Research Article Nonlinear Forced Vibration of a...

Research ArticleNonlinear Forced Vibration of a Viscoelastic BuckledBeam with 2 1 Internal Resonance

Liu-Yang Xiong1 Guo-Ce Zhang1 Hu Ding1 and Li-Qun Chen12

1 Shanghai Key Laboratory of Mechanics in Energy Engineering Shanghai Institute of Applied Mathematics and MechanicsShanghai University 149 Yan Chang Road Shanghai 200072 China

2Department of Mechanics Shanghai University Shanghai 200444 China

Correspondence should be addressed to Hu Ding dinghu3shueducn

Received 30 October 2013 Revised 1 December 2013 Accepted 2 December 2013 Published 8 January 2014

Academic Editor Sergio Preidikman

Copyright copy 2014 Liu-Yang Xiong et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Nonlinear dynamics of a viscoelastic buckled beam subjected to primary resonance in the presence of internal resonanceis investigated for the first time For appropriate choice of system parameters the natural frequency of the second mode isapproximately twice that of the first providing the condition for 2 1 internal resonance The ordinary differential equations ofthe two mode shapes are established using the Galerkin methodThe problem is replaced by two coupled second-order differentialequations with quadratic and cubic nonlinearitiesThemultiple scales method is applied to derive themodulation-phase equationsSteady-state solutions of the system as well as their stability are examined The frequency-amplitude curves exhibit the steady-state response in the directly excited and indirectly excited modes due to modal interaction The double-jump the saturationphenomenon and the nonperiodic region phenomena are observed illustrating the influence of internal resonance The validityrange of the analytical approximations is assessed by comparing the analytical approximate results with a numerical solution by theRunge-Kutta method The unstable regions in the internal resonance are explored via numerical simulations

1 Introduction

The main goal of this paper is to present an exhaustiveinvestigation for the forced vibration of a buckled beamwith quadratic and cubic nonlinearities in the equations ofmotion Buckled beams play an important role in the designof machines and structures Buckled beams have received agreat deal of attention from various scholars also due to theircomplex nonlinear behaviors [1] For example the nonlinearmodal interaction [2] and the internal resonance [3] arearising out of commensurable relationships of frequencies forspecific values of the system parameters Furthermore in thepresence of external excitation the internal resonance canhave possible influence on the buckled beam behavior whichneeds to be studied [4]

Lacarbonara et al [5] studied the frequency-responsecurves of a primary resonance of a buckled beamThe authorsfound that the response curves obtained using a single-mode approximation are in disagreementwith those obtainedby the experiment Emam and Nayfeh [6] focused on the

dynamics of a buckled beam subject to a primary-resonanceexcitation via the Galerkin truncation They revealed thatusing a single-mode truncation leads to errors in the staticand dynamic problems Nayfeh and Balachandran pointedout that those systems are characterized by quadratic non-linearities which may lead to two-to-one and combinationautoparametric resonances [7] Chin and Nayfeh considereda hinged-clamped beam with a cubic nonlinearity and thethree-to-one ratio of the second and first natural frequencies[8] The authors found that the beam exhibits a three-to-one internal resonance resulting in an energy exchangebetween these modes Based on an initially buckled beamwith clamped ends Afaneh and Ibrahim found that the firstmode transfers energy to the second mode in the presenceof the 1 1 internal resonance [9] Chin et al analyzed theresponse of a buckled beam possessing a two-to-one internalresonance to a principal parametric resonance of the highermode [10]The authors found that the response curves exhibitthe double-jump and the saturation phenomenon Based ona simply-supported beam under a vertical concentrated load

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 906324 14 pageshttpdxdoiorg1011552014906324

2 Mathematical Problems in Engineering

Y

XP

L

V(X)

F(X)cos(ΩT)Q(X T) =

Figure 1 A buckled beam with external harmonic excitation

Machado and Saravia [11] analyzed the resonance responsesin internal resonance conditions of the kind 2 3 1 Emamand Nayfeh [12] focused on a primary-resonance excitationin the presence of 1-1 and 3-1 internal resonances for abuckled beam with fixed ends The investigation shows thatthe interacting modes are nonlinearly coupled It is worthnoting that all above-mentioned researchers assumed that thebuckled beams under their consideration are elastic and didnot account for any internal damping factors On the otherhand it is well known that the dissipation of energy usingviscoelastic damping materials within vibrating structurescan reduce noise and vibration Due to their importancein the design of the continuous structures and systems theeffects of internal damping have been widely studied for thelast two decades A comprehensive overview can be foundin [13 14] Very recently Galuppi and Royer-Carfagni foundthat there are noteworthy differences between the quasielasticand the full viscoelastic model [15] So far from the literaturesurvey it is visible that the buckling and vibration behavior ofthe viscoelastic beam with internal resonances has not beeninvestigated To address the lack of research in this aspect thiswork is devoted to studying forced vibration of a viscoelasticbuckled beams in the presence of the 2 1 internal resonance

A nonlinear phenomenon has been found in the dynamicanalysis for the system with quadratic nonlinearities result-ing in a two-to-one internal resonance More precisely itwill be shown that there are regions in the frequency-response curves of first primary resonance without any stablesolution This phenomenon was first found by Nayfeh andhis coworkers by a two-degree-of-freedom discrete systemwith quadratic nonlinearities [16 17] Alasty and Shabanifound that the nonperiodic regions exist in discrete spring-pendulums [18] The authors found that the solutions arequasi-periodic inmost points of the unstable periodic regionsand only in special cases the chaotic solutions may occurVery recently based on the dynamics of a pipe conveyingfluid in the supercritical flow speed regime with external andinternal resonance Chen and his coworkers [19] found thatthe non-periodic region phenomena occur in a gyroscopicsystem From the review of literature it is found that the studyof the nonperiodic region in the internal resonance in the areaof continua systems such as buckled beams has not yet beenexplored so far Hence the present investigation proposes toinvestigate the same

In this work an analytical treatment is performed toinvestigate nonlinear dynamics of the viscoelastic buck-led beam with external and two-to-one internal resonance

around the stable curved equilibrium configuration Thestable steady-state periodic response curves from themultiplescales method are qualitatively and quantitatively confirmedby Runge-Kutta method The unstable regions are discussedby numerical simulationsThe present paper is organized intosix sections Section 2 describes the modeling of a buckledbeam subjected to an external distributed harmonic excita-tion Steady-state solutions and their stability are discussedin Section 3 Section 4 presents the results of approximatesolutions for illustrative examples The numerical verifica-tions are presented in Section 5 Section 6 ends the paperwithconcluding remarks

2 Mathematical Model

Consider a slender beam hinged to a base undergoingharmonic motion 119865(119883) cosΩ119879 as illustrated schematicallyin Figure 1 The beam is modeled as an Euler-Bernoulli beamwhich is made of the viscoelastic material The equationof transverse motion of the beam with external harmonicexcitation is given by [10 19]

119898119881119879119879

+ 119875119881119883119883

+ 119864119868119881119883119883119883119883

+ 119868Λ119881119883119883119883119883119879

=119864119860

2119871119881119883119883

int

119871

0

1198812

119883d119883 +

119860Λ

119871119881119883119883

times int

119871

0

119881119883119881119883119879

d119883 + 119865 (119883) cos (Ω119879)

(1)

where 119883 is the neutral axis coordinate along the undeflectedbeam and 119879 is the time A comma preceding 119883 or 119879 denotespartial derivatives with respect to 119883 or 119879 119881(119883 119879) is thetransverse displacement119871 is the length119860 is the cross-sectionarea 119898 is the mass per unit length 119868 is the area moment ofinertial and 119875 is the axial load The viscoelastic material isconstituted by the Kelvin relation with Youngrsquos modulus 119864

and the viscoelastic damping coefficient Λ [19] The effect ofthe external damping is neglected here The beam is simplysupported at both ends

119881 (0 119879) = 119881 (119871 119879) = 0

119881119883119883 (0 119879) = 119881

119883119883 (119871 119879) = 0

(2)

Mathematical Problems in Engineering 3

Incorporating the following dimensionless quantities

V =119881

119871 119909 =

119883

119871 119905 =

119879

119871

radic119875

119898

119891 (119909) =119871

119875119865 (119909) 120596 = Ω119871radic

119898

119875

120572 =119868Λ

1198713radic119898119875

1198962

1=

119864119860

119875 119896

2

119891=

119864119868

1198751198712

(3)

the equation of motion and the boundary conditions can benondimensionalized as

V119905119905

+ V119909119909

+ 1198962

119891V119909119909119909119909

+ 120572V119909119909119909119909119905

= 119891 (119909) cos (120596119905) +1198962

1

2V119909119909

times int

1

0

V2119909d119909 +

1205721198962

1

1198962

119891

V119909119909

int

1

0

V119909V119909119905d119909

(4)

V (0 119905) = V (1 119905) = 0

V119909119909 (0 119905) = V

119909119909 (1 119905) = 0

(5)

If the axial load is larger than the critical value thestraight equilibrium configuration becomes unstable Thestatic buckling deflection V(119909) is governed by

V10158401015840 + 1198962

119891V1015840101584010158401015840 =

1198962

1

2V10158401015840 int1

0

(V1015840)2

d119909 (6)

The first buckling mode of hinged-hinged beams is the onlystable equilibrium position [12] One assumes a solution inthe form

V (119909) = 119860119878sin (120587119909) (7)

Substituting (7) into (6) one obtains the equilibrium solution

119860119878=

2

1198961120587radic1 minus 119896

2

1198911205872 (8)

Therefore the transformation V(119909 119905) harr V(119909) + V(119909 119905) in(4) yields the governing equation of motion measured fromthe first buckling mode

V119905119905

+ 12058721198962

119891V119909119909

+ 1198962

119891V119909119909119909119909

+ 120572V119909119909119909119909119905

= minus1205721198962

1

1198962

119891

[V119909119909

minus 1198601198781205872 sin (120587119909)]

times int

1

0

[V + 119860119878sin (120587119909)] V119909119909119905d119909

+1198962

1

2[V119909119909

minus 1198601198781205872 sin (120587119909)]

times int

1

0

[V2119909+ 21205872119860119878V sin (120587119909)] d119909

+ 119891 (119909) cos (120596119905)

(9)

3 Schemes of Solution

31 Galerkin Discretization The Galerkin truncation will beproposed to discretize the equation ofmotion of a simply sup-ported viscoelastic buckled beam into ordinary differentialequations Suppose that the transverse displacement V(119909 119905) isapproximated by

V (119909 119905) =

119899

sum

119895=1

120601119895 (119909) 119902119895 (119905) (10)

where 120601119895(119909) (119895 = 1 2 119899) is the 119895th eigenfunction of the

free undamped vibration of the beamwith the hinged-hingedboundary conditions namely 120601

119895(119909) = sin(119895120587119909) and 119902

119895(119905)

(119895 = 1 2 119899) is the 119895th modal coordinatesThe second-order Galerkin truncation is investigated

first as the effect of higher-order mode will be discussedin Section 35 Substituting (10) (with 119899 = 2) into (9)multiplying the resulting equation by weighted function120601119895(119909) (119895 = 1 2) and integrating the product from 0 to 1 yield

1199021+ 1205721205874

1199021+

1

21198962

112058741198602

1198781199021

= 2 cos (120596119905) int

1

0

119891 (119909) sin (120587119909) d119909

minus1

21198962

112058741198601198781199022

1+

1

2(119860119878+ 1199021)

times [1199022

1+ 41199022

2+

2120572

1198962

119891

(119860119878

1199021+ 1199021

1199021+ 41199022

1199022)]

1199022+ 16120572120587

41199022+ 12120587

41198962

1198911199022

= 2 cos (120596119905) int

1

0

119891 (119909) sin (2120587119909) d119909

minus 1198962

112058741199022[21198601198781199021+ 1199022

1+ 41199022

2

+2120572

1198962

119891

(119860119878

1199021+ 1199021

1199021+ 41199022

1199022)]

(11)

32 2 1 Internal Resonance The method of multiple scaleswill be employed to seek an approximate solution to (11)


Introduce the fast and slow time scales1198790= 119905 and119879

1= 120576119905The

approximate expansions of the solutions to (11) are assumedto be

1199021(1198790 1198791) = 120576119902

11(1198790 1198791) + 120576211990212

(1198790 1198791) + sdot sdot sdot

1199022(1198790 1198791) = 120576119902

21(1198790 1198791) + 120576211990222

(1198790 1198791) + sdot sdot sdot

(12)

where 120576 is a small nondimensional bookkeeping parameterthat is used to distinguish different orders of magnitude Atthe end of analysis the bookkeeping parameterrsquos value is setto be equal to unity For weak external excitations120572 and119891 arescaled as 120572 harr 120576120572 and 119891 harr 120576

2119891 Substitution of (12) into (11)

and equalization of coefficients of 1205760 and 1205761 in the resulting

equations lead toOrder 120576

0

1199021111987901198790

+1

21198962

112058741198602

11987811990211

= 0

1199022111987901198790

+ 1212058741198962

11989111990221

= 0

(13)

Order 1205761

1199021211987901198790

+1

21198962

112058741198602

11987811990212

= 2 cos (120596119905) int

1

0

119891 (119909) sin (120587119909) d119909

minus 1198962

11205874119860119878(3

41199022

11+ 1199022

21)

minus 1205721205874(1 +

1198962

11198602

119878

21198962

119891

)119902111198790

minus 21199021111987901198791

1199022211987901198790

+ 1212058741198962

11989111990222

= 2 cos (120596119905) int

1

0

119891 (119909) sin (2120587119909) d119909

minus 21198962

112058741198601198781199021111990221

minus 161205721205874119902211198790

minus 21199022111987901198791

(14)

Equation (13) defines a two-degree-of-freedom linearsystem Its two natural frequencies are

1205961= min (120596

119886 120596119887) 120596

2= max (120596

119886 120596119887) (15)

where 120596119886

= 11989611205872119860119878radic2 and 120596

119887= 2radic3120587

2119896119891 Figure 2 shows

variation of the nondimensional natural frequencies with thenondimensional parameter 119896

119891 As can be noted from the

figure 2 1 internal resonance will be activated when 119896119891is

near 006366 or 020132The solution of (13) can be expressed in the following

form

11990211

(1198790 1198791) = 1198841(1198791) 1198901198941205961198861198790 + cc

11990221

(1198790 1198791) = 1198842(1198791) 1198901198941205961198871198790 + cc

(16)

10

5

0000 008 016 024 032

kf

1205962

1205962

120596n

1205961

1205961

Figure 2 Variation of the natural frequencies of a buckled beamwith the parameter 119896

119891

where cc stands for the complex conjugate of the precedingterms Substitution of (16) into (14) yields

1199021211987901198790

+ 1205962

11988611990212

= 1198901198941205961198790 int

1

0

119891 (119909) sin (120587119909) d119909 minus 1198962

112058741198601198781198842

211989021198941205961198871198790

minus 1205721205874(1 +

1198962

11198602

119878

21198962

119891

) 11989412059611988611988411198901198941205961198861198790

minus 21198941205961198861198841015840

11198901198941205961198861198790 + cc + NST

1199022211987901198790

+ 1205962

11988711990222

= 1198901198941205961198790 int

1

0

119891 (119909) sin (2120587119909) d119909

minus 21198962

1120587411986011987811988411198842119890119894(120596119886minus120596119887)1198790

minus 16120572120587411989412059611988711988421198901198941205961198871198790

minus 21198941205961198871198841015840

21198901198941205961198871198790 + cc + NST

(17)

where NST stands for all the other nonsecular terms If 120596119886=

2120596119887 some complex nonlinear behaviors may be observed

due to the resulting nonlinear secular terms Under thiscondition one obtains 120596

1= 120596119887 1205962

= 120596119886and 119896

119891= 006366

The chief aim of the present work is to investigate primaryresonance in the presence of 2 1 internal resonance

33 First Primary Resonance When 119891(119909) = 119887 sin(2120587119909)the primary resonance of the first mode in the presence of2 1 internal resonance will be investigated Introduce thedetuning parameters 120590 and 120590

0to describe the nearness of 120596

to 1205961and 120596

2to 21205961 respectively Thus

1205962= 21205961+ 1205761205900 120596 = 120596

1+ 120576120590 (18)


Substitution of (18) into (17) and equalization of coef-ficients of 119890

11989412059611198790 and 11989011989412059621198790 on both sides of the resulting

equation lead to

minus1198962

112058741198601198781198842

2119890minus11989412059001198791 minus 120572120587

4(1 +

1198962

11198602

119878

21198962

119891

) 11989412059621198841minus 211989412059621198841015840

1= 0

119887

21198901198941205901198791 minus 2119896

2

112058741198601198781198841119884211989011989412059001198791 minus 16120572120587

411989412059611198842minus 211989412059611198841015840

2= 0

(19)

Express the solution to (19) in the polar form

1198841(1198791) =

1

21198861(1198791) 1198901198941205741(1198791)

1198842(1198791) =

1

21198862(1198791) 1198901198941205742(1198791)

(20)

where 119886119899and 120574

119899are the real valued amplitude and phase

respectively Substituting (20) into (19) and separating theresulting equation into real and imaginary parts yield

1198861015840

1=

1

4120596minus1

21198862

21198962

11205874119860119878sin (2120579

2minus 1205791)

minus1

211988611205721205874(1 +

1198962

11198602

119878

21198962

119891

)

1198861015840

2=

119887

2120596minus1

1sin 1205792minus

1

2120596minus1

1119886111988621198962

11205874119860119878sin (2120579

2minus 1205791)

minus 812057212058741198862

11988611205791015840

1= 1198861(2120590 minus 120590

0) minus

1

4120596minus1

21198862

21198962

11205874119860119878cos (2120579

2minus 1205791)

11988621205791015840

2=

119887

2120596minus1

1cos 1205792minus

1

2120596minus1

1119886111988621198962

11205874119860119878cos (2120579

2minus 1205791)

+ 1198862120590

(21)

where 1205791= 2120590119879

1minus 12059001198791minus 1205741and 1205792= 1205901198791minus 1205742 Steady-state

responses occur when 119886119899and 120579

119899are constants Eliminating

1205791 1205792 1198861from (21) the frequency-response relationship is

obtained

1198872

41205962

1

=

12059621205721198962

11205878(1 + 119896

2

11198602

11987821198962

119891) 1198863

2

21205961[12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 4(2120590 minus 1205900)2]

+ 812057212058741198862

2

+

12059621198962

11205874(2120590 minus 120590

0) 1198863

2

1205961[12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 4(2120590 minus 1205900)2]

minus 1205901198862

2

(22)

Then the frequency-response relationship in the 1198861mode can

be obtained

1198861=

119896112058721198862

2

radic212057221205878(1 + 1198962

11198602

11987821198962

119891)

2

+ 8(2120590 minus 1205900)2

(23)

The stability of the steady-state responses can be deter-mined by the Routh-Hurwitz criterion The real parts ofeigenvalues of the Jacobian matrix of (21) reveal the stabilityof the fixed point

34 Second Primary Resonance When 119891(119909) = 119887 sin(120587119909)the primary resonance of the second mode in the presenceof 2 1 internal resonance will be investigated In this casethe frequency relations for the internal resonance and secondprimary resonance are introduced

1205962= 21205961+ 1205761205900 120596 = 120596

2+ 120576120590 (24)

where120590 and1205900are the detuning parameters Substituting (24)

into (17) and equating the coefficients of 11989011989412059611198790 and 11989011989412059621198790 on

both sides one obtains

119887

21198901198941205901198791 minus 119896

2

112058741198601198781198842

2119890minus11989412059001198791 minus 120572120587

4(1 +

1198962

11198602

119878

21198962

119891

) 11989412059621198841

minus 211989412059621198841015840

1= 0

minus 21198962

112058741198601198781198841119884211989011989412059001198791 minus 16120572120587

411989412059611198842minus 211989412059611198841015840

2= 0

(25)

The polar transformations for1198841and1198842are introduced in

(20) Substituting (20) into (25) and separating the resultingequation into real and imaginary parts yield

1198861015840

1=

119887

2120596minus1

2sin 1205791+

1

4120596minus1

21198862

21198962

11205874119860119878sin (2120579

2minus 1205791)

minus1

211988611205721205874(1 +

1198962

11198602

119878

21198962

119891

)

1198861015840

2= minus

1

2120596minus1

1119886111988621198962

11205874119860119878sin (2120579

2minus 1205791) minus 8120572120587

41198862

11988611205791015840

1=

119887

2120596minus1

2cos 1205791+ 1198861120590

minus1

4120596minus1

21198862

21198962

11205874119860119878cos (2120579

2minus 1205791)

11988621205791015840

2= minus

1

2120596minus1

1119886111988621198962

11205874119860119878cos (2120579

2minus 1205791) +

1

21198862(120590 + 120590

0)

(26)

where 1205791

= 1205901198791minus 1205741and 120579

2= 05120590119879

1+ 05120590

01198791minus 1205742 The

steady-state solutions are obtained by setting the right-handside of (26) equal to zero There are two possible solutionsThe first is a single-mode (119886

2= 0) steady-state solution given


by (27)This is the solution of local linearization (as indicatedby the subscript ldquo119897rdquo)

1198861= 1198861119897

=119887

1205962radic12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 41205902

1198862= 1198862119897

= 0

(27)

The other possibility is coupledmode (1198862

= 0) steady-statesolution This is the nonlinear solution (as indicated by thesubscript ldquo119899rdquo)

1198861= 1198861119899

=

119896119891

1205871198961

radic3(120590 + 120590

0)2+ 768120572

21205878

1 minus 12058721198962

119891

1198862= 1198862119899

=1

1205872radic

radic6Γ1plusmn (41198962

11198962

1198911198872minus 61205878Γ2

2)12

1198601198781198961198911198963

1

(28)

where

Γ1= 41198962

119891120590 (120590 + 120590

0) minus 16120587

81205722(21198962

119891+ 1198602

1198781198962

1)

Γ2= 120572 [(66120590 + 2120590

0) 1198962

119891+ 1198602

1198781198962

1(120590 + 120590

0)]

(29)

The stability of the nontrivial state can be determinedby the Routh-Hurwitz criterion but it is not suitable forthe single-mode state To determine the stability of the locallinear solution an alternative Cartesian formulation for thecomplex amplitude equations will be used as follows

1198841(1198791) =

1

2[1199091(1198791) minus 1198941199101(1198791)] 119890119894V11198791

1198842(1198791) =

1

2[1199092(1198791) minus 1198941199102(1198791)] 119890119894V21198791

(30)

where V1

= 120590 and V2

= (12)(120590 + 1205900) Substituting the new

definition (30) into (25) one finally has

1199091015840

1=

1198962

11205874119860119878

21205962

11990921199102minus

1

21205721205874(1 +

1198962

11198602

119878

21198962

119891

)1199091minus V11199101

1199101015840

1=

119887

21205962

minus1198962

11205874119860119878

41205962

(1199092

2minus 1199102

2) minus

1

21205721205874(1 +

1198962

11198602

119878

21198962

119891

)1199101

+ V11199091

1199091015840

2=

1198962

11205874119860119878

21205961

(11990921199101minus 11990911199102) minus 8120572120587

41199092minus V21199102

1199101015840

2= V21199092minus

1198962

11205874119860119878

21205961

(11990911199092+ 11991011199102) minus 8120572120587

41199102

(31)

Then the stability of the local linear solution is determined ina similar way to the previous case

35 Influence of Higher Truncation Order In the presenceof 2 1 internal resonance the effect of higher truncationorder on the dynamic responses of the buckled beam will bediscussed

Substituting (10) (with 119899 = 4) into (9) multiplying theresulting equation by weighted function 120601

119895(119909) (119895 = 1 2 3 4)

and integrating the product from 0 to 1 yield

1199021+ 1205721205874

1199021+

1

21198962

112058741198602

1198781199021

= 2 cos (120596119905) int

1

0

119891 (119909) sin (120587119909) d119909 minus1

21198962

112058741198601198781199022

1

minus1

41198962

11205874(119860119878+ 1199021)

times [1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022+ 91199023

1199023+ 16119902

41199024) ]

1199022+ 16120572120587

41199022+ 12120587

41198962

1198911199022

= 2 cos (120596119905) int

1

0

119891 (119909) sin (2120587119909) d119909

minus 1198962

112058741199022[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199023+ 81120572120587

41199023+ 72120587

41198962

1198911199023

= 2 cos (120596119905) int

1

0

119891 (119909) sin (3120587119909) d119909

minus9

41198962

112058741199023[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199024+ 256120572120587

41199024+ 240120587

41198962

1198911199024

= 2 cos (120596119905) int

1

0

119891 (119909) sin (4120587119909) d119909

minus 41198962

112058741199024[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

(32)


There exist four different natural frequencies

120596119886= 120587radic2 (1 minus 119896

2

1198911205872) 120596

119887= radic12120587

2119896119891

120596119888= radic72120587

2119896119891 120596

119889= radic240120587

2119896119891

(33)

Under the condition of 120596119886

= 2120596119887 namely 2 1 internal

resonance one obtains

1205961= radic12120587

2119896119891 120596

2= 120587radic2 (1 minus 119896

2

1198911205872)

1205963= radic72120587

2119896119891 120596

4= radic240120587

2119896119891

(34)

As compared with the second-order Galerkin truncation thethird and fourth natural frequencies are added and the firsttwo natural frequencies have no change

The first and second primary resonances under theharmonic load 119891(119909) = 119887 sin(2120587119909) and 119891(119909) = 119887 sin(120587119909) areinvestigated respectively Based on the orthogonal propertyof the trigonometric functions substitution of 119891(119909) into (10)leads to

1199023+ 81120572120587

41199023+ 72120587

41198962

1198911199023

= minus9

41198962

112058741199023[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199024+ 256120572120587

41199024+ 240120587

41198962

1198911199024

= minus41198962

112058741199024[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

(35)

There is no stimulus for modal coordinates 1199023and 119902

4 In

addition it could be deduced that the 1199023and 119902

4will not be

activated by internal resonance between the first two modessince the added natural frequencies and the first two naturalfrequencies are incommensurableThe steady-state responsesfor 1199023and 1199024only have trivial solutions Similar results will be

obtained for higher truncation order

4 Case Studies

In this section approximate analytical results will be illus-trated by numerical examples LC4 superduralumin ismainlyapplied to the structure suffered from large load in aircraftIn the following numerical case a superduralumin beam

with circular cross section will be considered Its physicalparameters are

119871 = 1m 119860 = 314159 eminus4m2

119864 = 74Gpa 119868 = 785398 eminus9m4(36)

Perfect 2 1 internal resonance is activated when 119896119891

=

0063662 In this case

119875 =119864119868

1198962

1198911198712

= 14340403KN (37)

The dimensionless parameters are given by

119896119891

= 006366 1198961= 1273239 119860

119904= 004899

120572 = 000005 119887 = 0006

1205961= 217656 120596

2= 435312

(38)

41 First Primary Resonance In the following investigationthe normal continuous lines represent stable solutions andthe broken lines represent the unstable solutions in all figures

For the investigation of the system subjected to theprincipal resonance of the first mode in the presence of2 1 internal resonance the amplitude-frequency responsecurves along with their stability are obtained as shown inFigures 3(a) and 3(b) The detuning parameter 120590 whichdescribes the nearness of 120596 to 120596

1is taken as the control

parameterThe amplitude-frequency response characteristicsof 1198862are investigated as the response of 119886

1is obtained from

the relationship between 1198861and 119886

2shown in (23) Figure 3

shows that the response curves have both soft and hardcharacteristics Therefore the double-jumping phenomenaare illustrated here Furthermore it is worth noting that thereis no stable solution near the perfect first primary resonanceat specific parameter combinations

Amplitude-frequency response curves with the differ-ent amplitude of the external excitation the viscoelasticcoefficient and the axial load are discussed in Figures 4and 5 Figure 4(a) shows the amplitude-frequency responsecurves of the first primary resonance with three differentexternal excitation amplitudes that is to say 119887 = 0004119887 = 0006 and 119887 = 0008 Clearly the height of thetwo resonance peaks and the bandwidth of the resonanceare increasing with larger external excitation amplitudeFigure 4(b) shows the amplitude-frequency response curve ofthe first primary resonance corresponding to three differentviscoelastic damping coefficients that is 120572 = 000001120572 = 000005 and 120572 = 00001 The numerical resultsillustrate that the increasing viscoelastic damping decreasesthe amplitude of the resonance and shrinks the bandwidth ofthe resonance Moreover it is also seen that the viscoelasticdamping has significant effect on the unstable regions Theresonance response may eventually degenerate into a singlestable solution with the large viscoelastic damping

When 119875 = 14340403KN 1205761205900

= 0 is obtained andthe system is considered to be perfectly tuned The detuningparameter 120590

0changes with the dimensionless parameter


minus2 0 2

120576120590

120576a1

0000

0005

0010

Stability boundary

Stability boundaryStability

boundary

(a)

minus2 0 2

120576120590

120576a2

0000

0005

0010

0015Stability boundary

Stability boundary

Stability boundary

(b)

Figure 3 Amplitude-frequency response curves and stability boundaries of the system subjected to principal resonance of the first mode inthe presence of 2 1 internal resonance (a) 119886

1mode (b) 119886

2mode

0000

0005

0010

0015

minus2 0 2

b = 0004

b = 0006

b = 0008

120576120590

120576a2

(a)

0000

0005

0010

0015

minus2 0 2

120572 = 00001

120572 = 000005

120572 = 000001

120576120590

120576a2

(b)

Figure 4 Amplitude-frequency response curves with different parameters (a) different external excitation amplitudes and (b) differentviscoelastic damping coefficients

119896119891which is dependent on the axial load If 120576120590

0= 05 and

1205761205900

= minus05 are taken into consideration one derives 119875 =

18125417KN and 119875 = 11638770KN respectively Figure 5shows the amplitude-frequency response curves of the firstprimary resonance with three different axial loads that isto say 119875 = 14340403KN 119875 = 18125417KN and 119875 =

11638770KN As the axial load is increased the height of thepeak bending to the left is decreased and that of peak bendingto the right is increased Contrary results are obtained whendecreasing the axial load

42 Second Primary Resonance In this section the systemsubjected to the principal resonance of the second mode inthe presence of 2 1 internal resonance will be discussed

The amplitude-frequency response curves and stabilityboundaries are illustrated in Figure 6 The detuning param-eter 120590 which describes the nearness of 120596 to 120596

2is taken as the

control parameter Different from the previous first primary

0000

0005

0010

0015

0020

minus2 minus1 0 1 2

P = 11638770KN

P = 18125417KN

P = 14340403KN

120576120590

120576a2

Figure 5 Amplitude-frequency response curves with different axialloads


0000

0002

0004

0006

0008

Stability boundary

minus1 0 1 2minus2

a1l

a2l

120576120590

120576an

(a)

0000

0005

0010

0015

Stability boundary

minus2 0 2

a1n

a2n120576an

120576120590

(b)

Figure 6 Amplitude-frequency response curves and stability boundaries of the system subjected to principal resonance of the second modein the presence of 2 1 internal resonance (a) local linear solution (b) nonlinear solution

0000

0005

0010

0015

minus1 0 1 2minus2

b = 0004

b = 0006

b = 0008

a1n

a2n

120576an

120576120590

(a)

0000 0001 0002

00000

00002

00004

00006

00008

00010

a1l

a2l

a2n

a1n

120576an

1205762b

(b)

Figure 7 Effect of external excitation amplitude (a) amplitude-frequency response curves with different amplitude of external excitation(b) saturation phenomenon

resonance the response curves have a single mode and a cou-pled mode Figure 6 shows the jump phenomenon betweenthe single mode and the coupled mode Nonlinear solution isactivated by the principal resonance due tomodal interactionAs local linear solution becomes unstable near the principalresonance the nonlinear solution will be investigated in thefollowing study The amplitude-frequency response curveswith the different amplitude of the external excitationsviscoelastic coefficients and axial loads are obtained

Figure 7(a) shows the amplitude-frequency responsecurve of the second primary resonance with three differentexternal excitation amplitudes that is to say 119887 = 0004119887 = 0006 and 119887 = 0008 The modal amplitude responseis depicted as function of the external excitation with theperfect primary resonance (120576120590 = 0) and internal resonance(1205761205900= 0) In Figure 7(a) as the external excitation amplitude

is increased the height of the peak of 1198862mode is increased

and the bandwidth of resonance is expanded On the otherhand the nonlinear stable solution of 119886

1mode is not changed

which becomes saturated As shown in Figure 7(b) theamplitude 119886

1does not change eventually with the growing

of excitation amplitude There is an energy transfer from 1198861

mode to 1198862mode Namely 119886

1mode activates 119886

2mode

Figure 8(a) shows the amplitude-frequency responsecurve of the second primary resonance with three differentviscoelastic damping coefficients that is to say 120572 = 000001120572 = 000005 120572 = 00001 Numerical results illustrate that theresonance peak of 119886

2mode decreaseswith the increasing axial

load near the principal resonance On the other hand thenonlinear stable solution of 119886

1mode interestingly increases

with the growing axial load Figure 8(b) shows the amplitude-frequency response curve of the second primary resonancewith different axial loads namely 119875 = 14340403KN 119875 =

18125417KN and 119875 = 11638770KN as the previous


00000

00025

00050

00075

00100

00125

a2n

a1n

minus1 0 1 2minus2

120576120590

120576an

120572 = 000001

120572 = 000005

120572 = 00001

120572 = 000001

120572 = 000005

120572 = 00001

(a)

0002

0004

0006

0008

0010

0012

P = 14340403KN

P = 14340403KN

P = 18125417KN

P = 18125417KN

P = 11638770KN

P = 11638770KN

a2n

a1n

0 2minus2

120576120590

120576an

0000

(b)

Figure 8 Amplitude-frequency response curves with different parameters (a) different external viscoelastic damping coefficients (b)different axial loads

0000

0005

0010

0015

0 2minus2

120576120590

120576an

a1

a2

(a)

0000

0005

0010120576an

a1a1

a2

a2

minus1 0 1 2minus2

120576120590

(b)

Figure 9 Comparison of amplitude-frequency response obtained by numerical method and approximate analytical method (a) first primaryresonance (b) second primary resonance

section The numerical results in Figure 8(b) depict that thenonlinear resonance response curvemoves to the leftwith theincreasing axial load

5 Numerical Verification

To verify the dynamic characteristics of the transverse weakforced vibration of the viscoelastic buckled beam the fourth-order Runge-Kutta method is used to numerically calculate(11) which is defined as a two-degree-of-freedom linearsystem with small time-dependent nonlinear perturbations

51 Comparison of Amplitude-Frequency Response The val-ues of the system parameters are still given by (38) Theamplitude-frequency responses are obtained from periodicsolutions with different excited frequencies The normalcontinuous lines represent stable approximate analytical solu-tions the broken lines represent the unstable approximate

analytical solutions and the triangle marks represent thenumerical solutions

Figure 9 shows that the results calculated by Runge-Kuttamethod and those obtained by the multiple scale method arein basic agreement The double-jump phenomenon in thefirst primary resonance and the complex jump phenomenonbetween linear and nonlinear mode in the second primaryresonance are verified Under these parameters the steady-state response near the perfect first primary resonancebecomes unstable as the approximate analytical solutionsshown in Figure 3

52 Comparison of Time History Curve The approximateanalytical solutions of the time history response are obtainedfrom (10) (12) (16) (20) (21) and (26) In order to verifythe amplitude and the phase of the steady-state responsethe four-order Runge-Kutta method is employed to obtainthe time history response from (11) The normal continuous


990 995 1000minus0006

minus0004

minus0002

0000

0002

0004

0006

t

Figure 10 Time history response of quarter point under the first primary resonance at 120576120590 = minus16

minus00006990 995 1000

minus00004

minus00002

00000

00002

00004

00006

t

(a)

990 995 1000

t

minus0006

minus0004

minus0002

0000

0002

0004

0006

(b)

Figure 11 Time history response of quarter point under second primary resonance (a) at 120576120590 = minus02 (b) at 120576120590 = minus16

lines represent stable approximate analytical solutions andthe triangle marks represent the numerical solutions

The dynamics of the quarter point of the beam is to beinvestigated Due to the unstable regions near the perfect firstprimary resonance 120576120590 = minus16 is taken to verify the timehistory response far away from the perfect first primary res-onance at first Moreover for the second primary resonance120576120590 = minus02 and 120576120590 = minus16 are taken to study the time historyresponse near and far away from the perfect primary reso-nance respectively Figures 10 and 11 illustrate that the timehistory responses obtained by means of approximate analyt-ical method are in basic agreement with numerical results inthe case of the primary resonance and 2 1 internal resonance

53 Verification for Dynamic Response of Higher Trunca-tion Order Equation (32) is defined as a four-degree-of-freedom linear system with small time-dependent non-linear perturbations Calculating (32) by the fourth-orderRunge-Kutta method yields the time history curve of the 119902

3

and 1199024

Figures 12 and 13 indicate the attenuation responses of1199023and 119902

4which are regarded as processes of free vibration

The higher modes which are not coupled with the first twomodes by internal resonance under the harmonic excitations119891(119909) = 119887 sin(2120587119909) and 119891(119909) = 119887 sin(120587119909) have no effect onthe steady-state response of the system

54 Motion State Near the Perfect First Primary ResonanceAs shown in Figures 3 4 and 9 the periodic solutionsare unstable in some regions near the perfect first primaryresonanceThese nonperiodic regions will be studied by con-structing Poincare maps for specific parameter combinationsthat are shown in Figure 14 The steady-state response ofthe quarter point of the beam is still selected to be dis-cussed Figures 14(a) and 14(b) show that quasiperiodic andperiod-doubling solutions are obtained with the variationof viscoelastic damping coefficient For the enough smallviscoelastic damping coefficient there is also the possibilitythat the chaotic solution occurs as shown in Figure 14(c)It is seen that the solutions are quasi-periodic under someexcitation amplitudes and in other cases the period-doublingsolutionsmay be observed in Figures 14(d) and 14(e) Enoughlarge excitation amplitude will eventually lead to chaos asshown in Figure 14(f)


0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)

Figure 12 Time history response under first primary resonance of (a) 1199023mode (b) 119902

4mode

0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)

Figure 13 Time history response under second primary resonance of (a) 1199023mode (b) 119902

4mode

6 Conclusions

The purpose of the present work is to study the influence ofinternal resonance on the dynamic response of the buckledbeam with external distributed excitation Due to the pres-ence of quadratic and cubic nonlinearities in the equationsof motion analytical and numerical results indicate thatthe buckled beam displays a wealth of phenomena whensubjected to dynamic loadsThe following major conclusionsare drawn from this study

(1) Application of the Galerkin method truncates thegoverning equation into a two-degree-of-freedomsystem The natural frequencies of the correspondinglinear system are obtained For specific axial loadtwo-to-one internal resonance in buckled beam maybe activated

(2) In the case of first primary resonance and 2 1 internalresonance the amplitude-frequency response curves

along with their stability are obtained The double-jumping phenomenon with the change of detuningparameters can be detected Response curves haveboth soft and hard characteristics

(3) Under the conditions of second primary resonanceand 2 1 internal resonance unlike the first primaryresonance two possible steady-state solutions arerecognized The jump phenomenon in two possiblemodes and the saturation phenomenon are detected

(4) The effects of external excitation amplitude the vis-coelastic damping and the axial load on the reso-nance peak and thewidth of the resonance regions arediscussed

(5) Approximate analytical results are compared with theresults obtained by Runge-Kutta numerical integra-tions The agreement between the approximate ana-lytical results and the numerical results is satisfactory


minus0010 minus0005 0000 0005

minus001

000

001

002

003Ve

loci

ty

Displacement

(a)

Velo

city

minus0010 minus0005 0000 0005 0010

minus001

000

001

002

003

Displacement

(b)

minus001 000 001

minus002

000

002

004

Displacement

Velo

city

(c)

minus0010 minus0005 0000 0005

0000

0005

0010

0015

0020

Displacement

Velo

city

(d)

minus0010 minus0005 0000 0005

000

001

002

003

Displacement

Velo

city

(e)

minus0015 minus0010 minus0005 0000 0005minus002

000

002

004

Displacement

Velo

city

(f)

Figure 14 Poincare maps for= (a) 120572 = 0000037 119887 = 0004 (b) 120572 = 0000032 119887 = 0004 (c) 120572 = 000002 119887 = 0004 (d) 120572 = 000005119887 = 0006 (e) 120572 = 000005 119887 = 00084 and (f) 120572 = 000005 119887 = 001

(6) Nonperiodic region near the first primary resonanceis discussed via numerical simulation Poincare mapsare employed to demonstrate the transition from

quasiperiodic and periodic doubling motions tochaos as the increase of the excitation amplitude andthe decrease of viscoelastic damping


Conflict of Interests

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

Acknowledgments

The authors gratefully acknowledge the support of theState Key Program of National Natural Science Foundationof China through Grant nos 10932006 and 11232009 theNationalNatural Science Foundation of China throughGrantno 11372171 and the Innovation Programof ShanghaiMunic-ipal Education Commission through Grant no 12YZ028

References

[1] W Y Tseng and J Dugundji ldquoNonlinear vibrations of a buckledbeamunder harmonic excitationrdquo Journal of AppliedMechanicsvol 38 no 2 pp 467ndash476 1971

[2] G-B Min and J G Eisley ldquoNonlinear vibrationsof buckledbeamsrdquo Journal of Engineering for Industry vol 94 no 2 pp637ndash646 1972

[3] A H Nayfeh W Lacarbonara and C M Chin ldquoNonlinearnormal modes of buckled beams three-to-one and one-to-oneinternal resonancesrdquoNonlinearDynamics vol 18 no 3 pp 253ndash273 1999

[4] C-M Chin and A H Nayfeh ldquoThree-to-one internal reso-nances in hinged-clamped beamsrdquoNonlinear Dynamics vol 12no 2 pp 129ndash154 1997

[5] W Lacarbonara A H Nayfeh and W Kreider ldquoExperimentalvalidation of reduction methods for nonlinear vibrations ofdistributed-parameter systems analysis of a buckled beamrdquoNonlinear Dynamics vol 17 no 2 pp 95ndash117 1998

[6] S A Emam and A H Nayfeh ldquoOn the nonlinear dynamics ofa buckled beam subjected to a primary-resonance excitationrdquoNonlinear Dynamics vol 35 no 1 pp 1ndash17 2004

[7] A H Nayfeh and B Balachandran ldquoModal interactions indynamical and structural systemsrdquo Applied Mechanics Reviewvol 42 no 11 pp 175ndash201 1989

[8] C-M Chin and A H Nayfeh ldquoThree-to-one internal res-onances in parametrically excited hinged-clamped beamsrdquoNonlinear Dynamics vol 20 no 2 pp 131ndash158 1999

[9] A A Afaneh and R A Ibrahim ldquoNonlinear response of aninitially buckled beam with 11 internal resonance to sinusoidalexcitationrdquoNonlinear Dynamics vol 4 no 6 pp 547ndash571 1993

[10] C Chin A H Nayfeh and W Lacarbonara ldquoTwo-to-oneinternal resonances in parametrically excited buckled beamsrdquoin Proceedings of the 38th Structures Structural Dynamics andMaterials no 97ndash1081 AiAA Kissmmee Fla USA 1997

[11] S P Machado and C M Saravia ldquoShear-deformable thin-walled composite Beams in internal and external resonancerdquoComposite Structures vol 97 pp 30ndash39 2013

[12] S A Emam and A H Nayfeh ldquoNon-linear response of buckledbeams to 11 and 31 internal resonancesrdquo International Journalof Non-Linear Mechanics vol 52 pp 12ndash25 2013

[13] C W de Silva Vibration Damping Control and Design CRCPress Taylor amp Francis Group 2007

[14] W-R Chen ldquoBending vibration of axially loaded Timoshenkobeamswith locally distributed KelvinmdashVoigt dampingrdquo Journalof Sound and Vibration vol 330 no 13 pp 3040ndash3056 2011

[15] L Galuppi and G Royer-Carfagni ldquoBuckling of three-layeredcomposite beams with viscoelastic interactionrdquo CompositeStructures vol 107 pp 512ndash521 2014

[16] A H Nayfeh D T Mook and L R Marshall ldquoNonlinearcoupling of pitch and roll modes in ship motionsrdquo Journal ofHydronautics vol 7 no 4 pp 145ndash152 1973

[17] A H Nayfeh and D T Mook Nonlinear Oscillations WileyInterscience New York NY USA 1979

[18] A Alasty and R Shabani ldquoChaotic motions and fractal basinboundaries in spring-pendulum systemrdquo Nonlinear AnalysisReal World Applications vol 7 no 1 pp 81ndash95 2006

[19] L Q Chen Y L Zhang G C Zhang and H Ding ldquoEvolutionof the double-jumping in pipes conveying fluid flowing atthe supercritical speedrdquo International Journal of Non-LinearMechanics vol 58 pp 11ndash21 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Y

XP

L

V(X)

F(X)cos(ΩT)Q(X T) =

Figure 1 A buckled beam with external harmonic excitation

Machado and Saravia [11] analyzed the resonance responsesin internal resonance conditions of the kind 2 3 1 Emamand Nayfeh [12] focused on a primary-resonance excitationin the presence of 1-1 and 3-1 internal resonances for abuckled beam with fixed ends The investigation shows thatthe interacting modes are nonlinearly coupled It is worthnoting that all above-mentioned researchers assumed that thebuckled beams under their consideration are elastic and didnot account for any internal damping factors On the otherhand it is well known that the dissipation of energy usingviscoelastic damping materials within vibrating structurescan reduce noise and vibration Due to their importancein the design of the continuous structures and systems theeffects of internal damping have been widely studied for thelast two decades A comprehensive overview can be foundin [13 14] Very recently Galuppi and Royer-Carfagni foundthat there are noteworthy differences between the quasielasticand the full viscoelastic model [15] So far from the literaturesurvey it is visible that the buckling and vibration behavior ofthe viscoelastic beam with internal resonances has not beeninvestigated To address the lack of research in this aspect thiswork is devoted to studying forced vibration of a viscoelasticbuckled beams in the presence of the 2 1 internal resonance

A nonlinear phenomenon has been found in the dynamicanalysis for the system with quadratic nonlinearities result-ing in a two-to-one internal resonance More precisely itwill be shown that there are regions in the frequency-response curves of first primary resonance without any stablesolution This phenomenon was first found by Nayfeh andhis coworkers by a two-degree-of-freedom discrete systemwith quadratic nonlinearities [16 17] Alasty and Shabanifound that the nonperiodic regions exist in discrete spring-pendulums [18] The authors found that the solutions arequasi-periodic inmost points of the unstable periodic regionsand only in special cases the chaotic solutions may occurVery recently based on the dynamics of a pipe conveyingfluid in the supercritical flow speed regime with external andinternal resonance Chen and his coworkers [19] found thatthe non-periodic region phenomena occur in a gyroscopicsystem From the review of literature it is found that the studyof the nonperiodic region in the internal resonance in the areaof continua systems such as buckled beams has not yet beenexplored so far Hence the present investigation proposes toinvestigate the same

In this work an analytical treatment is performed toinvestigate nonlinear dynamics of the viscoelastic buck-led beam with external and two-to-one internal resonance

around the stable curved equilibrium configuration Thestable steady-state periodic response curves from themultiplescales method are qualitatively and quantitatively confirmedby Runge-Kutta method The unstable regions are discussedby numerical simulationsThe present paper is organized intosix sections Section 2 describes the modeling of a buckledbeam subjected to an external distributed harmonic excita-tion Steady-state solutions and their stability are discussedin Section 3 Section 4 presents the results of approximatesolutions for illustrative examples The numerical verifica-tions are presented in Section 5 Section 6 ends the paperwithconcluding remarks

2 Mathematical Model

Consider a slender beam hinged to a base undergoingharmonic motion 119865(119883) cosΩ119879 as illustrated schematicallyin Figure 1 The beam is modeled as an Euler-Bernoulli beamwhich is made of the viscoelastic material The equationof transverse motion of the beam with external harmonicexcitation is given by [10 19]

119898119881119879119879

+ 119875119881119883119883

+ 119864119868119881119883119883119883119883

+ 119868Λ119881119883119883119883119883119879

=119864119860

2119871119881119883119883

int

119871

0

1198812

119883d119883 +

119860Λ

119871119881119883119883

times int

119871

0

119881119883119881119883119879

d119883 + 119865 (119883) cos (Ω119879)

(1)

where 119883 is the neutral axis coordinate along the undeflectedbeam and 119879 is the time A comma preceding 119883 or 119879 denotespartial derivatives with respect to 119883 or 119879 119881(119883 119879) is thetransverse displacement119871 is the length119860 is the cross-sectionarea 119898 is the mass per unit length 119868 is the area moment ofinertial and 119875 is the axial load The viscoelastic material isconstituted by the Kelvin relation with Youngrsquos modulus 119864

and the viscoelastic damping coefficient Λ [19] The effect ofthe external damping is neglected here The beam is simplysupported at both ends

119881 (0 119879) = 119881 (119871 119879) = 0

119881119883119883 (0 119879) = 119881

119883119883 (119871 119879) = 0

(2)



V =119881

119871 119909 =

119883

119871 119905 =

119879

119871

radic119875

119898

119891 (119909) =119871

119875119865 (119909) 120596 = Ω119871radic

119898

119875

120572 =119868Λ

1198713radic119898119875

1198962

1=

119864119860

119875 119896

2

119891=

119864119868

1198751198712

(3)


V119905119905

+ V119909119909

+ 1198962

119891V119909119909119909119909

+ 120572V119909119909119909119909119905

= 119891 (119909) cos (120596119905) +1198962

1

2V119909119909

times int

1

0

V2119909d119909 +

1205721198962

1

1198962

119891

V119909119909

int

1

0

V119909V119909119905d119909

(4)

V (0 119905) = V (1 119905) = 0

V119909119909 (0 119905) = V

119909119909 (1 119905) = 0

(5)


V10158401015840 + 1198962

119891V1015840101584010158401015840 =

1198962

1

2V10158401015840 int1

0

(V1015840)2

d119909 (6)


V (119909) = 119860119878sin (120587119909) (7)


119860119878=

2

1198961120587radic1 minus 119896

2

1198911205872 (8)


V119905119905

+ 12058721198962

119891V119909119909

+ 1198962

119891V119909119909119909119909

+ 120572V119909119909119909119909119905

= minus1205721198962

1

1198962

119891

[V119909119909

minus 1198601198781205872 sin (120587119909)]

times int

1

0

[V + 119860119878sin (120587119909)] V119909119909119905d119909

+1198962

1

2[V119909119909

minus 1198601198781205872 sin (120587119909)]

times int

1

0

[V2119909+ 21205872119860119878V sin (120587119909)] d119909

+ 119891 (119909) cos (120596119905)

(9)



V (119909 119905) =

119899

sum

119895=1

120601119895 (119909) 119902119895 (119905) (10)



119895(119909) = sin(119895120587119909) and 119902

119895(119905)



1199021+ 1205721205874

1199021+

1

21198962

112058741198602

1198781199021

= 2 cos (120596119905) int

1

0

119891 (119909) sin (120587119909) d119909

minus1

21198962

112058741198601198781199022

1+

1

2(119860119878+ 1199021)

times [1199022

1+ 41199022

2+

2120572

1198962

119891

(119860119878

1199021+ 1199021

1199021+ 41199022

1199022)]

1199022+ 16120572120587

41199022+ 12120587

41198962

1198911199022

= 2 cos (120596119905) int

1

0

119891 (119909) sin (2120587119909) d119909

minus 1198962

112058741199022[21198601198781199021+ 1199022

1+ 41199022

2

+2120572

1198962

119891

(119860119878

1199021+ 1199021

1199021+ 41199022

1199022)]

(11)




1= 120576119905The


1199021(1198790 1198791) = 120576119902

11(1198790 1198791) + 120576211990212

(1198790 1198791) + sdot sdot sdot

1199022(1198790 1198791) = 120576119902

21(1198790 1198791) + 120576211990222

(1198790 1198791) + sdot sdot sdot

(12)





0

1199021111987901198790

+1

21198962

112058741198602

11987811990211

= 0

1199022111987901198790

+ 1212058741198962

11989111990221

= 0

(13)

Order 1205761

1199021211987901198790

+1

21198962

112058741198602

11987811990212

= 2 cos (120596119905) int

1

0

119891 (119909) sin (120587119909) d119909

minus 1198962

11205874119860119878(3

41199022

11+ 1199022

21)

minus 1205721205874(1 +

1198962

11198602

119878

21198962

119891

)119902111198790

minus 21199021111987901198791

1199022211987901198790

+ 1212058741198962

11989111990222

= 2 cos (120596119905) int

1

0

119891 (119909) sin (2120587119909) d119909

minus 21198962

112058741198601198781199021111990221

minus 161205721205874119902211198790

minus 21199022111987901198791

(14)


1205961= min (120596

119886 120596119887) 120596

2= max (120596

119886 120596119887) (15)

where 120596119886

= 11989611205872119860119878radic2 and 120596

119887= 2radic3120587

2119896119891 Figure 2 shows





form

11990211

(1198790 1198791) = 1198841(1198791) 1198901198941205961198861198790 + cc

11990221

(1198790 1198791) = 1198842(1198791) 1198901198941205961198871198790 + cc

(16)

10

5

0000 008 016 024 032

kf

1205962

1205962

120596n

1205961

1205961


119891


1199021211987901198790

+ 1205962

11988611990212

= 1198901198941205961198790 int

1

0

119891 (119909) sin (120587119909) d119909 minus 1198962

112058741198601198781198842

211989021198941205961198871198790

minus 1205721205874(1 +

1198962

11198602

119878

21198962

119891

) 11989412059611988611988411198901198941205961198861198790

minus 21198941205961198861198841015840

11198901198941205961198861198790 + cc + NST

1199022211987901198790

+ 1205962

11988711990222

= 1198901198941205961198790 int

1

0

119891 (119909) sin (2120587119909) d119909

minus 21198962

1120587411986011987811988411198842119890119894(120596119886minus120596119887)1198790

minus 16120572120587411989412059611988711988421198901198941205961198871198790

minus 21198941205961198871198841015840

21198901198941205961198871198790 + cc + NST

(17)




1= 120596119887 1205962

= 120596119886and 119896

119891= 006366




to 1205961and 120596


1205962= 21205961+ 1205761205900 120596 = 120596

1+ 120576120590 (18)




equation lead to

minus1198962

112058741198601198781198842

2119890minus11989412059001198791 minus 120572120587

4(1 +

1198962

11198602

119878

21198962

119891

) 11989412059621198841minus 211989412059621198841015840

1= 0

119887

21198901198941205901198791 minus 2119896

2

112058741198601198781198841119884211989011989412059001198791 minus 16120572120587

411989412059611198842minus 211989412059611198841015840

2= 0

(19)


1198841(1198791) =

1

21198861(1198791) 1198901198941205741(1198791)

1198842(1198791) =

1

21198862(1198791) 1198901198941205742(1198791)

(20)

where 119886119899and 120574



1198861015840

1=

1

4120596minus1

21198862

21198962

11205874119860119878sin (2120579

2minus 1205791)

minus1

211988611205721205874(1 +

1198962

11198602

119878

21198962

119891

)

1198861015840

2=

119887

2120596minus1

1sin 1205792minus

1

2120596minus1

1119886111988621198962

11205874119860119878sin (2120579

2minus 1205791)

minus 812057212058741198862

11988611205791015840

1= 1198861(2120590 minus 120590

0) minus

1

4120596minus1

21198862

21198962

11205874119860119878cos (2120579

2minus 1205791)

11988621205791015840

2=

119887

2120596minus1

1cos 1205792minus

1

2120596minus1

1119886111988621198962

11205874119860119878cos (2120579

2minus 1205791)

+ 1198862120590

(21)

where 1205791= 2120590119879





obtained

1198872

41205962

1

=

12059621205721198962

11205878(1 + 119896

2

11198602

11987821198962

119891) 1198863

2

21205961[12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 4(2120590 minus 1205900)2]

+ 812057212058741198862

2

+

12059621198962

11205874(2120590 minus 120590

0) 1198863

2

1205961[12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 4(2120590 minus 1205900)2]

minus 1205901198862

2

(22)


be obtained

1198861=

119896112058721198862

2

radic212057221205878(1 + 1198962

11198602

11987821198962

119891)

2

+ 8(2120590 minus 1205900)2

(23)



1205962= 21205961+ 1205761205900 120596 = 120596

2+ 120576120590 (24)




119887

21198901198941205901198791 minus 119896

2

112058741198601198781198842

2119890minus11989412059001198791 minus 120572120587

4(1 +

1198962

11198602

119878

21198962

119891

) 11989412059621198841

minus 211989412059621198841015840

1= 0

minus 21198962

112058741198601198781198841119884211989011989412059001198791 minus 16120572120587

411989412059611198842minus 211989412059611198841015840

2= 0

(25)



1198861015840

1=

119887

2120596minus1

2sin 1205791+

1

4120596minus1

21198862

21198962

11205874119860119878sin (2120579

2minus 1205791)

minus1

211988611205721205874(1 +

1198962

11198602

119878

21198962

119891

)

1198861015840

2= minus

1

2120596minus1

1119886111988621198962

11205874119860119878sin (2120579

2minus 1205791) minus 8120572120587

41198862

11988611205791015840

1=

119887

2120596minus1

2cos 1205791+ 1198861120590

minus1

4120596minus1

21198862

21198962

11205874119860119878cos (2120579

2minus 1205791)

11988621205791015840

2= minus

1

2120596minus1

1119886111988621198962

11205874119860119878cos (2120579

2minus 1205791) +

1

21198862(120590 + 120590

0)

(26)

where 1205791

= 1205901198791minus 1205741and 120579

2= 05120590119879

1+ 05120590

01198791minus 1205742 The





1198861= 1198861119897

=119887

1205962radic12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 41205902

1198862= 1198862119897

= 0

(27)



1198861= 1198861119899

=

119896119891

1205871198961

radic3(120590 + 120590

0)2+ 768120572

21205878

1 minus 12058721198962

119891

1198862= 1198862119899

=1

1205872radic


11198962

1198911198872minus 61205878Γ2

2)12

1198601198781198961198911198963

1

(28)

where

Γ1= 41198962

119891120590 (120590 + 120590

0) minus 16120587

81205722(21198962

119891+ 1198602

1198781198962

1)

Γ2= 120572 [(66120590 + 2120590

0) 1198962

119891+ 1198602

1198781198962

1(120590 + 120590

0)]

(29)


1198841(1198791) =

1

2[1199091(1198791) minus 1198941199101(1198791)] 119890119894V11198791

1198842(1198791) =

1

2[1199092(1198791) minus 1198941199102(1198791)] 119890119894V21198791

(30)

where V1

= 120590 and V2



1199091015840

1=

1198962

11205874119860119878

21205962

11990921199102minus

1

21205721205874(1 +

1198962

11198602

119878

21198962

119891

)1199091minus V11199101

1199101015840

1=

119887

21205962

minus1198962

11205874119860119878

41205962

(1199092

2minus 1199102

2) minus

1

21205721205874(1 +

1198962

11198602

119878

21198962

119891

)1199101

+ V11199091

1199091015840

2=

1198962

11205874119860119878

21205961

(11990921199101minus 11990911199102) minus 8120572120587

41199092minus V21199102

1199101015840

2= V21199092minus

1198962

11205874119860119878

21205961

(11990911199092+ 11991011199102) minus 8120572120587

41199102

(31)




119895(119909) (119895 = 1 2 3 4)


1199021+ 1205721205874

1199021+

1

21198962

112058741198602

1198781199021

= 2 cos (120596119905) int

1

0

119891 (119909) sin (120587119909) d119909 minus1

21198962

112058741198601198781199022

1

minus1

41198962

11205874(119860119878+ 1199021)

times [1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022+ 91199023

1199023+ 16119902

41199024) ]

1199022+ 16120572120587

41199022+ 12120587

41198962

1198911199022

= 2 cos (120596119905) int

1

0

119891 (119909) sin (2120587119909) d119909

minus 1198962

112058741199022[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199023+ 81120572120587

41199023+ 72120587

41198962

1198911199023

= 2 cos (120596119905) int

1

0

119891 (119909) sin (3120587119909) d119909

minus9

41198962

112058741199023[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199024+ 256120572120587

41199024+ 240120587

41198962

1198911199024

= 2 cos (120596119905) int

1

0

119891 (119909) sin (4120587119909) d119909

minus 41198962

112058741199024[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

(32)



120596119886= 120587radic2 (1 minus 119896

2

1198911205872) 120596

119887= radic12120587

2119896119891

120596119888= radic72120587

2119896119891 120596

119889= radic240120587

2119896119891

(33)




1205961= radic12120587

2119896119891 120596

2= 120587radic2 (1 minus 119896

2

1198911205872)

1205963= radic72120587

2119896119891 120596

4= radic240120587

2119896119891

(34)



1199023+ 81120572120587

41199023+ 72120587

41198962

1198911199023

= minus9

41198962

112058741199023[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199024+ 256120572120587

41199024+ 240120587

41198962

1198911199024

= minus41198962

112058741199024[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

(35)


4 In


4will not be



4 Case Studies



119871 = 1m 119860 = 314159 eminus4m2

119864 = 74Gpa 119868 = 785398 eminus9m4(36)


=


119875 =119864119868

1198962

1198911198712

= 14340403KN (37)


119896119891

= 006366 1198961= 1273239 119860

119904= 004899

120572 = 000005 119887 = 0006

1205961= 217656 120596

2= 435312

(38)





1is obtained from





When 119875 = 14340403KN 1205761205900




minus2 0 2

120576120590

120576a1

0000

0005

0010

Stability boundary


boundary

(a)

minus2 0 2

120576120590

120576a2

0000

0005

0010


Stability boundary

Stability boundary

(b)


1mode (b) 119886

2mode

0000

0005

0010

0015

minus2 0 2

b = 0004

b = 0006

b = 0008

120576120590

120576a2

(a)

0000

0005

0010

0015

minus2 0 2

120572 = 00001

120572 = 000005

120572 = 000001

120576120590

120576a2

(b)



0= 05 and

1205761205900






2is taken as the


0000

0005

0010

0015

0020

minus2 minus1 0 1 2

P = 11638770KN

P = 18125417KN

P = 14340403KN

120576120590

120576a2



0000

0002

0004

0006

0008

Stability boundary

minus1 0 1 2minus2

a1l

a2l

120576120590

120576an

(a)

0000

0005

0010

0015

Stability boundary

minus2 0 2

a1n

a2n120576an

120576120590

(b)


0000

0005

0010

0015

minus1 0 1 2minus2

b = 0004

b = 0006

b = 0008

a1n

a2n

120576an

120576120590

(a)

0000 0001 0002

00000

00002

00004

00006

00008

00010

a1l

a2l

a2n

a1n

120576an

1205762b

(b)












2mode








00000

00025

00050

00075

00100

00125

a2n

a1n

minus1 0 1 2minus2

120576120590

120576an

120572 = 000001

120572 = 000005

120572 = 00001

120572 = 000001

120572 = 000005

120572 = 00001

(a)

0002

0004

0006

0008

0010

0012

P = 14340403KN

P = 14340403KN

P = 18125417KN

P = 18125417KN

P = 11638770KN

P = 11638770KN

a2n

a1n

0 2minus2

120576120590

120576an

0000

(b)


0000

0005

0010

0015

0 2minus2

120576120590

120576an

a1

a2

(a)

0000

0005

0010120576an

a1a1

a2

a2

minus1 0 1 2minus2

120576120590

(b)










990 995 1000minus0006

minus0004

minus0002

0000

0002

0004

0006

t


minus00006990 995 1000

minus00004

minus00002

00000

00002

00004

00006

t

(a)

990 995 1000

t

minus0006

minus0004

minus0002

0000

0002

0004

0006

(b)





3

and 1199024






0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

6 Conclusions









minus0010 minus0005 0000 0005

minus001

000

001

002

003Ve

loci

ty

Displacement

(a)

Velo

city

minus0010 minus0005 0000 0005 0010

minus001

000

001

002

003

Displacement

(b)

minus001 000 001

minus002

000

002

004

Displacement

Velo

city

(c)

minus0010 minus0005 0000 0005

0000

0005

0010

0015

0020

Displacement

Velo

city

(d)

minus0010 minus0005 0000 0005

000

001

002

003

Displacement

Velo

city

(e)


000

002

004

Displacement

Velo

city

(f)







Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











V =119881

119871 119909 =

119883

119871 119905 =

119879

119871

radic119875

119898

119891 (119909) =119871

119875119865 (119909) 120596 = Ω119871radic

119898

119875

120572 =119868Λ

1198713radic119898119875

1198962

1=

119864119860

119875 119896

2

119891=

119864119868

1198751198712

(3)


V119905119905

+ V119909119909

+ 1198962

119891V119909119909119909119909

+ 120572V119909119909119909119909119905

= 119891 (119909) cos (120596119905) +1198962

1

2V119909119909

times int

1

0

V2119909d119909 +

1205721198962

1

1198962

119891

V119909119909

int

1

0

V119909V119909119905d119909

(4)

V (0 119905) = V (1 119905) = 0

V119909119909 (0 119905) = V

119909119909 (1 119905) = 0

(5)


V10158401015840 + 1198962

119891V1015840101584010158401015840 =

1198962

1

2V10158401015840 int1

0

(V1015840)2

d119909 (6)


V (119909) = 119860119878sin (120587119909) (7)


119860119878=

2

1198961120587radic1 minus 119896

2

1198911205872 (8)


V119905119905

+ 12058721198962

119891V119909119909

+ 1198962

119891V119909119909119909119909

+ 120572V119909119909119909119909119905

= minus1205721198962

1

1198962

119891

[V119909119909

minus 1198601198781205872 sin (120587119909)]

times int

1

0

[V + 119860119878sin (120587119909)] V119909119909119905d119909

+1198962

1

2[V119909119909

minus 1198601198781205872 sin (120587119909)]

times int

1

0

[V2119909+ 21205872119860119878V sin (120587119909)] d119909

+ 119891 (119909) cos (120596119905)

(9)



V (119909 119905) =

119899

sum

119895=1

120601119895 (119909) 119902119895 (119905) (10)



119895(119909) = sin(119895120587119909) and 119902

119895(119905)



1199021+ 1205721205874

1199021+

1

21198962

112058741198602

1198781199021

= 2 cos (120596119905) int

1

0

119891 (119909) sin (120587119909) d119909

minus1

21198962

112058741198601198781199022

1+

1

2(119860119878+ 1199021)

times [1199022

1+ 41199022

2+

2120572

1198962

119891

(119860119878

1199021+ 1199021

1199021+ 41199022

1199022)]

1199022+ 16120572120587

41199022+ 12120587

41198962

1198911199022

= 2 cos (120596119905) int

1

0

119891 (119909) sin (2120587119909) d119909

minus 1198962

112058741199022[21198601198781199021+ 1199022

1+ 41199022

2

+2120572

1198962

119891

(119860119878

1199021+ 1199021

1199021+ 41199022

1199022)]

(11)




1= 120576119905The


1199021(1198790 1198791) = 120576119902

11(1198790 1198791) + 120576211990212

(1198790 1198791) + sdot sdot sdot

1199022(1198790 1198791) = 120576119902

21(1198790 1198791) + 120576211990222

(1198790 1198791) + sdot sdot sdot

(12)





0

1199021111987901198790

+1

21198962

112058741198602

11987811990211

= 0

1199022111987901198790

+ 1212058741198962

11989111990221

= 0

(13)

Order 1205761

1199021211987901198790

+1

21198962

112058741198602

11987811990212

= 2 cos (120596119905) int

1

0

119891 (119909) sin (120587119909) d119909

minus 1198962

11205874119860119878(3

41199022

11+ 1199022

21)

minus 1205721205874(1 +

1198962

11198602

119878

21198962

119891

)119902111198790

minus 21199021111987901198791

1199022211987901198790

+ 1212058741198962

11989111990222

= 2 cos (120596119905) int

1

0

119891 (119909) sin (2120587119909) d119909

minus 21198962

112058741198601198781199021111990221

minus 161205721205874119902211198790

minus 21199022111987901198791

(14)


1205961= min (120596

119886 120596119887) 120596

2= max (120596

119886 120596119887) (15)

where 120596119886

= 11989611205872119860119878radic2 and 120596

119887= 2radic3120587

2119896119891 Figure 2 shows





form

11990211

(1198790 1198791) = 1198841(1198791) 1198901198941205961198861198790 + cc

11990221

(1198790 1198791) = 1198842(1198791) 1198901198941205961198871198790 + cc

(16)

10

5

0000 008 016 024 032

kf

1205962

1205962

120596n

1205961

1205961


119891


1199021211987901198790

+ 1205962

11988611990212

= 1198901198941205961198790 int

1

0

119891 (119909) sin (120587119909) d119909 minus 1198962

112058741198601198781198842

211989021198941205961198871198790

minus 1205721205874(1 +

1198962

11198602

119878

21198962

119891

) 11989412059611988611988411198901198941205961198861198790

minus 21198941205961198861198841015840

11198901198941205961198861198790 + cc + NST

1199022211987901198790

+ 1205962

11988711990222

= 1198901198941205961198790 int

1

0

119891 (119909) sin (2120587119909) d119909

minus 21198962

1120587411986011987811988411198842119890119894(120596119886minus120596119887)1198790

minus 16120572120587411989412059611988711988421198901198941205961198871198790

minus 21198941205961198871198841015840

21198901198941205961198871198790 + cc + NST

(17)




1= 120596119887 1205962

= 120596119886and 119896

119891= 006366




to 1205961and 120596


1205962= 21205961+ 1205761205900 120596 = 120596

1+ 120576120590 (18)




equation lead to

minus1198962

112058741198601198781198842

2119890minus11989412059001198791 minus 120572120587

4(1 +

1198962

11198602

119878

21198962

119891

) 11989412059621198841minus 211989412059621198841015840

1= 0

119887

21198901198941205901198791 minus 2119896

2

112058741198601198781198841119884211989011989412059001198791 minus 16120572120587

411989412059611198842minus 211989412059611198841015840

2= 0

(19)


1198841(1198791) =

1

21198861(1198791) 1198901198941205741(1198791)

1198842(1198791) =

1

21198862(1198791) 1198901198941205742(1198791)

(20)

where 119886119899and 120574



1198861015840

1=

1

4120596minus1

21198862

21198962

11205874119860119878sin (2120579

2minus 1205791)

minus1

211988611205721205874(1 +

1198962

11198602

119878

21198962

119891

)

1198861015840

2=

119887

2120596minus1

1sin 1205792minus

1

2120596minus1

1119886111988621198962

11205874119860119878sin (2120579

2minus 1205791)

minus 812057212058741198862

11988611205791015840

1= 1198861(2120590 minus 120590

0) minus

1

4120596minus1

21198862

21198962

11205874119860119878cos (2120579

2minus 1205791)

11988621205791015840

2=

119887

2120596minus1

1cos 1205792minus

1

2120596minus1

1119886111988621198962

11205874119860119878cos (2120579

2minus 1205791)

+ 1198862120590

(21)

where 1205791= 2120590119879





obtained

1198872

41205962

1

=

12059621205721198962

11205878(1 + 119896

2

11198602

11987821198962

119891) 1198863

2

21205961[12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 4(2120590 minus 1205900)2]

+ 812057212058741198862

2

+

12059621198962

11205874(2120590 minus 120590

0) 1198863

2

1205961[12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 4(2120590 minus 1205900)2]

minus 1205901198862

2

(22)


be obtained

1198861=

119896112058721198862

2

radic212057221205878(1 + 1198962

11198602

11987821198962

119891)

2

+ 8(2120590 minus 1205900)2

(23)



1205962= 21205961+ 1205761205900 120596 = 120596

2+ 120576120590 (24)




119887

21198901198941205901198791 minus 119896

2

112058741198601198781198842

2119890minus11989412059001198791 minus 120572120587

4(1 +

1198962

11198602

119878

21198962

119891

) 11989412059621198841

minus 211989412059621198841015840

1= 0

minus 21198962

112058741198601198781198841119884211989011989412059001198791 minus 16120572120587

411989412059611198842minus 211989412059611198841015840

2= 0

(25)



1198861015840

1=

119887

2120596minus1

2sin 1205791+

1

4120596minus1

21198862

21198962

11205874119860119878sin (2120579

2minus 1205791)

minus1

211988611205721205874(1 +

1198962

11198602

119878

21198962

119891

)

1198861015840

2= minus

1

2120596minus1

1119886111988621198962

11205874119860119878sin (2120579

2minus 1205791) minus 8120572120587

41198862

11988611205791015840

1=

119887

2120596minus1

2cos 1205791+ 1198861120590

minus1

4120596minus1

21198862

21198962

11205874119860119878cos (2120579

2minus 1205791)

11988621205791015840

2= minus

1

2120596minus1

1119886111988621198962

11205874119860119878cos (2120579

2minus 1205791) +

1

21198862(120590 + 120590

0)

(26)

where 1205791

= 1205901198791minus 1205741and 120579

2= 05120590119879

1+ 05120590

01198791minus 1205742 The





1198861= 1198861119897

=119887

1205962radic12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 41205902

1198862= 1198862119897

= 0

(27)



1198861= 1198861119899

=

119896119891

1205871198961

radic3(120590 + 120590

0)2+ 768120572

21205878

1 minus 12058721198962

119891

1198862= 1198862119899

=1

1205872radic


11198962

1198911198872minus 61205878Γ2

2)12

1198601198781198961198911198963

1

(28)

where

Γ1= 41198962

119891120590 (120590 + 120590

0) minus 16120587

81205722(21198962

119891+ 1198602

1198781198962

1)

Γ2= 120572 [(66120590 + 2120590

0) 1198962

119891+ 1198602

1198781198962

1(120590 + 120590

0)]

(29)


1198841(1198791) =

1

2[1199091(1198791) minus 1198941199101(1198791)] 119890119894V11198791

1198842(1198791) =

1

2[1199092(1198791) minus 1198941199102(1198791)] 119890119894V21198791

(30)

where V1

= 120590 and V2



1199091015840

1=

1198962

11205874119860119878

21205962

11990921199102minus

1

21205721205874(1 +

1198962

11198602

119878

21198962

119891

)1199091minus V11199101

1199101015840

1=

119887

21205962

minus1198962

11205874119860119878

41205962

(1199092

2minus 1199102

2) minus

1

21205721205874(1 +

1198962

11198602

119878

21198962

119891

)1199101

+ V11199091

1199091015840

2=

1198962

11205874119860119878

21205961

(11990921199101minus 11990911199102) minus 8120572120587

41199092minus V21199102

1199101015840

2= V21199092minus

1198962

11205874119860119878

21205961

(11990911199092+ 11991011199102) minus 8120572120587

41199102

(31)




119895(119909) (119895 = 1 2 3 4)


1199021+ 1205721205874

1199021+

1

21198962

112058741198602

1198781199021

= 2 cos (120596119905) int

1

0

119891 (119909) sin (120587119909) d119909 minus1

21198962

112058741198601198781199022

1

minus1

41198962

11205874(119860119878+ 1199021)

times [1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022+ 91199023

1199023+ 16119902

41199024) ]

1199022+ 16120572120587

41199022+ 12120587

41198962

1198911199022

= 2 cos (120596119905) int

1

0

119891 (119909) sin (2120587119909) d119909

minus 1198962

112058741199022[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199023+ 81120572120587

41199023+ 72120587

41198962

1198911199023

= 2 cos (120596119905) int

1

0

119891 (119909) sin (3120587119909) d119909

minus9

41198962

112058741199023[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199024+ 256120572120587

41199024+ 240120587

41198962

1198911199024

= 2 cos (120596119905) int

1

0

119891 (119909) sin (4120587119909) d119909

minus 41198962

112058741199024[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

(32)



120596119886= 120587radic2 (1 minus 119896

2

1198911205872) 120596

119887= radic12120587

2119896119891

120596119888= radic72120587

2119896119891 120596

119889= radic240120587

2119896119891

(33)




1205961= radic12120587

2119896119891 120596

2= 120587radic2 (1 minus 119896

2

1198911205872)

1205963= radic72120587

2119896119891 120596

4= radic240120587

2119896119891

(34)



1199023+ 81120572120587

41199023+ 72120587

41198962

1198911199023

= minus9

41198962

112058741199023[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199024+ 256120572120587

41199024+ 240120587

41198962

1198911199024

= minus41198962

112058741199024[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

(35)


4 In


4will not be



4 Case Studies



119871 = 1m 119860 = 314159 eminus4m2

119864 = 74Gpa 119868 = 785398 eminus9m4(36)


=


119875 =119864119868

1198962

1198911198712

= 14340403KN (37)


119896119891

= 006366 1198961= 1273239 119860

119904= 004899

120572 = 000005 119887 = 0006

1205961= 217656 120596

2= 435312

(38)





1is obtained from





When 119875 = 14340403KN 1205761205900




minus2 0 2

120576120590

120576a1

0000

0005

0010

Stability boundary


boundary

(a)

minus2 0 2

120576120590

120576a2

0000

0005

0010


Stability boundary

Stability boundary

(b)


1mode (b) 119886

2mode

0000

0005

0010

0015

minus2 0 2

b = 0004

b = 0006

b = 0008

120576120590

120576a2

(a)

0000

0005

0010

0015

minus2 0 2

120572 = 00001

120572 = 000005

120572 = 000001

120576120590

120576a2

(b)



0= 05 and

1205761205900






2is taken as the


0000

0005

0010

0015

0020

minus2 minus1 0 1 2

P = 11638770KN

P = 18125417KN

P = 14340403KN

120576120590

120576a2



0000

0002

0004

0006

0008

Stability boundary

minus1 0 1 2minus2

a1l

a2l

120576120590

120576an

(a)

0000

0005

0010

0015

Stability boundary

minus2 0 2

a1n

a2n120576an

120576120590

(b)


0000

0005

0010

0015

minus1 0 1 2minus2

b = 0004

b = 0006

b = 0008

a1n

a2n

120576an

120576120590

(a)

0000 0001 0002

00000

00002

00004

00006

00008

00010

a1l

a2l

a2n

a1n

120576an

1205762b

(b)












2mode








00000

00025

00050

00075

00100

00125

a2n

a1n

minus1 0 1 2minus2

120576120590

120576an

120572 = 000001

120572 = 000005

120572 = 00001

120572 = 000001

120572 = 000005

120572 = 00001

(a)

0002

0004

0006

0008

0010

0012

P = 14340403KN

P = 14340403KN

P = 18125417KN

P = 18125417KN

P = 11638770KN

P = 11638770KN

a2n

a1n

0 2minus2

120576120590

120576an

0000

(b)


0000

0005

0010

0015

0 2minus2

120576120590

120576an

a1

a2

(a)

0000

0005

0010120576an

a1a1

a2

a2

minus1 0 1 2minus2

120576120590

(b)










990 995 1000minus0006

minus0004

minus0002

0000

0002

0004

0006

t


minus00006990 995 1000

minus00004

minus00002

00000

00002

00004

00006

t

(a)

990 995 1000

t

minus0006

minus0004

minus0002

0000

0002

0004

0006

(b)





3

and 1199024






0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

6 Conclusions









minus0010 minus0005 0000 0005

minus001

000

001

002

003Ve

loci

ty

Displacement

(a)

Velo

city

minus0010 minus0005 0000 0005 0010

minus001

000

001

002

003

Displacement

(b)

minus001 000 001

minus002

000

002

004

Displacement

Velo

city

(c)

minus0010 minus0005 0000 0005

0000

0005

0010

0015

0020

Displacement

Velo

city

(d)

minus0010 minus0005 0000 0005

000

001

002

003

Displacement

Velo

city

(e)


000

002

004

Displacement

Velo

city

(f)







Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1= 120576119905The


1199021(1198790 1198791) = 120576119902

11(1198790 1198791) + 120576211990212

(1198790 1198791) + sdot sdot sdot

1199022(1198790 1198791) = 120576119902

21(1198790 1198791) + 120576211990222

(1198790 1198791) + sdot sdot sdot

(12)





0

1199021111987901198790

+1

21198962

112058741198602

11987811990211

= 0

1199022111987901198790

+ 1212058741198962

11989111990221

= 0

(13)

Order 1205761

1199021211987901198790

+1

21198962

112058741198602

11987811990212

= 2 cos (120596119905) int

1

0

119891 (119909) sin (120587119909) d119909

minus 1198962

11205874119860119878(3

41199022

11+ 1199022

21)

minus 1205721205874(1 +

1198962

11198602

119878

21198962

119891

)119902111198790

minus 21199021111987901198791

1199022211987901198790

+ 1212058741198962

11989111990222

= 2 cos (120596119905) int

1

0

119891 (119909) sin (2120587119909) d119909

minus 21198962

112058741198601198781199021111990221

minus 161205721205874119902211198790

minus 21199022111987901198791

(14)


1205961= min (120596

119886 120596119887) 120596

2= max (120596

119886 120596119887) (15)

where 120596119886

= 11989611205872119860119878radic2 and 120596

119887= 2radic3120587

2119896119891 Figure 2 shows





form

11990211

(1198790 1198791) = 1198841(1198791) 1198901198941205961198861198790 + cc

11990221

(1198790 1198791) = 1198842(1198791) 1198901198941205961198871198790 + cc

(16)

10

5

0000 008 016 024 032

kf

1205962

1205962

120596n

1205961

1205961


119891


1199021211987901198790

+ 1205962

11988611990212

= 1198901198941205961198790 int

1

0

119891 (119909) sin (120587119909) d119909 minus 1198962

112058741198601198781198842

211989021198941205961198871198790

minus 1205721205874(1 +

1198962

11198602

119878

21198962

119891

) 11989412059611988611988411198901198941205961198861198790

minus 21198941205961198861198841015840

11198901198941205961198861198790 + cc + NST

1199022211987901198790

+ 1205962

11988711990222

= 1198901198941205961198790 int

1

0

119891 (119909) sin (2120587119909) d119909

minus 21198962

1120587411986011987811988411198842119890119894(120596119886minus120596119887)1198790

minus 16120572120587411989412059611988711988421198901198941205961198871198790

minus 21198941205961198871198841015840

21198901198941205961198871198790 + cc + NST

(17)




1= 120596119887 1205962

= 120596119886and 119896

119891= 006366




to 1205961and 120596


1205962= 21205961+ 1205761205900 120596 = 120596

1+ 120576120590 (18)




equation lead to

minus1198962

112058741198601198781198842

2119890minus11989412059001198791 minus 120572120587

4(1 +

1198962

11198602

119878

21198962

119891

) 11989412059621198841minus 211989412059621198841015840

1= 0

119887

21198901198941205901198791 minus 2119896

2

112058741198601198781198841119884211989011989412059001198791 minus 16120572120587

411989412059611198842minus 211989412059611198841015840

2= 0

(19)


1198841(1198791) =

1

21198861(1198791) 1198901198941205741(1198791)

1198842(1198791) =

1

21198862(1198791) 1198901198941205742(1198791)

(20)

where 119886119899and 120574



1198861015840

1=

1

4120596minus1

21198862

21198962

11205874119860119878sin (2120579

2minus 1205791)

minus1

211988611205721205874(1 +

1198962

11198602

119878

21198962

119891

)

1198861015840

2=

119887

2120596minus1

1sin 1205792minus

1

2120596minus1

1119886111988621198962

11205874119860119878sin (2120579

2minus 1205791)

minus 812057212058741198862

11988611205791015840

1= 1198861(2120590 minus 120590

0) minus

1

4120596minus1

21198862

21198962

11205874119860119878cos (2120579

2minus 1205791)

11988621205791015840

2=

119887

2120596minus1

1cos 1205792minus

1

2120596minus1

1119886111988621198962

11205874119860119878cos (2120579

2minus 1205791)

+ 1198862120590

(21)

where 1205791= 2120590119879





obtained

1198872

41205962

1

=

12059621205721198962

11205878(1 + 119896

2

11198602

11987821198962

119891) 1198863

2

21205961[12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 4(2120590 minus 1205900)2]

+ 812057212058741198862

2

+

12059621198962

11205874(2120590 minus 120590

0) 1198863

2

1205961[12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 4(2120590 minus 1205900)2]

minus 1205901198862

2

(22)


be obtained

1198861=

119896112058721198862

2

radic212057221205878(1 + 1198962

11198602

11987821198962

119891)

2

+ 8(2120590 minus 1205900)2

(23)



1205962= 21205961+ 1205761205900 120596 = 120596

2+ 120576120590 (24)




119887

21198901198941205901198791 minus 119896

2

112058741198601198781198842

2119890minus11989412059001198791 minus 120572120587

4(1 +

1198962

11198602

119878

21198962

119891

) 11989412059621198841

minus 211989412059621198841015840

1= 0

minus 21198962

112058741198601198781198841119884211989011989412059001198791 minus 16120572120587

411989412059611198842minus 211989412059611198841015840

2= 0

(25)



1198861015840

1=

119887

2120596minus1

2sin 1205791+

1

4120596minus1

21198862

21198962

11205874119860119878sin (2120579

2minus 1205791)

minus1

211988611205721205874(1 +

1198962

11198602

119878

21198962

119891

)

1198861015840

2= minus

1

2120596minus1

1119886111988621198962

11205874119860119878sin (2120579

2minus 1205791) minus 8120572120587

41198862

11988611205791015840

1=

119887

2120596minus1

2cos 1205791+ 1198861120590

minus1

4120596minus1

21198862

21198962

11205874119860119878cos (2120579

2minus 1205791)

11988621205791015840

2= minus

1

2120596minus1

1119886111988621198962

11205874119860119878cos (2120579

2minus 1205791) +

1

21198862(120590 + 120590

0)

(26)

where 1205791

= 1205901198791minus 1205741and 120579

2= 05120590119879

1+ 05120590

01198791minus 1205742 The





1198861= 1198861119897

=119887

1205962radic12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 41205902

1198862= 1198862119897

= 0

(27)



1198861= 1198861119899

=

119896119891

1205871198961

radic3(120590 + 120590

0)2+ 768120572

21205878

1 minus 12058721198962

119891

1198862= 1198862119899

=1

1205872radic


11198962

1198911198872minus 61205878Γ2

2)12

1198601198781198961198911198963

1

(28)

where

Γ1= 41198962

119891120590 (120590 + 120590

0) minus 16120587

81205722(21198962

119891+ 1198602

1198781198962

1)

Γ2= 120572 [(66120590 + 2120590

0) 1198962

119891+ 1198602

1198781198962

1(120590 + 120590

0)]

(29)


1198841(1198791) =

1

2[1199091(1198791) minus 1198941199101(1198791)] 119890119894V11198791

1198842(1198791) =

1

2[1199092(1198791) minus 1198941199102(1198791)] 119890119894V21198791

(30)

where V1

= 120590 and V2



1199091015840

1=

1198962

11205874119860119878

21205962

11990921199102minus

1

21205721205874(1 +

1198962

11198602

119878

21198962

119891

)1199091minus V11199101

1199101015840

1=

119887

21205962

minus1198962

11205874119860119878

41205962

(1199092

2minus 1199102

2) minus

1

21205721205874(1 +

1198962

11198602

119878

21198962

119891

)1199101

+ V11199091

1199091015840

2=

1198962

11205874119860119878

21205961

(11990921199101minus 11990911199102) minus 8120572120587

41199092minus V21199102

1199101015840

2= V21199092minus

1198962

11205874119860119878

21205961

(11990911199092+ 11991011199102) minus 8120572120587

41199102

(31)




119895(119909) (119895 = 1 2 3 4)


1199021+ 1205721205874

1199021+

1

21198962

112058741198602

1198781199021

= 2 cos (120596119905) int

1

0

119891 (119909) sin (120587119909) d119909 minus1

21198962

112058741198601198781199022

1

minus1

41198962

11205874(119860119878+ 1199021)

times [1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022+ 91199023

1199023+ 16119902

41199024) ]

1199022+ 16120572120587

41199022+ 12120587

41198962

1198911199022

= 2 cos (120596119905) int

1

0

119891 (119909) sin (2120587119909) d119909

minus 1198962

112058741199022[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199023+ 81120572120587

41199023+ 72120587

41198962

1198911199023

= 2 cos (120596119905) int

1

0

119891 (119909) sin (3120587119909) d119909

minus9

41198962

112058741199023[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199024+ 256120572120587

41199024+ 240120587

41198962

1198911199024

= 2 cos (120596119905) int

1

0

119891 (119909) sin (4120587119909) d119909

minus 41198962

112058741199024[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

(32)



120596119886= 120587radic2 (1 minus 119896

2

1198911205872) 120596

119887= radic12120587

2119896119891

120596119888= radic72120587

2119896119891 120596

119889= radic240120587

2119896119891

(33)




1205961= radic12120587

2119896119891 120596

2= 120587radic2 (1 minus 119896

2

1198911205872)

1205963= radic72120587

2119896119891 120596

4= radic240120587

2119896119891

(34)



1199023+ 81120572120587

41199023+ 72120587

41198962

1198911199023

= minus9

41198962

112058741199023[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199024+ 256120572120587

41199024+ 240120587

41198962

1198911199024

= minus41198962

112058741199024[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

(35)


4 In


4will not be



4 Case Studies



119871 = 1m 119860 = 314159 eminus4m2

119864 = 74Gpa 119868 = 785398 eminus9m4(36)


=


119875 =119864119868

1198962

1198911198712

= 14340403KN (37)


119896119891

= 006366 1198961= 1273239 119860

119904= 004899

120572 = 000005 119887 = 0006

1205961= 217656 120596

2= 435312

(38)





1is obtained from





When 119875 = 14340403KN 1205761205900




minus2 0 2

120576120590

120576a1

0000

0005

0010

Stability boundary


boundary

(a)

minus2 0 2

120576120590

120576a2

0000

0005

0010


Stability boundary

Stability boundary

(b)


1mode (b) 119886

2mode

0000

0005

0010

0015

minus2 0 2

b = 0004

b = 0006

b = 0008

120576120590

120576a2

(a)

0000

0005

0010

0015

minus2 0 2

120572 = 00001

120572 = 000005

120572 = 000001

120576120590

120576a2

(b)



0= 05 and

1205761205900






2is taken as the


0000

0005

0010

0015

0020

minus2 minus1 0 1 2

P = 11638770KN

P = 18125417KN

P = 14340403KN

120576120590

120576a2



0000

0002

0004

0006

0008

Stability boundary

minus1 0 1 2minus2

a1l

a2l

120576120590

120576an

(a)

0000

0005

0010

0015

Stability boundary

minus2 0 2

a1n

a2n120576an

120576120590

(b)


0000

0005

0010

0015

minus1 0 1 2minus2

b = 0004

b = 0006

b = 0008

a1n

a2n

120576an

120576120590

(a)

0000 0001 0002

00000

00002

00004

00006

00008

00010

a1l

a2l

a2n

a1n

120576an

1205762b

(b)












2mode








00000

00025

00050

00075

00100

00125

a2n

a1n

minus1 0 1 2minus2

120576120590

120576an

120572 = 000001

120572 = 000005

120572 = 00001

120572 = 000001

120572 = 000005

120572 = 00001

(a)

0002

0004

0006

0008

0010

0012

P = 14340403KN

P = 14340403KN

P = 18125417KN

P = 18125417KN

P = 11638770KN

P = 11638770KN

a2n

a1n

0 2minus2

120576120590

120576an

0000

(b)


0000

0005

0010

0015

0 2minus2

120576120590

120576an

a1

a2

(a)

0000

0005

0010120576an

a1a1

a2

a2

minus1 0 1 2minus2

120576120590

(b)










990 995 1000minus0006

minus0004

minus0002

0000

0002

0004

0006

t


minus00006990 995 1000

minus00004

minus00002

00000

00002

00004

00006

t

(a)

990 995 1000

t

minus0006

minus0004

minus0002

0000

0002

0004

0006

(b)





3

and 1199024






0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

6 Conclusions









minus0010 minus0005 0000 0005

minus001

000

001

002

003Ve

loci

ty

Displacement

(a)

Velo

city

minus0010 minus0005 0000 0005 0010

minus001

000

001

002

003

Displacement

(b)

minus001 000 001

minus002

000

002

004

Displacement

Velo

city

(c)

minus0010 minus0005 0000 0005

0000

0005

0010

0015

0020

Displacement

Velo

city

(d)

minus0010 minus0005 0000 0005

000

001

002

003

Displacement

Velo

city

(e)


000

002

004

Displacement

Velo

city

(f)







Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












equation lead to

minus1198962

112058741198601198781198842

2119890minus11989412059001198791 minus 120572120587

4(1 +

1198962

11198602

119878

21198962

119891

) 11989412059621198841minus 211989412059621198841015840

1= 0

119887

21198901198941205901198791 minus 2119896

2

112058741198601198781198841119884211989011989412059001198791 minus 16120572120587

411989412059611198842minus 211989412059611198841015840

2= 0

(19)


1198841(1198791) =

1

21198861(1198791) 1198901198941205741(1198791)

1198842(1198791) =

1

21198862(1198791) 1198901198941205742(1198791)

(20)

where 119886119899and 120574



1198861015840

1=

1

4120596minus1

21198862

21198962

11205874119860119878sin (2120579

2minus 1205791)

minus1

211988611205721205874(1 +

1198962

11198602

119878

21198962

119891

)

1198861015840

2=

119887

2120596minus1

1sin 1205792minus

1

2120596minus1

1119886111988621198962

11205874119860119878sin (2120579

2minus 1205791)

minus 812057212058741198862

11988611205791015840

1= 1198861(2120590 minus 120590

0) minus

1

4120596minus1

21198862

21198962

11205874119860119878cos (2120579

2minus 1205791)

11988621205791015840

2=

119887

2120596minus1

1cos 1205792minus

1

2120596minus1

1119886111988621198962

11205874119860119878cos (2120579

2minus 1205791)

+ 1198862120590

(21)

where 1205791= 2120590119879





obtained

1198872

41205962

1

=

12059621205721198962

11205878(1 + 119896

2

11198602

11987821198962

119891) 1198863

2

21205961[12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 4(2120590 minus 1205900)2]

+ 812057212058741198862

2

+

12059621198962

11205874(2120590 minus 120590

0) 1198863

2

1205961[12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 4(2120590 minus 1205900)2]

minus 1205901198862

2

(22)


be obtained

1198861=

119896112058721198862

2

radic212057221205878(1 + 1198962

11198602

11987821198962

119891)

2

+ 8(2120590 minus 1205900)2

(23)



1205962= 21205961+ 1205761205900 120596 = 120596

2+ 120576120590 (24)




119887

21198901198941205901198791 minus 119896

2

112058741198601198781198842

2119890minus11989412059001198791 minus 120572120587

4(1 +

1198962

11198602

119878

21198962

119891

) 11989412059621198841

minus 211989412059621198841015840

1= 0

minus 21198962

112058741198601198781198841119884211989011989412059001198791 minus 16120572120587

411989412059611198842minus 211989412059611198841015840

2= 0

(25)



1198861015840

1=

119887

2120596minus1

2sin 1205791+

1

4120596minus1

21198862

21198962

11205874119860119878sin (2120579

2minus 1205791)

minus1

211988611205721205874(1 +

1198962

11198602

119878

21198962

119891

)

1198861015840

2= minus

1

2120596minus1

1119886111988621198962

11205874119860119878sin (2120579

2minus 1205791) minus 8120572120587

41198862

11988611205791015840

1=

119887

2120596minus1

2cos 1205791+ 1198861120590

minus1

4120596minus1

21198862

21198962

11205874119860119878cos (2120579

2minus 1205791)

11988621205791015840

2= minus

1

2120596minus1

1119886111988621198962

11205874119860119878cos (2120579

2minus 1205791) +

1

21198862(120590 + 120590

0)

(26)

where 1205791

= 1205901198791minus 1205741and 120579

2= 05120590119879

1+ 05120590

01198791minus 1205742 The





1198861= 1198861119897

=119887

1205962radic12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 41205902

1198862= 1198862119897

= 0

(27)



1198861= 1198861119899

=

119896119891

1205871198961

radic3(120590 + 120590

0)2+ 768120572

21205878

1 minus 12058721198962

119891

1198862= 1198862119899

=1

1205872radic


11198962

1198911198872minus 61205878Γ2

2)12

1198601198781198961198911198963

1

(28)

where

Γ1= 41198962

119891120590 (120590 + 120590

0) minus 16120587

81205722(21198962

119891+ 1198602

1198781198962

1)

Γ2= 120572 [(66120590 + 2120590

0) 1198962

119891+ 1198602

1198781198962

1(120590 + 120590

0)]

(29)


1198841(1198791) =

1

2[1199091(1198791) minus 1198941199101(1198791)] 119890119894V11198791

1198842(1198791) =

1

2[1199092(1198791) minus 1198941199102(1198791)] 119890119894V21198791

(30)

where V1

= 120590 and V2



1199091015840

1=

1198962

11205874119860119878

21205962

11990921199102minus

1

21205721205874(1 +

1198962

11198602

119878

21198962

119891

)1199091minus V11199101

1199101015840

1=

119887

21205962

minus1198962

11205874119860119878

41205962

(1199092

2minus 1199102

2) minus

1

21205721205874(1 +

1198962

11198602

119878

21198962

119891

)1199101

+ V11199091

1199091015840

2=

1198962

11205874119860119878

21205961

(11990921199101minus 11990911199102) minus 8120572120587

41199092minus V21199102

1199101015840

2= V21199092minus

1198962

11205874119860119878

21205961

(11990911199092+ 11991011199102) minus 8120572120587

41199102

(31)




119895(119909) (119895 = 1 2 3 4)


1199021+ 1205721205874

1199021+

1

21198962

112058741198602

1198781199021

= 2 cos (120596119905) int

1

0

119891 (119909) sin (120587119909) d119909 minus1

21198962

112058741198601198781199022

1

minus1

41198962

11205874(119860119878+ 1199021)

times [1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022+ 91199023

1199023+ 16119902

41199024) ]

1199022+ 16120572120587

41199022+ 12120587

41198962

1198911199022

= 2 cos (120596119905) int

1

0

119891 (119909) sin (2120587119909) d119909

minus 1198962

112058741199022[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199023+ 81120572120587

41199023+ 72120587

41198962

1198911199023

= 2 cos (120596119905) int

1

0

119891 (119909) sin (3120587119909) d119909

minus9

41198962

112058741199023[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199024+ 256120572120587

41199024+ 240120587

41198962

1198911199024

= 2 cos (120596119905) int

1

0

119891 (119909) sin (4120587119909) d119909

minus 41198962

112058741199024[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

(32)



120596119886= 120587radic2 (1 minus 119896

2

1198911205872) 120596

119887= radic12120587

2119896119891

120596119888= radic72120587

2119896119891 120596

119889= radic240120587

2119896119891

(33)




1205961= radic12120587

2119896119891 120596

2= 120587radic2 (1 minus 119896

2

1198911205872)

1205963= radic72120587

2119896119891 120596

4= radic240120587

2119896119891

(34)



1199023+ 81120572120587

41199023+ 72120587

41198962

1198911199023

= minus9

41198962

112058741199023[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199024+ 256120572120587

41199024+ 240120587

41198962

1198911199024

= minus41198962

112058741199024[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

(35)


4 In


4will not be



4 Case Studies



119871 = 1m 119860 = 314159 eminus4m2

119864 = 74Gpa 119868 = 785398 eminus9m4(36)


=


119875 =119864119868

1198962

1198911198712

= 14340403KN (37)


119896119891

= 006366 1198961= 1273239 119860

119904= 004899

120572 = 000005 119887 = 0006

1205961= 217656 120596

2= 435312

(38)





1is obtained from





When 119875 = 14340403KN 1205761205900




minus2 0 2

120576120590

120576a1

0000

0005

0010

Stability boundary


boundary

(a)

minus2 0 2

120576120590

120576a2

0000

0005

0010


Stability boundary

Stability boundary

(b)


1mode (b) 119886

2mode

0000

0005

0010

0015

minus2 0 2

b = 0004

b = 0006

b = 0008

120576120590

120576a2

(a)

0000

0005

0010

0015

minus2 0 2

120572 = 00001

120572 = 000005

120572 = 000001

120576120590

120576a2

(b)



0= 05 and

1205761205900






2is taken as the


0000

0005

0010

0015

0020

minus2 minus1 0 1 2

P = 11638770KN

P = 18125417KN

P = 14340403KN

120576120590

120576a2



0000

0002

0004

0006

0008

Stability boundary

minus1 0 1 2minus2

a1l

a2l

120576120590

120576an

(a)

0000

0005

0010

0015

Stability boundary

minus2 0 2

a1n

a2n120576an

120576120590

(b)


0000

0005

0010

0015

minus1 0 1 2minus2

b = 0004

b = 0006

b = 0008

a1n

a2n

120576an

120576120590

(a)

0000 0001 0002

00000

00002

00004

00006

00008

00010

a1l

a2l

a2n

a1n

120576an

1205762b

(b)












2mode








00000

00025

00050

00075

00100

00125

a2n

a1n

minus1 0 1 2minus2

120576120590

120576an

120572 = 000001

120572 = 000005

120572 = 00001

120572 = 000001

120572 = 000005

120572 = 00001

(a)

0002

0004

0006

0008

0010

0012

P = 14340403KN

P = 14340403KN

P = 18125417KN

P = 18125417KN

P = 11638770KN

P = 11638770KN

a2n

a1n

0 2minus2

120576120590

120576an

0000

(b)


0000

0005

0010

0015

0 2minus2

120576120590

120576an

a1

a2

(a)

0000

0005

0010120576an

a1a1

a2

a2

minus1 0 1 2minus2

120576120590

(b)










990 995 1000minus0006

minus0004

minus0002

0000

0002

0004

0006

t


minus00006990 995 1000

minus00004

minus00002

00000

00002

00004

00006

t

(a)

990 995 1000

t

minus0006

minus0004

minus0002

0000

0002

0004

0006

(b)





3

and 1199024






0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

6 Conclusions









minus0010 minus0005 0000 0005

minus001

000

001

002

003Ve

loci

ty

Displacement

(a)

Velo

city

minus0010 minus0005 0000 0005 0010

minus001

000

001

002

003

Displacement

(b)

minus001 000 001

minus002

000

002

004

Displacement

Velo

city

(c)

minus0010 minus0005 0000 0005

0000

0005

0010

0015

0020

Displacement

Velo

city

(d)

minus0010 minus0005 0000 0005

000

001

002

003

Displacement

Velo

city

(e)


000

002

004

Displacement

Velo

city

(f)







Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198861= 1198861119897

=119887

1205962radic12057221205878(1 + 119896

2

11198602

11987821198962

119891)

2

+ 41205902

1198862= 1198862119897

= 0

(27)



1198861= 1198861119899

=

119896119891

1205871198961

radic3(120590 + 120590

0)2+ 768120572

21205878

1 minus 12058721198962

119891

1198862= 1198862119899

=1

1205872radic


11198962

1198911198872minus 61205878Γ2

2)12

1198601198781198961198911198963

1

(28)

where

Γ1= 41198962

119891120590 (120590 + 120590

0) minus 16120587

81205722(21198962

119891+ 1198602

1198781198962

1)

Γ2= 120572 [(66120590 + 2120590

0) 1198962

119891+ 1198602

1198781198962

1(120590 + 120590

0)]

(29)


1198841(1198791) =

1

2[1199091(1198791) minus 1198941199101(1198791)] 119890119894V11198791

1198842(1198791) =

1

2[1199092(1198791) minus 1198941199102(1198791)] 119890119894V21198791

(30)

where V1

= 120590 and V2



1199091015840

1=

1198962

11205874119860119878

21205962

11990921199102minus

1

21205721205874(1 +

1198962

11198602

119878

21198962

119891

)1199091minus V11199101

1199101015840

1=

119887

21205962

minus1198962

11205874119860119878

41205962

(1199092

2minus 1199102

2) minus

1

21205721205874(1 +

1198962

11198602

119878

21198962

119891

)1199101

+ V11199091

1199091015840

2=

1198962

11205874119860119878

21205961

(11990921199101minus 11990911199102) minus 8120572120587

41199092minus V21199102

1199101015840

2= V21199092minus

1198962

11205874119860119878

21205961

(11990911199092+ 11991011199102) minus 8120572120587

41199102

(31)




119895(119909) (119895 = 1 2 3 4)


1199021+ 1205721205874

1199021+

1

21198962

112058741198602

1198781199021

= 2 cos (120596119905) int

1

0

119891 (119909) sin (120587119909) d119909 minus1

21198962

112058741198601198781199022

1

minus1

41198962

11205874(119860119878+ 1199021)

times [1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022+ 91199023

1199023+ 16119902

41199024) ]

1199022+ 16120572120587

41199022+ 12120587

41198962

1198911199022

= 2 cos (120596119905) int

1

0

119891 (119909) sin (2120587119909) d119909

minus 1198962

112058741199022[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199023+ 81120572120587

41199023+ 72120587

41198962

1198911199023

= 2 cos (120596119905) int

1

0

119891 (119909) sin (3120587119909) d119909

minus9

41198962

112058741199023[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199024+ 256120572120587

41199024+ 240120587

41198962

1198911199024

= 2 cos (120596119905) int

1

0

119891 (119909) sin (4120587119909) d119909

minus 41198962

112058741199024[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

(32)



120596119886= 120587radic2 (1 minus 119896

2

1198911205872) 120596

119887= radic12120587

2119896119891

120596119888= radic72120587

2119896119891 120596

119889= radic240120587

2119896119891

(33)




1205961= radic12120587

2119896119891 120596

2= 120587radic2 (1 minus 119896

2

1198911205872)

1205963= radic72120587

2119896119891 120596

4= radic240120587

2119896119891

(34)



1199023+ 81120572120587

41199023+ 72120587

41198962

1198911199023

= minus9

41198962

112058741199023[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199024+ 256120572120587

41199024+ 240120587

41198962

1198911199024

= minus41198962

112058741199024[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

(35)


4 In


4will not be



4 Case Studies



119871 = 1m 119860 = 314159 eminus4m2

119864 = 74Gpa 119868 = 785398 eminus9m4(36)


=


119875 =119864119868

1198962

1198911198712

= 14340403KN (37)


119896119891

= 006366 1198961= 1273239 119860

119904= 004899

120572 = 000005 119887 = 0006

1205961= 217656 120596

2= 435312

(38)





1is obtained from





When 119875 = 14340403KN 1205761205900




minus2 0 2

120576120590

120576a1

0000

0005

0010

Stability boundary


boundary

(a)

minus2 0 2

120576120590

120576a2

0000

0005

0010


Stability boundary

Stability boundary

(b)


1mode (b) 119886

2mode

0000

0005

0010

0015

minus2 0 2

b = 0004

b = 0006

b = 0008

120576120590

120576a2

(a)

0000

0005

0010

0015

minus2 0 2

120572 = 00001

120572 = 000005

120572 = 000001

120576120590

120576a2

(b)



0= 05 and

1205761205900






2is taken as the


0000

0005

0010

0015

0020

minus2 minus1 0 1 2

P = 11638770KN

P = 18125417KN

P = 14340403KN

120576120590

120576a2



0000

0002

0004

0006

0008

Stability boundary

minus1 0 1 2minus2

a1l

a2l

120576120590

120576an

(a)

0000

0005

0010

0015

Stability boundary

minus2 0 2

a1n

a2n120576an

120576120590

(b)


0000

0005

0010

0015

minus1 0 1 2minus2

b = 0004

b = 0006

b = 0008

a1n

a2n

120576an

120576120590

(a)

0000 0001 0002

00000

00002

00004

00006

00008

00010

a1l

a2l

a2n

a1n

120576an

1205762b

(b)












2mode








00000

00025

00050

00075

00100

00125

a2n

a1n

minus1 0 1 2minus2

120576120590

120576an

120572 = 000001

120572 = 000005

120572 = 00001

120572 = 000001

120572 = 000005

120572 = 00001

(a)

0002

0004

0006

0008

0010

0012

P = 14340403KN

P = 14340403KN

P = 18125417KN

P = 18125417KN

P = 11638770KN

P = 11638770KN

a2n

a1n

0 2minus2

120576120590

120576an

0000

(b)


0000

0005

0010

0015

0 2minus2

120576120590

120576an

a1

a2

(a)

0000

0005

0010120576an

a1a1

a2

a2

minus1 0 1 2minus2

120576120590

(b)










990 995 1000minus0006

minus0004

minus0002

0000

0002

0004

0006

t


minus00006990 995 1000

minus00004

minus00002

00000

00002

00004

00006

t

(a)

990 995 1000

t

minus0006

minus0004

minus0002

0000

0002

0004

0006

(b)





3

and 1199024






0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

6 Conclusions









minus0010 minus0005 0000 0005

minus001

000

001

002

003Ve

loci

ty

Displacement

(a)

Velo

city

minus0010 minus0005 0000 0005 0010

minus001

000

001

002

003

Displacement

(b)

minus001 000 001

minus002

000

002

004

Displacement

Velo

city

(c)

minus0010 minus0005 0000 0005

0000

0005

0010

0015

0020

Displacement

Velo

city

(d)

minus0010 minus0005 0000 0005

000

001

002

003

Displacement

Velo

city

(e)


000

002

004

Displacement

Velo

city

(f)







Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120596119886= 120587radic2 (1 minus 119896

2

1198911205872) 120596

119887= radic12120587

2119896119891

120596119888= radic72120587

2119896119891 120596

119889= radic240120587

2119896119891

(33)




1205961= radic12120587

2119896119891 120596

2= 120587radic2 (1 minus 119896

2

1198911205872)

1205963= radic72120587

2119896119891 120596

4= radic240120587

2119896119891

(34)



1199023+ 81120572120587

41199023+ 72120587

41198962

1198911199023

= minus9

41198962

112058741199023[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

1199024+ 256120572120587

41199024+ 240120587

41198962

1198911199024

= minus41198962

112058741199024[21198601198781199021+ 1199022

1+ 41199022

2+ 91199022

3+ 16119902

2

4+

2120572

1198962

119891

times (119860119878

1199021+ 1199021

1199021+ 41199022

1199022

+91199023

1199023+ 16119902

41199024) ]

(35)


4 In


4will not be



4 Case Studies



119871 = 1m 119860 = 314159 eminus4m2

119864 = 74Gpa 119868 = 785398 eminus9m4(36)


=


119875 =119864119868

1198962

1198911198712

= 14340403KN (37)


119896119891

= 006366 1198961= 1273239 119860

119904= 004899

120572 = 000005 119887 = 0006

1205961= 217656 120596

2= 435312

(38)





1is obtained from





When 119875 = 14340403KN 1205761205900




minus2 0 2

120576120590

120576a1

0000

0005

0010

Stability boundary


boundary

(a)

minus2 0 2

120576120590

120576a2

0000

0005

0010


Stability boundary

Stability boundary

(b)


1mode (b) 119886

2mode

0000

0005

0010

0015

minus2 0 2

b = 0004

b = 0006

b = 0008

120576120590

120576a2

(a)

0000

0005

0010

0015

minus2 0 2

120572 = 00001

120572 = 000005

120572 = 000001

120576120590

120576a2

(b)



0= 05 and

1205761205900






2is taken as the


0000

0005

0010

0015

0020

minus2 minus1 0 1 2

P = 11638770KN

P = 18125417KN

P = 14340403KN

120576120590

120576a2



0000

0002

0004

0006

0008

Stability boundary

minus1 0 1 2minus2

a1l

a2l

120576120590

120576an

(a)

0000

0005

0010

0015

Stability boundary

minus2 0 2

a1n

a2n120576an

120576120590

(b)


0000

0005

0010

0015

minus1 0 1 2minus2

b = 0004

b = 0006

b = 0008

a1n

a2n

120576an

120576120590

(a)

0000 0001 0002

00000

00002

00004

00006

00008

00010

a1l

a2l

a2n

a1n

120576an

1205762b

(b)












2mode








00000

00025

00050

00075

00100

00125

a2n

a1n

minus1 0 1 2minus2

120576120590

120576an

120572 = 000001

120572 = 000005

120572 = 00001

120572 = 000001

120572 = 000005

120572 = 00001

(a)

0002

0004

0006

0008

0010

0012

P = 14340403KN

P = 14340403KN

P = 18125417KN

P = 18125417KN

P = 11638770KN

P = 11638770KN

a2n

a1n

0 2minus2

120576120590

120576an

0000

(b)


0000

0005

0010

0015

0 2minus2

120576120590

120576an

a1

a2

(a)

0000

0005

0010120576an

a1a1

a2

a2

minus1 0 1 2minus2

120576120590

(b)










990 995 1000minus0006

minus0004

minus0002

0000

0002

0004

0006

t


minus00006990 995 1000

minus00004

minus00002

00000

00002

00004

00006

t

(a)

990 995 1000

t

minus0006

minus0004

minus0002

0000

0002

0004

0006

(b)





3

and 1199024






0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

6 Conclusions









minus0010 minus0005 0000 0005

minus001

000

001

002

003Ve

loci

ty

Displacement

(a)

Velo

city

minus0010 minus0005 0000 0005 0010

minus001

000

001

002

003

Displacement

(b)

minus001 000 001

minus002

000

002

004

Displacement

Velo

city

(c)

minus0010 minus0005 0000 0005

0000

0005

0010

0015

0020

Displacement

Velo

city

(d)

minus0010 minus0005 0000 0005

000

001

002

003

Displacement

Velo

city

(e)


000

002

004

Displacement

Velo

city

(f)







Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus2 0 2

120576120590

120576a1

0000

0005

0010

Stability boundary


boundary

(a)

minus2 0 2

120576120590

120576a2

0000

0005

0010


Stability boundary

Stability boundary

(b)


1mode (b) 119886

2mode

0000

0005

0010

0015

minus2 0 2

b = 0004

b = 0006

b = 0008

120576120590

120576a2

(a)

0000

0005

0010

0015

minus2 0 2

120572 = 00001

120572 = 000005

120572 = 000001

120576120590

120576a2

(b)



0= 05 and

1205761205900






2is taken as the


0000

0005

0010

0015

0020

minus2 minus1 0 1 2

P = 11638770KN

P = 18125417KN

P = 14340403KN

120576120590

120576a2



0000

0002

0004

0006

0008

Stability boundary

minus1 0 1 2minus2

a1l

a2l

120576120590

120576an

(a)

0000

0005

0010

0015

Stability boundary

minus2 0 2

a1n

a2n120576an

120576120590

(b)


0000

0005

0010

0015

minus1 0 1 2minus2

b = 0004

b = 0006

b = 0008

a1n

a2n

120576an

120576120590

(a)

0000 0001 0002

00000

00002

00004

00006

00008

00010

a1l

a2l

a2n

a1n

120576an

1205762b

(b)












2mode








00000

00025

00050

00075

00100

00125

a2n

a1n

minus1 0 1 2minus2

120576120590

120576an

120572 = 000001

120572 = 000005

120572 = 00001

120572 = 000001

120572 = 000005

120572 = 00001

(a)

0002

0004

0006

0008

0010

0012

P = 14340403KN

P = 14340403KN

P = 18125417KN

P = 18125417KN

P = 11638770KN

P = 11638770KN

a2n

a1n

0 2minus2

120576120590

120576an

0000

(b)


0000

0005

0010

0015

0 2minus2

120576120590

120576an

a1

a2

(a)

0000

0005

0010120576an

a1a1

a2

a2

minus1 0 1 2minus2

120576120590

(b)










990 995 1000minus0006

minus0004

minus0002

0000

0002

0004

0006

t


minus00006990 995 1000

minus00004

minus00002

00000

00002

00004

00006

t

(a)

990 995 1000

t

minus0006

minus0004

minus0002

0000

0002

0004

0006

(b)





3

and 1199024






0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

6 Conclusions









minus0010 minus0005 0000 0005

minus001

000

001

002

003Ve

loci

ty

Displacement

(a)

Velo

city

minus0010 minus0005 0000 0005 0010

minus001

000

001

002

003

Displacement

(b)

minus001 000 001

minus002

000

002

004

Displacement

Velo

city

(c)

minus0010 minus0005 0000 0005

0000

0005

0010

0015

0020

Displacement

Velo

city

(d)

minus0010 minus0005 0000 0005

000

001

002

003

Displacement

Velo

city

(e)


000

002

004

Displacement

Velo

city

(f)







Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0000

0002

0004

0006

0008

Stability boundary

minus1 0 1 2minus2

a1l

a2l

120576120590

120576an

(a)

0000

0005

0010

0015

Stability boundary

minus2 0 2

a1n

a2n120576an

120576120590

(b)


0000

0005

0010

0015

minus1 0 1 2minus2

b = 0004

b = 0006

b = 0008

a1n

a2n

120576an

120576120590

(a)

0000 0001 0002

00000

00002

00004

00006

00008

00010

a1l

a2l

a2n

a1n

120576an

1205762b

(b)












2mode








00000

00025

00050

00075

00100

00125

a2n

a1n

minus1 0 1 2minus2

120576120590

120576an

120572 = 000001

120572 = 000005

120572 = 00001

120572 = 000001

120572 = 000005

120572 = 00001

(a)

0002

0004

0006

0008

0010

0012

P = 14340403KN

P = 14340403KN

P = 18125417KN

P = 18125417KN

P = 11638770KN

P = 11638770KN

a2n

a1n

0 2minus2

120576120590

120576an

0000

(b)


0000

0005

0010

0015

0 2minus2

120576120590

120576an

a1

a2

(a)

0000

0005

0010120576an

a1a1

a2

a2

minus1 0 1 2minus2

120576120590

(b)










990 995 1000minus0006

minus0004

minus0002

0000

0002

0004

0006

t


minus00006990 995 1000

minus00004

minus00002

00000

00002

00004

00006

t

(a)

990 995 1000

t

minus0006

minus0004

minus0002

0000

0002

0004

0006

(b)





3

and 1199024






0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

6 Conclusions









minus0010 minus0005 0000 0005

minus001

000

001

002

003Ve

loci

ty

Displacement

(a)

Velo

city

minus0010 minus0005 0000 0005 0010

minus001

000

001

002

003

Displacement

(b)

minus001 000 001

minus002

000

002

004

Displacement

Velo

city

(c)

minus0010 minus0005 0000 0005

0000

0005

0010

0015

0020

Displacement

Velo

city

(d)

minus0010 minus0005 0000 0005

000

001

002

003

Displacement

Velo

city

(e)


000

002

004

Displacement

Velo

city

(f)







Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00000

00025

00050

00075

00100

00125

a2n

a1n

minus1 0 1 2minus2

120576120590

120576an

120572 = 000001

120572 = 000005

120572 = 00001

120572 = 000001

120572 = 000005

120572 = 00001

(a)

0002

0004

0006

0008

0010

0012

P = 14340403KN

P = 14340403KN

P = 18125417KN

P = 18125417KN

P = 11638770KN

P = 11638770KN

a2n

a1n

0 2minus2

120576120590

120576an

0000

(b)


0000

0005

0010

0015

0 2minus2

120576120590

120576an

a1

a2

(a)

0000

0005

0010120576an

a1a1

a2

a2

minus1 0 1 2minus2

120576120590

(b)










990 995 1000minus0006

minus0004

minus0002

0000

0002

0004

0006

t


minus00006990 995 1000

minus00004

minus00002

00000

00002

00004

00006

t

(a)

990 995 1000

t

minus0006

minus0004

minus0002

0000

0002

0004

0006

(b)





3

and 1199024






0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

6 Conclusions









minus0010 minus0005 0000 0005

minus001

000

001

002

003Ve

loci

ty

Displacement

(a)

Velo

city

minus0010 minus0005 0000 0005 0010

minus001

000

001

002

003

Displacement

(b)

minus001 000 001

minus002

000

002

004

Displacement

Velo

city

(c)

minus0010 minus0005 0000 0005

0000

0005

0010

0015

0020

Displacement

Velo

city

(d)

minus0010 minus0005 0000 0005

000

001

002

003

Displacement

Velo

city

(e)


000

002

004

Displacement

Velo

city

(f)







Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










990 995 1000minus0006

minus0004

minus0002

0000

0002

0004

0006

t


minus00006990 995 1000

minus00004

minus00002

00000

00002

00004

00006

t

(a)

990 995 1000

t

minus0006

minus0004

minus0002

0000

0002

0004

0006

(b)





3

and 1199024






0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

6 Conclusions









minus0010 minus0005 0000 0005

minus001

000

001

002

003Ve

loci

ty

Displacement

(a)

Velo

city

minus0010 minus0005 0000 0005 0010

minus001

000

001

002

003

Displacement

(b)

minus001 000 001

minus002

000

002

004

Displacement

Velo

city

(c)

minus0010 minus0005 0000 0005

0000

0005

0010

0015

0020

Displacement

Velo

city

(d)

minus0010 minus0005 0000 0005

000

001

002

003

Displacement

Velo

city

(e)


000

002

004

Displacement

Velo

city

(f)







Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

0 10 20 30 40 50minus00010

minus00005

00000

00005

00010

t

q3

(a)

t

0 10 20 30minus00010

minus00005

00000

00005

00010

q4

(b)


4mode

6 Conclusions









minus0010 minus0005 0000 0005

minus001

000

001

002

003Ve

loci

ty

Displacement

(a)

Velo

city

minus0010 minus0005 0000 0005 0010

minus001

000

001

002

003

Displacement

(b)

minus001 000 001

minus002

000

002

004

Displacement

Velo

city

(c)

minus0010 minus0005 0000 0005

0000

0005

0010

0015

0020

Displacement

Velo

city

(d)

minus0010 minus0005 0000 0005

000

001

002

003

Displacement

Velo

city

(e)


000

002

004

Displacement

Velo

city

(f)







Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus0010 minus0005 0000 0005

minus001

000

001

002

003Ve

loci

ty

Displacement

(a)

Velo

city

minus0010 minus0005 0000 0005 0010

minus001

000

001

002

003

Displacement

(b)

minus001 000 001

minus002

000

002

004

Displacement

Velo

city

(c)

minus0010 minus0005 0000 0005

0000

0005

0010

0015

0020

Displacement

Velo

city

(d)

minus0010 minus0005 0000 0005

000

001

002

003

Displacement

Velo

city

(e)


000

002

004

Displacement

Velo

city

(f)







Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Nonlinear Forced Vibration of a...

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Transcript of Research Article Nonlinear Forced Vibration of a...