119892 minus

119868

sum

119894=1

[

[

2

sum

119895=1

(119902119894

119911119895+

120597119898119894

119901119909119895

120597119910+

120597119898119894

119901119910119895

120597119909minus 119902119894

119909119895

120597119908119894

119901119895

120597119909

minus119902119894

119910119895

120597119908119894

119901119895

120597119910)120575 (119910 minus 119910119895)

]

]

(7c)


1198861199011119906119901119899 + 1198861199012119873119901119899 = 1198861199013 (8a)

1205731199011119906119901119905 + 1205731199012119873119901119905 = 1205731199013 (8b)

1205741199011119908119901 + 1205741199012119877119901119899 = 1205741199013 (8c)

1205751199011

120597119908119901

120597119899+ 1205751199012119872119901119899 = 1205751199013

(8d)

1205761119896119908119901 + 1205762119896

10038171003817100381710038171003817119879119908119901

10038171003817100381710038171003817119896= 1205763119896 1205762119896 = 0 (8e)


119908119901 (x 0) = 1199081199010 (x) (9a)

119901 (x 0) = 1199081199010 (x) (9b)




119877119901119899 = minus119863[120597


120597

120597119904(

1205972119908119901

120597119904120597119899minus 120581

120597119908119901

120597119904)]

+ 119873119901119899

120597119908119901

120597119899+ 119873119901119905

120597119908119901

120597119904

(10a)


1205972119908119901

1205971199042+ 120581

120597119908119901

120597119899)] (10b)



119879119908119901 = 119863(]119901 minus 1)(

1205972119908119901

120597119904120597119899minus 120581

120597119908119901

120597119904) (11)


1205741199012 = 1205741199013 = 1205751199012 = 1205751199013 = 0)





119906119894

119887(119909119894 119910119894 119911119894 119905) = 119906

119894

119887(119909119894 119905) minus 119910

119894120579119894

119887119911(119909119894 119905)

+ 119911119894120579119894

119887119910(119909119894 119905) +

120597120579119894

119887119909

120597119909119894120593119875119894

119878(119910119894 119911119894 119905)

(12a)

V119894119887(119909119894 119910119894 119911119894 119905) = V119894

119887(119909119894 119905) minus 119911

119894120579119894

119887119909(119909119894 119905) (12b)

119908119894

119887(119909119894 119910119894 119911119894 119905) = 119908

119894

119887(119909119894 119905) + 119910

119894120579119894

119887119909(119909119894 119905) (12c)

120579119894

119887119910(119909119894 119905) = minus

120597119908119894

119887(119909119894 119905)

120597119909119894 (12d)

120579119894

119887119911(119909119894 119905) =

120597V119894119887(119909119894 119905)

120597119909119894 (12e)

where 119906119894119887 V119894119887 and119908

119894



119887=

119906119894

119887(119909119894) V119894119887= V119894119887(119909119894) and 119908

119894

119887= 119908119894



= 120579119894

119887119910(119909119894) and 120579

119894

119887119911=

120579119894




119887119909(119909119894) regarded as the




120576119909119909 =120597119906119894

119887

120597119909119894+

1

2

[

[

(120597V119894119887

120597119909119894)

2

+ (120597119908119894

119887

120597119909119894)

2

]

]

(13a)

120574119909119911 =120597119908119894

119887

120597119909119894+

120597119906119894

119887

120597119911119894+ (

120597V119894119887

120597119909119894

120597V119894119887

120597119911119894+

120597119908119894

119887

120597119909119894

120597119908119894

119887

120597119911119894) (13b)

120574119909119910 =120597V119894119887

120597119909119894+

120597119906119894

119887

120597119910119894+ (

120597V119894119887

120597119909119894

120597V119894119887

120597119910119894+

120597119908119894

119887

120597119909119894

120597119908119894

119887

120597119910119894) (13c)

120576119910119910 = 120576119911119911 = 120574119910119911 = 0 (13d)


119873119894

119887= 119864119894

119887119860119894

119887[120597119906119894

119887

120597119909119894+

1

2((

120597V119894119887

120597119909119894)

2

+ (120597119908119894

119887

120597119909119894)

2

)] (14a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (14b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (14c)

119872119875119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894 (14d)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (14e)

where 119872119875119894



119887119908is the



119910and 119868

119894


119894

119878is the


119905and 119862

119894



119868119894

119905= intΩ

(1199101198942+ 1199111198942+ 119910119894 120597120593119875119894

119878

120597119911119894minus 119911119894 120597120593119875119894

119878

120597119910119894)119889Ω (15a)

119862119894

119878= intΩ

(120593119875119894

119878)2

119889Ω (15b)


minus120597119873119894

119887

120597119909119894+ 120588119894

119887Α119894

119887119894

119887=

2

sum

119895=1

119902119894

119909119895 (16a)

minus 119873119894

119887

1205972V119894119887

1205971199091198942+

1205972119872119894

119887119911

1205971199091198942

+ 120588119894

119887Α119894

119887V119894119887minus 120588119894

119887119868119894

119911

1205972V119894119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(16b)

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942minus

1205972119872119894

119887119910

1205971199091198942

+ 120588119894

119887Α119894

119887119894

119887minus 120588119894

119887119868119894

119910

1205972119894

119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(16c)

minus120597119872119894

119887119905

120597119909119894minus

1205972119872119894

119887119908

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(16d)


minus 119864119894

119887119860119894

119887(1205972119906119894

119887

1205971199091198942+

120597119908119894

119887

120597119909119894

1205972119908119894

119887

1205971199091198942

+120597V119894119887

120597119909119894

1205972V119894119887

1205971199091198942) + 120588

119894

119887Α119894

119887119894

119887

=

2

sum

119895=1

119902119894

119909119895

(17a)

119864119894

119887119868119894

119911

1205974V119894119887

1205971199091198944minus 119873119894

119887

1205972V119894119887

1205971199091198942minus 120588119894

119887119868119894

119911

1205972V1198871205971199092

+ 120588119894

119887Α119894

119887V119887 minus 120588

119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(17b)

119864119894

119887119868119894

119910

1205974119908119894

119887

1205971199091198944

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942

minus 120588119894

119887119868119894

119910

1205972119887

1205971199092+ 120588119894

119887Α119894

119887119887 minus 120588

119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(17c)

119864119894

119887119862119894

119878

1205974120579119894

119887119909

1205971199091198944

minus 119866119894

119887119868119894

119905

1205972120579119894

119887119909

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(17d)


119894 are given as

119886119894

1198871119906119894

119887+ 120572119894

1198872119873119894

119887= 120572119894

1198873 (18)

120573119894

1198871V119894119887+ 120573119894

1198872119877119894

119887119910= 120573119894

1198873 (19a)

120573119894

1198871120579119894

119887119911+ 120573119894

1198872119872119894

119887119911= 120573119894

1198873 (19b)

120574119894

1198871119908119894

119887+ 120574119894

1198872119877119894

119887119911= 120574119894

1198873 (20a)

120574119894

1198871120579119894

119887119910+ 120574119894

1198872119872119894

119887119910= 120574119894

1198873 (20b)

120575119894

1198871120579119894

119887119909+ 120575119894

1198872119872119894

119887119905= 120575119894

1198873 (21a)

120575119894

1198871

119889120579119894

119887119909

119889119909119894+ 120575119894

1198872119872119894

119887119908= 120575119894

1198873(21b)


119908119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22a)

119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22b)


119894

119887119910 120579119894


119894

119887119910 119877119894

119887119911

and 119872119894

119887119911 119872119894119887119910



119877119894

119887119910= 119873119894

119887

120597V119894119887

120597119909119894minus 119864119894

119887119868119894

119911

1205973V119894119887

1205971199091198943 (23a)

119877119894

119887119911= 119873119894

119887

120597119908119894

119887

120597119909119894minus 119864119894

119887119868119894

119910

1205973119908119894

119887

1205971199091198943 (23b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (24a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (24b)


and 119872119894

119887119905 119872119894119887119908


119872119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894minus 119864119894

119887119862119894

119878

1205973120579119894

119887119909

1205971199091198943 (25a)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (25b)

Finally 120572119894119887119896 120573119894

119887119896 120573119894

119887119896 120574119894

119887119896 120574119894

119887119896 120575119894

119887119896 120575119894

119887119896(119896 = 1 2 3) are func-




119894

119887 V119894119887119908119894119887 120579119894119887119909 1199021198941199091 1199021198941199101 1199021198941199111 119902119894

1199092

119902119894

1199102 and 119902

119894





119906119894

1199011minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199011

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894+

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198911

+119902119894

1199091

1198961198941199091

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=1)

(26a)

119906119894

1199012minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199012

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894minus

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198912

+119902119894

1199092

1198961198941199092

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=2)

(26b)


V1198941199011

minus V119894119887=

ℎ119901

2

120597119908119894

1199011

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199101

1198961198941199101

[1 minus1

2(120579119894

119887119909)2

]


119895=1)

(27a)

V1198941199012

minus V119894119887=

ℎ119901

2

120597119908119894

1199012

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199102

1198961198941199102

[1 minus1

2(120579119894

119887119909)2

]


119895=2)

(27b)


119908119894

1199011minus 119908119894

119887= minus

119887119894

119891

4120579119894


119894

119895=1) (28a)

119908119894

1199012minus 119908119894

119887=

119887119894

119891

4120579119894


119894

119895=2) (28b)

where (120593119875119894

119878)119891119895


119894

119895and 119896

119894

119909119895 119896119894119910119895



=

119896119894

119909119895(119904119894

119909119895) and 119896

119894

119910119895= 119896119894

119910119895(119904119894



119894



119875119894









nabla2119906119901119894 = 119901119901119894 (119909 119910 119905) (119894 = 1 2) (29a)

nabla41199061199013 = 1199011199013 (119909 119910 119905) (29b)






[

[

E11990111 0 00 E11990122 00 0 E11990133

]

]

d1199011d1199012d1199013

+

0D1198991198971199011

0D1198991198971199012

00

D1198991198971199013

=

01205721199013

01205731199013

001205741199013

1205751199013

(30)

where

E11990111 = [A1 H1 H20 D11990122 D11990123

] (31a)

E11990122 = [A1 H1 H20 D11990144 D11990145

] (31b)

E11990133 =[[[

[

A2 H1 H2 G1 G2A1 0 0 H1 H20 D11990178 D11990179 0 D1199017110 D11990188 D11990189 D119901810 0

]]]

]

(31c)



119901119894(119894 = 1 2) are 119873 times 1 and

D1198991198971199013


d119901119894 = p119901119894 u119901119894 u119901119894119899119879 (119894 = 1 2) (32a)


2u1199013)119899119879 (32b)


u119901119894 = (119906119901119894)1(119906119901119894)2

sdot sdot sdot (119906119901119894)119873119879

(119894 = 1 2 3)

(33a)

u119901119894119899 = (

120597119906119901119894

120597119899)

1

(

120597119906119901119894

120597119899)

2

sdot sdot sdot (

120597119906119901119894

120597119899)

119873

119879

(119894 = 1 2 3)

(33b)


21199061199013)1

(nabla21199061199013)2


119879

(33c)

(nabla2u1199013)119899

= (

120597nabla21199061199013

120597119899)

1

(

120597nabla21199061199013

120597119899)

2

sdot sdot sdot (

120597nabla21199061199013

120597119899)

119873

119879

(33d)


(119901119901119894)1(119901119901119894)2

sdot sdot sdot (119901119901119894)119872119879



u119901119894 = B119901119894d119901119894 (119894 = 1 2 3) (34a)u119901119894119909 = B119901119894119909d119901119894 (119894 = 1 2 3) (34b)

u119901119894119910 = B119901119894119910d119901119894 (119894 = 1 2 3) (34c)


u119901119894119909119909 = B119901119894119909119909d119901119894 (119894 = 1 2 3) (34d)

u119901119894119910119910 = B119901119894119910119910d119901119894 (119894 = 1 2 3) (34e)

u119901119894119909119910 = B119901119894119909119910d119901119894 (119894 = 1 2 3) (34f)


119866119901ℎ119901 [p1199011 +1 + ]1199011 minus ]119901

(B1199011119909119909d1199011 + B1199012119909119910d1199012)

+ 2

1 minus ]119901(B1199013119909119909d1199013)119889119892B1199013119909d1199013

+(B1199013119910119910d1199013)119889119892 sdot B1199013119909d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119910d1199013]

minus 120588119901ℎ119901B1199011d1199011 = Zq119909(35a)

119866119901ℎ119901 [p1199012 +1 + ]1199011 minus ]119901

(B1199011119909119910d1199011 + B1199012119910119910d1199012)

+ 2

1 minus ]119901(B1199013119910119910d1199013)119889119892B1199013119910d1199013

+ (B1199013119909119909d1199013)119889119892 sdot B1199013119910d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119909d1199013]

minus 120588119901ℎ119901B1199012d1199012 = Zq119910

(35b)

119863p1199013 minus 119862

times [(B1199011119909d1199011)119889119892B1199013119909119909d1199013

+1

2(B1199013119909d1199013)119889119892(B1199013119909d1199013)119889119892B1199013119909119909d1199013

+ ]119901(B1199012119910d1199012)119889119892B1199013119909119909d1199013 +1

2]119901(B1199013119910d1199013)119889119892

times (B1199013119910d1199013)119889119892B1199013119909119909d1199013]

+ (1 minus ]119901)

times [(B1199011119910d1199011)119889119892B1199013119909119910d1199013

+ (B1199012119909d1199012)119889119892B1199013119909119910d1199013

+(B1199013119909d1199013)119889119892 sdot (B1199013119910d1199013)119889119892B1199013119909119910d1199013]

+ [(B1199012119910d1199012)119889119892B1199013119910119910d1199013 +1

2(B1199013119910d1199013)119889119892

sdot (B1199013119910d1199013)119889119892B1199013119910119910d1199013

+ ]119901(B1199011119909d1199011)119889119892B1199013119910119910d1199013

+1

2]119901(B1199013119909d1199013)119889119892

sdot (B1199013119909d1199013)119889119892B1199013119910119910d1199013]

+ 120588119901ℎ119901B1199013d1199013 minus 120588119901ℎ119901(B1199011d1199011)119889119892B1199013119909d1199013


119901

12B1199013119909119909d1199013

minus

120588119901ℎ3

119901

12B1199013119910119910d1199013


+ (Zq119910)119889119892B1199013119910d1199013

(35c)

where q119909 = q1199091 q1199092119879 q119910 = q1199101 q1199102

119879 andq119911 = q1199111 q1199112


119894



119901119910119895= 119902119894

119909119895ℎ1199012 119898

119894

119901119909119895= 119902119894

119910119895ℎ1199012 respectively



119887 V119894119887 and 119908

119894

119887




119887(119909119894 119905) and V119894

119887(119909119894 119905) 119908119894





Let 119906119894

1198871= 119906119894

119887(119909119894 119905) 119906

119894

1198872= V119894119887(119909119894 119905) 119906

119894

1198873= 119908119894

119887(119909119894 119905)

and 119906119894

1198874= 120579119894




1205972119906119894

1198871

1205971199091198942

= 119901119894

1198871(119909119894 119905) (36a)

1205974119906119894

119887119895

1205971199091198944= 119901119894

119887119895(119909119894 119905) (119895 = 2 3 4) (36b)




119894





119887119895(120577119894) 119906119894119887119895119909

(120577119894) 119906119894119887119895119909119909

(120577119894) and 119906

119894

119887119895119909119909119909(120577119894)

(120577119894 = 0 119897119894) in terms of 119901119894



[[[

[

E11989411988711

0 0 00 E119894

119887220 0

0 0 E11989411988733

00 0 0 E119894

11988744

]]]

]

d1198941198871

d1198941198872

d1198941198873

d1198941198874

+

0D119894 1198991198971198871

00

D119894 1198991198971198872

00

D119894 1198991198971198873

000

=

0120572119894

1198873

00120573119894

1198873

00120574119894

1198873

00120575119894

1198873

(37)

where E11989411988711

and E11989411988722

E11989411988733

E11989411988744


1198871is

a 2 times 1 and D119894 119899119897119887119895


1198873and

120573119894

1198873 1205741198941198873 1205751198941198873



1198873and 120573

119894

1198873 120573119894

1198873 120574119894

1198873 120574119894

1198873 120575119894

1198873 120575119894


d1198941198871

= p1198941198871

u1198941198871

u1198941198871119909

119879

(38a)

d119894119887119895

= p119894119887119895 u119894119887119895

u119894119887119895119909

u119894119887119895119909119909

u119894119887119895119909119909119909

119879

(119895 = 2 3 4)

(38b)


u119894119887119895

= 119906119894

119887119895(0 119905) 119906

119894

119887119895(119897119894 119905)119879

(119895 = 1 2 3 4) (39a)

u119894119887119895119909

= 120597119906119894

119887119895(0 119905)

120597119909119894

120597119906119894

119887119895(119897119894 119905)

120597119909119894

119879

(119895 = 1 2 3 4) (39b)

u119894119887119895119909119909

= 1205972119906119894

119887119895(0 119905)

1205971199091198942

1205972119906119894

119887119895(119897119894 119905)

1205971199091198942

119879

(119895 = 2 3 4)

(39c)

u119894119887119895119909119909119909

= 1205973119906119894

119887119895(0 119905)

1205971199091198943

1205973119906119894

119887119895(119897119894 119905)

1205971199091198943

119879

(119895 = 2 3 4)

(39d)


119887119895=

(119901119894

119887119895)1

(119901119894

119887119895)2


119887119895)119871119879



119894



119894isin (0 119897

119894) and


u1198941198871

= B1198941198871d1198941198871 (40a)

u1198941198871119909

= B1198941198871119909

d1198941198871 (40b)

u119894119887119895

= B119894119887119895d119894119887119895 (119895 = 2 3 4) (40c)

u119894119887119895119909

= B119894119887119895119909

d119894119887119895 (119895 = 2 3 4) (40d)

u119894119887119895119909119909

= B119894119887119895119909119909

d119894119887119895 (119895 = 2 3 4) (40e)

u119894119887119895119909119909119909

= B119894119887119895119909119909119909

d119894119887119895 (119895 = 2 3 4) (40f)

where B1198941198871B1198941198871119909

are 119871 times (119871 + 4) and B119894119887119895B119894119887119895119909

B119894119887119895119909119909

B119894119887119895119909119909119909



x

y

a

a

Fr

Fr

ClCl

Lx = 18 cm

Ly=9

cm A

(a)

g

02 cmhb = 05 cm

1 cm 5 cm3 cm

Ly = 9 cm

(b)


Time (ms)

060

040

020

000

minus020

000 050 100 150 200 250

Disp

lace

men

tw (c

m)





minus 119864119894

119887119860119894

119887[p1198941198871

+ (B1198941198872119909

d1198941198872)119889119892B1198941198872119909119909

d1198941198872

+ (B1198941198873119909

d1198941198873)119889119892B1198941198873119909119909

d1198941198873]

+ 120588119894

119887Α119894

119887T1q1 = q119894

1199091+ q1198941199092

(41a)

119864119894

119887119868119894

119911p1198941198872

minus (N119894119887)119889119892B1198941198872119909119909

d1198941198872

minus 120588119894

119887119868119894

119911B1198941198872119909119909

d1198941198872

+ 120588119894

119887Α119894

119887B1198941198872d1198941198872

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198872119909

d1198941198872

= q1198941199101

+ q1198941199102

minus (B1198941198872119909

d1198941198872)119889119892

(q1198941199091

+ q1198941199092) minus X119894119887119910

(q1198941199091

+ q1198941199092)

(41b)

119864119894

119887119868119894

119910p1198941198873

minus (N119894119887)119889119892B1198941198873119909119909

d1198941198873

minus 120588119894

119887119868119894

119910B1198941198873119909119909

d1198941198873

+ 120588119894

119887Α119894

119887B1198941198873d1198941198873

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198873119909

d1198941198873

= q1198941199111

+ q1198941199112

minus (B1198941198873119909

d1198941198873)119889119892

(q1198941199091

+ q1198941199092) + X119894119887119911

(q1198941199091

+ q1198941199092)

(41c)

119864119894

119887119862119894

119878p1198941198874

minus 119866119894

119887119868119894

119905B1198941198874119909119909

d1198941198874

+ 120588119894

119887119868119894

119901B1198941198874d1198941198874

minus 120588119894

119887119862119894

119878B1198941198874119909119909

d1198941198874

= e1198941199101q1198941199111

+ e1198941199102q1198941199112

minus e1198941199111q1198941199101

minus e1198941199112q1198941199102

+ Χ119894

119887119908(q1198941199091

+ q1198941199092)

(41d)

where (N119894119887)119889119892


119887119911 X119894119887119910 X119894119887119908


119898119894119887119911119895 119898119894119887119908119895


1199101 e1198941199102 e1198941199111

e1198941199112


119894

119910119895 119890119894


119894

119911119895 119902119894

119910119895with



Y1B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y1B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

+

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198911

B1198941198874119909

d1198874

+ K1198941199091

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42a)


008

008

018

018

028

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036040044048


000067300301006080091501220153018402140245027603070337036803990430460491


008

008

008

00801

8018

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036


000064600236004780072009620120145016901930217024102660290314033803620387



Y2B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y2B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

minus

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198912

B1198941198874119909

d1198874

+ K1198941199092

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42b)

Y1B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y1B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199101

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43a)

Y2B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y2B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199102

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43b)

Y1B1199013d1199013 minus B1198941198873d1198941198873

= minus

119887119894

119891

4B1198941198874d1198941198874 (44a)

Y2B1199013d1199013 minus B1198941198873d1198941198873

=

119887119894

119891

4B1198941198874d1198941198874 (44b)

000 050 100 150 200 250

060

040

020

000

Disp

lace

men

tw (c

m)

Time (ms)



minus020



119887119895(119894 = 1 sdot sdot sdot 119868)



000015030045060075

9

6

3

0

1815

12

9

6

3

0


000010020030040050

9

6

3

0

1815

12

9

6

3

0


000010020030040050060

9

6

3

0

1815

12

9

6

3

0


00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0


00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000


00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000




0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119875119894





119906119901 (119909 119910 119911 119905) = 119906119901 (119909 119910 119905) minus 119911

120597119908119901 (119909 119910 119905)

120597119909 (1a)

V119901 (119909 119910 119911 119905) = V119901 (119909 119910 119905) minus 119911

120597119908119901 (119909 119910 119905)

120597119910 (1b)

119908119901 (119909 119910 119911 119905) = 119908119901 (119909 119910 119905) (1c)



120576119909119909 =

120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

(2a)

120576119910119910 =

120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

(2b)

120574119909119910 =

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119909

120597119908119901

120597119910 (2c)

120576119911119911 = 120574119909119911 = 120574119910119911 = 0 (2d)


119878119909119909

119878119910119910

119878119909119910

=

[[[[[[[[[[

[

119864119901

(1 minus ]119901)2

119864119901]119901

(1 minus ]119901)2

0

119864119901]119901

(1 minus ]119901)2

119864119901

(1 minus ]119901)2

0

0 0

119864119901

2 (1 + ]119901)

]]]]]]]]]]

]

120576119909119909

120576119910119910

120574119909119910

(3)


119878119909119909 =

119864119901

(1 minus ]2119901)

times [

120597119906119901

120597119909minus 119911

1205972119908119901

1205971199092+

1

2(

120597119908119901

120597119909)

2

]

+]119901 [120597V119901120597119910

minus 119911

1205972119908119901

1205971199102+

1

2(

120597119908119901

120597119910)

2

]

(4a)

119878119910119910 =

119864119901

(1 minus ]2119901)

]119901 [120597119906119901

120597119909minus 119911

1205972119908119901

1205971199092+

1

2(

120597119908119901

120597119909)

2

]

+[

120597V119901120597119910

minus 119911

1205972119908119901

1205971199102+

1

2(

120597119908119901

120597119910)

2

]

(4b)

119878119909119910 =

119864119901

2 (1 + ]119901)(

120597119906119901

120597119910+

120597V119901120597119909

minus 2119911

1205972119908119901

120597119909120597119910+

120597119908119901

120597119909

120597119908119901

120597119910)

(4c)


119873119901119909 = 119862[

120597119906119901

120597119909+ ]119901

120597V119901120597119909

+1

2(

120597119908119901

120597119909)

2

+1

2]119901(

120597119908119901

120597119910)

2

]

(5a)

119873119901119910 = 119862[

120597V119901120597119909

+ ]119901120597119906119901

120597119909+

1

2(

120597119908119901

120597119910)

2

+1

2]119901(

120597119908119901

120597119909)

2

]

(5b)

119873119901119909119910 = 119862

1 minus ]1199012

(

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119910

120597119908119901

120597119909) (5c)

119872119901119909 = minus119863(

1205972119908119901

1205971199092+ ]119901

1205972119908119901

1205971199102) (5d)

119872119901119910 = minus119863(

1205972119908119901

1205971199102+ ]119901

1205972119908119901

1205971199092) (5e)

119872119901119909119910 = minus119863(1 minus ]119901)1205972119908119901

120597119909120597119910 (5f)


3

11990112(1 minus ]2

119901) are the




120597119873119901119909

120597119909+

120597119873119901119909119910

120597119910minus 120588119901ℎ119901119901 =

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119909119895120575 (119910 minus 119910119895)) (6a)

120597119873119901119910

120597119910+

120597119873119901119909119910

120597119909minus 120588119901ℎ119901V119901 =

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119910119895120575 (119910 minus 119910119895)) (6b)

minus

1205972119872119901119909

1205971199092minus 2

1205972119872119901119909119910

120597119909120597119910minus

1205972119872119901119910

1205971199102minus 119873119901119909

1205972119908119901

1205971199092

minus 2119873119901119909119910

1205972119908119901

120597119909120597119910minus 119873119901119910

1205972119908119901

1205971199102

+ 120588119901ℎ119901119901 minus 120588119901ℎ119901119901

120597119908119901

120597119909minus 120588119901ℎ119901V119901

120597119908119901

120597119910

minus

120588119901ℎ3

119901

12

1205972119901

1205971199092minus

120588119901ℎ3

119901

12

1205972119901

1205971199102

= 119892 minus

119868

sum

119894=1

[

[

2

sum

119895=1

(119902119894

119911119895+

120597119898119894

119901119909119895

120597119910+

120597119898119894

119901119910119895

120597119909minus 119902119894

119909119895

120597119908119894

119901119895

120597119909

minus119902119894

119910119895

120597119908119894

119901119895

120597119910)120575 (119910 minus 119910119895)

]

]

(6c)


119866119901ℎ119901 [nabla2119906119901 +

1 + ]1199011 minus ]119901

120597

120597119909(

120597119906119901

120597119909+

120597V119901120597119910

)

+(2

1 minus ]119901

1205972119908119901

1205971199092+

1205972119908119901

1205971199102)

120597119908119901

120597119909

+

1 + ]1199011 minus ]119901

1205972119908119901

120597119909120597119910

120597119908119901

120597119910] minus 120588119901ℎ119901119901

=

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119909119895120575 (119910 minus 119910119895))

(7a)

119866119901ℎ119901 [nabla2V119901 +

1 + ]1199011 minus ]119901

120597

120597119910(

120597119906119901

120597119909+

120597V119901120597119910

)

+ (2

1 minus ]119901

1205972119908119901

1205971199102+

1205972119908119901

1205971199092)

120597119908119901

120597119910

+

1 + ]1199011 minus ]119901

1205972119908119901

120597119909120597119910

120597119908119901

120597119909] minus 120588119901ℎ119901V119901

=

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119910119895120575 (119910 minus 119910119895))

(7b)

119863nabla4119908119901 minus 119862

times [(

120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

)

+]119901(120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

)]

times

1205972119908119901

1205971199092+ (1 minus ]119901)

sdot (

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119909

120597119908119901

120597119910)

1205972119908119901

120597119909120597119910

+ [(

120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

)

+]119901(120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

)] sdot

1205972119908119901

1205971199102

+ 120588119901ℎ119901119901 minus 120588119901ℎ119901119901

120597119908119901

120597119909minus 120588119901ℎ119901V119901

120597119908119901

120597119910

minus

120588119901ℎ3

119901

12

1205972119901

1205971199092minus

120588119901ℎ3

119901

12

1205972119901

1205971199102

= 119892 minus

119868

sum

119894=1

[

[

2

sum

119895=1

(119902119894

119911119895+

120597119898119894

119901119909119895

120597119910+

120597119898119894

119901119910119895

120597119909minus 119902119894

119909119895

120597119908119894

119901119895

120597119909

minus119902119894

119910119895

120597119908119894

119901119895

120597119910)120575 (119910 minus 119910119895)

]

]

(7c)


1198861199011119906119901119899 + 1198861199012119873119901119899 = 1198861199013 (8a)

1205731199011119906119901119905 + 1205731199012119873119901119905 = 1205731199013 (8b)

1205741199011119908119901 + 1205741199012119877119901119899 = 1205741199013 (8c)

1205751199011

120597119908119901

120597119899+ 1205751199012119872119901119899 = 1205751199013

(8d)

1205761119896119908119901 + 1205762119896

10038171003817100381710038171003817119879119908119901

10038171003817100381710038171003817119896= 1205763119896 1205762119896 = 0 (8e)


119908119901 (x 0) = 1199081199010 (x) (9a)

119901 (x 0) = 1199081199010 (x) (9b)




119877119901119899 = minus119863[120597


120597

120597119904(

1205972119908119901

120597119904120597119899minus 120581

120597119908119901

120597119904)]

+ 119873119901119899

120597119908119901

120597119899+ 119873119901119905

120597119908119901

120597119904

(10a)


1205972119908119901

1205971199042+ 120581

120597119908119901

120597119899)] (10b)



119879119908119901 = 119863(]119901 minus 1)(

1205972119908119901

120597119904120597119899minus 120581

120597119908119901

120597119904) (11)


1205741199012 = 1205741199013 = 1205751199012 = 1205751199013 = 0)





119906119894

119887(119909119894 119910119894 119911119894 119905) = 119906

119894

119887(119909119894 119905) minus 119910

119894120579119894

119887119911(119909119894 119905)

+ 119911119894120579119894

119887119910(119909119894 119905) +

120597120579119894

119887119909

120597119909119894120593119875119894

119878(119910119894 119911119894 119905)

(12a)

V119894119887(119909119894 119910119894 119911119894 119905) = V119894

119887(119909119894 119905) minus 119911

119894120579119894

119887119909(119909119894 119905) (12b)

119908119894

119887(119909119894 119910119894 119911119894 119905) = 119908

119894

119887(119909119894 119905) + 119910

119894120579119894

119887119909(119909119894 119905) (12c)

120579119894

119887119910(119909119894 119905) = minus

120597119908119894

119887(119909119894 119905)

120597119909119894 (12d)

120579119894

119887119911(119909119894 119905) =

120597V119894119887(119909119894 119905)

120597119909119894 (12e)

where 119906119894119887 V119894119887 and119908

119894



119887=

119906119894

119887(119909119894) V119894119887= V119894119887(119909119894) and 119908

119894

119887= 119908119894



= 120579119894

119887119910(119909119894) and 120579

119894

119887119911=

120579119894




119887119909(119909119894) regarded as the




120576119909119909 =120597119906119894

119887

120597119909119894+

1

2

[

[

(120597V119894119887

120597119909119894)

2

+ (120597119908119894

119887

120597119909119894)

2

]

]

(13a)

120574119909119911 =120597119908119894

119887

120597119909119894+

120597119906119894

119887

120597119911119894+ (

120597V119894119887

120597119909119894

120597V119894119887

120597119911119894+

120597119908119894

119887

120597119909119894

120597119908119894

119887

120597119911119894) (13b)

120574119909119910 =120597V119894119887

120597119909119894+

120597119906119894

119887

120597119910119894+ (

120597V119894119887

120597119909119894

120597V119894119887

120597119910119894+

120597119908119894

119887

120597119909119894

120597119908119894

119887

120597119910119894) (13c)

120576119910119910 = 120576119911119911 = 120574119910119911 = 0 (13d)


119873119894

119887= 119864119894

119887119860119894

119887[120597119906119894

119887

120597119909119894+

1

2((

120597V119894119887

120597119909119894)

2

+ (120597119908119894

119887

120597119909119894)

2

)] (14a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (14b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (14c)

119872119875119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894 (14d)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (14e)

where 119872119875119894



119887119908is the



119910and 119868

119894


119894

119878is the


119905and 119862

119894



119868119894

119905= intΩ

(1199101198942+ 1199111198942+ 119910119894 120597120593119875119894

119878

120597119911119894minus 119911119894 120597120593119875119894

119878

120597119910119894)119889Ω (15a)

119862119894

119878= intΩ

(120593119875119894

119878)2

119889Ω (15b)


minus120597119873119894

119887

120597119909119894+ 120588119894

119887Α119894

119887119894

119887=

2

sum

119895=1

119902119894

119909119895 (16a)

minus 119873119894

119887

1205972V119894119887

1205971199091198942+

1205972119872119894

119887119911

1205971199091198942

+ 120588119894

119887Α119894

119887V119894119887minus 120588119894

119887119868119894

119911

1205972V119894119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(16b)

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942minus

1205972119872119894

119887119910

1205971199091198942

+ 120588119894

119887Α119894

119887119894

119887minus 120588119894

119887119868119894

119910

1205972119894

119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(16c)

minus120597119872119894

119887119905

120597119909119894minus

1205972119872119894

119887119908

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(16d)


minus 119864119894

119887119860119894

119887(1205972119906119894

119887

1205971199091198942+

120597119908119894

119887

120597119909119894

1205972119908119894

119887

1205971199091198942

+120597V119894119887

120597119909119894

1205972V119894119887

1205971199091198942) + 120588

119894

119887Α119894

119887119894

119887

=

2

sum

119895=1

119902119894

119909119895

(17a)

119864119894

119887119868119894

119911

1205974V119894119887

1205971199091198944minus 119873119894

119887

1205972V119894119887

1205971199091198942minus 120588119894

119887119868119894

119911

1205972V1198871205971199092

+ 120588119894

119887Α119894

119887V119887 minus 120588

119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(17b)

119864119894

119887119868119894

119910

1205974119908119894

119887

1205971199091198944

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942

minus 120588119894

119887119868119894

119910

1205972119887

1205971199092+ 120588119894

119887Α119894

119887119887 minus 120588

119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(17c)

119864119894

119887119862119894

119878

1205974120579119894

119887119909

1205971199091198944

minus 119866119894

119887119868119894

119905

1205972120579119894

119887119909

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(17d)


119894 are given as

119886119894

1198871119906119894

119887+ 120572119894

1198872119873119894

119887= 120572119894

1198873 (18)

120573119894

1198871V119894119887+ 120573119894

1198872119877119894

119887119910= 120573119894

1198873 (19a)

120573119894

1198871120579119894

119887119911+ 120573119894

1198872119872119894

119887119911= 120573119894

1198873 (19b)

120574119894

1198871119908119894

119887+ 120574119894

1198872119877119894

119887119911= 120574119894

1198873 (20a)

120574119894

1198871120579119894

119887119910+ 120574119894

1198872119872119894

119887119910= 120574119894

1198873 (20b)

120575119894

1198871120579119894

119887119909+ 120575119894

1198872119872119894

119887119905= 120575119894

1198873 (21a)

120575119894

1198871

119889120579119894

119887119909

119889119909119894+ 120575119894

1198872119872119894

119887119908= 120575119894

1198873(21b)


119908119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22a)

119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22b)


119894

119887119910 120579119894


119894

119887119910 119877119894

119887119911

and 119872119894

119887119911 119872119894119887119910



119877119894

119887119910= 119873119894

119887

120597V119894119887

120597119909119894minus 119864119894

119887119868119894

119911

1205973V119894119887

1205971199091198943 (23a)

119877119894

119887119911= 119873119894

119887

120597119908119894

119887

120597119909119894minus 119864119894

119887119868119894

119910

1205973119908119894

119887

1205971199091198943 (23b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (24a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (24b)


and 119872119894

119887119905 119872119894119887119908


119872119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894minus 119864119894

119887119862119894

119878

1205973120579119894

119887119909

1205971199091198943 (25a)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (25b)

Finally 120572119894119887119896 120573119894

119887119896 120573119894

119887119896 120574119894

119887119896 120574119894

119887119896 120575119894

119887119896 120575119894

119887119896(119896 = 1 2 3) are func-




119894

119887 V119894119887119908119894119887 120579119894119887119909 1199021198941199091 1199021198941199101 1199021198941199111 119902119894

1199092

119902119894

1199102 and 119902

119894





119906119894

1199011minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199011

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894+

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198911

+119902119894

1199091

1198961198941199091

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=1)

(26a)

119906119894

1199012minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199012

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894minus

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198912

+119902119894

1199092

1198961198941199092

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=2)

(26b)


V1198941199011

minus V119894119887=

ℎ119901

2

120597119908119894

1199011

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199101

1198961198941199101

[1 minus1

2(120579119894

119887119909)2

]


119895=1)

(27a)

V1198941199012

minus V119894119887=

ℎ119901

2

120597119908119894

1199012

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199102

1198961198941199102

[1 minus1

2(120579119894

119887119909)2

]


119895=2)

(27b)


119908119894

1199011minus 119908119894

119887= minus

119887119894

119891

4120579119894


119894

119895=1) (28a)

119908119894

1199012minus 119908119894

119887=

119887119894

119891

4120579119894


119894

119895=2) (28b)

where (120593119875119894

119878)119891119895


119894

119895and 119896

119894

119909119895 119896119894119910119895



=

119896119894

119909119895(119904119894

119909119895) and 119896

119894

119910119895= 119896119894

119910119895(119904119894



119894



119875119894









nabla2119906119901119894 = 119901119901119894 (119909 119910 119905) (119894 = 1 2) (29a)

nabla41199061199013 = 1199011199013 (119909 119910 119905) (29b)






[

[

E11990111 0 00 E11990122 00 0 E11990133

]

]

d1199011d1199012d1199013

+

0D1198991198971199011

0D1198991198971199012

00

D1198991198971199013

=

01205721199013

01205731199013

001205741199013

1205751199013

(30)

where

E11990111 = [A1 H1 H20 D11990122 D11990123

] (31a)

E11990122 = [A1 H1 H20 D11990144 D11990145

] (31b)

E11990133 =[[[

[

A2 H1 H2 G1 G2A1 0 0 H1 H20 D11990178 D11990179 0 D1199017110 D11990188 D11990189 D119901810 0

]]]

]

(31c)



119901119894(119894 = 1 2) are 119873 times 1 and

D1198991198971199013


d119901119894 = p119901119894 u119901119894 u119901119894119899119879 (119894 = 1 2) (32a)


2u1199013)119899119879 (32b)


u119901119894 = (119906119901119894)1(119906119901119894)2

sdot sdot sdot (119906119901119894)119873119879

(119894 = 1 2 3)

(33a)

u119901119894119899 = (

120597119906119901119894

120597119899)

1

(

120597119906119901119894

120597119899)

2

sdot sdot sdot (

120597119906119901119894

120597119899)

119873

119879

(119894 = 1 2 3)

(33b)


21199061199013)1

(nabla21199061199013)2


119879

(33c)

(nabla2u1199013)119899

= (

120597nabla21199061199013

120597119899)

1

(

120597nabla21199061199013

120597119899)

2

sdot sdot sdot (

120597nabla21199061199013

120597119899)

119873

119879

(33d)


(119901119901119894)1(119901119901119894)2

sdot sdot sdot (119901119901119894)119872119879



u119901119894 = B119901119894d119901119894 (119894 = 1 2 3) (34a)u119901119894119909 = B119901119894119909d119901119894 (119894 = 1 2 3) (34b)

u119901119894119910 = B119901119894119910d119901119894 (119894 = 1 2 3) (34c)


u119901119894119909119909 = B119901119894119909119909d119901119894 (119894 = 1 2 3) (34d)

u119901119894119910119910 = B119901119894119910119910d119901119894 (119894 = 1 2 3) (34e)

u119901119894119909119910 = B119901119894119909119910d119901119894 (119894 = 1 2 3) (34f)


119866119901ℎ119901 [p1199011 +1 + ]1199011 minus ]119901

(B1199011119909119909d1199011 + B1199012119909119910d1199012)

+ 2

1 minus ]119901(B1199013119909119909d1199013)119889119892B1199013119909d1199013

+(B1199013119910119910d1199013)119889119892 sdot B1199013119909d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119910d1199013]

minus 120588119901ℎ119901B1199011d1199011 = Zq119909(35a)

119866119901ℎ119901 [p1199012 +1 + ]1199011 minus ]119901

(B1199011119909119910d1199011 + B1199012119910119910d1199012)

+ 2

1 minus ]119901(B1199013119910119910d1199013)119889119892B1199013119910d1199013

+ (B1199013119909119909d1199013)119889119892 sdot B1199013119910d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119909d1199013]

minus 120588119901ℎ119901B1199012d1199012 = Zq119910

(35b)

119863p1199013 minus 119862

times [(B1199011119909d1199011)119889119892B1199013119909119909d1199013

+1

2(B1199013119909d1199013)119889119892(B1199013119909d1199013)119889119892B1199013119909119909d1199013

+ ]119901(B1199012119910d1199012)119889119892B1199013119909119909d1199013 +1

2]119901(B1199013119910d1199013)119889119892

times (B1199013119910d1199013)119889119892B1199013119909119909d1199013]

+ (1 minus ]119901)

times [(B1199011119910d1199011)119889119892B1199013119909119910d1199013

+ (B1199012119909d1199012)119889119892B1199013119909119910d1199013

+(B1199013119909d1199013)119889119892 sdot (B1199013119910d1199013)119889119892B1199013119909119910d1199013]

+ [(B1199012119910d1199012)119889119892B1199013119910119910d1199013 +1

2(B1199013119910d1199013)119889119892

sdot (B1199013119910d1199013)119889119892B1199013119910119910d1199013

+ ]119901(B1199011119909d1199011)119889119892B1199013119910119910d1199013

+1

2]119901(B1199013119909d1199013)119889119892

sdot (B1199013119909d1199013)119889119892B1199013119910119910d1199013]

+ 120588119901ℎ119901B1199013d1199013 minus 120588119901ℎ119901(B1199011d1199011)119889119892B1199013119909d1199013


119901

12B1199013119909119909d1199013

minus

120588119901ℎ3

119901

12B1199013119910119910d1199013


+ (Zq119910)119889119892B1199013119910d1199013

(35c)

where q119909 = q1199091 q1199092119879 q119910 = q1199101 q1199102

119879 andq119911 = q1199111 q1199112


119894



119901119910119895= 119902119894

119909119895ℎ1199012 119898

119894

119901119909119895= 119902119894

119910119895ℎ1199012 respectively



119887 V119894119887 and 119908

119894

119887




119887(119909119894 119905) and V119894

119887(119909119894 119905) 119908119894





Let 119906119894

1198871= 119906119894

119887(119909119894 119905) 119906

119894

1198872= V119894119887(119909119894 119905) 119906

119894

1198873= 119908119894

119887(119909119894 119905)

and 119906119894

1198874= 120579119894




1205972119906119894

1198871

1205971199091198942

= 119901119894

1198871(119909119894 119905) (36a)

1205974119906119894

119887119895

1205971199091198944= 119901119894

119887119895(119909119894 119905) (119895 = 2 3 4) (36b)




119894





119887119895(120577119894) 119906119894119887119895119909

(120577119894) 119906119894119887119895119909119909

(120577119894) and 119906

119894

119887119895119909119909119909(120577119894)

(120577119894 = 0 119897119894) in terms of 119901119894



[[[

[

E11989411988711

0 0 00 E119894

119887220 0

0 0 E11989411988733

00 0 0 E119894

11988744

]]]

]

d1198941198871

d1198941198872

d1198941198873

d1198941198874

+

0D119894 1198991198971198871

00

D119894 1198991198971198872

00

D119894 1198991198971198873

000

=

0120572119894

1198873

00120573119894

1198873

00120574119894

1198873

00120575119894

1198873

(37)

where E11989411988711

and E11989411988722

E11989411988733

E11989411988744


1198871is

a 2 times 1 and D119894 119899119897119887119895


1198873and

120573119894

1198873 1205741198941198873 1205751198941198873



1198873and 120573

119894

1198873 120573119894

1198873 120574119894

1198873 120574119894

1198873 120575119894

1198873 120575119894


d1198941198871

= p1198941198871

u1198941198871

u1198941198871119909

119879

(38a)

d119894119887119895

= p119894119887119895 u119894119887119895

u119894119887119895119909

u119894119887119895119909119909

u119894119887119895119909119909119909

119879

(119895 = 2 3 4)

(38b)


u119894119887119895

= 119906119894

119887119895(0 119905) 119906

119894

119887119895(119897119894 119905)119879

(119895 = 1 2 3 4) (39a)

u119894119887119895119909

= 120597119906119894

119887119895(0 119905)

120597119909119894

120597119906119894

119887119895(119897119894 119905)

120597119909119894

119879

(119895 = 1 2 3 4) (39b)

u119894119887119895119909119909

= 1205972119906119894

119887119895(0 119905)

1205971199091198942

1205972119906119894

119887119895(119897119894 119905)

1205971199091198942

119879

(119895 = 2 3 4)

(39c)

u119894119887119895119909119909119909

= 1205973119906119894

119887119895(0 119905)

1205971199091198943

1205973119906119894

119887119895(119897119894 119905)

1205971199091198943

119879

(119895 = 2 3 4)

(39d)


119887119895=

(119901119894

119887119895)1

(119901119894

119887119895)2


119887119895)119871119879



119894



119894isin (0 119897

119894) and


u1198941198871

= B1198941198871d1198941198871 (40a)

u1198941198871119909

= B1198941198871119909

d1198941198871 (40b)

u119894119887119895

= B119894119887119895d119894119887119895 (119895 = 2 3 4) (40c)

u119894119887119895119909

= B119894119887119895119909

d119894119887119895 (119895 = 2 3 4) (40d)

u119894119887119895119909119909

= B119894119887119895119909119909

d119894119887119895 (119895 = 2 3 4) (40e)

u119894119887119895119909119909119909

= B119894119887119895119909119909119909

d119894119887119895 (119895 = 2 3 4) (40f)

where B1198941198871B1198941198871119909

are 119871 times (119871 + 4) and B119894119887119895B119894119887119895119909

B119894119887119895119909119909

B119894119887119895119909119909119909



x

y

a

a

Fr

Fr

ClCl

Lx = 18 cm

Ly=9

cm A

(a)

g

02 cmhb = 05 cm

1 cm 5 cm3 cm

Ly = 9 cm

(b)


Time (ms)

060

040

020

000

minus020

000 050 100 150 200 250

Disp

lace

men

tw (c

m)





minus 119864119894

119887119860119894

119887[p1198941198871

+ (B1198941198872119909

d1198941198872)119889119892B1198941198872119909119909

d1198941198872

+ (B1198941198873119909

d1198941198873)119889119892B1198941198873119909119909

d1198941198873]

+ 120588119894

119887Α119894

119887T1q1 = q119894

1199091+ q1198941199092

(41a)

119864119894

119887119868119894

119911p1198941198872

minus (N119894119887)119889119892B1198941198872119909119909

d1198941198872

minus 120588119894

119887119868119894

119911B1198941198872119909119909

d1198941198872

+ 120588119894

119887Α119894

119887B1198941198872d1198941198872

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198872119909

d1198941198872

= q1198941199101

+ q1198941199102

minus (B1198941198872119909

d1198941198872)119889119892

(q1198941199091

+ q1198941199092) minus X119894119887119910

(q1198941199091

+ q1198941199092)

(41b)

119864119894

119887119868119894

119910p1198941198873

minus (N119894119887)119889119892B1198941198873119909119909

d1198941198873

minus 120588119894

119887119868119894

119910B1198941198873119909119909

d1198941198873

+ 120588119894

119887Α119894

119887B1198941198873d1198941198873

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198873119909

d1198941198873

= q1198941199111

+ q1198941199112

minus (B1198941198873119909

d1198941198873)119889119892

(q1198941199091

+ q1198941199092) + X119894119887119911

(q1198941199091

+ q1198941199092)

(41c)

119864119894

119887119862119894

119878p1198941198874

minus 119866119894

119887119868119894

119905B1198941198874119909119909

d1198941198874

+ 120588119894

119887119868119894

119901B1198941198874d1198941198874

minus 120588119894

119887119862119894

119878B1198941198874119909119909

d1198941198874

= e1198941199101q1198941199111

+ e1198941199102q1198941199112

minus e1198941199111q1198941199101

minus e1198941199112q1198941199102

+ Χ119894

119887119908(q1198941199091

+ q1198941199092)

(41d)

where (N119894119887)119889119892


119887119911 X119894119887119910 X119894119887119908


119898119894119887119911119895 119898119894119887119908119895


1199101 e1198941199102 e1198941199111

e1198941199112


119894

119910119895 119890119894


119894

119911119895 119902119894

119910119895with



Y1B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y1B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

+

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198911

B1198941198874119909

d1198874

+ K1198941199091

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42a)


008

008

018

018

028

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036040044048


000067300301006080091501220153018402140245027603070337036803990430460491


008

008

008

00801

8018

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036


000064600236004780072009620120145016901930217024102660290314033803620387



Y2B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y2B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

minus

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198912

B1198941198874119909

d1198874

+ K1198941199092

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42b)

Y1B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y1B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199101

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43a)

Y2B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y2B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199102

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43b)

Y1B1199013d1199013 minus B1198941198873d1198941198873

= minus

119887119894

119891

4B1198941198874d1198941198874 (44a)

Y2B1199013d1199013 minus B1198941198873d1198941198873

=

119887119894

119891

4B1198941198874d1198941198874 (44b)

000 050 100 150 200 250

060

040

020

000

Disp

lace

men

tw (c

m)

Time (ms)



minus020



119887119895(119894 = 1 sdot sdot sdot 119868)



000015030045060075

9

6

3

0

1815

12

9

6

3

0


000010020030040050

9

6

3

0

1815

12

9

6

3

0


000010020030040050060

9

6

3

0

1815

12

9

6

3

0


00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0


00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000


00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000




0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120597119873119901119909

120597119909+

120597119873119901119909119910

120597119910minus 120588119901ℎ119901119901 =

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119909119895120575 (119910 minus 119910119895)) (6a)

120597119873119901119910

120597119910+

120597119873119901119909119910

120597119909minus 120588119901ℎ119901V119901 =

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119910119895120575 (119910 minus 119910119895)) (6b)

minus

1205972119872119901119909

1205971199092minus 2

1205972119872119901119909119910

120597119909120597119910minus

1205972119872119901119910

1205971199102minus 119873119901119909

1205972119908119901

1205971199092

minus 2119873119901119909119910

1205972119908119901

120597119909120597119910minus 119873119901119910

1205972119908119901

1205971199102

+ 120588119901ℎ119901119901 minus 120588119901ℎ119901119901

120597119908119901

120597119909minus 120588119901ℎ119901V119901

120597119908119901

120597119910

minus

120588119901ℎ3

119901

12

1205972119901

1205971199092minus

120588119901ℎ3

119901

12

1205972119901

1205971199102

= 119892 minus

119868

sum

119894=1

[

[

2

sum

119895=1

(119902119894

119911119895+

120597119898119894

119901119909119895

120597119910+

120597119898119894

119901119910119895

120597119909minus 119902119894

119909119895

120597119908119894

119901119895

120597119909

minus119902119894

119910119895

120597119908119894

119901119895

120597119910)120575 (119910 minus 119910119895)

]

]

(6c)


119866119901ℎ119901 [nabla2119906119901 +

1 + ]1199011 minus ]119901

120597

120597119909(

120597119906119901

120597119909+

120597V119901120597119910

)

+(2

1 minus ]119901

1205972119908119901

1205971199092+

1205972119908119901

1205971199102)

120597119908119901

120597119909

+

1 + ]1199011 minus ]119901

1205972119908119901

120597119909120597119910

120597119908119901

120597119910] minus 120588119901ℎ119901119901

=

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119909119895120575 (119910 minus 119910119895))

(7a)

119866119901ℎ119901 [nabla2V119901 +

1 + ]1199011 minus ]119901

120597

120597119910(

120597119906119901

120597119909+

120597V119901120597119910

)

+ (2

1 minus ]119901

1205972119908119901

1205971199102+

1205972119908119901

1205971199092)

120597119908119901

120597119910

+

1 + ]1199011 minus ]119901

1205972119908119901

120597119909120597119910

120597119908119901

120597119909] minus 120588119901ℎ119901V119901

=

119868

sum

119894=1

(

2

sum

119895=1

119902119894

119910119895120575 (119910 minus 119910119895))

(7b)

119863nabla4119908119901 minus 119862

times [(

120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

)

+]119901(120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

)]

times

1205972119908119901

1205971199092+ (1 minus ]119901)

sdot (

120597119906119901

120597119910+

120597V119901120597119909

+

120597119908119901

120597119909

120597119908119901

120597119910)

1205972119908119901

120597119909120597119910

+ [(

120597V119901120597119910

+1

2(

120597119908119901

120597119910)

2

)

+]119901(120597119906119901

120597119909+

1

2(

120597119908119901

120597119909)

2

)] sdot

1205972119908119901

1205971199102

+ 120588119901ℎ119901119901 minus 120588119901ℎ119901119901

120597119908119901

120597119909minus 120588119901ℎ119901V119901

120597119908119901

120597119910

minus

120588119901ℎ3

119901

12

1205972119901

1205971199092minus

120588119901ℎ3

119901

12

1205972119901

1205971199102

= 119892 minus

119868

sum

119894=1

[

[

2

sum

119895=1

(119902119894

119911119895+

120597119898119894

119901119909119895

120597119910+

120597119898119894

119901119910119895

120597119909minus 119902119894

119909119895

120597119908119894

119901119895

120597119909

minus119902119894

119910119895

120597119908119894

119901119895

120597119910)120575 (119910 minus 119910119895)

]

]

(7c)


1198861199011119906119901119899 + 1198861199012119873119901119899 = 1198861199013 (8a)

1205731199011119906119901119905 + 1205731199012119873119901119905 = 1205731199013 (8b)

1205741199011119908119901 + 1205741199012119877119901119899 = 1205741199013 (8c)

1205751199011

120597119908119901

120597119899+ 1205751199012119872119901119899 = 1205751199013

(8d)

1205761119896119908119901 + 1205762119896

10038171003817100381710038171003817119879119908119901

10038171003817100381710038171003817119896= 1205763119896 1205762119896 = 0 (8e)


119908119901 (x 0) = 1199081199010 (x) (9a)

119901 (x 0) = 1199081199010 (x) (9b)




119877119901119899 = minus119863[120597


120597

120597119904(

1205972119908119901

120597119904120597119899minus 120581

120597119908119901

120597119904)]

+ 119873119901119899

120597119908119901

120597119899+ 119873119901119905

120597119908119901

120597119904

(10a)


1205972119908119901

1205971199042+ 120581

120597119908119901

120597119899)] (10b)



119879119908119901 = 119863(]119901 minus 1)(

1205972119908119901

120597119904120597119899minus 120581

120597119908119901

120597119904) (11)


1205741199012 = 1205741199013 = 1205751199012 = 1205751199013 = 0)





119906119894

119887(119909119894 119910119894 119911119894 119905) = 119906

119894

119887(119909119894 119905) minus 119910

119894120579119894

119887119911(119909119894 119905)

+ 119911119894120579119894

119887119910(119909119894 119905) +

120597120579119894

119887119909

120597119909119894120593119875119894

119878(119910119894 119911119894 119905)

(12a)

V119894119887(119909119894 119910119894 119911119894 119905) = V119894

119887(119909119894 119905) minus 119911

119894120579119894

119887119909(119909119894 119905) (12b)

119908119894

119887(119909119894 119910119894 119911119894 119905) = 119908

119894

119887(119909119894 119905) + 119910

119894120579119894

119887119909(119909119894 119905) (12c)

120579119894

119887119910(119909119894 119905) = minus

120597119908119894

119887(119909119894 119905)

120597119909119894 (12d)

120579119894

119887119911(119909119894 119905) =

120597V119894119887(119909119894 119905)

120597119909119894 (12e)

where 119906119894119887 V119894119887 and119908

119894



119887=

119906119894

119887(119909119894) V119894119887= V119894119887(119909119894) and 119908

119894

119887= 119908119894



= 120579119894

119887119910(119909119894) and 120579

119894

119887119911=

120579119894




119887119909(119909119894) regarded as the




120576119909119909 =120597119906119894

119887

120597119909119894+

1

2

[

[

(120597V119894119887

120597119909119894)

2

+ (120597119908119894

119887

120597119909119894)

2

]

]

(13a)

120574119909119911 =120597119908119894

119887

120597119909119894+

120597119906119894

119887

120597119911119894+ (

120597V119894119887

120597119909119894

120597V119894119887

120597119911119894+

120597119908119894

119887

120597119909119894

120597119908119894

119887

120597119911119894) (13b)

120574119909119910 =120597V119894119887

120597119909119894+

120597119906119894

119887

120597119910119894+ (

120597V119894119887

120597119909119894

120597V119894119887

120597119910119894+

120597119908119894

119887

120597119909119894

120597119908119894

119887

120597119910119894) (13c)

120576119910119910 = 120576119911119911 = 120574119910119911 = 0 (13d)


119873119894

119887= 119864119894

119887119860119894

119887[120597119906119894

119887

120597119909119894+

1

2((

120597V119894119887

120597119909119894)

2

+ (120597119908119894

119887

120597119909119894)

2

)] (14a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (14b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (14c)

119872119875119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894 (14d)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (14e)

where 119872119875119894



119887119908is the



119910and 119868

119894


119894

119878is the


119905and 119862

119894



119868119894

119905= intΩ

(1199101198942+ 1199111198942+ 119910119894 120597120593119875119894

119878

120597119911119894minus 119911119894 120597120593119875119894

119878

120597119910119894)119889Ω (15a)

119862119894

119878= intΩ

(120593119875119894

119878)2

119889Ω (15b)


minus120597119873119894

119887

120597119909119894+ 120588119894

119887Α119894

119887119894

119887=

2

sum

119895=1

119902119894

119909119895 (16a)

minus 119873119894

119887

1205972V119894119887

1205971199091198942+

1205972119872119894

119887119911

1205971199091198942

+ 120588119894

119887Α119894

119887V119894119887minus 120588119894

119887119868119894

119911

1205972V119894119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(16b)

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942minus

1205972119872119894

119887119910

1205971199091198942

+ 120588119894

119887Α119894

119887119894

119887minus 120588119894

119887119868119894

119910

1205972119894

119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(16c)

minus120597119872119894

119887119905

120597119909119894minus

1205972119872119894

119887119908

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(16d)


minus 119864119894

119887119860119894

119887(1205972119906119894

119887

1205971199091198942+

120597119908119894

119887

120597119909119894

1205972119908119894

119887

1205971199091198942

+120597V119894119887

120597119909119894

1205972V119894119887

1205971199091198942) + 120588

119894

119887Α119894

119887119894

119887

=

2

sum

119895=1

119902119894

119909119895

(17a)

119864119894

119887119868119894

119911

1205974V119894119887

1205971199091198944minus 119873119894

119887

1205972V119894119887

1205971199091198942minus 120588119894

119887119868119894

119911

1205972V1198871205971199092

+ 120588119894

119887Α119894

119887V119887 minus 120588

119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(17b)

119864119894

119887119868119894

119910

1205974119908119894

119887

1205971199091198944

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942

minus 120588119894

119887119868119894

119910

1205972119887

1205971199092+ 120588119894

119887Α119894

119887119887 minus 120588

119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(17c)

119864119894

119887119862119894

119878

1205974120579119894

119887119909

1205971199091198944

minus 119866119894

119887119868119894

119905

1205972120579119894

119887119909

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(17d)


119894 are given as

119886119894

1198871119906119894

119887+ 120572119894

1198872119873119894

119887= 120572119894

1198873 (18)

120573119894

1198871V119894119887+ 120573119894

1198872119877119894

119887119910= 120573119894

1198873 (19a)

120573119894

1198871120579119894

119887119911+ 120573119894

1198872119872119894

119887119911= 120573119894

1198873 (19b)

120574119894

1198871119908119894

119887+ 120574119894

1198872119877119894

119887119911= 120574119894

1198873 (20a)

120574119894

1198871120579119894

119887119910+ 120574119894

1198872119872119894

119887119910= 120574119894

1198873 (20b)

120575119894

1198871120579119894

119887119909+ 120575119894

1198872119872119894

119887119905= 120575119894

1198873 (21a)

120575119894

1198871

119889120579119894

119887119909

119889119909119894+ 120575119894

1198872119872119894

119887119908= 120575119894

1198873(21b)


119908119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22a)

119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22b)


119894

119887119910 120579119894


119894

119887119910 119877119894

119887119911

and 119872119894

119887119911 119872119894119887119910



119877119894

119887119910= 119873119894

119887

120597V119894119887

120597119909119894minus 119864119894

119887119868119894

119911

1205973V119894119887

1205971199091198943 (23a)

119877119894

119887119911= 119873119894

119887

120597119908119894

119887

120597119909119894minus 119864119894

119887119868119894

119910

1205973119908119894

119887

1205971199091198943 (23b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (24a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (24b)


and 119872119894

119887119905 119872119894119887119908


119872119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894minus 119864119894

119887119862119894

119878

1205973120579119894

119887119909

1205971199091198943 (25a)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (25b)

Finally 120572119894119887119896 120573119894

119887119896 120573119894

119887119896 120574119894

119887119896 120574119894

119887119896 120575119894

119887119896 120575119894

119887119896(119896 = 1 2 3) are func-




119894

119887 V119894119887119908119894119887 120579119894119887119909 1199021198941199091 1199021198941199101 1199021198941199111 119902119894

1199092

119902119894

1199102 and 119902

119894





119906119894

1199011minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199011

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894+

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198911

+119902119894

1199091

1198961198941199091

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=1)

(26a)

119906119894

1199012minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199012

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894minus

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198912

+119902119894

1199092

1198961198941199092

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=2)

(26b)


V1198941199011

minus V119894119887=

ℎ119901

2

120597119908119894

1199011

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199101

1198961198941199101

[1 minus1

2(120579119894

119887119909)2

]


119895=1)

(27a)

V1198941199012

minus V119894119887=

ℎ119901

2

120597119908119894

1199012

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199102

1198961198941199102

[1 minus1

2(120579119894

119887119909)2

]


119895=2)

(27b)


119908119894

1199011minus 119908119894

119887= minus

119887119894

119891

4120579119894


119894

119895=1) (28a)

119908119894

1199012minus 119908119894

119887=

119887119894

119891

4120579119894


119894

119895=2) (28b)

where (120593119875119894

119878)119891119895


119894

119895and 119896

119894

119909119895 119896119894119910119895



=

119896119894

119909119895(119904119894

119909119895) and 119896

119894

119910119895= 119896119894

119910119895(119904119894



119894



119875119894









nabla2119906119901119894 = 119901119901119894 (119909 119910 119905) (119894 = 1 2) (29a)

nabla41199061199013 = 1199011199013 (119909 119910 119905) (29b)






[

[

E11990111 0 00 E11990122 00 0 E11990133

]

]

d1199011d1199012d1199013

+

0D1198991198971199011

0D1198991198971199012

00

D1198991198971199013

=

01205721199013

01205731199013

001205741199013

1205751199013

(30)

where

E11990111 = [A1 H1 H20 D11990122 D11990123

] (31a)

E11990122 = [A1 H1 H20 D11990144 D11990145

] (31b)

E11990133 =[[[

[

A2 H1 H2 G1 G2A1 0 0 H1 H20 D11990178 D11990179 0 D1199017110 D11990188 D11990189 D119901810 0

]]]

]

(31c)



119901119894(119894 = 1 2) are 119873 times 1 and

D1198991198971199013


d119901119894 = p119901119894 u119901119894 u119901119894119899119879 (119894 = 1 2) (32a)


2u1199013)119899119879 (32b)


u119901119894 = (119906119901119894)1(119906119901119894)2

sdot sdot sdot (119906119901119894)119873119879

(119894 = 1 2 3)

(33a)

u119901119894119899 = (

120597119906119901119894

120597119899)

1

(

120597119906119901119894

120597119899)

2

sdot sdot sdot (

120597119906119901119894

120597119899)

119873

119879

(119894 = 1 2 3)

(33b)


21199061199013)1

(nabla21199061199013)2


119879

(33c)

(nabla2u1199013)119899

= (

120597nabla21199061199013

120597119899)

1

(

120597nabla21199061199013

120597119899)

2

sdot sdot sdot (

120597nabla21199061199013

120597119899)

119873

119879

(33d)


(119901119901119894)1(119901119901119894)2

sdot sdot sdot (119901119901119894)119872119879



u119901119894 = B119901119894d119901119894 (119894 = 1 2 3) (34a)u119901119894119909 = B119901119894119909d119901119894 (119894 = 1 2 3) (34b)

u119901119894119910 = B119901119894119910d119901119894 (119894 = 1 2 3) (34c)


u119901119894119909119909 = B119901119894119909119909d119901119894 (119894 = 1 2 3) (34d)

u119901119894119910119910 = B119901119894119910119910d119901119894 (119894 = 1 2 3) (34e)

u119901119894119909119910 = B119901119894119909119910d119901119894 (119894 = 1 2 3) (34f)


119866119901ℎ119901 [p1199011 +1 + ]1199011 minus ]119901

(B1199011119909119909d1199011 + B1199012119909119910d1199012)

+ 2

1 minus ]119901(B1199013119909119909d1199013)119889119892B1199013119909d1199013

+(B1199013119910119910d1199013)119889119892 sdot B1199013119909d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119910d1199013]

minus 120588119901ℎ119901B1199011d1199011 = Zq119909(35a)

119866119901ℎ119901 [p1199012 +1 + ]1199011 minus ]119901

(B1199011119909119910d1199011 + B1199012119910119910d1199012)

+ 2

1 minus ]119901(B1199013119910119910d1199013)119889119892B1199013119910d1199013

+ (B1199013119909119909d1199013)119889119892 sdot B1199013119910d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119909d1199013]

minus 120588119901ℎ119901B1199012d1199012 = Zq119910

(35b)

119863p1199013 minus 119862

times [(B1199011119909d1199011)119889119892B1199013119909119909d1199013

+1

2(B1199013119909d1199013)119889119892(B1199013119909d1199013)119889119892B1199013119909119909d1199013

+ ]119901(B1199012119910d1199012)119889119892B1199013119909119909d1199013 +1

2]119901(B1199013119910d1199013)119889119892

times (B1199013119910d1199013)119889119892B1199013119909119909d1199013]

+ (1 minus ]119901)

times [(B1199011119910d1199011)119889119892B1199013119909119910d1199013

+ (B1199012119909d1199012)119889119892B1199013119909119910d1199013

+(B1199013119909d1199013)119889119892 sdot (B1199013119910d1199013)119889119892B1199013119909119910d1199013]

+ [(B1199012119910d1199012)119889119892B1199013119910119910d1199013 +1

2(B1199013119910d1199013)119889119892

sdot (B1199013119910d1199013)119889119892B1199013119910119910d1199013

+ ]119901(B1199011119909d1199011)119889119892B1199013119910119910d1199013

+1

2]119901(B1199013119909d1199013)119889119892

sdot (B1199013119909d1199013)119889119892B1199013119910119910d1199013]

+ 120588119901ℎ119901B1199013d1199013 minus 120588119901ℎ119901(B1199011d1199011)119889119892B1199013119909d1199013


119901

12B1199013119909119909d1199013

minus

120588119901ℎ3

119901

12B1199013119910119910d1199013


+ (Zq119910)119889119892B1199013119910d1199013

(35c)

where q119909 = q1199091 q1199092119879 q119910 = q1199101 q1199102

119879 andq119911 = q1199111 q1199112


119894



119901119910119895= 119902119894

119909119895ℎ1199012 119898

119894

119901119909119895= 119902119894

119910119895ℎ1199012 respectively



119887 V119894119887 and 119908

119894

119887




119887(119909119894 119905) and V119894

119887(119909119894 119905) 119908119894





Let 119906119894

1198871= 119906119894

119887(119909119894 119905) 119906

119894

1198872= V119894119887(119909119894 119905) 119906

119894

1198873= 119908119894

119887(119909119894 119905)

and 119906119894

1198874= 120579119894




1205972119906119894

1198871

1205971199091198942

= 119901119894

1198871(119909119894 119905) (36a)

1205974119906119894

119887119895

1205971199091198944= 119901119894

119887119895(119909119894 119905) (119895 = 2 3 4) (36b)




119894





119887119895(120577119894) 119906119894119887119895119909

(120577119894) 119906119894119887119895119909119909

(120577119894) and 119906

119894

119887119895119909119909119909(120577119894)

(120577119894 = 0 119897119894) in terms of 119901119894



[[[

[

E11989411988711

0 0 00 E119894

119887220 0

0 0 E11989411988733

00 0 0 E119894

11988744

]]]

]

d1198941198871

d1198941198872

d1198941198873

d1198941198874

+

0D119894 1198991198971198871

00

D119894 1198991198971198872

00

D119894 1198991198971198873

000

=

0120572119894

1198873

00120573119894

1198873

00120574119894

1198873

00120575119894

1198873

(37)

where E11989411988711

and E11989411988722

E11989411988733

E11989411988744


1198871is

a 2 times 1 and D119894 119899119897119887119895


1198873and

120573119894

1198873 1205741198941198873 1205751198941198873



1198873and 120573

119894

1198873 120573119894

1198873 120574119894

1198873 120574119894

1198873 120575119894

1198873 120575119894


d1198941198871

= p1198941198871

u1198941198871

u1198941198871119909

119879

(38a)

d119894119887119895

= p119894119887119895 u119894119887119895

u119894119887119895119909

u119894119887119895119909119909

u119894119887119895119909119909119909

119879

(119895 = 2 3 4)

(38b)


u119894119887119895

= 119906119894

119887119895(0 119905) 119906

119894

119887119895(119897119894 119905)119879

(119895 = 1 2 3 4) (39a)

u119894119887119895119909

= 120597119906119894

119887119895(0 119905)

120597119909119894

120597119906119894

119887119895(119897119894 119905)

120597119909119894

119879

(119895 = 1 2 3 4) (39b)

u119894119887119895119909119909

= 1205972119906119894

119887119895(0 119905)

1205971199091198942

1205972119906119894

119887119895(119897119894 119905)

1205971199091198942

119879

(119895 = 2 3 4)

(39c)

u119894119887119895119909119909119909

= 1205973119906119894

119887119895(0 119905)

1205971199091198943

1205973119906119894

119887119895(119897119894 119905)

1205971199091198943

119879

(119895 = 2 3 4)

(39d)


119887119895=

(119901119894

119887119895)1

(119901119894

119887119895)2


119887119895)119871119879



119894



119894isin (0 119897

119894) and


u1198941198871

= B1198941198871d1198941198871 (40a)

u1198941198871119909

= B1198941198871119909

d1198941198871 (40b)

u119894119887119895

= B119894119887119895d119894119887119895 (119895 = 2 3 4) (40c)

u119894119887119895119909

= B119894119887119895119909

d119894119887119895 (119895 = 2 3 4) (40d)

u119894119887119895119909119909

= B119894119887119895119909119909

d119894119887119895 (119895 = 2 3 4) (40e)

u119894119887119895119909119909119909

= B119894119887119895119909119909119909

d119894119887119895 (119895 = 2 3 4) (40f)

where B1198941198871B1198941198871119909

are 119871 times (119871 + 4) and B119894119887119895B119894119887119895119909

B119894119887119895119909119909

B119894119887119895119909119909119909



x

y

a

a

Fr

Fr

ClCl

Lx = 18 cm

Ly=9

cm A

(a)

g

02 cmhb = 05 cm

1 cm 5 cm3 cm

Ly = 9 cm

(b)


Time (ms)

060

040

020

000

minus020

000 050 100 150 200 250

Disp

lace

men

tw (c

m)





minus 119864119894

119887119860119894

119887[p1198941198871

+ (B1198941198872119909

d1198941198872)119889119892B1198941198872119909119909

d1198941198872

+ (B1198941198873119909

d1198941198873)119889119892B1198941198873119909119909

d1198941198873]

+ 120588119894

119887Α119894

119887T1q1 = q119894

1199091+ q1198941199092

(41a)

119864119894

119887119868119894

119911p1198941198872

minus (N119894119887)119889119892B1198941198872119909119909

d1198941198872

minus 120588119894

119887119868119894

119911B1198941198872119909119909

d1198941198872

+ 120588119894

119887Α119894

119887B1198941198872d1198941198872

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198872119909

d1198941198872

= q1198941199101

+ q1198941199102

minus (B1198941198872119909

d1198941198872)119889119892

(q1198941199091

+ q1198941199092) minus X119894119887119910

(q1198941199091

+ q1198941199092)

(41b)

119864119894

119887119868119894

119910p1198941198873

minus (N119894119887)119889119892B1198941198873119909119909

d1198941198873

minus 120588119894

119887119868119894

119910B1198941198873119909119909

d1198941198873

+ 120588119894

119887Α119894

119887B1198941198873d1198941198873

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198873119909

d1198941198873

= q1198941199111

+ q1198941199112

minus (B1198941198873119909

d1198941198873)119889119892

(q1198941199091

+ q1198941199092) + X119894119887119911

(q1198941199091

+ q1198941199092)

(41c)

119864119894

119887119862119894

119878p1198941198874

minus 119866119894

119887119868119894

119905B1198941198874119909119909

d1198941198874

+ 120588119894

119887119868119894

119901B1198941198874d1198941198874

minus 120588119894

119887119862119894

119878B1198941198874119909119909

d1198941198874

= e1198941199101q1198941199111

+ e1198941199102q1198941199112

minus e1198941199111q1198941199101

minus e1198941199112q1198941199102

+ Χ119894

119887119908(q1198941199091

+ q1198941199092)

(41d)

where (N119894119887)119889119892


119887119911 X119894119887119910 X119894119887119908


119898119894119887119911119895 119898119894119887119908119895


1199101 e1198941199102 e1198941199111

e1198941199112


119894

119910119895 119890119894


119894

119911119895 119902119894

119910119895with



Y1B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y1B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

+

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198911

B1198941198874119909

d1198874

+ K1198941199091

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42a)


008

008

018

018

028

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036040044048


000067300301006080091501220153018402140245027603070337036803990430460491


008

008

008

00801

8018

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036


000064600236004780072009620120145016901930217024102660290314033803620387



Y2B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y2B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

minus

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198912

B1198941198874119909

d1198874

+ K1198941199092

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42b)

Y1B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y1B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199101

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43a)

Y2B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y2B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199102

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43b)

Y1B1199013d1199013 minus B1198941198873d1198941198873

= minus

119887119894

119891

4B1198941198874d1198941198874 (44a)

Y2B1199013d1199013 minus B1198941198873d1198941198873

=

119887119894

119891

4B1198941198874d1198941198874 (44b)

000 050 100 150 200 250

060

040

020

000

Disp

lace

men

tw (c

m)

Time (ms)



minus020



119887119895(119894 = 1 sdot sdot sdot 119868)



000015030045060075

9

6

3

0

1815

12

9

6

3

0


000010020030040050

9

6

3

0

1815

12

9

6

3

0


000010020030040050060

9

6

3

0

1815

12

9

6

3

0


00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0


00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000


00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000




0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119877119901119899 = minus119863[120597


120597

120597119904(

1205972119908119901

120597119904120597119899minus 120581

120597119908119901

120597119904)]

+ 119873119901119899

120597119908119901

120597119899+ 119873119901119905

120597119908119901

120597119904

(10a)


1205972119908119901

1205971199042+ 120581

120597119908119901

120597119899)] (10b)



119879119908119901 = 119863(]119901 minus 1)(

1205972119908119901

120597119904120597119899minus 120581

120597119908119901

120597119904) (11)


1205741199012 = 1205741199013 = 1205751199012 = 1205751199013 = 0)





119906119894

119887(119909119894 119910119894 119911119894 119905) = 119906

119894

119887(119909119894 119905) minus 119910

119894120579119894

119887119911(119909119894 119905)

+ 119911119894120579119894

119887119910(119909119894 119905) +

120597120579119894

119887119909

120597119909119894120593119875119894

119878(119910119894 119911119894 119905)

(12a)

V119894119887(119909119894 119910119894 119911119894 119905) = V119894

119887(119909119894 119905) minus 119911

119894120579119894

119887119909(119909119894 119905) (12b)

119908119894

119887(119909119894 119910119894 119911119894 119905) = 119908

119894

119887(119909119894 119905) + 119910

119894120579119894

119887119909(119909119894 119905) (12c)

120579119894

119887119910(119909119894 119905) = minus

120597119908119894

119887(119909119894 119905)

120597119909119894 (12d)

120579119894

119887119911(119909119894 119905) =

120597V119894119887(119909119894 119905)

120597119909119894 (12e)

where 119906119894119887 V119894119887 and119908

119894



119887=

119906119894

119887(119909119894) V119894119887= V119894119887(119909119894) and 119908

119894

119887= 119908119894



= 120579119894

119887119910(119909119894) and 120579

119894

119887119911=

120579119894




119887119909(119909119894) regarded as the




120576119909119909 =120597119906119894

119887

120597119909119894+

1

2

[

[

(120597V119894119887

120597119909119894)

2

+ (120597119908119894

119887

120597119909119894)

2

]

]

(13a)

120574119909119911 =120597119908119894

119887

120597119909119894+

120597119906119894

119887

120597119911119894+ (

120597V119894119887

120597119909119894

120597V119894119887

120597119911119894+

120597119908119894

119887

120597119909119894

120597119908119894

119887

120597119911119894) (13b)

120574119909119910 =120597V119894119887

120597119909119894+

120597119906119894

119887

120597119910119894+ (

120597V119894119887

120597119909119894

120597V119894119887

120597119910119894+

120597119908119894

119887

120597119909119894

120597119908119894

119887

120597119910119894) (13c)

120576119910119910 = 120576119911119911 = 120574119910119911 = 0 (13d)


119873119894

119887= 119864119894

119887119860119894

119887[120597119906119894

119887

120597119909119894+

1

2((

120597V119894119887

120597119909119894)

2

+ (120597119908119894

119887

120597119909119894)

2

)] (14a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (14b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (14c)

119872119875119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894 (14d)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (14e)

where 119872119875119894



119887119908is the



119910and 119868

119894


119894

119878is the


119905and 119862

119894



119868119894

119905= intΩ

(1199101198942+ 1199111198942+ 119910119894 120597120593119875119894

119878

120597119911119894minus 119911119894 120597120593119875119894

119878

120597119910119894)119889Ω (15a)

119862119894

119878= intΩ

(120593119875119894

119878)2

119889Ω (15b)


minus120597119873119894

119887

120597119909119894+ 120588119894

119887Α119894

119887119894

119887=

2

sum

119895=1

119902119894

119909119895 (16a)

minus 119873119894

119887

1205972V119894119887

1205971199091198942+

1205972119872119894

119887119911

1205971199091198942

+ 120588119894

119887Α119894

119887V119894119887minus 120588119894

119887119868119894

119911

1205972V119894119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(16b)

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942minus

1205972119872119894

119887119910

1205971199091198942

+ 120588119894

119887Α119894

119887119894

119887minus 120588119894

119887119868119894

119910

1205972119894

119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(16c)

minus120597119872119894

119887119905

120597119909119894minus

1205972119872119894

119887119908

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(16d)


minus 119864119894

119887119860119894

119887(1205972119906119894

119887

1205971199091198942+

120597119908119894

119887

120597119909119894

1205972119908119894

119887

1205971199091198942

+120597V119894119887

120597119909119894

1205972V119894119887

1205971199091198942) + 120588

119894

119887Α119894

119887119894

119887

=

2

sum

119895=1

119902119894

119909119895

(17a)

119864119894

119887119868119894

119911

1205974V119894119887

1205971199091198944minus 119873119894

119887

1205972V119894119887

1205971199091198942minus 120588119894

119887119868119894

119911

1205972V1198871205971199092

+ 120588119894

119887Α119894

119887V119887 minus 120588

119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(17b)

119864119894

119887119868119894

119910

1205974119908119894

119887

1205971199091198944

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942

minus 120588119894

119887119868119894

119910

1205972119887

1205971199092+ 120588119894

119887Α119894

119887119887 minus 120588

119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(17c)

119864119894

119887119862119894

119878

1205974120579119894

119887119909

1205971199091198944

minus 119866119894

119887119868119894

119905

1205972120579119894

119887119909

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(17d)


119894 are given as

119886119894

1198871119906119894

119887+ 120572119894

1198872119873119894

119887= 120572119894

1198873 (18)

120573119894

1198871V119894119887+ 120573119894

1198872119877119894

119887119910= 120573119894

1198873 (19a)

120573119894

1198871120579119894

119887119911+ 120573119894

1198872119872119894

119887119911= 120573119894

1198873 (19b)

120574119894

1198871119908119894

119887+ 120574119894

1198872119877119894

119887119911= 120574119894

1198873 (20a)

120574119894

1198871120579119894

119887119910+ 120574119894

1198872119872119894

119887119910= 120574119894

1198873 (20b)

120575119894

1198871120579119894

119887119909+ 120575119894

1198872119872119894

119887119905= 120575119894

1198873 (21a)

120575119894

1198871

119889120579119894

119887119909

119889119909119894+ 120575119894

1198872119872119894

119887119908= 120575119894

1198873(21b)


119908119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22a)

119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22b)


119894

119887119910 120579119894


119894

119887119910 119877119894

119887119911

and 119872119894

119887119911 119872119894119887119910



119877119894

119887119910= 119873119894

119887

120597V119894119887

120597119909119894minus 119864119894

119887119868119894

119911

1205973V119894119887

1205971199091198943 (23a)

119877119894

119887119911= 119873119894

119887

120597119908119894

119887

120597119909119894minus 119864119894

119887119868119894

119910

1205973119908119894

119887

1205971199091198943 (23b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (24a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (24b)


and 119872119894

119887119905 119872119894119887119908


119872119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894minus 119864119894

119887119862119894

119878

1205973120579119894

119887119909

1205971199091198943 (25a)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (25b)

Finally 120572119894119887119896 120573119894

119887119896 120573119894

119887119896 120574119894

119887119896 120574119894

119887119896 120575119894

119887119896 120575119894

119887119896(119896 = 1 2 3) are func-




119894

119887 V119894119887119908119894119887 120579119894119887119909 1199021198941199091 1199021198941199101 1199021198941199111 119902119894

1199092

119902119894

1199102 and 119902

119894





119906119894

1199011minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199011

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894+

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198911

+119902119894

1199091

1198961198941199091

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=1)

(26a)

119906119894

1199012minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199012

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894minus

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198912

+119902119894

1199092

1198961198941199092

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=2)

(26b)


V1198941199011

minus V119894119887=

ℎ119901

2

120597119908119894

1199011

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199101

1198961198941199101

[1 minus1

2(120579119894

119887119909)2

]


119895=1)

(27a)

V1198941199012

minus V119894119887=

ℎ119901

2

120597119908119894

1199012

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199102

1198961198941199102

[1 minus1

2(120579119894

119887119909)2

]


119895=2)

(27b)


119908119894

1199011minus 119908119894

119887= minus

119887119894

119891

4120579119894


119894

119895=1) (28a)

119908119894

1199012minus 119908119894

119887=

119887119894

119891

4120579119894


119894

119895=2) (28b)

where (120593119875119894

119878)119891119895


119894

119895and 119896

119894

119909119895 119896119894119910119895



=

119896119894

119909119895(119904119894

119909119895) and 119896

119894

119910119895= 119896119894

119910119895(119904119894



119894



119875119894









nabla2119906119901119894 = 119901119901119894 (119909 119910 119905) (119894 = 1 2) (29a)

nabla41199061199013 = 1199011199013 (119909 119910 119905) (29b)






[

[

E11990111 0 00 E11990122 00 0 E11990133

]

]

d1199011d1199012d1199013

+

0D1198991198971199011

0D1198991198971199012

00

D1198991198971199013

=

01205721199013

01205731199013

001205741199013

1205751199013

(30)

where

E11990111 = [A1 H1 H20 D11990122 D11990123

] (31a)

E11990122 = [A1 H1 H20 D11990144 D11990145

] (31b)

E11990133 =[[[

[

A2 H1 H2 G1 G2A1 0 0 H1 H20 D11990178 D11990179 0 D1199017110 D11990188 D11990189 D119901810 0

]]]

]

(31c)



119901119894(119894 = 1 2) are 119873 times 1 and

D1198991198971199013


d119901119894 = p119901119894 u119901119894 u119901119894119899119879 (119894 = 1 2) (32a)


2u1199013)119899119879 (32b)


u119901119894 = (119906119901119894)1(119906119901119894)2

sdot sdot sdot (119906119901119894)119873119879

(119894 = 1 2 3)

(33a)

u119901119894119899 = (

120597119906119901119894

120597119899)

1

(

120597119906119901119894

120597119899)

2

sdot sdot sdot (

120597119906119901119894

120597119899)

119873

119879

(119894 = 1 2 3)

(33b)


21199061199013)1

(nabla21199061199013)2


119879

(33c)

(nabla2u1199013)119899

= (

120597nabla21199061199013

120597119899)

1

(

120597nabla21199061199013

120597119899)

2

sdot sdot sdot (

120597nabla21199061199013

120597119899)

119873

119879

(33d)


(119901119901119894)1(119901119901119894)2

sdot sdot sdot (119901119901119894)119872119879



u119901119894 = B119901119894d119901119894 (119894 = 1 2 3) (34a)u119901119894119909 = B119901119894119909d119901119894 (119894 = 1 2 3) (34b)

u119901119894119910 = B119901119894119910d119901119894 (119894 = 1 2 3) (34c)


u119901119894119909119909 = B119901119894119909119909d119901119894 (119894 = 1 2 3) (34d)

u119901119894119910119910 = B119901119894119910119910d119901119894 (119894 = 1 2 3) (34e)

u119901119894119909119910 = B119901119894119909119910d119901119894 (119894 = 1 2 3) (34f)


119866119901ℎ119901 [p1199011 +1 + ]1199011 minus ]119901

(B1199011119909119909d1199011 + B1199012119909119910d1199012)

+ 2

1 minus ]119901(B1199013119909119909d1199013)119889119892B1199013119909d1199013

+(B1199013119910119910d1199013)119889119892 sdot B1199013119909d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119910d1199013]

minus 120588119901ℎ119901B1199011d1199011 = Zq119909(35a)

119866119901ℎ119901 [p1199012 +1 + ]1199011 minus ]119901

(B1199011119909119910d1199011 + B1199012119910119910d1199012)

+ 2

1 minus ]119901(B1199013119910119910d1199013)119889119892B1199013119910d1199013

+ (B1199013119909119909d1199013)119889119892 sdot B1199013119910d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119909d1199013]

minus 120588119901ℎ119901B1199012d1199012 = Zq119910

(35b)

119863p1199013 minus 119862

times [(B1199011119909d1199011)119889119892B1199013119909119909d1199013

+1

2(B1199013119909d1199013)119889119892(B1199013119909d1199013)119889119892B1199013119909119909d1199013

+ ]119901(B1199012119910d1199012)119889119892B1199013119909119909d1199013 +1

2]119901(B1199013119910d1199013)119889119892

times (B1199013119910d1199013)119889119892B1199013119909119909d1199013]

+ (1 minus ]119901)

times [(B1199011119910d1199011)119889119892B1199013119909119910d1199013

+ (B1199012119909d1199012)119889119892B1199013119909119910d1199013

+(B1199013119909d1199013)119889119892 sdot (B1199013119910d1199013)119889119892B1199013119909119910d1199013]

+ [(B1199012119910d1199012)119889119892B1199013119910119910d1199013 +1

2(B1199013119910d1199013)119889119892

sdot (B1199013119910d1199013)119889119892B1199013119910119910d1199013

+ ]119901(B1199011119909d1199011)119889119892B1199013119910119910d1199013

+1

2]119901(B1199013119909d1199013)119889119892

sdot (B1199013119909d1199013)119889119892B1199013119910119910d1199013]

+ 120588119901ℎ119901B1199013d1199013 minus 120588119901ℎ119901(B1199011d1199011)119889119892B1199013119909d1199013


119901

12B1199013119909119909d1199013

minus

120588119901ℎ3

119901

12B1199013119910119910d1199013


+ (Zq119910)119889119892B1199013119910d1199013

(35c)

where q119909 = q1199091 q1199092119879 q119910 = q1199101 q1199102

119879 andq119911 = q1199111 q1199112


119894



119901119910119895= 119902119894

119909119895ℎ1199012 119898

119894

119901119909119895= 119902119894

119910119895ℎ1199012 respectively



119887 V119894119887 and 119908

119894

119887




119887(119909119894 119905) and V119894

119887(119909119894 119905) 119908119894





Let 119906119894

1198871= 119906119894

119887(119909119894 119905) 119906

119894

1198872= V119894119887(119909119894 119905) 119906

119894

1198873= 119908119894

119887(119909119894 119905)

and 119906119894

1198874= 120579119894




1205972119906119894

1198871

1205971199091198942

= 119901119894

1198871(119909119894 119905) (36a)

1205974119906119894

119887119895

1205971199091198944= 119901119894

119887119895(119909119894 119905) (119895 = 2 3 4) (36b)




119894





119887119895(120577119894) 119906119894119887119895119909

(120577119894) 119906119894119887119895119909119909

(120577119894) and 119906

119894

119887119895119909119909119909(120577119894)

(120577119894 = 0 119897119894) in terms of 119901119894



[[[

[

E11989411988711

0 0 00 E119894

119887220 0

0 0 E11989411988733

00 0 0 E119894

11988744

]]]

]

d1198941198871

d1198941198872

d1198941198873

d1198941198874

+

0D119894 1198991198971198871

00

D119894 1198991198971198872

00

D119894 1198991198971198873

000

=

0120572119894

1198873

00120573119894

1198873

00120574119894

1198873

00120575119894

1198873

(37)

where E11989411988711

and E11989411988722

E11989411988733

E11989411988744


1198871is

a 2 times 1 and D119894 119899119897119887119895


1198873and

120573119894

1198873 1205741198941198873 1205751198941198873



1198873and 120573

119894

1198873 120573119894

1198873 120574119894

1198873 120574119894

1198873 120575119894

1198873 120575119894


d1198941198871

= p1198941198871

u1198941198871

u1198941198871119909

119879

(38a)

d119894119887119895

= p119894119887119895 u119894119887119895

u119894119887119895119909

u119894119887119895119909119909

u119894119887119895119909119909119909

119879

(119895 = 2 3 4)

(38b)


u119894119887119895

= 119906119894

119887119895(0 119905) 119906

119894

119887119895(119897119894 119905)119879

(119895 = 1 2 3 4) (39a)

u119894119887119895119909

= 120597119906119894

119887119895(0 119905)

120597119909119894

120597119906119894

119887119895(119897119894 119905)

120597119909119894

119879

(119895 = 1 2 3 4) (39b)

u119894119887119895119909119909

= 1205972119906119894

119887119895(0 119905)

1205971199091198942

1205972119906119894

119887119895(119897119894 119905)

1205971199091198942

119879

(119895 = 2 3 4)

(39c)

u119894119887119895119909119909119909

= 1205973119906119894

119887119895(0 119905)

1205971199091198943

1205973119906119894

119887119895(119897119894 119905)

1205971199091198943

119879

(119895 = 2 3 4)

(39d)


119887119895=

(119901119894

119887119895)1

(119901119894

119887119895)2


119887119895)119871119879



119894



119894isin (0 119897

119894) and


u1198941198871

= B1198941198871d1198941198871 (40a)

u1198941198871119909

= B1198941198871119909

d1198941198871 (40b)

u119894119887119895

= B119894119887119895d119894119887119895 (119895 = 2 3 4) (40c)

u119894119887119895119909

= B119894119887119895119909

d119894119887119895 (119895 = 2 3 4) (40d)

u119894119887119895119909119909

= B119894119887119895119909119909

d119894119887119895 (119895 = 2 3 4) (40e)

u119894119887119895119909119909119909

= B119894119887119895119909119909119909

d119894119887119895 (119895 = 2 3 4) (40f)

where B1198941198871B1198941198871119909

are 119871 times (119871 + 4) and B119894119887119895B119894119887119895119909

B119894119887119895119909119909

B119894119887119895119909119909119909



x

y

a

a

Fr

Fr

ClCl

Lx = 18 cm

Ly=9

cm A

(a)

g

02 cmhb = 05 cm

1 cm 5 cm3 cm

Ly = 9 cm

(b)


Time (ms)

060

040

020

000

minus020

000 050 100 150 200 250

Disp

lace

men

tw (c

m)





minus 119864119894

119887119860119894

119887[p1198941198871

+ (B1198941198872119909

d1198941198872)119889119892B1198941198872119909119909

d1198941198872

+ (B1198941198873119909

d1198941198873)119889119892B1198941198873119909119909

d1198941198873]

+ 120588119894

119887Α119894

119887T1q1 = q119894

1199091+ q1198941199092

(41a)

119864119894

119887119868119894

119911p1198941198872

minus (N119894119887)119889119892B1198941198872119909119909

d1198941198872

minus 120588119894

119887119868119894

119911B1198941198872119909119909

d1198941198872

+ 120588119894

119887Α119894

119887B1198941198872d1198941198872

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198872119909

d1198941198872

= q1198941199101

+ q1198941199102

minus (B1198941198872119909

d1198941198872)119889119892

(q1198941199091

+ q1198941199092) minus X119894119887119910

(q1198941199091

+ q1198941199092)

(41b)

119864119894

119887119868119894

119910p1198941198873

minus (N119894119887)119889119892B1198941198873119909119909

d1198941198873

minus 120588119894

119887119868119894

119910B1198941198873119909119909

d1198941198873

+ 120588119894

119887Α119894

119887B1198941198873d1198941198873

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198873119909

d1198941198873

= q1198941199111

+ q1198941199112

minus (B1198941198873119909

d1198941198873)119889119892

(q1198941199091

+ q1198941199092) + X119894119887119911

(q1198941199091

+ q1198941199092)

(41c)

119864119894

119887119862119894

119878p1198941198874

minus 119866119894

119887119868119894

119905B1198941198874119909119909

d1198941198874

+ 120588119894

119887119868119894

119901B1198941198874d1198941198874

minus 120588119894

119887119862119894

119878B1198941198874119909119909

d1198941198874

= e1198941199101q1198941199111

+ e1198941199102q1198941199112

minus e1198941199111q1198941199101

minus e1198941199112q1198941199102

+ Χ119894

119887119908(q1198941199091

+ q1198941199092)

(41d)

where (N119894119887)119889119892


119887119911 X119894119887119910 X119894119887119908


119898119894119887119911119895 119898119894119887119908119895


1199101 e1198941199102 e1198941199111

e1198941199112


119894

119910119895 119890119894


119894

119911119895 119902119894

119910119895with



Y1B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y1B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

+

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198911

B1198941198874119909

d1198874

+ K1198941199091

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42a)


008

008

018

018

028

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036040044048


000067300301006080091501220153018402140245027603070337036803990430460491


008

008

008

00801

8018

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036


000064600236004780072009620120145016901930217024102660290314033803620387



Y2B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y2B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

minus

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198912

B1198941198874119909

d1198874

+ K1198941199092

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42b)

Y1B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y1B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199101

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43a)

Y2B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y2B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199102

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43b)

Y1B1199013d1199013 minus B1198941198873d1198941198873

= minus

119887119894

119891

4B1198941198874d1198941198874 (44a)

Y2B1199013d1199013 minus B1198941198873d1198941198873

=

119887119894

119891

4B1198941198874d1198941198874 (44b)

000 050 100 150 200 250

060

040

020

000

Disp

lace

men

tw (c

m)

Time (ms)



minus020



119887119895(119894 = 1 sdot sdot sdot 119868)



000015030045060075

9

6

3

0

1815

12

9

6

3

0


000010020030040050

9

6

3

0

1815

12

9

6

3

0


000010020030040050060

9

6

3

0

1815

12

9

6

3

0


00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0


00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000


00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000




0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119910and 119868

119894


119894

119878is the


119905and 119862

119894



119868119894

119905= intΩ

(1199101198942+ 1199111198942+ 119910119894 120597120593119875119894

119878

120597119911119894minus 119911119894 120597120593119875119894

119878

120597119910119894)119889Ω (15a)

119862119894

119878= intΩ

(120593119875119894

119878)2

119889Ω (15b)


minus120597119873119894

119887

120597119909119894+ 120588119894

119887Α119894

119887119894

119887=

2

sum

119895=1

119902119894

119909119895 (16a)

minus 119873119894

119887

1205972V119894119887

1205971199091198942+

1205972119872119894

119887119911

1205971199091198942

+ 120588119894

119887Α119894

119887V119894119887minus 120588119894

119887119868119894

119911

1205972V119894119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(16b)

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942minus

1205972119872119894

119887119910

1205971199091198942

+ 120588119894

119887Α119894

119887119894

119887minus 120588119894

119887119868119894

119910

1205972119894

119887

1205971199092minus 120588119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(16c)

minus120597119872119894

119887119905

120597119909119894minus

1205972119872119894

119887119908

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(16d)


minus 119864119894

119887119860119894

119887(1205972119906119894

119887

1205971199091198942+

120597119908119894

119887

120597119909119894

1205972119908119894

119887

1205971199091198942

+120597V119894119887

120597119909119894

1205972V119894119887

1205971199091198942) + 120588

119894

119887Α119894

119887119894

119887

=

2

sum

119895=1

119902119894

119909119895

(17a)

119864119894

119887119868119894

119911

1205974V119894119887

1205971199091198944minus 119873119894

119887

1205972V119894119887

1205971199091198942minus 120588119894

119887119868119894

119911

1205972V1198871205971199092

+ 120588119894

119887Α119894

119887V119887 minus 120588

119894

119887Α119894

119887119894

119887

120597V119894119887

120597119909119894

=

2

sum

119895=1

(119902119894

119910119895minus 119902119894

119909119895

120597V119894119887

120597119909119894minus

120597119898119894

119887119911119895

120597119909119894)

(17b)

119864119894

119887119868119894

119910

1205974119908119894

119887

1205971199091198944

minus 119873119894

119887

1205972119908119894

119887

1205971199091198942

minus 120588119894

119887119868119894

119910

1205972119887

1205971199092+ 120588119894

119887Α119894

119887119887 minus 120588

119894

119887Α119894

119887119894

119887

120597119908119894

119887

120597119909119894

=

2

sum

119895=1

(119902119894

119911119895minus 119902119894

119909119895

120597119908119894

119887

120597119909119894+

120597119898119894

119887119910119895

120597119909119894)

(17c)

119864119894

119887119862119894

119878

1205974120579119894

119887119909

1205971199091198944

minus 119866119894

119887119868119894

119905

1205972120579119894

119887119909

1205971199091198942

+ 120588119894

119887119868119894

119878120579119894

119887119909minus 120588119894

119887119862119894

119878

1205972 120579119894

119887119909

1205971199092

=

2

sum

119895=1

[119898119894

119887119909119895+

120597119898119894

119887119908119895

120597119909119894]

(17d)


119894 are given as

119886119894

1198871119906119894

119887+ 120572119894

1198872119873119894

119887= 120572119894

1198873 (18)

120573119894

1198871V119894119887+ 120573119894

1198872119877119894

119887119910= 120573119894

1198873 (19a)

120573119894

1198871120579119894

119887119911+ 120573119894

1198872119872119894

119887119911= 120573119894

1198873 (19b)

120574119894

1198871119908119894

119887+ 120574119894

1198872119877119894

119887119911= 120574119894

1198873 (20a)

120574119894

1198871120579119894

119887119910+ 120574119894

1198872119872119894

119887119910= 120574119894

1198873 (20b)

120575119894

1198871120579119894

119887119909+ 120575119894

1198872119872119894

119887119905= 120575119894

1198873 (21a)

120575119894

1198871

119889120579119894

119887119909

119889119909119894+ 120575119894

1198872119872119894

119887119908= 120575119894

1198873(21b)


119908119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22a)

119894

119887(119909119894 0) = 119908

119894

1198870(119909119894) (22b)


119894

119887119910 120579119894


119894

119887119910 119877119894

119887119911

and 119872119894

119887119911 119872119894119887119910



119877119894

119887119910= 119873119894

119887

120597V119894119887

120597119909119894minus 119864119894

119887119868119894

119911

1205973V119894119887

1205971199091198943 (23a)

119877119894

119887119911= 119873119894

119887

120597119908119894

119887

120597119909119894minus 119864119894

119887119868119894

119910

1205973119908119894

119887

1205971199091198943 (23b)

119872119894

119887119911= 119864119894

119887119868119894

119911

1205972V119894119887

1205971199091198942 (24a)

119872119894

119887119910= minus119864119894

119887119868119894

119910

1205972119908119894

119887

1205971199091198942 (24b)


and 119872119894

119887119905 119872119894119887119908


119872119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894minus 119864119894

119887119862119894

119878

1205973120579119894

119887119909

1205971199091198943 (25a)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (25b)

Finally 120572119894119887119896 120573119894

119887119896 120573119894

119887119896 120574119894

119887119896 120574119894

119887119896 120575119894

119887119896 120575119894

119887119896(119896 = 1 2 3) are func-




119894

119887 V119894119887119908119894119887 120579119894119887119909 1199021198941199091 1199021198941199101 1199021198941199111 119902119894

1199092

119902119894

1199102 and 119902

119894





119906119894

1199011minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199011

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894+

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198911

+119902119894

1199091

1198961198941199091

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=1)

(26a)

119906119894

1199012minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199012

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894minus

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198912

+119902119894

1199092

1198961198941199092

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=2)

(26b)


V1198941199011

minus V119894119887=

ℎ119901

2

120597119908119894

1199011

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199101

1198961198941199101

[1 minus1

2(120579119894

119887119909)2

]


119895=1)

(27a)

V1198941199012

minus V119894119887=

ℎ119901

2

120597119908119894

1199012

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199102

1198961198941199102

[1 minus1

2(120579119894

119887119909)2

]


119895=2)

(27b)


119908119894

1199011minus 119908119894

119887= minus

119887119894

119891

4120579119894


119894

119895=1) (28a)

119908119894

1199012minus 119908119894

119887=

119887119894

119891

4120579119894


119894

119895=2) (28b)

where (120593119875119894

119878)119891119895


119894

119895and 119896

119894

119909119895 119896119894119910119895



=

119896119894

119909119895(119904119894

119909119895) and 119896

119894

119910119895= 119896119894

119910119895(119904119894



119894



119875119894









nabla2119906119901119894 = 119901119901119894 (119909 119910 119905) (119894 = 1 2) (29a)

nabla41199061199013 = 1199011199013 (119909 119910 119905) (29b)






[

[

E11990111 0 00 E11990122 00 0 E11990133

]

]

d1199011d1199012d1199013

+

0D1198991198971199011

0D1198991198971199012

00

D1198991198971199013

=

01205721199013

01205731199013

001205741199013

1205751199013

(30)

where

E11990111 = [A1 H1 H20 D11990122 D11990123

] (31a)

E11990122 = [A1 H1 H20 D11990144 D11990145

] (31b)

E11990133 =[[[

[

A2 H1 H2 G1 G2A1 0 0 H1 H20 D11990178 D11990179 0 D1199017110 D11990188 D11990189 D119901810 0

]]]

]

(31c)



119901119894(119894 = 1 2) are 119873 times 1 and

D1198991198971199013


d119901119894 = p119901119894 u119901119894 u119901119894119899119879 (119894 = 1 2) (32a)


2u1199013)119899119879 (32b)


u119901119894 = (119906119901119894)1(119906119901119894)2

sdot sdot sdot (119906119901119894)119873119879

(119894 = 1 2 3)

(33a)

u119901119894119899 = (

120597119906119901119894

120597119899)

1

(

120597119906119901119894

120597119899)

2

sdot sdot sdot (

120597119906119901119894

120597119899)

119873

119879

(119894 = 1 2 3)

(33b)


21199061199013)1

(nabla21199061199013)2


119879

(33c)

(nabla2u1199013)119899

= (

120597nabla21199061199013

120597119899)

1

(

120597nabla21199061199013

120597119899)

2

sdot sdot sdot (

120597nabla21199061199013

120597119899)

119873

119879

(33d)


(119901119901119894)1(119901119901119894)2

sdot sdot sdot (119901119901119894)119872119879



u119901119894 = B119901119894d119901119894 (119894 = 1 2 3) (34a)u119901119894119909 = B119901119894119909d119901119894 (119894 = 1 2 3) (34b)

u119901119894119910 = B119901119894119910d119901119894 (119894 = 1 2 3) (34c)


u119901119894119909119909 = B119901119894119909119909d119901119894 (119894 = 1 2 3) (34d)

u119901119894119910119910 = B119901119894119910119910d119901119894 (119894 = 1 2 3) (34e)

u119901119894119909119910 = B119901119894119909119910d119901119894 (119894 = 1 2 3) (34f)


119866119901ℎ119901 [p1199011 +1 + ]1199011 minus ]119901

(B1199011119909119909d1199011 + B1199012119909119910d1199012)

+ 2

1 minus ]119901(B1199013119909119909d1199013)119889119892B1199013119909d1199013

+(B1199013119910119910d1199013)119889119892 sdot B1199013119909d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119910d1199013]

minus 120588119901ℎ119901B1199011d1199011 = Zq119909(35a)

119866119901ℎ119901 [p1199012 +1 + ]1199011 minus ]119901

(B1199011119909119910d1199011 + B1199012119910119910d1199012)

+ 2

1 minus ]119901(B1199013119910119910d1199013)119889119892B1199013119910d1199013

+ (B1199013119909119909d1199013)119889119892 sdot B1199013119910d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119909d1199013]

minus 120588119901ℎ119901B1199012d1199012 = Zq119910

(35b)

119863p1199013 minus 119862

times [(B1199011119909d1199011)119889119892B1199013119909119909d1199013

+1

2(B1199013119909d1199013)119889119892(B1199013119909d1199013)119889119892B1199013119909119909d1199013

+ ]119901(B1199012119910d1199012)119889119892B1199013119909119909d1199013 +1

2]119901(B1199013119910d1199013)119889119892

times (B1199013119910d1199013)119889119892B1199013119909119909d1199013]

+ (1 minus ]119901)

times [(B1199011119910d1199011)119889119892B1199013119909119910d1199013

+ (B1199012119909d1199012)119889119892B1199013119909119910d1199013

+(B1199013119909d1199013)119889119892 sdot (B1199013119910d1199013)119889119892B1199013119909119910d1199013]

+ [(B1199012119910d1199012)119889119892B1199013119910119910d1199013 +1

2(B1199013119910d1199013)119889119892

sdot (B1199013119910d1199013)119889119892B1199013119910119910d1199013

+ ]119901(B1199011119909d1199011)119889119892B1199013119910119910d1199013

+1

2]119901(B1199013119909d1199013)119889119892

sdot (B1199013119909d1199013)119889119892B1199013119910119910d1199013]

+ 120588119901ℎ119901B1199013d1199013 minus 120588119901ℎ119901(B1199011d1199011)119889119892B1199013119909d1199013


119901

12B1199013119909119909d1199013

minus

120588119901ℎ3

119901

12B1199013119910119910d1199013


+ (Zq119910)119889119892B1199013119910d1199013

(35c)

where q119909 = q1199091 q1199092119879 q119910 = q1199101 q1199102

119879 andq119911 = q1199111 q1199112


119894



119901119910119895= 119902119894

119909119895ℎ1199012 119898

119894

119901119909119895= 119902119894

119910119895ℎ1199012 respectively



119887 V119894119887 and 119908

119894

119887




119887(119909119894 119905) and V119894

119887(119909119894 119905) 119908119894





Let 119906119894

1198871= 119906119894

119887(119909119894 119905) 119906

119894

1198872= V119894119887(119909119894 119905) 119906

119894

1198873= 119908119894

119887(119909119894 119905)

and 119906119894

1198874= 120579119894




1205972119906119894

1198871

1205971199091198942

= 119901119894

1198871(119909119894 119905) (36a)

1205974119906119894

119887119895

1205971199091198944= 119901119894

119887119895(119909119894 119905) (119895 = 2 3 4) (36b)




119894





119887119895(120577119894) 119906119894119887119895119909

(120577119894) 119906119894119887119895119909119909

(120577119894) and 119906

119894

119887119895119909119909119909(120577119894)

(120577119894 = 0 119897119894) in terms of 119901119894



[[[

[

E11989411988711

0 0 00 E119894

119887220 0

0 0 E11989411988733

00 0 0 E119894

11988744

]]]

]

d1198941198871

d1198941198872

d1198941198873

d1198941198874

+

0D119894 1198991198971198871

00

D119894 1198991198971198872

00

D119894 1198991198971198873

000

=

0120572119894

1198873

00120573119894

1198873

00120574119894

1198873

00120575119894

1198873

(37)

where E11989411988711

and E11989411988722

E11989411988733

E11989411988744


1198871is

a 2 times 1 and D119894 119899119897119887119895


1198873and

120573119894

1198873 1205741198941198873 1205751198941198873



1198873and 120573

119894

1198873 120573119894

1198873 120574119894

1198873 120574119894

1198873 120575119894

1198873 120575119894


d1198941198871

= p1198941198871

u1198941198871

u1198941198871119909

119879

(38a)

d119894119887119895

= p119894119887119895 u119894119887119895

u119894119887119895119909

u119894119887119895119909119909

u119894119887119895119909119909119909

119879

(119895 = 2 3 4)

(38b)


u119894119887119895

= 119906119894

119887119895(0 119905) 119906

119894

119887119895(119897119894 119905)119879

(119895 = 1 2 3 4) (39a)

u119894119887119895119909

= 120597119906119894

119887119895(0 119905)

120597119909119894

120597119906119894

119887119895(119897119894 119905)

120597119909119894

119879

(119895 = 1 2 3 4) (39b)

u119894119887119895119909119909

= 1205972119906119894

119887119895(0 119905)

1205971199091198942

1205972119906119894

119887119895(119897119894 119905)

1205971199091198942

119879

(119895 = 2 3 4)

(39c)

u119894119887119895119909119909119909

= 1205973119906119894

119887119895(0 119905)

1205971199091198943

1205973119906119894

119887119895(119897119894 119905)

1205971199091198943

119879

(119895 = 2 3 4)

(39d)


119887119895=

(119901119894

119887119895)1

(119901119894

119887119895)2


119887119895)119871119879



119894



119894isin (0 119897

119894) and


u1198941198871

= B1198941198871d1198941198871 (40a)

u1198941198871119909

= B1198941198871119909

d1198941198871 (40b)

u119894119887119895

= B119894119887119895d119894119887119895 (119895 = 2 3 4) (40c)

u119894119887119895119909

= B119894119887119895119909

d119894119887119895 (119895 = 2 3 4) (40d)

u119894119887119895119909119909

= B119894119887119895119909119909

d119894119887119895 (119895 = 2 3 4) (40e)

u119894119887119895119909119909119909

= B119894119887119895119909119909119909

d119894119887119895 (119895 = 2 3 4) (40f)

where B1198941198871B1198941198871119909

are 119871 times (119871 + 4) and B119894119887119895B119894119887119895119909

B119894119887119895119909119909

B119894119887119895119909119909119909



x

y

a

a

Fr

Fr

ClCl

Lx = 18 cm

Ly=9

cm A

(a)

g

02 cmhb = 05 cm

1 cm 5 cm3 cm

Ly = 9 cm

(b)


Time (ms)

060

040

020

000

minus020

000 050 100 150 200 250

Disp

lace

men

tw (c

m)





minus 119864119894

119887119860119894

119887[p1198941198871

+ (B1198941198872119909

d1198941198872)119889119892B1198941198872119909119909

d1198941198872

+ (B1198941198873119909

d1198941198873)119889119892B1198941198873119909119909

d1198941198873]

+ 120588119894

119887Α119894

119887T1q1 = q119894

1199091+ q1198941199092

(41a)

119864119894

119887119868119894

119911p1198941198872

minus (N119894119887)119889119892B1198941198872119909119909

d1198941198872

minus 120588119894

119887119868119894

119911B1198941198872119909119909

d1198941198872

+ 120588119894

119887Α119894

119887B1198941198872d1198941198872

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198872119909

d1198941198872

= q1198941199101

+ q1198941199102

minus (B1198941198872119909

d1198941198872)119889119892

(q1198941199091

+ q1198941199092) minus X119894119887119910

(q1198941199091

+ q1198941199092)

(41b)

119864119894

119887119868119894

119910p1198941198873

minus (N119894119887)119889119892B1198941198873119909119909

d1198941198873

minus 120588119894

119887119868119894

119910B1198941198873119909119909

d1198941198873

+ 120588119894

119887Α119894

119887B1198941198873d1198941198873

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198873119909

d1198941198873

= q1198941199111

+ q1198941199112

minus (B1198941198873119909

d1198941198873)119889119892

(q1198941199091

+ q1198941199092) + X119894119887119911

(q1198941199091

+ q1198941199092)

(41c)

119864119894

119887119862119894

119878p1198941198874

minus 119866119894

119887119868119894

119905B1198941198874119909119909

d1198941198874

+ 120588119894

119887119868119894

119901B1198941198874d1198941198874

minus 120588119894

119887119862119894

119878B1198941198874119909119909

d1198941198874

= e1198941199101q1198941199111

+ e1198941199102q1198941199112

minus e1198941199111q1198941199101

minus e1198941199112q1198941199102

+ Χ119894

119887119908(q1198941199091

+ q1198941199092)

(41d)

where (N119894119887)119889119892


119887119911 X119894119887119910 X119894119887119908


119898119894119887119911119895 119898119894119887119908119895


1199101 e1198941199102 e1198941199111

e1198941199112


119894

119910119895 119890119894


119894

119911119895 119902119894

119910119895with



Y1B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y1B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

+

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198911

B1198941198874119909

d1198874

+ K1198941199091

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42a)


008

008

018

018

028

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036040044048


000067300301006080091501220153018402140245027603070337036803990430460491


008

008

008

00801

8018

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036


000064600236004780072009620120145016901930217024102660290314033803620387



Y2B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y2B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

minus

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198912

B1198941198874119909

d1198874

+ K1198941199092

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42b)

Y1B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y1B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199101

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43a)

Y2B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y2B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199102

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43b)

Y1B1199013d1199013 minus B1198941198873d1198941198873

= minus

119887119894

119891

4B1198941198874d1198941198874 (44a)

Y2B1199013d1199013 minus B1198941198873d1198941198873

=

119887119894

119891

4B1198941198874d1198941198874 (44b)

000 050 100 150 200 250

060

040

020

000

Disp

lace

men

tw (c

m)

Time (ms)



minus020



119887119895(119894 = 1 sdot sdot sdot 119868)



000015030045060075

9

6

3

0

1815

12

9

6

3

0


000010020030040050

9

6

3

0

1815

12

9

6

3

0


000010020030040050060

9

6

3

0

1815

12

9

6

3

0


00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0


00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000


00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000




0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










and 119872119894

119887119905 119872119894119887119908


119872119894

119887119905= 119866119894

119887119868119894

119905

120597120579119894

119887119909

120597119909119894minus 119864119894

119887119862119894

119878

1205973120579119894

119887119909

1205971199091198943 (25a)

119872119894

119887119908= minus119864119894

119887119862119894

119878

1205972120579119894

119887119909

1205971199091198942 (25b)

Finally 120572119894119887119896 120573119894

119887119896 120573119894

119887119896 120574119894

119887119896 120574119894

119887119896 120575119894

119887119896 120575119894

119887119896(119896 = 1 2 3) are func-




119894

119887 V119894119887119908119894119887 120579119894119887119909 1199021198941199091 1199021198941199101 1199021198941199111 119902119894

1199092

119902119894

1199102 and 119902

119894





119906119894

1199011minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199011

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894+

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198911

+119902119894

1199091

1198961198941199091

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=1)

(26a)

119906119894

1199012minus 119906119894

119887=

ℎ119901

2

120597119908119894

1199012

120597119909+

ℎ119894

119887

2

120597119908119894

119887

120597119909119894minus

119887119894

119891

4

120597V119894119887

120597119909119894

+120597120579119894

119887119909

120597119909119894(120593119875119894

119878)1198912

+119902119894

1199092

1198961198941199092

[1 minus1

2(120597119908119894

119887

120597119909119894)

2

]


119895=2)

(26b)


V1198941199011

minus V119894119887=

ℎ119901

2

120597119908119894

1199011

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199101

1198961198941199101

[1 minus1

2(120579119894

119887119909)2

]


119895=1)

(27a)

V1198941199012

minus V119894119887=

ℎ119901

2

120597119908119894

1199012

120597119910+

ℎ119894

119887

2120579119894

119887119909+

119902119894

1199102

1198961198941199102

[1 minus1

2(120579119894

119887119909)2

]


119895=2)

(27b)


119908119894

1199011minus 119908119894

119887= minus

119887119894

119891

4120579119894


119894

119895=1) (28a)

119908119894

1199012minus 119908119894

119887=

119887119894

119891

4120579119894


119894

119895=2) (28b)

where (120593119875119894

119878)119891119895


119894

119895and 119896

119894

119909119895 119896119894119910119895



=

119896119894

119909119895(119904119894

119909119895) and 119896

119894

119910119895= 119896119894

119910119895(119904119894



119894



119875119894









nabla2119906119901119894 = 119901119901119894 (119909 119910 119905) (119894 = 1 2) (29a)

nabla41199061199013 = 1199011199013 (119909 119910 119905) (29b)






[

[

E11990111 0 00 E11990122 00 0 E11990133

]

]

d1199011d1199012d1199013

+

0D1198991198971199011

0D1198991198971199012

00

D1198991198971199013

=

01205721199013

01205731199013

001205741199013

1205751199013

(30)

where

E11990111 = [A1 H1 H20 D11990122 D11990123

] (31a)

E11990122 = [A1 H1 H20 D11990144 D11990145

] (31b)

E11990133 =[[[

[

A2 H1 H2 G1 G2A1 0 0 H1 H20 D11990178 D11990179 0 D1199017110 D11990188 D11990189 D119901810 0

]]]

]

(31c)



119901119894(119894 = 1 2) are 119873 times 1 and

D1198991198971199013


d119901119894 = p119901119894 u119901119894 u119901119894119899119879 (119894 = 1 2) (32a)


2u1199013)119899119879 (32b)


u119901119894 = (119906119901119894)1(119906119901119894)2

sdot sdot sdot (119906119901119894)119873119879

(119894 = 1 2 3)

(33a)

u119901119894119899 = (

120597119906119901119894

120597119899)

1

(

120597119906119901119894

120597119899)

2

sdot sdot sdot (

120597119906119901119894

120597119899)

119873

119879

(119894 = 1 2 3)

(33b)


21199061199013)1

(nabla21199061199013)2


119879

(33c)

(nabla2u1199013)119899

= (

120597nabla21199061199013

120597119899)

1

(

120597nabla21199061199013

120597119899)

2

sdot sdot sdot (

120597nabla21199061199013

120597119899)

119873

119879

(33d)


(119901119901119894)1(119901119901119894)2

sdot sdot sdot (119901119901119894)119872119879



u119901119894 = B119901119894d119901119894 (119894 = 1 2 3) (34a)u119901119894119909 = B119901119894119909d119901119894 (119894 = 1 2 3) (34b)

u119901119894119910 = B119901119894119910d119901119894 (119894 = 1 2 3) (34c)


u119901119894119909119909 = B119901119894119909119909d119901119894 (119894 = 1 2 3) (34d)

u119901119894119910119910 = B119901119894119910119910d119901119894 (119894 = 1 2 3) (34e)

u119901119894119909119910 = B119901119894119909119910d119901119894 (119894 = 1 2 3) (34f)


119866119901ℎ119901 [p1199011 +1 + ]1199011 minus ]119901

(B1199011119909119909d1199011 + B1199012119909119910d1199012)

+ 2

1 minus ]119901(B1199013119909119909d1199013)119889119892B1199013119909d1199013

+(B1199013119910119910d1199013)119889119892 sdot B1199013119909d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119910d1199013]

minus 120588119901ℎ119901B1199011d1199011 = Zq119909(35a)

119866119901ℎ119901 [p1199012 +1 + ]1199011 minus ]119901

(B1199011119909119910d1199011 + B1199012119910119910d1199012)

+ 2

1 minus ]119901(B1199013119910119910d1199013)119889119892B1199013119910d1199013

+ (B1199013119909119909d1199013)119889119892 sdot B1199013119910d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119909d1199013]

minus 120588119901ℎ119901B1199012d1199012 = Zq119910

(35b)

119863p1199013 minus 119862

times [(B1199011119909d1199011)119889119892B1199013119909119909d1199013

+1

2(B1199013119909d1199013)119889119892(B1199013119909d1199013)119889119892B1199013119909119909d1199013

+ ]119901(B1199012119910d1199012)119889119892B1199013119909119909d1199013 +1

2]119901(B1199013119910d1199013)119889119892

times (B1199013119910d1199013)119889119892B1199013119909119909d1199013]

+ (1 minus ]119901)

times [(B1199011119910d1199011)119889119892B1199013119909119910d1199013

+ (B1199012119909d1199012)119889119892B1199013119909119910d1199013

+(B1199013119909d1199013)119889119892 sdot (B1199013119910d1199013)119889119892B1199013119909119910d1199013]

+ [(B1199012119910d1199012)119889119892B1199013119910119910d1199013 +1

2(B1199013119910d1199013)119889119892

sdot (B1199013119910d1199013)119889119892B1199013119910119910d1199013

+ ]119901(B1199011119909d1199011)119889119892B1199013119910119910d1199013

+1

2]119901(B1199013119909d1199013)119889119892

sdot (B1199013119909d1199013)119889119892B1199013119910119910d1199013]

+ 120588119901ℎ119901B1199013d1199013 minus 120588119901ℎ119901(B1199011d1199011)119889119892B1199013119909d1199013


119901

12B1199013119909119909d1199013

minus

120588119901ℎ3

119901

12B1199013119910119910d1199013


+ (Zq119910)119889119892B1199013119910d1199013

(35c)

where q119909 = q1199091 q1199092119879 q119910 = q1199101 q1199102

119879 andq119911 = q1199111 q1199112


119894



119901119910119895= 119902119894

119909119895ℎ1199012 119898

119894

119901119909119895= 119902119894

119910119895ℎ1199012 respectively



119887 V119894119887 and 119908

119894

119887




119887(119909119894 119905) and V119894

119887(119909119894 119905) 119908119894





Let 119906119894

1198871= 119906119894

119887(119909119894 119905) 119906

119894

1198872= V119894119887(119909119894 119905) 119906

119894

1198873= 119908119894

119887(119909119894 119905)

and 119906119894

1198874= 120579119894




1205972119906119894

1198871

1205971199091198942

= 119901119894

1198871(119909119894 119905) (36a)

1205974119906119894

119887119895

1205971199091198944= 119901119894

119887119895(119909119894 119905) (119895 = 2 3 4) (36b)




119894





119887119895(120577119894) 119906119894119887119895119909

(120577119894) 119906119894119887119895119909119909

(120577119894) and 119906

119894

119887119895119909119909119909(120577119894)

(120577119894 = 0 119897119894) in terms of 119901119894



[[[

[

E11989411988711

0 0 00 E119894

119887220 0

0 0 E11989411988733

00 0 0 E119894

11988744

]]]

]

d1198941198871

d1198941198872

d1198941198873

d1198941198874

+

0D119894 1198991198971198871

00

D119894 1198991198971198872

00

D119894 1198991198971198873

000

=

0120572119894

1198873

00120573119894

1198873

00120574119894

1198873

00120575119894

1198873

(37)

where E11989411988711

and E11989411988722

E11989411988733

E11989411988744


1198871is

a 2 times 1 and D119894 119899119897119887119895


1198873and

120573119894

1198873 1205741198941198873 1205751198941198873



1198873and 120573

119894

1198873 120573119894

1198873 120574119894

1198873 120574119894

1198873 120575119894

1198873 120575119894


d1198941198871

= p1198941198871

u1198941198871

u1198941198871119909

119879

(38a)

d119894119887119895

= p119894119887119895 u119894119887119895

u119894119887119895119909

u119894119887119895119909119909

u119894119887119895119909119909119909

119879

(119895 = 2 3 4)

(38b)


u119894119887119895

= 119906119894

119887119895(0 119905) 119906

119894

119887119895(119897119894 119905)119879

(119895 = 1 2 3 4) (39a)

u119894119887119895119909

= 120597119906119894

119887119895(0 119905)

120597119909119894

120597119906119894

119887119895(119897119894 119905)

120597119909119894

119879

(119895 = 1 2 3 4) (39b)

u119894119887119895119909119909

= 1205972119906119894

119887119895(0 119905)

1205971199091198942

1205972119906119894

119887119895(119897119894 119905)

1205971199091198942

119879

(119895 = 2 3 4)

(39c)

u119894119887119895119909119909119909

= 1205973119906119894

119887119895(0 119905)

1205971199091198943

1205973119906119894

119887119895(119897119894 119905)

1205971199091198943

119879

(119895 = 2 3 4)

(39d)


119887119895=

(119901119894

119887119895)1

(119901119894

119887119895)2


119887119895)119871119879



119894



119894isin (0 119897

119894) and


u1198941198871

= B1198941198871d1198941198871 (40a)

u1198941198871119909

= B1198941198871119909

d1198941198871 (40b)

u119894119887119895

= B119894119887119895d119894119887119895 (119895 = 2 3 4) (40c)

u119894119887119895119909

= B119894119887119895119909

d119894119887119895 (119895 = 2 3 4) (40d)

u119894119887119895119909119909

= B119894119887119895119909119909

d119894119887119895 (119895 = 2 3 4) (40e)

u119894119887119895119909119909119909

= B119894119887119895119909119909119909

d119894119887119895 (119895 = 2 3 4) (40f)

where B1198941198871B1198941198871119909

are 119871 times (119871 + 4) and B119894119887119895B119894119887119895119909

B119894119887119895119909119909

B119894119887119895119909119909119909



x

y

a

a

Fr

Fr

ClCl

Lx = 18 cm

Ly=9

cm A

(a)

g

02 cmhb = 05 cm

1 cm 5 cm3 cm

Ly = 9 cm

(b)


Time (ms)

060

040

020

000

minus020

000 050 100 150 200 250

Disp

lace

men

tw (c

m)





minus 119864119894

119887119860119894

119887[p1198941198871

+ (B1198941198872119909

d1198941198872)119889119892B1198941198872119909119909

d1198941198872

+ (B1198941198873119909

d1198941198873)119889119892B1198941198873119909119909

d1198941198873]

+ 120588119894

119887Α119894

119887T1q1 = q119894

1199091+ q1198941199092

(41a)

119864119894

119887119868119894

119911p1198941198872

minus (N119894119887)119889119892B1198941198872119909119909

d1198941198872

minus 120588119894

119887119868119894

119911B1198941198872119909119909

d1198941198872

+ 120588119894

119887Α119894

119887B1198941198872d1198941198872

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198872119909

d1198941198872

= q1198941199101

+ q1198941199102

minus (B1198941198872119909

d1198941198872)119889119892

(q1198941199091

+ q1198941199092) minus X119894119887119910

(q1198941199091

+ q1198941199092)

(41b)

119864119894

119887119868119894

119910p1198941198873

minus (N119894119887)119889119892B1198941198873119909119909

d1198941198873

minus 120588119894

119887119868119894

119910B1198941198873119909119909

d1198941198873

+ 120588119894

119887Α119894

119887B1198941198873d1198941198873

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198873119909

d1198941198873

= q1198941199111

+ q1198941199112

minus (B1198941198873119909

d1198941198873)119889119892

(q1198941199091

+ q1198941199092) + X119894119887119911

(q1198941199091

+ q1198941199092)

(41c)

119864119894

119887119862119894

119878p1198941198874

minus 119866119894

119887119868119894

119905B1198941198874119909119909

d1198941198874

+ 120588119894

119887119868119894

119901B1198941198874d1198941198874

minus 120588119894

119887119862119894

119878B1198941198874119909119909

d1198941198874

= e1198941199101q1198941199111

+ e1198941199102q1198941199112

minus e1198941199111q1198941199101

minus e1198941199112q1198941199102

+ Χ119894

119887119908(q1198941199091

+ q1198941199092)

(41d)

where (N119894119887)119889119892


119887119911 X119894119887119910 X119894119887119908


119898119894119887119911119895 119898119894119887119908119895


1199101 e1198941199102 e1198941199111

e1198941199112


119894

119910119895 119890119894


119894

119911119895 119902119894

119910119895with



Y1B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y1B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

+

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198911

B1198941198874119909

d1198874

+ K1198941199091

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42a)


008

008

018

018

028

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036040044048


000067300301006080091501220153018402140245027603070337036803990430460491


008

008

008

00801

8018

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036


000064600236004780072009620120145016901930217024102660290314033803620387



Y2B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y2B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

minus

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198912

B1198941198874119909

d1198874

+ K1198941199092

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42b)

Y1B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y1B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199101

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43a)

Y2B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y2B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199102

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43b)

Y1B1199013d1199013 minus B1198941198873d1198941198873

= minus

119887119894

119891

4B1198941198874d1198941198874 (44a)

Y2B1199013d1199013 minus B1198941198873d1198941198873

=

119887119894

119891

4B1198941198874d1198941198874 (44b)

000 050 100 150 200 250

060

040

020

000

Disp

lace

men

tw (c

m)

Time (ms)



minus020



119887119895(119894 = 1 sdot sdot sdot 119868)



000015030045060075

9

6

3

0

1815

12

9

6

3

0


000010020030040050

9

6

3

0

1815

12

9

6

3

0


000010020030040050060

9

6

3

0

1815

12

9

6

3

0


00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0


00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000


00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000




0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











nabla2119906119901119894 = 119901119901119894 (119909 119910 119905) (119894 = 1 2) (29a)

nabla41199061199013 = 1199011199013 (119909 119910 119905) (29b)






[

[

E11990111 0 00 E11990122 00 0 E11990133

]

]

d1199011d1199012d1199013

+

0D1198991198971199011

0D1198991198971199012

00

D1198991198971199013

=

01205721199013

01205731199013

001205741199013

1205751199013

(30)

where

E11990111 = [A1 H1 H20 D11990122 D11990123

] (31a)

E11990122 = [A1 H1 H20 D11990144 D11990145

] (31b)

E11990133 =[[[

[

A2 H1 H2 G1 G2A1 0 0 H1 H20 D11990178 D11990179 0 D1199017110 D11990188 D11990189 D119901810 0

]]]

]

(31c)



119901119894(119894 = 1 2) are 119873 times 1 and

D1198991198971199013


d119901119894 = p119901119894 u119901119894 u119901119894119899119879 (119894 = 1 2) (32a)


2u1199013)119899119879 (32b)


u119901119894 = (119906119901119894)1(119906119901119894)2

sdot sdot sdot (119906119901119894)119873119879

(119894 = 1 2 3)

(33a)

u119901119894119899 = (

120597119906119901119894

120597119899)

1

(

120597119906119901119894

120597119899)

2

sdot sdot sdot (

120597119906119901119894

120597119899)

119873

119879

(119894 = 1 2 3)

(33b)


21199061199013)1

(nabla21199061199013)2


119879

(33c)

(nabla2u1199013)119899

= (

120597nabla21199061199013

120597119899)

1

(

120597nabla21199061199013

120597119899)

2

sdot sdot sdot (

120597nabla21199061199013

120597119899)

119873

119879

(33d)


(119901119901119894)1(119901119901119894)2

sdot sdot sdot (119901119901119894)119872119879



u119901119894 = B119901119894d119901119894 (119894 = 1 2 3) (34a)u119901119894119909 = B119901119894119909d119901119894 (119894 = 1 2 3) (34b)

u119901119894119910 = B119901119894119910d119901119894 (119894 = 1 2 3) (34c)


u119901119894119909119909 = B119901119894119909119909d119901119894 (119894 = 1 2 3) (34d)

u119901119894119910119910 = B119901119894119910119910d119901119894 (119894 = 1 2 3) (34e)

u119901119894119909119910 = B119901119894119909119910d119901119894 (119894 = 1 2 3) (34f)


119866119901ℎ119901 [p1199011 +1 + ]1199011 minus ]119901

(B1199011119909119909d1199011 + B1199012119909119910d1199012)

+ 2

1 minus ]119901(B1199013119909119909d1199013)119889119892B1199013119909d1199013

+(B1199013119910119910d1199013)119889119892 sdot B1199013119909d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119910d1199013]

minus 120588119901ℎ119901B1199011d1199011 = Zq119909(35a)

119866119901ℎ119901 [p1199012 +1 + ]1199011 minus ]119901

(B1199011119909119910d1199011 + B1199012119910119910d1199012)

+ 2

1 minus ]119901(B1199013119910119910d1199013)119889119892B1199013119910d1199013

+ (B1199013119909119909d1199013)119889119892 sdot B1199013119910d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119909d1199013]

minus 120588119901ℎ119901B1199012d1199012 = Zq119910

(35b)

119863p1199013 minus 119862

times [(B1199011119909d1199011)119889119892B1199013119909119909d1199013

+1

2(B1199013119909d1199013)119889119892(B1199013119909d1199013)119889119892B1199013119909119909d1199013

+ ]119901(B1199012119910d1199012)119889119892B1199013119909119909d1199013 +1

2]119901(B1199013119910d1199013)119889119892

times (B1199013119910d1199013)119889119892B1199013119909119909d1199013]

+ (1 minus ]119901)

times [(B1199011119910d1199011)119889119892B1199013119909119910d1199013

+ (B1199012119909d1199012)119889119892B1199013119909119910d1199013

+(B1199013119909d1199013)119889119892 sdot (B1199013119910d1199013)119889119892B1199013119909119910d1199013]

+ [(B1199012119910d1199012)119889119892B1199013119910119910d1199013 +1

2(B1199013119910d1199013)119889119892

sdot (B1199013119910d1199013)119889119892B1199013119910119910d1199013

+ ]119901(B1199011119909d1199011)119889119892B1199013119910119910d1199013

+1

2]119901(B1199013119909d1199013)119889119892

sdot (B1199013119909d1199013)119889119892B1199013119910119910d1199013]

+ 120588119901ℎ119901B1199013d1199013 minus 120588119901ℎ119901(B1199011d1199011)119889119892B1199013119909d1199013


119901

12B1199013119909119909d1199013

minus

120588119901ℎ3

119901

12B1199013119910119910d1199013


+ (Zq119910)119889119892B1199013119910d1199013

(35c)

where q119909 = q1199091 q1199092119879 q119910 = q1199101 q1199102

119879 andq119911 = q1199111 q1199112


119894



119901119910119895= 119902119894

119909119895ℎ1199012 119898

119894

119901119909119895= 119902119894

119910119895ℎ1199012 respectively



119887 V119894119887 and 119908

119894

119887




119887(119909119894 119905) and V119894

119887(119909119894 119905) 119908119894





Let 119906119894

1198871= 119906119894

119887(119909119894 119905) 119906

119894

1198872= V119894119887(119909119894 119905) 119906

119894

1198873= 119908119894

119887(119909119894 119905)

and 119906119894

1198874= 120579119894




1205972119906119894

1198871

1205971199091198942

= 119901119894

1198871(119909119894 119905) (36a)

1205974119906119894

119887119895

1205971199091198944= 119901119894

119887119895(119909119894 119905) (119895 = 2 3 4) (36b)




119894





119887119895(120577119894) 119906119894119887119895119909

(120577119894) 119906119894119887119895119909119909

(120577119894) and 119906

119894

119887119895119909119909119909(120577119894)

(120577119894 = 0 119897119894) in terms of 119901119894



[[[

[

E11989411988711

0 0 00 E119894

119887220 0

0 0 E11989411988733

00 0 0 E119894

11988744

]]]

]

d1198941198871

d1198941198872

d1198941198873

d1198941198874

+

0D119894 1198991198971198871

00

D119894 1198991198971198872

00

D119894 1198991198971198873

000

=

0120572119894

1198873

00120573119894

1198873

00120574119894

1198873

00120575119894

1198873

(37)

where E11989411988711

and E11989411988722

E11989411988733

E11989411988744


1198871is

a 2 times 1 and D119894 119899119897119887119895


1198873and

120573119894

1198873 1205741198941198873 1205751198941198873



1198873and 120573

119894

1198873 120573119894

1198873 120574119894

1198873 120574119894

1198873 120575119894

1198873 120575119894


d1198941198871

= p1198941198871

u1198941198871

u1198941198871119909

119879

(38a)

d119894119887119895

= p119894119887119895 u119894119887119895

u119894119887119895119909

u119894119887119895119909119909

u119894119887119895119909119909119909

119879

(119895 = 2 3 4)

(38b)


u119894119887119895

= 119906119894

119887119895(0 119905) 119906

119894

119887119895(119897119894 119905)119879

(119895 = 1 2 3 4) (39a)

u119894119887119895119909

= 120597119906119894

119887119895(0 119905)

120597119909119894

120597119906119894

119887119895(119897119894 119905)

120597119909119894

119879

(119895 = 1 2 3 4) (39b)

u119894119887119895119909119909

= 1205972119906119894

119887119895(0 119905)

1205971199091198942

1205972119906119894

119887119895(119897119894 119905)

1205971199091198942

119879

(119895 = 2 3 4)

(39c)

u119894119887119895119909119909119909

= 1205973119906119894

119887119895(0 119905)

1205971199091198943

1205973119906119894

119887119895(119897119894 119905)

1205971199091198943

119879

(119895 = 2 3 4)

(39d)


119887119895=

(119901119894

119887119895)1

(119901119894

119887119895)2


119887119895)119871119879



119894



119894isin (0 119897

119894) and


u1198941198871

= B1198941198871d1198941198871 (40a)

u1198941198871119909

= B1198941198871119909

d1198941198871 (40b)

u119894119887119895

= B119894119887119895d119894119887119895 (119895 = 2 3 4) (40c)

u119894119887119895119909

= B119894119887119895119909

d119894119887119895 (119895 = 2 3 4) (40d)

u119894119887119895119909119909

= B119894119887119895119909119909

d119894119887119895 (119895 = 2 3 4) (40e)

u119894119887119895119909119909119909

= B119894119887119895119909119909119909

d119894119887119895 (119895 = 2 3 4) (40f)

where B1198941198871B1198941198871119909

are 119871 times (119871 + 4) and B119894119887119895B119894119887119895119909

B119894119887119895119909119909

B119894119887119895119909119909119909



x

y

a

a

Fr

Fr

ClCl

Lx = 18 cm

Ly=9

cm A

(a)

g

02 cmhb = 05 cm

1 cm 5 cm3 cm

Ly = 9 cm

(b)


Time (ms)

060

040

020

000

minus020

000 050 100 150 200 250

Disp

lace

men

tw (c

m)





minus 119864119894

119887119860119894

119887[p1198941198871

+ (B1198941198872119909

d1198941198872)119889119892B1198941198872119909119909

d1198941198872

+ (B1198941198873119909

d1198941198873)119889119892B1198941198873119909119909

d1198941198873]

+ 120588119894

119887Α119894

119887T1q1 = q119894

1199091+ q1198941199092

(41a)

119864119894

119887119868119894

119911p1198941198872

minus (N119894119887)119889119892B1198941198872119909119909

d1198941198872

minus 120588119894

119887119868119894

119911B1198941198872119909119909

d1198941198872

+ 120588119894

119887Α119894

119887B1198941198872d1198941198872

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198872119909

d1198941198872

= q1198941199101

+ q1198941199102

minus (B1198941198872119909

d1198941198872)119889119892

(q1198941199091

+ q1198941199092) minus X119894119887119910

(q1198941199091

+ q1198941199092)

(41b)

119864119894

119887119868119894

119910p1198941198873

minus (N119894119887)119889119892B1198941198873119909119909

d1198941198873

minus 120588119894

119887119868119894

119910B1198941198873119909119909

d1198941198873

+ 120588119894

119887Α119894

119887B1198941198873d1198941198873

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198873119909

d1198941198873

= q1198941199111

+ q1198941199112

minus (B1198941198873119909

d1198941198873)119889119892

(q1198941199091

+ q1198941199092) + X119894119887119911

(q1198941199091

+ q1198941199092)

(41c)

119864119894

119887119862119894

119878p1198941198874

minus 119866119894

119887119868119894

119905B1198941198874119909119909

d1198941198874

+ 120588119894

119887119868119894

119901B1198941198874d1198941198874

minus 120588119894

119887119862119894

119878B1198941198874119909119909

d1198941198874

= e1198941199101q1198941199111

+ e1198941199102q1198941199112

minus e1198941199111q1198941199101

minus e1198941199112q1198941199102

+ Χ119894

119887119908(q1198941199091

+ q1198941199092)

(41d)

where (N119894119887)119889119892


119887119911 X119894119887119910 X119894119887119908


119898119894119887119911119895 119898119894119887119908119895


1199101 e1198941199102 e1198941199111

e1198941199112


119894

119910119895 119890119894


119894

119911119895 119902119894

119910119895with



Y1B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y1B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

+

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198911

B1198941198874119909

d1198874

+ K1198941199091

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42a)


008

008

018

018

028

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036040044048


000067300301006080091501220153018402140245027603070337036803990430460491


008

008

008

00801

8018

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036


000064600236004780072009620120145016901930217024102660290314033803620387



Y2B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y2B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

minus

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198912

B1198941198874119909

d1198874

+ K1198941199092

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42b)

Y1B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y1B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199101

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43a)

Y2B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y2B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199102

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43b)

Y1B1199013d1199013 minus B1198941198873d1198941198873

= minus

119887119894

119891

4B1198941198874d1198941198874 (44a)

Y2B1199013d1199013 minus B1198941198873d1198941198873

=

119887119894

119891

4B1198941198874d1198941198874 (44b)

000 050 100 150 200 250

060

040

020

000

Disp

lace

men

tw (c

m)

Time (ms)



minus020



119887119895(119894 = 1 sdot sdot sdot 119868)



000015030045060075

9

6

3

0

1815

12

9

6

3

0


000010020030040050

9

6

3

0

1815

12

9

6

3

0


000010020030040050060

9

6

3

0

1815

12

9

6

3

0


00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0


00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000


00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000




0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










u119901119894119909119909 = B119901119894119909119909d119901119894 (119894 = 1 2 3) (34d)

u119901119894119910119910 = B119901119894119910119910d119901119894 (119894 = 1 2 3) (34e)

u119901119894119909119910 = B119901119894119909119910d119901119894 (119894 = 1 2 3) (34f)


119866119901ℎ119901 [p1199011 +1 + ]1199011 minus ]119901

(B1199011119909119909d1199011 + B1199012119909119910d1199012)

+ 2

1 minus ]119901(B1199013119909119909d1199013)119889119892B1199013119909d1199013

+(B1199013119910119910d1199013)119889119892 sdot B1199013119909d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119910d1199013]

minus 120588119901ℎ119901B1199011d1199011 = Zq119909(35a)

119866119901ℎ119901 [p1199012 +1 + ]1199011 minus ]119901

(B1199011119909119910d1199011 + B1199012119910119910d1199012)

+ 2

1 minus ]119901(B1199013119910119910d1199013)119889119892B1199013119910d1199013

+ (B1199013119909119909d1199013)119889119892 sdot B1199013119910d1199013

+

1 + ]1199011 minus ]119901

(B1199013119909119910d1199013)119889119892B1199013119909d1199013]

minus 120588119901ℎ119901B1199012d1199012 = Zq119910

(35b)

119863p1199013 minus 119862

times [(B1199011119909d1199011)119889119892B1199013119909119909d1199013

+1

2(B1199013119909d1199013)119889119892(B1199013119909d1199013)119889119892B1199013119909119909d1199013

+ ]119901(B1199012119910d1199012)119889119892B1199013119909119909d1199013 +1

2]119901(B1199013119910d1199013)119889119892

times (B1199013119910d1199013)119889119892B1199013119909119909d1199013]

+ (1 minus ]119901)

times [(B1199011119910d1199011)119889119892B1199013119909119910d1199013

+ (B1199012119909d1199012)119889119892B1199013119909119910d1199013

+(B1199013119909d1199013)119889119892 sdot (B1199013119910d1199013)119889119892B1199013119909119910d1199013]

+ [(B1199012119910d1199012)119889119892B1199013119910119910d1199013 +1

2(B1199013119910d1199013)119889119892

sdot (B1199013119910d1199013)119889119892B1199013119910119910d1199013

+ ]119901(B1199011119909d1199011)119889119892B1199013119910119910d1199013

+1

2]119901(B1199013119909d1199013)119889119892

sdot (B1199013119909d1199013)119889119892B1199013119910119910d1199013]

+ 120588119901ℎ119901B1199013d1199013 minus 120588119901ℎ119901(B1199011d1199011)119889119892B1199013119909d1199013


119901

12B1199013119909119909d1199013

minus

120588119901ℎ3

119901

12B1199013119910119910d1199013


+ (Zq119910)119889119892B1199013119910d1199013

(35c)

where q119909 = q1199091 q1199092119879 q119910 = q1199101 q1199102

119879 andq119911 = q1199111 q1199112


119894



119901119910119895= 119902119894

119909119895ℎ1199012 119898

119894

119901119909119895= 119902119894

119910119895ℎ1199012 respectively



119887 V119894119887 and 119908

119894

119887




119887(119909119894 119905) and V119894

119887(119909119894 119905) 119908119894





Let 119906119894

1198871= 119906119894

119887(119909119894 119905) 119906

119894

1198872= V119894119887(119909119894 119905) 119906

119894

1198873= 119908119894

119887(119909119894 119905)

and 119906119894

1198874= 120579119894




1205972119906119894

1198871

1205971199091198942

= 119901119894

1198871(119909119894 119905) (36a)

1205974119906119894

119887119895

1205971199091198944= 119901119894

119887119895(119909119894 119905) (119895 = 2 3 4) (36b)




119894





119887119895(120577119894) 119906119894119887119895119909

(120577119894) 119906119894119887119895119909119909

(120577119894) and 119906

119894

119887119895119909119909119909(120577119894)

(120577119894 = 0 119897119894) in terms of 119901119894



[[[

[

E11989411988711

0 0 00 E119894

119887220 0

0 0 E11989411988733

00 0 0 E119894

11988744

]]]

]

d1198941198871

d1198941198872

d1198941198873

d1198941198874

+

0D119894 1198991198971198871

00

D119894 1198991198971198872

00

D119894 1198991198971198873

000

=

0120572119894

1198873

00120573119894

1198873

00120574119894

1198873

00120575119894

1198873

(37)

where E11989411988711

and E11989411988722

E11989411988733

E11989411988744


1198871is

a 2 times 1 and D119894 119899119897119887119895


1198873and

120573119894

1198873 1205741198941198873 1205751198941198873



1198873and 120573

119894

1198873 120573119894

1198873 120574119894

1198873 120574119894

1198873 120575119894

1198873 120575119894


d1198941198871

= p1198941198871

u1198941198871

u1198941198871119909

119879

(38a)

d119894119887119895

= p119894119887119895 u119894119887119895

u119894119887119895119909

u119894119887119895119909119909

u119894119887119895119909119909119909

119879

(119895 = 2 3 4)

(38b)


u119894119887119895

= 119906119894

119887119895(0 119905) 119906

119894

119887119895(119897119894 119905)119879

(119895 = 1 2 3 4) (39a)

u119894119887119895119909

= 120597119906119894

119887119895(0 119905)

120597119909119894

120597119906119894

119887119895(119897119894 119905)

120597119909119894

119879

(119895 = 1 2 3 4) (39b)

u119894119887119895119909119909

= 1205972119906119894

119887119895(0 119905)

1205971199091198942

1205972119906119894

119887119895(119897119894 119905)

1205971199091198942

119879

(119895 = 2 3 4)

(39c)

u119894119887119895119909119909119909

= 1205973119906119894

119887119895(0 119905)

1205971199091198943

1205973119906119894

119887119895(119897119894 119905)

1205971199091198943

119879

(119895 = 2 3 4)

(39d)


119887119895=

(119901119894

119887119895)1

(119901119894

119887119895)2


119887119895)119871119879



119894



119894isin (0 119897

119894) and


u1198941198871

= B1198941198871d1198941198871 (40a)

u1198941198871119909

= B1198941198871119909

d1198941198871 (40b)

u119894119887119895

= B119894119887119895d119894119887119895 (119895 = 2 3 4) (40c)

u119894119887119895119909

= B119894119887119895119909

d119894119887119895 (119895 = 2 3 4) (40d)

u119894119887119895119909119909

= B119894119887119895119909119909

d119894119887119895 (119895 = 2 3 4) (40e)

u119894119887119895119909119909119909

= B119894119887119895119909119909119909

d119894119887119895 (119895 = 2 3 4) (40f)

where B1198941198871B1198941198871119909

are 119871 times (119871 + 4) and B119894119887119895B119894119887119895119909

B119894119887119895119909119909

B119894119887119895119909119909119909



x

y

a

a

Fr

Fr

ClCl

Lx = 18 cm

Ly=9

cm A

(a)

g

02 cmhb = 05 cm

1 cm 5 cm3 cm

Ly = 9 cm

(b)


Time (ms)

060

040

020

000

minus020

000 050 100 150 200 250

Disp

lace

men

tw (c

m)





minus 119864119894

119887119860119894

119887[p1198941198871

+ (B1198941198872119909

d1198941198872)119889119892B1198941198872119909119909

d1198941198872

+ (B1198941198873119909

d1198941198873)119889119892B1198941198873119909119909

d1198941198873]

+ 120588119894

119887Α119894

119887T1q1 = q119894

1199091+ q1198941199092

(41a)

119864119894

119887119868119894

119911p1198941198872

minus (N119894119887)119889119892B1198941198872119909119909

d1198941198872

minus 120588119894

119887119868119894

119911B1198941198872119909119909

d1198941198872

+ 120588119894

119887Α119894

119887B1198941198872d1198941198872

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198872119909

d1198941198872

= q1198941199101

+ q1198941199102

minus (B1198941198872119909

d1198941198872)119889119892

(q1198941199091

+ q1198941199092) minus X119894119887119910

(q1198941199091

+ q1198941199092)

(41b)

119864119894

119887119868119894

119910p1198941198873

minus (N119894119887)119889119892B1198941198873119909119909

d1198941198873

minus 120588119894

119887119868119894

119910B1198941198873119909119909

d1198941198873

+ 120588119894

119887Α119894

119887B1198941198873d1198941198873

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198873119909

d1198941198873

= q1198941199111

+ q1198941199112

minus (B1198941198873119909

d1198941198873)119889119892

(q1198941199091

+ q1198941199092) + X119894119887119911

(q1198941199091

+ q1198941199092)

(41c)

119864119894

119887119862119894

119878p1198941198874

minus 119866119894

119887119868119894

119905B1198941198874119909119909

d1198941198874

+ 120588119894

119887119868119894

119901B1198941198874d1198941198874

minus 120588119894

119887119862119894

119878B1198941198874119909119909

d1198941198874

= e1198941199101q1198941199111

+ e1198941199102q1198941199112

minus e1198941199111q1198941199101

minus e1198941199112q1198941199102

+ Χ119894

119887119908(q1198941199091

+ q1198941199092)

(41d)

where (N119894119887)119889119892


119887119911 X119894119887119910 X119894119887119908


119898119894119887119911119895 119898119894119887119908119895


1199101 e1198941199102 e1198941199111

e1198941199112


119894

119910119895 119890119894


119894

119911119895 119902119894

119910119895with



Y1B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y1B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

+

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198911

B1198941198874119909

d1198874

+ K1198941199091

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42a)


008

008

018

018

028

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036040044048


000067300301006080091501220153018402140245027603070337036803990430460491


008

008

008

00801

8018

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036


000064600236004780072009620120145016901930217024102660290314033803620387



Y2B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y2B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

minus

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198912

B1198941198874119909

d1198874

+ K1198941199092

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42b)

Y1B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y1B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199101

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43a)

Y2B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y2B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199102

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43b)

Y1B1199013d1199013 minus B1198941198873d1198941198873

= minus

119887119894

119891

4B1198941198874d1198941198874 (44a)

Y2B1199013d1199013 minus B1198941198873d1198941198873

=

119887119894

119891

4B1198941198874d1198941198874 (44b)

000 050 100 150 200 250

060

040

020

000

Disp

lace

men

tw (c

m)

Time (ms)



minus020



119887119895(119894 = 1 sdot sdot sdot 119868)



000015030045060075

9

6

3

0

1815

12

9

6

3

0


000010020030040050

9

6

3

0

1815

12

9

6

3

0


000010020030040050060

9

6

3

0

1815

12

9

6

3

0


00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0


00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000


00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000




0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Let 119906119894

1198871= 119906119894

119887(119909119894 119905) 119906

119894

1198872= V119894119887(119909119894 119905) 119906

119894

1198873= 119908119894

119887(119909119894 119905)

and 119906119894

1198874= 120579119894




1205972119906119894

1198871

1205971199091198942

= 119901119894

1198871(119909119894 119905) (36a)

1205974119906119894

119887119895

1205971199091198944= 119901119894

119887119895(119909119894 119905) (119895 = 2 3 4) (36b)




119894





119887119895(120577119894) 119906119894119887119895119909

(120577119894) 119906119894119887119895119909119909

(120577119894) and 119906

119894

119887119895119909119909119909(120577119894)

(120577119894 = 0 119897119894) in terms of 119901119894



[[[

[

E11989411988711

0 0 00 E119894

119887220 0

0 0 E11989411988733

00 0 0 E119894

11988744

]]]

]

d1198941198871

d1198941198872

d1198941198873

d1198941198874

+

0D119894 1198991198971198871

00

D119894 1198991198971198872

00

D119894 1198991198971198873

000

=

0120572119894

1198873

00120573119894

1198873

00120574119894

1198873

00120575119894

1198873

(37)

where E11989411988711

and E11989411988722

E11989411988733

E11989411988744


1198871is

a 2 times 1 and D119894 119899119897119887119895


1198873and

120573119894

1198873 1205741198941198873 1205751198941198873



1198873and 120573

119894

1198873 120573119894

1198873 120574119894

1198873 120574119894

1198873 120575119894

1198873 120575119894


d1198941198871

= p1198941198871

u1198941198871

u1198941198871119909

119879

(38a)

d119894119887119895

= p119894119887119895 u119894119887119895

u119894119887119895119909

u119894119887119895119909119909

u119894119887119895119909119909119909

119879

(119895 = 2 3 4)

(38b)


u119894119887119895

= 119906119894

119887119895(0 119905) 119906

119894

119887119895(119897119894 119905)119879

(119895 = 1 2 3 4) (39a)

u119894119887119895119909

= 120597119906119894

119887119895(0 119905)

120597119909119894

120597119906119894

119887119895(119897119894 119905)

120597119909119894

119879

(119895 = 1 2 3 4) (39b)

u119894119887119895119909119909

= 1205972119906119894

119887119895(0 119905)

1205971199091198942

1205972119906119894

119887119895(119897119894 119905)

1205971199091198942

119879

(119895 = 2 3 4)

(39c)

u119894119887119895119909119909119909

= 1205973119906119894

119887119895(0 119905)

1205971199091198943

1205973119906119894

119887119895(119897119894 119905)

1205971199091198943

119879

(119895 = 2 3 4)

(39d)


119887119895=

(119901119894

119887119895)1

(119901119894

119887119895)2


119887119895)119871119879



119894



119894isin (0 119897

119894) and


u1198941198871

= B1198941198871d1198941198871 (40a)

u1198941198871119909

= B1198941198871119909

d1198941198871 (40b)

u119894119887119895

= B119894119887119895d119894119887119895 (119895 = 2 3 4) (40c)

u119894119887119895119909

= B119894119887119895119909

d119894119887119895 (119895 = 2 3 4) (40d)

u119894119887119895119909119909

= B119894119887119895119909119909

d119894119887119895 (119895 = 2 3 4) (40e)

u119894119887119895119909119909119909

= B119894119887119895119909119909119909

d119894119887119895 (119895 = 2 3 4) (40f)

where B1198941198871B1198941198871119909

are 119871 times (119871 + 4) and B119894119887119895B119894119887119895119909

B119894119887119895119909119909

B119894119887119895119909119909119909



x

y

a

a

Fr

Fr

ClCl

Lx = 18 cm

Ly=9

cm A

(a)

g

02 cmhb = 05 cm

1 cm 5 cm3 cm

Ly = 9 cm

(b)


Time (ms)

060

040

020

000

minus020

000 050 100 150 200 250

Disp

lace

men

tw (c

m)





minus 119864119894

119887119860119894

119887[p1198941198871

+ (B1198941198872119909

d1198941198872)119889119892B1198941198872119909119909

d1198941198872

+ (B1198941198873119909

d1198941198873)119889119892B1198941198873119909119909

d1198941198873]

+ 120588119894

119887Α119894

119887T1q1 = q119894

1199091+ q1198941199092

(41a)

119864119894

119887119868119894

119911p1198941198872

minus (N119894119887)119889119892B1198941198872119909119909

d1198941198872

minus 120588119894

119887119868119894

119911B1198941198872119909119909

d1198941198872

+ 120588119894

119887Α119894

119887B1198941198872d1198941198872

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198872119909

d1198941198872

= q1198941199101

+ q1198941199102

minus (B1198941198872119909

d1198941198872)119889119892

(q1198941199091

+ q1198941199092) minus X119894119887119910

(q1198941199091

+ q1198941199092)

(41b)

119864119894

119887119868119894

119910p1198941198873

minus (N119894119887)119889119892B1198941198873119909119909

d1198941198873

minus 120588119894

119887119868119894

119910B1198941198873119909119909

d1198941198873

+ 120588119894

119887Α119894

119887B1198941198873d1198941198873

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198873119909

d1198941198873

= q1198941199111

+ q1198941199112

minus (B1198941198873119909

d1198941198873)119889119892

(q1198941199091

+ q1198941199092) + X119894119887119911

(q1198941199091

+ q1198941199092)

(41c)

119864119894

119887119862119894

119878p1198941198874

minus 119866119894

119887119868119894

119905B1198941198874119909119909

d1198941198874

+ 120588119894

119887119868119894

119901B1198941198874d1198941198874

minus 120588119894

119887119862119894

119878B1198941198874119909119909

d1198941198874

= e1198941199101q1198941199111

+ e1198941199102q1198941199112

minus e1198941199111q1198941199101

minus e1198941199112q1198941199102

+ Χ119894

119887119908(q1198941199091

+ q1198941199092)

(41d)

where (N119894119887)119889119892


119887119911 X119894119887119910 X119894119887119908


119898119894119887119911119895 119898119894119887119908119895


1199101 e1198941199102 e1198941199111

e1198941199112


119894

119910119895 119890119894


119894

119911119895 119902119894

119910119895with



Y1B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y1B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

+

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198911

B1198941198874119909

d1198874

+ K1198941199091

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42a)


008

008

018

018

028

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036040044048


000067300301006080091501220153018402140245027603070337036803990430460491


008

008

008

00801

8018

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036


000064600236004780072009620120145016901930217024102660290314033803620387



Y2B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y2B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

minus

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198912

B1198941198874119909

d1198874

+ K1198941199092

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42b)

Y1B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y1B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199101

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43a)

Y2B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y2B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199102

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43b)

Y1B1199013d1199013 minus B1198941198873d1198941198873

= minus

119887119894

119891

4B1198941198874d1198941198874 (44a)

Y2B1199013d1199013 minus B1198941198873d1198941198873

=

119887119894

119891

4B1198941198874d1198941198874 (44b)

000 050 100 150 200 250

060

040

020

000

Disp

lace

men

tw (c

m)

Time (ms)



minus020



119887119895(119894 = 1 sdot sdot sdot 119868)



000015030045060075

9

6

3

0

1815

12

9

6

3

0


000010020030040050

9

6

3

0

1815

12

9

6

3

0


000010020030040050060

9

6

3

0

1815

12

9

6

3

0


00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0


00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000


00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000




0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x

y

a

a

Fr

Fr

ClCl

Lx = 18 cm

Ly=9

cm A

(a)

g

02 cmhb = 05 cm

1 cm 5 cm3 cm

Ly = 9 cm

(b)


Time (ms)

060

040

020

000

minus020

000 050 100 150 200 250

Disp

lace

men

tw (c

m)





minus 119864119894

119887119860119894

119887[p1198941198871

+ (B1198941198872119909

d1198941198872)119889119892B1198941198872119909119909

d1198941198872

+ (B1198941198873119909

d1198941198873)119889119892B1198941198873119909119909

d1198941198873]

+ 120588119894

119887Α119894

119887T1q1 = q119894

1199091+ q1198941199092

(41a)

119864119894

119887119868119894

119911p1198941198872

minus (N119894119887)119889119892B1198941198872119909119909

d1198941198872

minus 120588119894

119887119868119894

119911B1198941198872119909119909

d1198941198872

+ 120588119894

119887Α119894

119887B1198941198872d1198941198872

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198872119909

d1198941198872

= q1198941199101

+ q1198941199102

minus (B1198941198872119909

d1198941198872)119889119892

(q1198941199091

+ q1198941199092) minus X119894119887119910

(q1198941199091

+ q1198941199092)

(41b)

119864119894

119887119868119894

119910p1198941198873

minus (N119894119887)119889119892B1198941198873119909119909

d1198941198873

minus 120588119894

119887119868119894

119910B1198941198873119909119909

d1198941198873

+ 120588119894

119887Α119894

119887B1198941198873d1198941198873

minus 120588119894

119887Α119894

119887(B1198941198871d1198941198871)119889119892B1198941198873119909

d1198941198873

= q1198941199111

+ q1198941199112

minus (B1198941198873119909

d1198941198873)119889119892

(q1198941199091

+ q1198941199092) + X119894119887119911

(q1198941199091

+ q1198941199092)

(41c)

119864119894

119887119862119894

119878p1198941198874

minus 119866119894

119887119868119894

119905B1198941198874119909119909

d1198941198874

+ 120588119894

119887119868119894

119901B1198941198874d1198941198874

minus 120588119894

119887119862119894

119878B1198941198874119909119909

d1198941198874

= e1198941199101q1198941199111

+ e1198941199102q1198941199112

minus e1198941199111q1198941199101

minus e1198941199112q1198941199102

+ Χ119894

119887119908(q1198941199091

+ q1198941199092)

(41d)

where (N119894119887)119889119892


119887119911 X119894119887119910 X119894119887119908


119898119894119887119911119895 119898119894119887119908119895


1199101 e1198941199102 e1198941199111

e1198941199112


119894

119910119895 119890119894


119894

119911119895 119902119894

119910119895with



Y1B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y1B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

+

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198911

B1198941198874119909

d1198874

+ K1198941199091

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42a)


008

008

018

018

028

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036040044048


000067300301006080091501220153018402140245027603070337036803990430460491


008

008

008

00801

8018

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036


000064600236004780072009620120145016901930217024102660290314033803620387



Y2B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y2B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

minus

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198912

B1198941198874119909

d1198874

+ K1198941199092

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42b)

Y1B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y1B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199101

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43a)

Y2B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y2B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199102

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43b)

Y1B1199013d1199013 minus B1198941198873d1198941198873

= minus

119887119894

119891

4B1198941198874d1198941198874 (44a)

Y2B1199013d1199013 minus B1198941198873d1198941198873

=

119887119894

119891

4B1198941198874d1198941198874 (44b)

000 050 100 150 200 250

060

040

020

000

Disp

lace

men

tw (c

m)

Time (ms)



minus020



119887119895(119894 = 1 sdot sdot sdot 119868)



000015030045060075

9

6

3

0

1815

12

9

6

3

0


000010020030040050

9

6

3

0

1815

12

9

6

3

0


000010020030040050060

9

6

3

0

1815

12

9

6

3

0


00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0


00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000


00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000




0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










008

008

018

018

028

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036040044048


000067300301006080091501220153018402140245027603070337036803990430460491


008

008

008

00801

8018

0 3 6 9 12 15 180

3

6

9

000004008012016020024028032036


000064600236004780072009620120145016901930217024102660290314033803620387



Y2B1199011d1199011 minus B1198941198871d1198941198871

=

ℎ119901

2Y2B1199013119909d1199013 +

ℎ119894

119887

2B1198941198873119909

d1198873

minus

119887119894

119891

4B1198941198872119909

d1198872 + (120593119875119894

119878)1198912

B1198941198874119909

d1198874

+ K1198941199092

[1 minus1

2(B1198941198873119909

d1198873)119889119892(B119894

1198873119909d1198873)119889119892] (q119894

1199091+ q1198941199092)

(42b)

Y1B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y1B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199101

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43a)

Y2B1199012d1199012 minus B1198941198872d1198941198872

=

ℎ119901

2Y2B1199013119910d1199013 +

ℎ119894

119887

2B1198941198874d1198941198874

+ K1198941199102

[1 minus1

2(B1198941198874119909

d1198874)119889119892(B119894

1198874119909d1198874)119889119892] (q119894

1199091+ q1198941199092)

(43b)

Y1B1199013d1199013 minus B1198941198873d1198941198873

= minus

119887119894

119891

4B1198941198874d1198941198874 (44a)

Y2B1199013d1199013 minus B1198941198873d1198941198873

=

119887119894

119891

4B1198941198874d1198941198874 (44b)

000 050 100 150 200 250

060

040

020

000

Disp

lace

men

tw (c

m)

Time (ms)



minus020



119887119895(119894 = 1 sdot sdot sdot 119868)



000015030045060075

9

6

3

0

1815

12

9

6

3

0


000010020030040050

9

6

3

0

1815

12

9

6

3

0


000010020030040050060

9

6

3

0

1815

12

9

6

3

0


00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0


00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000


00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000




0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










000015030045060075

9

6

3

0

1815

12

9

6

3

0


000010020030040050

9

6

3

0

1815

12

9

6

3

0


000010020030040050060

9

6

3

0

1815

12

9

6

3

0


00001002003004018

1512

9

6

3

0 9

6

3

0

080

1512

9 0


00001002003004005018

15

12

9

6

3

0 9

6

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000


00001002003004018

15

12

9

6

3

0

6

9

3

0

070

065

060

050

055

045

040

035

030

025

020

015

010

005

000




0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

0

0

00

00005

00005

0001

0 3 6 9 12 15 180

3

6

9 400E minus 003

300E minus 003

200E minus 003

100E minus 003

000E + 000

minus400E minus 003

minus300E minus 003

minus200E minus 003

minus100E minus 003

minus00005minus

00005minus0001


00005

00005

0 3 6 9 12 15 180

3

6


minus0002

minus00045

minus000

2


00005

00005

0 3 6 9 12 15 180

3

6

9



minus0002


00010001

00035

00035

0006

0 3 6 9 12 15 180

3

6

9 00065000550004500035000250001500005


minus00015


0001

0001

0001

0 3 6 9 12 15 180

3

6

9

0

0001

0002

0003

0004

minus0004

minus0003

minus0002

minus0001

minus00015


00015

00015 00015

0004

0004

0 3 6 9 12 15 180

3

6

9

000000001000020000300004000050


minus0001



x

y a

A

a

SS

SSSS

SS

Lx = 203 cm

Ly=20

3cm 5075 cm

(a)

g

0137 cm

98325 cm 0635 cm 98325 cm

Ly = 203 cm

hb = 1133 cm

(b)



119894

119895=1 119891119894119895=2


K1198941199101 and K119894



119909119895and 1119896

119894

119910119895(119895 = 1 2) of the arbitrary





3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










3000

2000

1000

000

000 040 080 120 160 200


Time (ms)

Disp

lace

men

tw (m

m)

Proposed method-AEM

(a)

600

400

200

000

000 040 080 120

Time (ms)

Disp

lace

men

tw (m

m)


Proposed method-AEM

(b)


x

y

a

a

A

Cl

Cl

Cl

Ly=09

m

Cl

Lx = 18m

(a)

g

002m

03m 01m 05m

hb = 01m

Ly = 09m

(b)



119909119895 119902119894119910119895 and 119902

119894





010m

001m

001m

0006m

010m



3 times 107 kNm2 120588119901 = 120588119887 = 25 tnm3 ]119901 = ]119887 = 02)




ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











ℎ119887 (cm) 119868119905 (cm4) 119862119878 (cm

6) (120601119875

119878)1198911

(cm2) (120601119875

119878)1198912


3000

2000

1000

000

000 200 400 600

Time (ms)

Disp

lace

men

tw (c

m)




000 030 060 090 120 150 180000

030

060

090

001

001 001

001002

002

0000000200040006000800100012001400160018002000220024


000000013400029700046000624000787000950011100128001440016001770019300209002260024200258


001

001 001

002

002

000 030 060 090 120 150 180000

030

060

090

000000020004000600080010001200140016001800200022


00000001110002540003970005400068300082500096800111001250014001540016800183001970021100226




2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2

2

2 2

2

6

6

66

6

610

10

14

14

18

030 060 090 120 150

030

060

minus2

minus2


2

2

2

2

6

6

6

6

6

610

10

10

14

1418

030 060 090 120 150

030

060

1800

1400

1000

600

200

minus200

minus600

minus1000

minus1400

minus2

minus2


5

5

5

5 5

5

5

15

15 15

15

15

25

25

35

35

030 060 090 120 150

030

060


5

5 5

5

5 5

5

15

15 15

15

15

25

25

35

030 060 090 120 150

030

060

4500








119875




119896119894

119909119895= 119896119894

119910119895= 250119870

100381610038161003816100381610038161003816100381610038161003816

119909119894minus

119897119894

119887119909

2

100381610038161003816100381610038161003816100381610038161003816

kNm2

(119895 = 1 2)

(45)



119868119905 (m4) 119862119878 (m

6) (120601119875

119878)1198911

(m2) (120601119875

119878)1198912










6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















6 kNm2 120588119901 =



119875

119878)119895(119895 = 1 2) at the
















Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Acknowledgment


References


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Nonlinear Dynamic Analysis of Plates Stiffened … · 2019. 7. 31. · Research...

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Transcript of Research Article Nonlinear Dynamic Analysis of Plates Stiffened … · 2019. 7. 31. · Research...