PseudodynamicBearingCapacityAnalysisofShallowStrip...

Research ArticlePseudodynamic Bearing Capacity Analysis of Shallow StripFooting Using the Advanced Optimization TechniqueldquoHybrid Symbiosis Organisms Search Algorithmrdquo withNumerical Validation

Arijit Saha 1 Apu Kumar Saha2 and Sima Ghosh1

1Department of Civil Engineering National Institute of Technology Agartala Barjala Jirania 799046 West Tripura India2Department of Mathematics National Institute of Technology Agartala Barjala Jirania 799046 West Tripura India

Correspondence should be addressed to Arijit Saha sahaarijit20gmailcom

Received 26 July 2017 Revised 25 September 2017 Accepted 9 October 2017 Published 5 April 2018

Academic Editor Moacir Kripka

Copyright copy 2018 Arijit Saha et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e analysis of shallow foundations subjected to seismic loading has been an important area of research for civil engineers ispaper presents an upper-bound solution for bearing capacity of shallow strip footing considering composite failure mechanismsby the pseudodynamic approach A recently developed hybrid symbiosis organisms search (HSOS) algorithm has been used tosolve this problem In the HSOS method the exploration capability of SQI and the exploitation potential of SOS have beencombined to increase the robustness of the algorithmis combination can improve the searching capability of the algorithm forattaining the global optimum Numerical analysis is also done using dynamic modules of PLAXIS-86v for the validation of thisanalytical solutione results obtained from the present analysis using HSOS are thoroughly compared with the existing availableliterature and also with the other optimization techniques e significance of the present methodology to analyze the bearingcapacity is discussed and the acceptability of HSOS technique is justified to solve such type of engineering problems

1 Introduction

e subject of bearing capacity is one of the importantaspects of geotechnical engineering problems Loads frombuildings are transmitted to the foundation by columns orby load-bearing walls of the structures Many researcherslike Prandtl [1] Terzaghi [2] Meyerhof [3 4] Vesic [5 6]and many more have investigated the mechanisms ofbearing capacity of foundation under a static loading con-dition Due to seismic loading foundations may experiencea reduction in bearing capacity and an increase in settle-ment Two sources of loading must be taken into consid-eration initial loading due to lateral forces imposed onsuperstructure and kinematic loading due to groundmovements developed during the earthquake e pio-neering works in determining the seismic bearing capacity ofshallow strip footings were done by Budhu and Al-Karni [7]

Dormieux and Pecker [8] Soubra [9ndash11] Richards et al [12]Choudhury and Subha Rao [13] Kumar and Ghosh [14]and many others using pseudostatic approach with thehelp of different solution techniques such as method ofslices limit equilibrium method of stress characteristicsand upper bound limit analysis Apart from these analyticalresearchers Shafiee and Jahanandish [15] and Chakrabortyand Kumar [16] used finite element method to estimate theseismic bearing capacity of strip footings on the soil usingPLAXIS-2D considering the pseudostatic approach Since inthe pseudostatic method the dynamic loading induced bythe earthquake is considered as time-independent whichultimately assumes that the magnitude and phase of ac-celeration are uniform through the soil layer pseudody-namic analysis is developed where the effects of both shearand primary waves are considered along with the periodof lateral shaking Ghosh [17] and Saha and Ghosh [18]

HindawiAdvances in Civil EngineeringVolume 2018 Article ID 3729360 18 pageshttpsdoiorg10115520183729360

evaluated pseudodynamic bearing capacity using limitanalysis method and limit equilibrium method respectivelyconsidering the linear failure surface In the earlier analysesthe resistance of unit weight surcharge and cohesion isconsidered separatelyerefore if the solution was done forshallow strip footing resting on c-Φ soil there will be threeseparate coefficients one for unit weight another for sur-charge and the other for cohesion But in a practical situ-ation there will be a single failure mechanism for thesimultaneous resistance of unit weight surcharge and co-hesion us an attempt is made to present a single seismicbearing capacity coefficient for the simultaneous resistanceof unit weight surcharge and cohesion Here in this paperthe pseudodynamic bearing capacity of shallow strip footingconsidering composite failure mechanism resting on c-Φ soilis solved using the upper-bound limit analysis method Arelative ease in solving geometrically complex multidi-mensional problem renders limit analysis attractive as analternative to numerical codes e kinematic method oflimit analysis hinges on constructing a velocity field that isadmissible for a rigid-perfect plastic material obeying theassociative flow rule

Nowadays nature-based global optimization algo-rithms such as genetic algorithms (GA) particle swarmoptimization (PSO) algorithm and many other algorithmshave been successfully applied to solve different scienceand engineering complex optimization problems espe-cially civil engineering problems such as slope stability[19 20 21ndash28] retaining walls [29ndash31] and structuraldesign [32] Cheng and Prayogo [33] introduced a newnature-based optimization technique called symbioticorganisms search (SOS) algorithm is technique is basedon the interactive relationship among the organism in theecosystem It has no algorithm-specific control parameterse SOS algorithm has been successfully applied to solvedifferent engineering optimization problems [34ndash38] Re-cently Nama et al [39] proposed a hybrid algorithm calledhybrid symbiotic organisms search (HSOS) algorithmwhich is the combination of SOS algorithm and simplequadratic interpolation method [40] Here in this paperHSOS algorithm is used to optimise the pseudodynamicbearing capacity of shallow strip footing considering upperbound limit analysis method Mathematically the problemcan be represented as a nonlinear hard optimizationproblem which can be solved by the HSOS algorithmwhich is found to be a more satisfactory optimum solutionand can be used for designing the shallow strip footing Inthe HSOS algorithm failure surface angle (α β) and tT areconsidered as the search variables So it can be applied toobtain optimal solutions in the different fields of scienceand engineering Numerical analysis is also done usingdynamic module of PLAXIS-86v software to validate thisanalytical solution Results are presented in tabular formincluding comparison with other available analyses Effectsof a wide range of variation of parameters like soil frictionangle (Φ) cohesion factor (2ccB0) depth factor (DfB0)and horizontal and vertical seismic accelerations (kh kv) onthe normalized reduction factor (NγeNγs) have beenstudied

erefore the main contributions of this paper aresummarized as follows

(i) Evaluation of pseudodynamic bearing capacitycoefficient of shallow strip footing resting on c-Φsoil considering composite failure surface usingupper bound limit analysis method

(ii) A single pseudodynamic bearing capacity coefficientis presented here considering the simultaneousresistance of unit weight surcharge and cohesion

(iii) A recent hybrid optimization algorithm (calledHSOS) is used to solve the pseudodynamic bearingcapacity minimization optimization problem

(iv) PLAXIS-86v software is used to solve this above-mentioned problem numerically for the validationof the analytical formulation

(v) e obtained results are compared with the otherresults which are available in literature and theresults obtained by other state-of-the-artalgorithms

e remaining part of the paper is organized as followsSection 2 discusses the formulation of the real-world geo-technical earthquake engineering optimization problemsuch as the pseudodynamic bearing capacity of a shallowfoundation e overview of the optimization algorithmHSOS is presented in Section 3 Section 4 presents discus-sions of the results obtained by the HSOS algorithm to showthe efficiency and accuracy of this hybrid algorithm forsolving this engineering optimization problem Numericalanalysis of shallow strip footing using the dynamic moduleof PLAXIS-86v software and the validation of analyticalformulation are discussed in Section 5 and finally Section 6presents the conclusion and the summary of the outcome ofthe paper

2 Formulation of Pseudodynamic BearingCapacity Coefficient

21 Consideration of Model Let us consider a shallow stripfooting of width (B0) resting below the ground surface ata depth of Df over which a load (P) of column acts ehomogeneous soil of effective unit weight c has MohrndashCoulomb characteristic c-Φ and can be considered asa rigid plastic body For shallow foundation (D f leB0) theoverburden pressure is idealized as a surcharge (q cDf)which acts over the length of BC e classical two-dimensional slip line field obtained by Prandtl [1] is thetraditional failure mechanism which has three regionssuch as active zone passive zone and logarithmic radial-fan transition zone In this composite failure mechanismhalf of the failure is assumed to occur along the surfaceAEDC which is composed of a triangular elastic zoneABE triangular passive Rankine zone BDC and in be-tween them a log spiral radial shear zone BDE shown inFigure 1(a) [41] It is a composite mechanism that is de-fined by the angular parameters α and β in which the log-spiral slip surface ED is a tangent to lines AE and DC at Eand D respectively Figures 2 and 3 show the detailed free

2 Advances in Civil Engineering

body diagram of the elastic zone ABE and compositepassive Rankine zone and the log-spiral shear zone BEDCrespectively

22 CollapseMechanism At collapse it is assumed that thefooting and the underlying zone ABE moves in phase witheach other at the same absolute velocity V1 making anangle Φ with the discontinuity line AE in order to rep-resent the normality condition for an associated ow ruleCoulomb material Hence there is no dissipation of en-ergy along the soil-structure interface Whereas the radiallog-spiral shearing zone BED is bounded by a log-spiralcurve ED e equation for the curve in polar coordinates(r θ) is r r0eθ tan ϕ e centre of this log-spiral ED is atpoint B and the radius r0 is the length of the line BE wherer0 B0 sin αcos ϕ and the width of the footing AB B0Note that in this mechanism we have assumed thatthe line AE is a tangent to the log-spiral curve at point Ehence there is no velocity discontinuity along BE eradial shear zone BED may be considered to be compo-sed of a sequence of rigid triangles as in the investiga-tions by Chen using the symmetrical Hill and Prandtlrsquos

mechanisms All the small triangles move as rigid bodiesin directions which make an angle Φ with the disconti-nuity line ED e velocity of each small triangle is de-termined by the condition that the relative velocitybetween the triangles in contact has the direction whichmakes an angle Φ to the contact surface It has been shown

A

Zone I

BC

P

Df q = γDf

Zone IIZone III

V

V1

V1

V2

Vs Vp D

E

B0

(a)

V

V1

V2

( minus + )

( minus )

( minus )( + minus )

( minus )

(b)

Figure 1 Composite failure mechanism [41] (a) Collapse mechanism (b) Velocity hodograph

WminusQv

Qh

AB

E

c

PL

dz

Vs Vp

z

B0

Figure 2 Elastic wedge

Advances in Civil Engineering 3

that the velocity V of each triangle is V1 V0eθ tan ϕ e

log-spiral curve ED is assumed to be tangent to the line DEat D hence there is no velocity discontinuity along theline BD Finally the triangular wedge BCD is assumed tobe rigid moving with velocity V2 V(β) V1e

β tanϕerefore the velocities so determined constitute a kine-matically admissible velocity eld Velocity hodograph ofthis composite failure mechanism is shown in Figure 1(b)Having established the velocity eld of the kinematicallyadmissible failure mechanism the incremental externalwork done and the incremental internal energy dissipa-tion are calculated following the procedure as mentionedin [42]

23 Analysis of Bearing Capacity

231 Elastic Wedge Weight of the wedge ABE

W c

2B20

sin α sin ρcos ϕ

(1)

where ρ (π2)minus (αminus ϕ)If the base of the wedge is subjected to harmonic

horizontal and vertical seismic accelerations of ampli-tude ahg and avg respectively the acceleration at anydepth z and time t below the top of the surfacecan be expressed as

ah(z t) ah sinω tminusr0 sin ρminus z

Vs( ) (2)

av(z t) av sinω tminusr0 sin ρminus z

Vp( ) (3)

e mass of a thin element of the elastic wedge atdepth z is

m(z) c

g

1tan α

+1

tan ρ( ) r0 sin ρminus z( )dz (4)

e total horizontal and vertical inertia forces actingwithin the elastic zone can be expressed as follows

Qh ckh1

tan α+

1tan ρ

( )intr0 sin ρ

0r0 sin ρminus z( ) sinω tminus

r0 sin ρminus zVs

( )dz

cB20λkh sin 2ρ sin 2α2πr0 sin ρ cos 2ϕ

1tan α

+1

tan ρ( ) cos 2π

t

Tminusr0 sin ρ

λ( ) +

r0 sin ρ2πλ

sin 2πt

Tminusr0 sin ρ

λ( )minus sin 2π

t

T [ ]

c

2B20qh

(5)

Qv ckv1

tan α+

1tan ρ

( )intr0 sin ρ


r0 sin ρminus zVp

( )dz

cB2

0ηkv sin2ρ sin 2α

2πr0 sin ρ cos 2ϕ1

tan α+

1tan ρ

( ) cos 2πt

Tminusr0 sin ρ

η( ) +

r0 sin ρ2πη

sin 2πt

Tminusr0 sin ρ

η( )minus sin 2π

t

T [ ]

c

2B20qv

(6)

d

q = Dfq = Df

Qh1

W1minusQv1

-

D

B

E

c

C

dz1

c

Vs Vp

z1

Qh2

W2minusQv2

Figure 3 Log-spiral zone


232 Passive Rankine Zone Weight of the wedge BCD

W1 c

2B20e

2β tanϕ sin 2α cos(βminus α + ϕ)

cosϕ sin ξ (7)

e mass of a thin element of the elastic wedge at depthz1 is

m z1( 1113857 c

gtan(β + ϕminus α) +

1tan(βminus α)

1113896 11138971113890

times r0eβ tanϕ cos(β + ϕminus α)minus z1113966 11139671113891dz (8)

e acceleration at any depth z1 and time t below the topof the surface can be expressed as

ah z1 t( 1113857 ah sinω tminusr0e

β tanϕ cos(β + ϕminus α)minus z1

Vs

1113888 1113889

(9)

av z1 t( 1113857 av sinω tminusr0e

β tan ϕ cos(β + ϕminus α)minus z1

Vp

1113888 1113889

(10)

e total horizontal and vertical inertia force actingwithin the passive Rankine zone can be expressed as follows

Qh1 ckh tan(β+ϕ minus α)+1

tan(β minus α)1113896 11138971113946

r0eβtanϕ cos(β+ϕminus α)

0r0e

βtanϕ cos(β+ϕ minus α) minus z1113966 1113967sinω t minusr0e

βtanϕcos(β+ϕ minus α) minus z1

Vs

1113888 11138891113890 1113891dz1

cB2

0λkhe2βtanϕ cos2(β+ϕ minus α)sin2α2πr0e

βtanϕ cos(β+ϕ minus α)cos2ϕtan(β+ϕ minus α)+

1tan(β minus α)

1113896 11138971113890cos2πt

Tminus

r0eβtanϕ cos(β+ϕ minus α)

λ1113888 1113889

+r0e

βtanϕ cos(β+ϕ minus α)

2πλsin2π

t

Tminus


λ1113888 1113889 minus sin2π

t

T1113896 11138971113891

c

2B20qh1

(11)

Qv1 ckv tan(β+ϕ minus α)+1

tan(β minus α)1113896 11138971113946


0r0e

βtanϕcos(β+ϕ minus α) minus z1113966 1113967sinω t minusr0e

βtanϕ cos(β+ϕ minus α) minus z1

Vp

1113888 11138891113890 1113891dz1

cB2

0ηkve2βtanϕ cos2(β+ϕ minus α)sin2α2πr0e


1tan(β minus α)

1113896 11138971113890cos2πt

Tminus


η1113888 1113889

+r0e

βtanϕcos(β+ϕ minus α)

2πηsin2π

t

Tminus


η1113888 1113889 minus sin2π

t

T1113896 11138971113891

c

2B20qv1

(12)

233 Log-Spiral Shear Zone Weight of the log-spiral shearzone BDE

W2 12

1113946β

0r20dθ

r20 e2β tanϕ minus 1( 1113857

4 tanϕ

B20 sin 2α e2β tan ϕ minus 1( 1113857

2 sin 2ϕ

(13)

e log-spiral zone BDE is divided into ldquonrdquo number ofslices which makes the angle of log-spiral center β into ldquonrdquonumber of dβ angles that is β ndβ as shown in Figure 4

Mass of strip on the ith slice of the log-spiral zone BDE

m(z)i c

g

2πdβ360deg sin 2(ρ +(iminus 05)dβ)

1113946Hi

0zi(dz)i (14)

where Hi (r02)(e(iminus1)dβ tanϕ + eidβ tanϕ) sin(ρ+(iminus 05)dβ)e acceleration at any depth zi and time t of any ith slice

of the log-spiral shear zone below the top of the surface canbe expressed as

ah(z t)i khg sinω tminusHi minus zi

Vs

1113888 1113889 (15)

av(z t)i kvg sinω tminusHi minus zi

Vp

1113888 1113889 (16)

e total horizontal and vertical inertia force actingwithin this ith slice can be expressed as follows


Qh( )i 2πcdβkh

360deg sin 2(ρ +(iminus 05)dβ)intHi

0zi sinω tminus

Hi minus ziVs

( )(dz)i

cdβkhλB2

0 sin 2α e(iminus 1)dβ tanϕ + eidβ tanϕ( )2

360deglowast4Hi cos 2ϕλ

Hi2πsin 2π

t

Tminus sin 2π

t

TminusHi

λ( ) minus cos 2π

t

T[ ]

c

2B20qh2 e

(iminus 1)dβ tan ϕ + eidβ tanϕ( )2

(17)

Qv( )i 2πcdβkv


0zi sinω tminus

Hi minus ziVp

( )(dz)i

cdβkvηB2

0 sin2α e(iminus 1)dβ tanϕ + eidβ tanϕ( )2

360deglowast4Hi cos 2ϕη

Hi2πsin 2π

t

Tminus sin 2π

t

TminusHi

η( ) minus cos 2π

t

T[ ]

c

2B20qv2 e


(18)

Now the total horizontal and vertical inertia force actingon log-spiral shear zone is expressed as

Qh2 sumn

i1Qh( )i

c

2B20qh2sum

n

i1e(iminus 1)dβ tanϕ + eidβ tanϕ( )

2

(19)

Qv2 sumn

i1Qv( )i

c

2B20qv2sum

n


2 (20)

e incremental external works due to the foundationload P surcharge load q the weight of the soil wedges ABEBCD and BDE and their corresponding inertial forces are

WP PLB0V1 sin (α minus ϕ) + kh sin 2πt

TminusH

λ( ) cos (α minus ϕ) minus kv sin 2π

t

TminusH

η( ) sin (α minus ϕ)[ ] (21)

WABE c

4B0

2 sin α sin 2ρcosϕ

V1 (22)

WBCD minusc

4B20e3β tanϕ sin2 α sin 2(β minus α + ϕ)

cosϕ sin ξV1 (23)

WQ minus cDfB0 sin αe2β tanϕ

sin ξminus kh sin 2π

t

TminusH

λ( ) cos (β minus α + ϕ) + 1 minus kv sin 2π

t

TminusH

η( )( ) sin (β minus α + ϕ) [ ]V1

(24)

WBDE minusc

2intβ

0r20dθVθ sin (θ + ϕ minus α)

minusc

2B20

sin2 αcos2 ϕ 1 + 9 tan2 ϕ( )

[3 tanϕ e3β tanϕ sin (β + ϕ minus α) minus sin (ϕ minus α) minus e3β tanϕ cos (β + ϕ minus α) + cos (ϕ minus α)]V1

(25)

d

Hizi

Qh2

W2 minus Qv2 (dz)isin( + (i minus 05)d)

ith slice

B

D

E

Figure 4 Generalized slice and centre of gravity of the log-spiral zone


WQh QhV1 cos (α minus ϕ)

WQv minus QvV1 sin (α minus ϕ)(26)

WQh1 Qh1V1eβ tanϕ cos (β minus α + ϕ)

WQv1 Qv1V1eβ tanϕ sin (β minus α + ϕ)

(27)

WQh2 c

2B20qh2 sum

n

i1e

(iminus 1)dβ tanϕ+ e

idβ tanϕ1113872 1113873

21113946

idβ

(iminus 1)dβVθ cos (θ + ϕ minus α)dθ1113890 1113891

c

2B20qh2

V1

(1 + tanϕ)sumn

in

1113876 e(iminus 1)dβ tanϕ

+ eidβ tan ϕ

1113872 111387321113882e

idβ tanϕ(sin (idβ + ϕ minus α) + tanϕ cos (idβ + ϕ minus α))

minus e(iminus 1)dβ tanϕ

(sin ((i minus 1)dβ + ϕ minus α) + tanϕ cos ((i minus 1)dβ + ϕ minus α))11138831113877

(28)

WQv2 c

2B20qv2sum

ni1 e


idβ tan ϕ1113872 1113873

21113946

idβ

(iminus 1)dβVθ sin (θ + ϕ minus α)dθ1113890 1113891

c

2B20qv2

V1

1 + tan2 ϕ( 1113857sumn

i11113876 e(iminus 1)dβ tanϕ

+ eidβ tan ϕ

1113872 111387321113882e

idβ tan ϕ(minus cos (idβ + ϕ minus α) + tanϕ sin (idβ + ϕ minus α))


(minus cos ((i minus 1)dβ + ϕ minus α) + tanϕ sin ((i minus 1)dβ + ϕ minus α))11138831113877

(29)

e incremental internal energy dissipation along thevelocity discontinuities AE and CD and the radial line DE is

DAE cACV1 cos ϕ cB0 sin ρV1 (30)

DCD cCDV2 cosϕ cB0e

2β tanϕ sin α cos(βminus α + ϕ)

sin(βminus α)V1

(31)

DDE c1113946β

0r0dθVθ cosϕ

cB0 sin α e2β tanϕ minus 1( 1113857

2 tanϕV1 (32)

Equating the work expended by the external loads to thepower dissipated internally for a kinematically admissiblevelocity field we can get the expression of pseudodynamicultimate bearing capacity of shallow strip footinge classicalultimate bearing capacity equation of shallow strip footing

PL Qult cNc + cDfNq + 05cB0Nc (33)

After solving the above equations the simplified form ofthe bearing capacity coefficients is as follows

Nc sinα2tanϕ

e2βtanϕ

minus 11113872 1113873+ cos(α minus ϕ)+e2βtanϕ sinαcos(β minus α+ϕ)

sin(β minus α)1113890 1113891 (34)

Nq sinαe2βtanϕ

sinξminus kh sin2π

t

Tminus

H

λ1113874 1113875cos(β minus α+ϕ)1113882 1113883+ 1 minus kv sin2π

t

Tminus

H

η1113888 11138891113888 1113889sin(β minus α+ϕ)1113896 11138971113890 11138911113890 1113891 (35)

Nc 1113890sin2α

cos2ϕ 1+9tan2ϕ( 11138573tanϕ e

3βtanϕ sin(β+ϕ minus α) minus sin(ϕ minus α)1113966 1113967 minus e3βtanϕcos(β+ϕ minus α)+ cos(ϕ minus α)1113960 1113961

minussinαsin2ρ2cosϕ

+e3βtanϕ sin2αsin2(β minus α+ϕ)

2cosϕsinξminus qhcos(α minus ϕ)+qv sin(α minus ϕ) minus qh1e

βtanϕcos(β+ϕ minus α) minus qv1eβtanϕ sin(β+ϕ minus α)

minussumn

i1 e(iminus 1)dβtanϕ+eidβtanϕ( 11138572

1+ tan2ϕ( 11138571113876qh2sum

ni11113882e

idβtanϕ(sin(idβ+ϕ minus α)+ tanϕcos(idβ+ϕ minus α))minus e

(iminus 1)dβtanϕ(sin((i minus 1)dβ+ϕ minus α)

+ tanϕcos((i minus 1)dβ+ϕ minus α))1113883+qv2sumni11113882e

idβtanϕ(sin(idβ+ϕ minus α)+ tanϕcos(idβ+ϕ minus α))

minus e(iminus 1)dβtanϕ

(sin((i minus 1)dβ+ϕ minus α)+ tanϕcos((i minus 1)dβ+ϕ minus α))111388311138771113891

(36)


An attampt is made to present lsquosingle seismic bearingcapacity coefficientrsquo for simultaneous resistance of unitweight surcharge and cohesion as in a practical situationthere will be a single failure mechanism for the simultaneousresistance of unit weight surcharge and cohesion So we get

PL c

2B0N (37)

After simplification of equations the expression of N isgiven below

N 1113890sin2 α

cos2 ϕ 1 + 9 tan2 ϕ( 11138573 tanϕ e

3β tan ϕ sin (β + ϕ minus α) minus sin (ϕ minus α)1113966 1113967 minus e3β tan ϕ cos (β + ϕ minus α) + cos (ϕ minus α)1113960 1113961

+e3β tanϕ sin2 α sin 2(β minus α + ϕ)

2 cos ϕ sin ξminussin α sin 2ρ2 cosϕ

+ 2Df

B0

sin αe2β tan ϕ

sin ξ⎡⎣ minus kh sin 2π

t

Tminus

H

λ1113874 1113875 cos (β minus α + ϕ)1113882 1113883

+ 1 minus kv sin 2πt

Tminus

H

η1113888 11138891113888 1113889 sin (β minus α + ϕ)1113896 1113897⎤⎦ minus qh cos (α minus ϕ) + qv sin (α minus ϕ) minus qh1e

β tanϕ cos (β + ϕ minus α)

minus qv1eβ tan ϕ sin (β + ϕ minus α) minus

sumni1 e(iminus 1)dβ tanϕ + eidβ tanϕ( 1113857

2

1 + tan2 ϕ( 11138571113876qh2sum

ni1e

idβ tanϕ(sin (idβ + ϕ minus α)

+ tanϕ cos (idβ + ϕ minus α)) minus e(iminus 1)dβ tanϕ

(sin ((i minus 1)dβ + ϕ minus α) + tanϕ cos ((i minus 1)dβ + ϕ minus α))

+ qv2sumni11113882e

idβ tanϕ(sin (idβ + ϕ minus α) + tanϕ cos (idβ + ϕ minus α)) minus e

(iminus 1)dβ tanϕ( sin ((i minus 1)dβ + ϕ minus α)

+ tanϕ cos ((i minus 1)dβ + ϕ minus α)111385711138831113877 +2c

cB0

sin α2 tanϕ

e2β tan ϕ

minus 11113872 1113873 + cos (α minus ϕ) +e2β tanϕ sin α cos (β minus α + ϕ)

sin (β minus α)1113896 11138971113891

(38)

Here N is the single pseudodynamic bearing capacitycoefficient of shallow strip footing under seismic loadingcondition In this formulation the objective functionpseudodynamic bearing capacity coefficient depends onthese Φ c α β tT kh kv Hλ and Hη functions Fora particular soil and seismic condition all other termsare constant except α β and tT So optimization ofpseudodynamic bearing capacity coefficient is done withrespect to α β and tT using the HSOS algorithm eadvantage of this HSOS algorithm is that it can improvethe searching capability of the algorithm for attaining theglobal optimization Here the optimum value of N isrepresented as NγeNow pseudodynamic ultimate bearingcapacity

Qult 05cB0Nce (39)

3 The Hybrid Symbiosis OrganismsSearch Algorithm

e hybrid symbiosis organisms search (HSOS) algorithm isa recently developed hybrid optimization algorithm which isused to solve this pseudodynamic bearing capacity ofshallow strip footing minimal optimization problem

31 e Symbiosis Organisms Search Algorithm Symbiosisorganisms search (SOS) algorithm is a population-based it-erative global optimization algorithm for solving global op-timization problems proposed by Cheng and Prayogo [33]

is algorithm is based on the basic concept of symbioticrelationships among the organisms in nature (ecosystem)ree types of symbiotic relationships are occurring in anecosystem ese are mutualism relationship commensal-ism relationship and parasitism relationship Mutualismrelationship describes the relationship where both organ-isms get benefits from the interaction Commensalism re-lationship is a symbiotic relationship between two differentorganisms where one organism gets the benefit and the otheris not significantly affected In the symbiotic parasitismrelationship one organism gets the benefit and the other isharmed but not always killed Based on the concept of threerelationships the SOS algorithm is executed In the SOSalgorithm a group of organisms in an ecosystem is con-sidered as a population size of the solution Each organism isanalogous to one solution vector and the fitness value ofeach organism represents the degree of adaptation to thedesired objective Initially a set of organisms in the eco-system is generated randomly within the search region enew candidate solution is generated through the biologicalinteraction between two organisms in the ecosystem whichcontains the mutualism commensalism and parasitismphases and the process of interaction is continued until thetermination criterion is satisfied A detailed description ofSOS algorithm can be seen in [33]

32e SimpleQuadratic Interpolation (SQI)Method In thissection the three-point quadratic interpolation is discussed


Considering the two organisms Orgj and Orgk where Orgj

(org1j org2j org3j orgDj ) and Orgk (org1k org2k org3k

orgDk ) from the ecosystem the organism Orgi is updated

according to the three-point quadratic interpolation [40] ethree-point approximate minimal point for organism Orgi isdetermined by the following equation

Orgm

i 05Orgm

i( 11138572 minus Orgm

j1113872 11138732

1113874 1113875fk + Orgmj1113872 1113873

2minus Orgm

k( 11138572

1113874 1113875fi + Orgmk( 1113857

2 minus Orgmi( 1113857

21113872 1113873fj

Orgmi minusOrg

ml( 1113857fk + Orgm

j minusOrgmk1113872 1113873fi + Orgm

k minusOrgmi( 1113857fj

(40)

where m 1 2 3 De SQI is intended to enhance the entire search ca-

pability of the algorithm Here fi fj and fk are the fitnessvalues of ith jth and kth organisms respectively

33 e Hybrid Symbiosis Organisms Search Algorithm Inthe development of heuristic global optimization algorithmthe balance of exploration and exploitation capability playsa major role [43] where ldquoExploration is the process of visitingentirely new regions of a search space whilst exploitation isthe process of visiting those regions of a search space withinthe neighborhood of previously visited pointsrdquo [43] As dis-cussed above the SQI method may be used for the betterexploration when executing the optimization process On theother hand Cheng and Prayoga [33] have elaborately dis-cussed the better exploitation ability of SOS for global opti-mization To balance the exploration capability of SQI and theexploitation potential of SOS the hybrid symbiosis organismssearch (HSOS) algorithm has been proposed is hybridmethod can increase the robustness as well as the searchingcapability of the algorithm for attaining the global optimi-zation By incorporating the SQI into the SOS algorithm theHSOS algorithm is developed and the flowchart of the HSOSalgorithm is shown in Figure 5e HSOS algorithm is able toexplore the new search region with the SOS algorithm and toexploit the population information with the SQI

If an organism is going to an infeasible region then theorganism is reflected back to the feasible region using thefollowing equation [44]

Orgi UBi + rand(0 1)lowast UBi minus LBi( 1113857 if Orgi lt LBi

UBi minus rand(0 1)lowast UBi minus LBi( 1113857 if Orgi gtUBi1113896

(41)

where LBi andUBi are respectively the lower and upperbounds of the ith organism

e algorithmic steps of hsos are given below

Step 1 Ecosystem initialization initialize the algorithmparameters and ecosystem organisms and evaluate the fit-ness value for each corresponding organism

Step 2 Main loop

Step 21 Mutualism phase select one organism Orgj ran-domly from the ecosysteme organism Orgi intersects with

the organism Orgj and then they try to improve the survivalcapabilities in the ecosystem e new organism for each ofOrgi and Orgj is calculated by the following equations

Orgnewi Orgi + rand(0 1)lowast Orgbest minusMutualVectorlowastBF1( 1113857

(42)Orgnewj Orgj + rand(0 1)lowast Orgbest minusMutualVectorlowastBF2( 1113857

(43)

whereMutualVector (Orgi + Orgj)2 Here BF1 and BF2 arecalled the benefit factors the value of which be either 1 or 2e level of benefits of the organism represents these factorsthat is whether an organism gets respectively partial or fullbenefit from the interaction Orgbest is the best organism in theecosystem MutualVector represents the relationship charac-teristic between the organisms Orgi and Orgj

Step 22 Commensalism phase between the interaction ofthe organisms Orgi and Orgj the organism Orgi gets benefitby the organism Orgj and try to improve the beneficialadvantage in the ecosystem to the higher degree of adaptione new organism Orgi is determined by the followingequation

Orgnewi Orgi + rand minus1 1( )lowast Orgbest minusOrgj1113872 1113873 (44)

where ine j and Orgbest is the best organism in the ecosystem

Step 23 Parasitism phase by duplicating randomly selecteddimensions of the organism Orgi an artificial parasite (Par-asite_Vector) is created From the ecosystem another or-ganismOrgj is selected randomly that is treated as a host to theParasite_Vector If the objective function value of the Para-site_Vector is better than the organism Orgj it can kill theorganism Orgj and adopt its position in the ecosystem If theobjective function value of Orgj is better than the Para-site_Vector Orgj will have resistance to the parasite and theParasite_Vector will not be able to reside in that ecosystem

Step 24 Simple quadratic interpolation the two organismsOrgj and Orgk (jne k) are selected randomly from the eco-system and then the organism Orgi is updated by quadraticinterpolation passing through these three organisms whichcan be expressed by (40)

Step 3 If the stopping criteria are not satisfied to go to Step 2then it will proceed until the best objective function value isobtained


4 Discussion on Results Obtained bythe HSOS Algorithm

e pseudodynamic bearing capacity coecient (Nγe) hasbeen optimized using the HSOS algorithm with respect to αβ and tT variables e algorithm was performed with 1000tness evaluations 30 independent runs and 50 eco-sizese best result has been taken among these 30 results isoptimized single seismic bearing capacity coecient (Nγe) ispresented in Tables 1 and 2 for static and seismic conditions(kh 01 02 and 03) respectively which can be used by theeld engineers in earthquake-prone areas for the simulta-neous resistance of unit weight surcharge and cohesion

41Parametric Study In this section a brief parametric studyand a comparative study have been presented e epoundect ofsoil friction angle (Φ) depth factor (DfB0) cohesion factor(2ccB0) and seismic accelerations (kh and kv) on normalizedreduction factor (NγeNγs) is discussed Normalized reductionfactor (NγeNγs) is the ratio of optimized seismic and staticbearing capacity coecient e variations of parameters areas follows Φ 20deg 30deg and 40deg kh 01 02 and 03 kv 0

kh2 and kh 2ccB0 0 025 and 05 and DfB0 025 07505 and 1 A detailed comparative study with other availableprevious research is also discussed in this section

411 Eect on NceNcs due to Variation of Φ Figure 6 showsthe variations of the normalized reduction factor (NγeNγs)with respect to horizontal seismic acceleration (kh) atdipounderent soil friction angles (Φ 20deg 30deg and 40deg) at

FEs = FEs + 1

Start

Initialized algorithm control parameters setof ecosystem organism and evaluation of

seismic bearing capacity coefficient foreach organism

Main loop

Apply mutualism phase to update eachorganism

Apply commensalism phase to update eachorganism

Apply parasitism phase to update eachorganism

Check stoppingcriteria

YesNo

Return optimumseismic bearing

capacity coefficient

Stop

Apply SQI method to update each organism

Figure 5 Flowchart of the HSOS algorithm

Table 1 Static condition

Φ 2ccB0DfB0

025 05 075 1

20deg0 8349 11756 15087 18377

025 11175 14488 17769 2102905 13886 17155 20407 23649

30deg0 30439 40168 49719 59177

025 35903 45497 54975 6439105 41263 50771 60189 69557

40deg0 14424 17837 21178 24489

025 15748 19117 2245 257505 17047 20394 2371 2701


2ccB0 025 Df 05 and kv kh2 It is seen that thenormalized reduction factor (NγeNγs) increases with in-crease in soil friction angle (Φ) Due to an increase in Φ theinternal resistance of the soil particles will be increasedwhich resembles the fact that there is an increase in theseismic bearing capacity factor

412 Effect on NγeNγs due to Variation of 2ccB0 Figure 7shows the variations of the normalized reduction factor(NγeNγs) with respect to seismic acceleration (kh) at dif-ferent cohesion factors (2ccB0 0 025 and 05) at Φ 30degDfB0 05 and kv kh2 It is seen that the normalizedreduction factor (NγeNγs) increases with an increase in thecohesion factor (2ccB0) Due to an increase in cohesion theseismic bearing capacity factor will be increased as the in-crease in cohesion causes an increase in intermolecular

attraction among the soil particle which offers more re-sistance against the shearing failure of the foundation

413 Effect on NγeNγs due to Variation of DfB0 Figure 8shows the variations of the normalized reduction factor(NγeNγs) with respect to seismic acceleration (kh) for differentdepth factors (DfB0 025 05 and 1) at Φ 30deg 2ccB0

025 and kv kh2 It is seen that the normalized reductionfactor (NγeNγs) increases with an increase in the depth factor(DfB0) Due to an increase in the depth factor (DfB0) sur-charge weight increases which increases the passive resistanceand hence increases the seismic bearing capacity factor

414 Effect on NγeNγs due to Variation of Seismic Accel-erations (kh and kv) From Figures 6ndash9 it is seen that thenormalized reduction factor (NγeNγs) decreases along

Table 2 Pseudodynamic bearing capacity coefficient (Nγe) for kh 01 02 and 03

kh 01

Φ 2ccB0

kv 0 kv kh2 kv khDfB0

025 05 075 1 025 05 075 1 025 05 075 1

20deg0 5881 8559 11172 13753 5882 8538 11128 13687 5878 851 11079 13614

025 8589 1117 1373 16277 869 11247 13782 16303 8797 11323 13832 1632905 11146 13688 16224 18754 11331 13851 16359 18865 11523 14015 16502 1898

30deg0 2198 29575 37021 44385 22107 29638 37028 44331 22223 29691 3707 44243

025 2685 34306 41675 48996 27128 34529 41834 49081 27415 34739 41979 4916905 3159 38965 46287 53564 32016 39325 46573 53793 32447 39688 46879 54028

40deg0 10141 12717 15255 17772 10232 12786 15302 17789 10317 12849 15341 17811

025 11236 1379 16311 18816 11351 13882 16385 18869 11474 13983 16461 1891605 12312 1485 17367 19858 12463 14979 17465 19942 12615 15103 17572 20021

kh 02

Φ 2ccB0kv 0 kv kh2 kv kh

DfB0025 05 075 1 025 05 075 1 025 05 075 1

20deg0 3627 5554 7437 9299 3495 5257 7019 8781 345 5049 6644 824

025 6254 8093 9926 11754 6276 801 9739 11464 6345 794 9535 1113105 8672 10493 12312 14129 8889 10605 12321 14036 9201 10801 12394 13987

30deg0 14856 20524 26077 3156 14451 19881 25193 3044 1388 19024 24045 29002

025 19156 24697 30169 35606 19026 24314 29539 34734 18803 23785 28716 3362105 23312 28779 34201 39615 23416 28634 33811 3898 23505 28422 33304 38169

40deg0 68468 87465 10613 12461 67336 85521 1034 12114 6574 82948 99846 11659

025 77438 96227 11479 13322 76783 94766 11251 13011 75728 92708 10951 126105 86315 10498 12341 14179 86076 10387 12154 13908 8549 10238 11902 13561

kh 03


DfB0025 05 075 1 025 05 075 1 025 05 075 1

20deg0 2843 4058 5268 6471 4291 5476 6642 7792 mdash mdash mdash mdash

025 5178 6389 7592 8794 9036 10181 11325 12468 mdash mdash mdash mdash05 751 8713 9915 11118 13713 14848 15984 17119 mdash mdash mdash mdash

30deg0 93 13312 1724 21118 8356 11737 15116 18496 815 10853 13557 16261

025 13109 17 20856 24684 12336 15716 19085 22433 12366 15074 17783 2049205 16736 2057 24389 28195 16285 19633 2296 26286 16541 1925 21958 24667

40deg0 44498 58106 71465 84714 40605 52714 64622 76375 35 45195 55199 65068

025 51859 65283 78574 917 48428 60359 72112 83809 43394 53371 63261 7302105 59 72396 8556 98686 56083 6785 79556 91165 51543 61416 71164 80912


with an increase in horizontal seismic acceleration (kh) AndFigure 9 shows the variations of the normalized reductionfactor (NγeNγs) with respect to seismic acceleration (kh) atdipounderent vertical seismic accelerations (kv 0 kh2 and kh) forΦ 30deg Df 05 and 2ccB0 025 It is seen that the nor-malized reduction factor (NγeNγs) decreases with the increase

in vertical seismic acceleration (kv) also Due to an increase inseismic acceleration and due to the sudden movement ofdipounderent waves the disturbance in the soil particles increaseswhich allows more soil mass to participate in the vibrationand hence decreases its resistance against bearing capacity

03

046

062

078

01 02 03

Nγe

Nγs

= 20 = 30 = 40

kh

Figure 6 e variations of the bearing capacity coecient withrespect to seismic acceleration (kh) at dipounderent soil friction angles(Φ 20deg 30deg and 40deg) for 2cB0 025 DfB0 05 and kv kh2

028

041

054

067

08

01 02 03

NγeN

γs

kh

2cB0 = 02cB0 = 0252cB0 = 05

Figure 7 e variations of the bearing capacity coecient withrespect to seismic acceleration (kh) at dipounderent cohesion factors(2cB0 0 025 and 05) for Φ 30deg δΦ2 DfB0 05 andkv kh2

NγeN

γs

kh

033

044

055

066

077

01 02 03

DfB0 = 025DfB0 = 05DfB0 = 1

Figure 8 e variations of the bearing capacity coecient withrespect to seismic acceleration (kh) at dipounderent depth factors(DfB0 025 05 and 1) for Φ 30deg δ Φ2 2cB0 025 andkv kh2NγeN

γs

kh

032

043

054

065

076

01 02 03

kv = 0kv = kh2kv = kh

Figure 9 e variations of the bearing capacity coecient withrespect to seismic acceleration (kh) at dipounderent vertical seismic ac-celerations (kv 0 kh2 and kh) for Φ 30deg 2cB0 025 DfB0 05and δ Φ2


415 Comparison of Result A detailed comparative study ofthe present analysis with previous research on similar type ofworks with dipounderent approaches is done here Figure 10 andTable 3 show the comparison of a pseudodynamic bearingcapacity coecient obtained from the present analysis withprevious seismic analyses with respect to dipounderent seismicaccelerations (kh 01 02 and 03) forΦ 30deg It is seen thatfor the lower value of seismic accelerations here in Figure 10kh 02 the values obtained from the present study are lessthan the values obtained from Soubra [10] (M1 andM2) [17]But when horizontal seismic acceleration increases from 02the bearing capacity coecient also increases gradually andat kh 03 the present analysis provides greater value incomparison to all the comparedmethods At kh 01 around75 24 and 29 decrease in Nγe coecient and atkh 02 around 2 15 and 12 decrease inNγe coecientin comparison to that in Soubra [10] (M1 and M2) andGhosh [17] respectively But at kh 03 it increases around26 16 and 48 respectively in comparison with therespective analyses

e performance results that is pseudodynamic bearingcapacity coecients obtained by the HSOS algorithm arecompared with other metaheuristic optimization algo-rithms Table 4 shows the performance result obtained byDE [45] PSO [46] ABC [47] HS [48] BSA [49] ABSA [50]SOS [33] and HSOS [39] algorithms at dipounderent condi-tions that are compared here From this table it is ob-served that the performance result that is pseudodynamicbearing capacity coecient (Nγe) obtained from this HSOSalgorithm is lesser than the other compared algorithmsin dipounderent soil and seismic conditions From the aboveinvestigations it can be said that HSOS algorithm cansatisfactorily be used to evaluate the seismic bearing capacityof shallow strip footing suggested here

5 Numerical Analysis

e numerical modeling of dynamic analysis of shallowstrip footing is performed using a nite element softwarePLAXIS 2D (v-86) which is equipped with features to dealwith various aspects of complex structures and study thesoil-structure interaction epoundect In addition to static loadsthe dynamic module of PLAXIS also provides a powerfultool for modeling the dynamic response of a soil structureduring an earthquake

51 Numerical Modeling A two-dimensional geometricalmodel is prepared that is to be composed of points lines andother components in the x-y plane e PLAXIS meshgenerator based on the input of the geometry model au-tomatically performs the generation of a mesh at an elementlevele shallow strip footing wasmodeled as a plane strainand 15 nodded triangular elements are used to simulate thefoundation soil e extension of the mesh was taken 100mwide and 30m depth as the earthquake forces cannot apoundectthe vertical boundaries Standard earthquake boundaries areapplied for earthquake loading conditions using SMC lesand then the mesh is generated Cluster renement of the

mesh is followed to obtain precise medium-sized mesh HSsmall model was used to incorporate dynamic soil propertiesof the soil samples Two dipounderent soil samples were used toanalyze the shallow strip footing under seismic loadingcondition as shown in Table 5 A uniformly distributed loadof 100 kNm applying on the section of the foundation alongwith dipounderent surcharge loads to represent the load comingfrom superstructure is analyzed in this paper as shown inFigure 11 Initial stresses are generated after turning opound theinitial pore water pressure tool

52 Calculation During the calculation stage three stepsare adopted where in the rst step calculations are donefor plastic analysis where applied vertical load and weightof soil are activated In the second step calculations aremade for dynamic analysis where earthquake data areincorporated as SMS le And in the nal step FOS isdetermined by the c-Φ reduction method El Salvador 2001earthquake data (moment magnitudeMw 76) are givenas input in the dynamic calculation as SMC le as shownin Figure 12 e vertical settlement of the foundation andthe corresponding factor at safety of each condition ob-tained from the numerical modeling are obtained Figures13 and 14 show the deformed mesh and vertical dis-placement contour respectively after undergoing stagedcalculations

53 Numerical Validation Finite element model of shallowstrip footing embedded in c-Φ soil is analyzed in PLAXIS-86v for the validation of the analytical solution e resultsobtained from this analytical analysis are compared with thenumerical solutions to validate the analysis At rst

3

9

15

21

01 02 03

Ghosh [16]Soubra [9] (M2)Soubra [9] (M1)Present study

Nγe

kh

Figure 10 e comparison of pseudodynamic bearing capacitycoecient obtained from present analysis with previous seismicanalyses with respect to dipounderent seismic accelerations (kh 01 02and 03) for Φ 30deg


settlement of foundation is calculated analytically using twoclassical equations such as

Richards et alrsquos [12] seismic settlement equation

se 0174V2

Ag

kh

A1113890 1113891

minus4

tan αAE (45)

where V is the peak velocity for the design earthquake(msec) A is the acceleration coefficient for the designearthquake and g is the acceleration due to gravityand the value of αAE depends on Φ and critical acceler-ation khlowast

Terzaghirsquos [2] immediate settlement equation

si qnB01minus ]2( 1113857

EIf (46)

where qn is the net foundation pressure qn lfloorQult minus cDfrfloorFOS ] is the Poisson ratio E is the Young modulus of soil andIf is the influence factor for shallow strip footing Here Qult isthe pseudodynamic ultimate bearing capacity which is ob-tained from (39)

e dynamic soil properties taken in numerical modeling[Plaxis-86v] are used same in the analytical formulation to

validate it Results obtained from the analytical solutionand numerical modeling have been tabulated in Table 6Two different types of soil models have been analyzedSettlement of shallow foundation for the correspondingsoil model is calculated using (41) and (42) Settlementvalues obtained from the finite element model in PLAXISare also tabulated It is seen that settlement obtainedfrom analytical solution is slightly in lower side in com-parison with the settlement obtained from PLAXIS-86v asin the analytical settlement calculation the only initialsettlement is considered So the formulation of pseudo-dynamic bearing capacity is well justified after the nu-merical validation

6 Conclusion

Using the pseudodynamic approach the effect of the shearwave and primary wave velocities traveling through the soillayer and the time and phase difference along with thehorizontal and vertical seismic accelerations are used toevaluate the seismic bearing capacity of the shallow stripfooting A mathematical formulation is suggested for si-multaneous resistance of unit weight surcharge and cohesion

Table 4 Comparison of seismic bearing capacity coefficient (Nγe) obtained by different standard algorithms

Φ kh DE PSO ABC HS BSA ABSA SOS HSOS(a) 2ccB0 025 DfB0 05 and kv kh2

20deg 01 1154 11771 11255 1162 1125 11284 11248 1124702 8714 8524 811 873 81 851 802 801

30deg 01 34681 34854 34535 34942 3453 34591 3453 3452902 24514 24641 24319 2443 24319 24361 2432 24314

40deg 01 13890 13912 13885 13954 13883 13899 1389 1388202 94768 95546 9478 9489 9477 9482 9477 94766

(b) 2ccB0 05 DfB0 075 and kv kh2

20deg 01 1646 16363 16359 16512 1637 1645 16361 1635902 1253 12325 12581 12524 1233 12812 12324 12321

30deg 01 4691 46579 46942 4676 4659 46751 46575 4657302 33931 33823 3415 3399 3383 3397 33816 33811

40deg 01 17476 17468 174691 17493 17467 17481 17469 1746502 12169 12159 12161 12159 12158 121561 12158 12154

Table 5 HS small model soil parameters for PLAXIS-86v

Sample c (kPa) c (kPa) Φ ψ ] E50ref (kPa) Eoedref (kPa) Eurref (kPa) m G0 (kPa) c07

S1 209 05 32 2 02 100E+ 4 100E+ 4 300E+ 4 05 100E+ 5 100Eminus 4S2 199 02 28 0 02 125E+ 4 100E+ 4 375E+ 4 05 130E+ 5 125Eminus 4

Table 3 Comparison of seismic bearing capacity coefficient (Nγe) for different values of kh and kv with Φ 30deg

khPresent study Ghosh [17] Budhu and Al-Karni

[7]Choudhury andSubba Rao [13] Soubra [10]

kv kh2 kv kh kv kh2 kv kh kv kh2 kv kh kv kh2 kv kh M1 M201 1443 1423 2039 2004 1021 946 84 776 156 18902 878 865 998 882 381 286 285 2 89 10303 568 567 385 235 121 056 098 029 45 49


using upper-bound limit analysis method A compositefailure mechanism which includes both planer and log-spiralzone is considered here to develop this mathematical modelfor the shallow strip footing resting on c-Φ soil e HSOSalgorithm is used to solve this problem e advantage ofthis HSOS algorithm is that it can improve the searchingcapability of the algorithm for attaining the global opti-mum From the comparison the results obtained by theHSOS algorithm with other standard algorithms show theacceptability of the results in all soil and seismic conditionsHence using the HSOS algorithm the coecient of seismicbearing capacity is presented in a tabular form Numericalmodeling of shallow strip footing is also analyzed usingPLAXIS-86v software for the validation of the analyticalsolution It is observed that the results obtained from this

analytical analysis are well justied with the numericalsolutions e epoundect of various parameters such as soilfriction angle (Φ) seismic accelerations (kh and kv) co-hesion factor (2ccB0) and depth factor (DfB0) is studiedhere It is seen that the pseudodynamic bearing capacitycoecient (Nγe) increases with the increase in Φ 2ccB0and DfB0 but it decreases with the increase in horizontaland vertical seismic accelerations (kh and kv) e valuesobtained from the present analysis are thoroughly com-pared with the available pseudostatic analysis as well aspseudodynamic analysis values and it is seen that thevalues obtained from the present study are comparablereasonably Using the values as provided in Tables 1 and 2the ultimate bearing capacity of foundation under seismicloading condition can be evaluated

000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000

Figure 11 Finite element geometry and foundation load along with surcharge load

700EL salvador earthquake (13-01-2001 173300)

600500400300200100

minus100minus200minus300minus400minus500minus600

Moment magnitude 760Surface-wave magnitude 780Local magnitude NA

Epicentral distance 9800 kmPeak value 76140 cms2

Sample rate 20000 Hz

Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0

Figure 12 El Salvador 2001 earthquake data


000 1000 2000 3000

3000

2000

1000

000

Deformed meshExtreme total displacement 495710minus3 m

(displacements scaled up 10000 times)Plane strain

4000 5000 6000 7000 8000 9000 10000 11000

Figure 13 e deformed mesh of model after calculation

000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000

Vertical displacements (Uy)Extreme Uy minus495710minus3 m

Plane strain(31000 34300)

Figure 14 Vertical displacement contour after calculation

Table 6 Comparison of settlements obtained from numerical and analytical analyses

Soil samples Depth factor (DfB0)Numerical solution Analytical solution

PLAXIS-86v Richards et al [12] Present analysis Terzaghi [2]FOS Settlement (mm) klowasth Settlement (mm) Nγe Settlement (mm)

Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature

2ccB0 Cohesion factorB0 Width of the footingC Cohesion of soilDf Depth of footing below ground surfaceDf B0 Depth factorg Acceleration due to gravityG Shear modulus of soilkh kv Horizontal and vertical seismic accelerationsNc Nq Nc Bearing capacity coefficientsNγe Optimized single seismic bearing capacity

coefficientNγs Optimized single static bearing capacity

coefficientNγeNγs Normalized reduction factorPL Uniformly distributed column loadq Surcharge loadingsr0 r Initial and final radii of the log-spiral zone

(ie BE and BD) respectivelyt Time of vibrationT Period of lateral shakingV1 V2 and Vθ Absolute and relative velocities

respectivelyVp Primary wave velocityVs Shear wave velocityα1 α2 Base angles of triangular elastic zone under

the foundationβ Angle that makes the log-spiral part in log-

spiral mechanismc Unit weight of soil mediumλ η Lamersquos constantυ Poissonrsquos ratio of the soil mediumΦ Angle of internal friction of the soilω Angular frequencySOS Symbiosis organisms searchSQI Simple quadratic interpolationHSOS Hybrid symbiosis organisms search

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] L Prandtl ldquoUber die Eindringungstestigkeit PlastisherBaustoffe und Die Festigkeit von Schneidenrdquo Zeitschrift FurAngewandte Mathematik und Mechanik vol 1 no 1pp 15ndash30 1921 (in German)

[2] K Terzaghi eoretical Soil Mechanics John Wiley amp SonsNewYork NY USA 1943

[3] G G Meyerhof ldquoe ultimate bearing capacity of founda-tions on slopesrdquo in Proceedings of the 4th InternationalConference on Soil Mechanics and Foundation Engineeringpp 384ndash386 London UK August 1957

[4] G G Meyerhof ldquoSome recent research on the bearing ca-pacity of foundationsrdquo Canadian Geotechnical Journal vol 1no 1 pp 16ndash26 1963

[5] A S Vesic ldquoAnalysis of ultimate loads of shallow foundationsrdquoJournal of the Soil Mechanics and Foundations Division vol 99no 1 pp 45ndash43 1973

[6] S Saran and R K Agarwal ldquoBearing capacity of eccentricallyobliquely loaded footingrdquo Journal of Geotechnical Engineer-ing vol 117 no 11 pp 1669ndash1690 1991

[7] M Budhu and A Al-Karni ldquoSeismic bearing capacity ofsoilsrdquo Geotechnique vol 43 no 1 pp 181ndash187 1993

[8] L Dormieux and A Pecker ldquoSeismic bearing capacity offoundation on cohesionless soilrdquo Journal of GeotechnicalEngineering vol 121 no 3 pp 300ndash303 1995

[9] A H Soubra ldquoDiscussion of ldquoseismic bearing capacity andsettlements of foundationsrdquordquo Journal of Geotechnical Engineeringvol 120 no 9 pp 1634ndash1636 1994

[10] A H Soubra ldquoSeismic bearing capacity of shallow stripfootings in seismic conditionsrdquo Proceedings of the Institutionof Civil EngineersndashGeotechnical Engineering vol 125 no 4pp 230ndash241 1997

[11] A H Soubra ldquoUpper bound solutions for bearing capacity offoundationsrdquo Journal of Geotechnical and GeoenvironmentalEngineering vol 125 no 1 pp 59ndash69 1999

[12] R Richards D G Elms and M Budhu ldquoSeismic bearingcapacity and settlements of foundationsrdquo Journal of Geo-technical Engineering vol 119 no 4 pp 662ndash674 1993

[13] D Choudhury and K S Subba Rao ldquoSeismic bearing ca-pacity of shallow strip footingsrdquo Geotechnical and GeologicalEngineering vol 23 no 4 pp 403ndash418 2005

[14] J Kumar and P Ghosh ldquoSeismic bearing capacity for em-bedded footing on sloping groundrdquo Geotechnique vol 56no 2 pp 133ndash140 2006

[15] A H Shafiee and M Jahanandish ldquoSeismic bearing capacityfactors for strip footingsrdquo in Proceedings of the NationalCongress on Civil Engineering Ferdowsi University ofMashhad Mashhad Iran 2010

[16] D Chakraborty and J Kumar ldquoSeismic bearing capacity ofshallow embedded foundations on a sloping ground surfacerdquoInternational Journal of Geomechanics vol 15 no 1p 04014035 2014

[17] P Ghosh ldquoUpper bound solutions of bearing capacity of stripfooting by pseudo-dynamic approachrdquo Acta Geotechnicavol 3 pp 115ndash123 2008

[18] A Saha and S Ghosh ldquoPseudo-dynamic analysis for bearingcapacity of foundation resting on c-Φ soilrdquo InternationalJournal of Geotechnical Engineering vol 9 no 4 pp 379ndash3872014

[19] A Sengupta and A Upadhyay ldquoLocating the critical failuresurface in a slope stability analysis by genetic algorithmrdquoApplied Soft Computing vol 9 no 1 pp 387ndash392 2009

[20] Y M Cheng L Chi S Li and W B Wei ldquoParticle swarmoptimization algorithm for location of critical non-circularfailure surface in two-dimensional slope stability analysisrdquoComputer and Geotechnics vol 34 no 2 pp 92ndash103 2007

[21] S K Das ldquoSlope stability analysis using genetic algorithmrdquoElectronic Journal of Geotechnical Engineering vol 10A2005

[22] K Deb and M Goyal ldquoOptimizing engineering designsusing combined genetic searchrdquo in Proceedings of SeventhInternational Conference on Genetic Algorithms pp 512ndash528Michigan State University East Lansing MI 1997

[23] A H Gandomi A R Kashani M Mousavi andM Jalalvandi ldquoSlope stability analyzing using recent swarmintelligence techniquesrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 39 no 3pp 295ndash309 2015


[24] A T C Goh ldquoGenetic algorithm search for critical slipsurface in multiple-wedge stability analysisrdquo CanadianGeotechnical Journal vol 36 pp 382ndash391 1999

[25] K S Kahatadeniya P Nanakorn and K M NeaupaneldquoDetermination of the critical failure surface for slope stabilityanalysis using ant colony optimizationrdquo Engineering Geologyvol 108 no 1-2 pp 133ndash141 2009

[26] Y-C Li Y-M Chen T L T Zhan D-S Ling and P J CleallldquoAn efficient approach for locating the critical slip surface inslope stability analysis using a real-coded genetic algorithmrdquoCanadian Geotechnical Journal vol 47 no 7 pp 806ndash8202010

[27] P McCombie and P Wilkinson ldquoe use of the simple ge-netic algorithm in finding the critical factor of safety in slopestability analysisrdquo Computers and Geotechnics vol 29 no 8pp 699ndash714 2002

[28] A R Zolfaghari A C Heath and P F McCombie ldquoSimplegenetic algorithm search for critical non-circular failuresurface in slope stability analysisrdquo Computer and Geotechnicsvol 32 no 3 pp 139ndash152 2005

[29] M Ghazavi and S Bazzazian Bonab Optimization of rein-forced concrete retaining walls using ant colony method inProceedings of the 3rd International Symposium on GeotechnicalSafety and Risk (ISGSR 2011) N Vogt B Schuppener DStraub and G Brau Eds Bundesanstalt fur WasserbauMunich Germany June 2011

[30] S Nama A K Saha and S Ghosh ldquoParameters optimizationof geotechnical problem using different optimization algo-rithmrdquoGeotechnical and Geological Engineering vol 33 no 5pp 1235ndash1253 2015

[31] S Nama A K Saha and S Ghosh ldquoImproved backtrackingsearch algorithm for pseudo dynamic active earth pressure onretaining wall supporting c-Φ backfillrdquo Applied Soft ComputingJournal vol 52 pp 885ndash897 2017

[32] L Li and F Liu Group Search Optimization for Applications inStructural Design Springer-Verlag Berlin Heidelberg Germany2011

[33] M-Y Cheng and D Prayogo ldquoSymbiotic organisms searcha new metaheuristic optimization algorithmrdquo Computers ampStructures vol 139 pp 98ndash112 2014

[34] S Nama A K Saha and S Ghosh ldquoImproved symbioticorganisms search algorithm for solving unconstrainedfunction optimizationrdquo Decision Science Letters vol 5no 2016 pp 361ndash380 2016

[35] M Abdullahi M A Ngadi and S M Abdulhamid ldquoSym-biotic organism search optimization based task scheduling incloud computing environmentrdquo Future Generation ComputerSystems vol 56 pp 640ndash650 2016

[36] G G Tejani V J Savsanin and V K Patel ldquoAdaptivesymbiotic organisms search (SOS) algorithm for structuraldesign optimizationrdquo Journal of Computational Design andEngineering vol 3 no 3 pp 226ndash249 2016

[37] M Y Cheng D Prayogo and D H Tran ldquoOptimizingmultiple-resources leveling in multiple projects using discretesymbiotic organisms searchrdquo Journal of Computing in CivilEngineering vol 30 no 3 2016

[38] D Prasad and V Mukherjee ldquoA novel symbiotic organ-isms search algorithm for optimal power flow of powersystem with FACTS devicesrdquo Engineering Science andTechnology An International Journal vol 19 no 1pp 79ndash89 2016

[39] S Nama A K Saha and S Ghosh ldquoA hybrid symbiosisorganisms search algorithm and its application to real worldproblemsrdquoMemetic Computing vol 9 no 3 pp 261ndash280 2017

[40] K Deep and K N Das ldquoQuadratic approximation basedhybrid genetic algorithm for function optimizationrdquo AppliedMathematics and Computation vol 203 no 1 pp 86ndash982008

[41] A Saha and S Ghosh ldquoPseudo-dynamic bearing capacity ofshallow strip footing resting on c-Φ soil considering com-posite failure surface bearing capacity analysis using pseudo-dynamic methodrdquo International Journal of GeotechnicalEarthquake Engineering vol 6 no 2 pp 12ndash34 2015

[42] W F Chen and X L Liu Limit Analysis in Soil MechanicsElsevier Science Publications Company New York NY USA1990

[43] M Crepinsek S H Liu and M Mernik ldquoExploration andexploitation in evolutionary algorithms a surveyrdquo ACMComputing Surveys vol 45 no 3 p 33 2013

[44] W Gong Z Cai and C X Ling ldquoDEBBO a hybrid dif-ferential evolution with biogeography-based optimization forglobal numerical optimizationrdquo Soft Computing vol 15pp 645ndash665 2011

[45] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4pp 341ndash359 1997

[46] Y Shi and R Eberhart ldquoA modified particle swarm optimizerrdquoin Proceedings of the IEEE World Congress on ComputationalIntelligence Evolutionary Computation Anchorage AK USA1998

[47] D Karaboga and B Basturk ldquoA powerful and efficient al-gorithm for numerical function optimization artificial beecolony (ABC) algorithmrdquo Journal of Global Optimizationvol 39 no 3 pp 459ndash471 2007

[48] M Mahdavi M Fesanghary and E Damangir ldquoAn improvedharmony search algorithm for solving optimization problemsrdquoApplied Mathematics and Computation vol 188 no 2pp 1567ndash1579 2007

[49] P Civicioglu ldquoBacktracking search optimization algorithmfor numerical optimization problemsrdquo Applied Mathematicsand Computation vol 219 no 15 pp 8121ndash8144 2013

[50] H Duan and Q Luo ldquoAdaptive backtracking search algo-rithm for induction magnetometer optimizationrdquo IEEETransactions on Magnetics vol 50 no 12 pp 1ndash6 2014


International Journal of

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Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Journal of

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Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

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Control Scienceand Engineering

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Journal ofEngineeringVolume 2018

SensorsJournal of



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Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


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wwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom

evaluated pseudodynamic bearing capacity using limitanalysis method and limit equilibrium method respectivelyconsidering the linear failure surface In the earlier analysesthe resistance of unit weight surcharge and cohesion isconsidered separatelyerefore if the solution was done forshallow strip footing resting on c-Φ soil there will be threeseparate coefficients one for unit weight another for sur-charge and the other for cohesion But in a practical situ-ation there will be a single failure mechanism for thesimultaneous resistance of unit weight surcharge and co-hesion us an attempt is made to present a single seismicbearing capacity coefficient for the simultaneous resistanceof unit weight surcharge and cohesion Here in this paperthe pseudodynamic bearing capacity of shallow strip footingconsidering composite failure mechanism resting on c-Φ soilis solved using the upper-bound limit analysis method Arelative ease in solving geometrically complex multidi-mensional problem renders limit analysis attractive as analternative to numerical codes e kinematic method oflimit analysis hinges on constructing a velocity field that isadmissible for a rigid-perfect plastic material obeying theassociative flow rule

Nowadays nature-based global optimization algo-rithms such as genetic algorithms (GA) particle swarmoptimization (PSO) algorithm and many other algorithmshave been successfully applied to solve different scienceand engineering complex optimization problems espe-cially civil engineering problems such as slope stability[19 20 21ndash28] retaining walls [29ndash31] and structuraldesign [32] Cheng and Prayogo [33] introduced a newnature-based optimization technique called symbioticorganisms search (SOS) algorithm is technique is basedon the interactive relationship among the organism in theecosystem It has no algorithm-specific control parameterse SOS algorithm has been successfully applied to solvedifferent engineering optimization problems [34ndash38] Re-cently Nama et al [39] proposed a hybrid algorithm calledhybrid symbiotic organisms search (HSOS) algorithmwhich is the combination of SOS algorithm and simplequadratic interpolation method [40] Here in this paperHSOS algorithm is used to optimise the pseudodynamicbearing capacity of shallow strip footing considering upperbound limit analysis method Mathematically the problemcan be represented as a nonlinear hard optimizationproblem which can be solved by the HSOS algorithmwhich is found to be a more satisfactory optimum solutionand can be used for designing the shallow strip footing Inthe HSOS algorithm failure surface angle (α β) and tT areconsidered as the search variables So it can be applied toobtain optimal solutions in the different fields of scienceand engineering Numerical analysis is also done usingdynamic module of PLAXIS-86v software to validate thisanalytical solution Results are presented in tabular formincluding comparison with other available analyses Effectsof a wide range of variation of parameters like soil frictionangle (Φ) cohesion factor (2ccB0) depth factor (DfB0)and horizontal and vertical seismic accelerations (kh kv) onthe normalized reduction factor (NγeNγs) have beenstudied

erefore the main contributions of this paper aresummarized as follows

(i) Evaluation of pseudodynamic bearing capacitycoefficient of shallow strip footing resting on c-Φsoil considering composite failure surface usingupper bound limit analysis method

(ii) A single pseudodynamic bearing capacity coefficientis presented here considering the simultaneousresistance of unit weight surcharge and cohesion

(iii) A recent hybrid optimization algorithm (calledHSOS) is used to solve the pseudodynamic bearingcapacity minimization optimization problem

(iv) PLAXIS-86v software is used to solve this above-mentioned problem numerically for the validationof the analytical formulation

(v) e obtained results are compared with the otherresults which are available in literature and theresults obtained by other state-of-the-artalgorithms

e remaining part of the paper is organized as followsSection 2 discusses the formulation of the real-world geo-technical earthquake engineering optimization problemsuch as the pseudodynamic bearing capacity of a shallowfoundation e overview of the optimization algorithmHSOS is presented in Section 3 Section 4 presents discus-sions of the results obtained by the HSOS algorithm to showthe efficiency and accuracy of this hybrid algorithm forsolving this engineering optimization problem Numericalanalysis of shallow strip footing using the dynamic moduleof PLAXIS-86v software and the validation of analyticalformulation are discussed in Section 5 and finally Section 6presents the conclusion and the summary of the outcome ofthe paper

2 Formulation of Pseudodynamic BearingCapacity Coefficient

21 Consideration of Model Let us consider a shallow stripfooting of width (B0) resting below the ground surface ata depth of Df over which a load (P) of column acts ehomogeneous soil of effective unit weight c has MohrndashCoulomb characteristic c-Φ and can be considered asa rigid plastic body For shallow foundation (D f leB0) theoverburden pressure is idealized as a surcharge (q cDf)which acts over the length of BC e classical two-dimensional slip line field obtained by Prandtl [1] is thetraditional failure mechanism which has three regionssuch as active zone passive zone and logarithmic radial-fan transition zone In this composite failure mechanismhalf of the failure is assumed to occur along the surfaceAEDC which is composed of a triangular elastic zoneABE triangular passive Rankine zone BDC and in be-tween them a log spiral radial shear zone BDE shown inFigure 1(a) [41] It is a composite mechanism that is de-fined by the angular parameters α and β in which the log-spiral slip surface ED is a tangent to lines AE and DC at Eand D respectively Figures 2 and 3 show the detailed free





A

Zone I

BC

P

Df q = γDf

Zone IIZone III

V

V1

V1

V2

Vs Vp D

E

B0

(a)

V

V1

V2

( minus + )

( minus )


( minus )

(b)


WminusQv

Qh

AB

E

c

PL

dz

Vs Vp

z

B0








W c

2B20

sin α sin ρcos ϕ

(1)




Vs( ) (2)


Vp( ) (3)


m(z) c

g

1tan α

+1



Qh ckh1

tan α+

1tan ρ

( )intr0 sin ρ


r0 sin ρminus zVs

( )dz


1tan α

+1

tan ρ( ) cos 2π

t

Tminusr0 sin ρ

λ( ) +

r0 sin ρ2πλ

sin 2πt

Tminusr0 sin ρ

λ( )minus sin 2π

t

T [ ]

c

2B20qh

(5)

Qv ckv1

tan α+

1tan ρ

( )intr0 sin ρ


r0 sin ρminus zVp

( )dz

cB2



tan α+

1tan ρ

( ) cos 2πt

Tminusr0 sin ρ

η( ) +

r0 sin ρ2πη

sin 2πt

Tminusr0 sin ρ

η( )minus sin 2π

t

T [ ]

c

2B20qv

(6)

d

q = Dfq = Df

Qh1

W1minusQv1

-

D

B

E

c

C

dz1

c

Vs Vp

z1

Qh2

W2minusQv2




W1 c

2B20e


cosϕ sin ξ (7)


m z1( 1113857 c


1tan(βminus α)

1113896 11138971113890





Vs

1113888 1113889

(9)



Vp

1113888 1113889

(10)



tan(β minus α)1113896 11138971113946


0r0e



Vs

1113888 11138891113890 1113891dz1

cB2



1tan(β minus α)

1113896 11138971113890cos2πt

Tminus


λ1113888 1113889

+r0e


2πλsin2π

t

Tminus


λ1113888 1113889 minus sin2π

t

T1113896 11138971113891

c

2B20qh1

(11)


tan(β minus α)1113896 11138971113946


0r0e



Vp

1113888 11138891113890 1113891dz1

cB2



1tan(β minus α)

1113896 11138971113890cos2πt

Tminus


η1113888 1113889

+r0e


2πηsin2π

t

Tminus


η1113888 1113889 minus sin2π

t

T1113896 11138971113891

c

2B20qv1

(12)


W2 12

1113946β

0r20dθ


4 tanϕ


2 sin 2ϕ

(13)



m(z)i c

g


1113946Hi

0zi(dz)i (14)




Vs

1113888 1113889 (15)


Vp

1113888 1113889 (16)



Qh( )i 2πcdβkh


0zi sinω tminus

Hi minus ziVs

( )(dz)i

cdβkhλB2



Hi2πsin 2π

t

Tminus sin 2π

t

TminusHi

λ( ) minus cos 2π

t

T[ ]

c

2B20qh2 e


(17)

Qv( )i 2πcdβkv


0zi sinω tminus

Hi minus ziVp

( )(dz)i

cdβkvηB2



Hi2πsin 2π

t

Tminus sin 2π

t

TminusHi

η( ) minus cos 2π

t

T[ ]

c

2B20qv2 e


(18)


Qh2 sumn

i1Qh( )i

c

2B20qh2sum

n


2

(19)

Qv2 sumn

i1Qv( )i

c

2B20qv2sum

n


2 (20)



TminusH


t

TminusH

η( ) sin (α minus ϕ)[ ] (21)

WABE c

4B0


V1 (22)

WBCD minusc


cosϕ sin ξV1 (23)



t

TminusH


t

TminusH

η( )( ) sin (β minus α + ϕ) [ ]V1

(24)

WBDE minusc

2intβ


minusc

2B20



(25)

d

Hizi

Qh2


ith slice

B

D

E







(27)

WQh2 c

2B20qh2 sum

n

i1e


idβ tanϕ1113872 1113873

21113946

idβ


c

2B20qh2

V1

(1 + tanϕ)sumn

in


+ eidβ tan ϕ

1113872 111387321113882e




(28)

WQv2 c

2B20qv2sum

ni1 e


idβ tan ϕ1113872 1113873

21113946

idβ


c

2B20qv2

V1

1 + tan2 ϕ( 1113857sumn


+ eidβ tan ϕ

1113872 111387321113882e




(29)





sin(βminus α)V1

(31)

DDE c1113946β

0r0dθVθ cosϕ


2 tanϕV1 (32)




Nc sinα2tanϕ

e2βtanϕ


sin(β minus α)1113890 1113891 (34)

Nq sinαe2βtanϕ


t

Tminus

H


t

Tminus

H

η1113888 11138891113888 1113889sin(β minus α+ϕ)1113896 11138971113890 11138911113890 1113891 (35)

Nc 1113890sin2α







minussumn


1+ tan2ϕ( 11138571113876qh2sum

ni11113882e







(36)



PL c

2B0N (37)


N 1113890sin2 α

cos2 ϕ 1 + 9 tan2 ϕ( 11138573 tanϕ e




+ 2Df

B0

sin αe2β tan ϕ


t

Tminus

H

λ1113874 1113875 cos (β minus α + ϕ)1113882 1113883


Tminus

H





2

1 + tan2 ϕ( 11138571113876qh2sum

ni1e




+ qv2sumni11113882e




cB0

sin α2 tanϕ

e2β tan ϕ


sin (β minus α)1113896 11138971113891

(38)


Qult 05cB0Nce (39)











Orgm

i 05Orgm


j1113872 11138732

1113874 1113875fk + Orgmj1113872 1113873

2minus Orgm

k( 11138572

1113874 1113875fi + Orgmk( 1113857


21113872 1113873fj

Orgmi minusOrg




(40)







(41)




Step 2 Main loop





(43)














FEs = FEs + 1

Start



Main loop





YesNo



Stop




Φ 2ccB0DfB0

025 05 075 1

20deg0 8349 11756 15087 18377

025 11175 14488 17769 2102905 13886 17155 20407 23649

30deg0 30439 40168 49719 59177

025 35903 45497 54975 6439105 41263 50771 60189 69557

40deg0 14424 17837 21178 24489

025 15748 19117 2245 257505 17047 20394 2371 2701









kh 01

Φ 2ccB0


025 05 075 1 025 05 075 1 025 05 075 1

20deg0 5881 8559 11172 13753 5882 8538 11128 13687 5878 851 11079 13614

025 8589 1117 1373 16277 869 11247 13782 16303 8797 11323 13832 1632905 11146 13688 16224 18754 11331 13851 16359 18865 11523 14015 16502 1898

30deg0 2198 29575 37021 44385 22107 29638 37028 44331 22223 29691 3707 44243

025 2685 34306 41675 48996 27128 34529 41834 49081 27415 34739 41979 4916905 3159 38965 46287 53564 32016 39325 46573 53793 32447 39688 46879 54028

40deg0 10141 12717 15255 17772 10232 12786 15302 17789 10317 12849 15341 17811

025 11236 1379 16311 18816 11351 13882 16385 18869 11474 13983 16461 1891605 12312 1485 17367 19858 12463 14979 17465 19942 12615 15103 17572 20021

kh 02


DfB0025 05 075 1 025 05 075 1 025 05 075 1

20deg0 3627 5554 7437 9299 3495 5257 7019 8781 345 5049 6644 824

025 6254 8093 9926 11754 6276 801 9739 11464 6345 794 9535 1113105 8672 10493 12312 14129 8889 10605 12321 14036 9201 10801 12394 13987

30deg0 14856 20524 26077 3156 14451 19881 25193 3044 1388 19024 24045 29002

025 19156 24697 30169 35606 19026 24314 29539 34734 18803 23785 28716 3362105 23312 28779 34201 39615 23416 28634 33811 3898 23505 28422 33304 38169

40deg0 68468 87465 10613 12461 67336 85521 1034 12114 6574 82948 99846 11659

025 77438 96227 11479 13322 76783 94766 11251 13011 75728 92708 10951 126105 86315 10498 12341 14179 86076 10387 12154 13908 8549 10238 11902 13561

kh 03


DfB0025 05 075 1 025 05 075 1 025 05 075 1



30deg0 93 13312 1724 21118 8356 11737 15116 18496 815 10853 13557 16261

025 13109 17 20856 24684 12336 15716 19085 22433 12366 15074 17783 2049205 16736 2057 24389 28195 16285 19633 2296 26286 16541 1925 21958 24667

40deg0 44498 58106 71465 84714 40605 52714 64622 76375 35 45195 55199 65068

025 51859 65283 78574 917 48428 60359 72112 83809 43394 53371 63261 7302105 59 72396 8556 98686 56083 6785 79556 91165 51543 61416 71164 80912




03

046

062

078

01 02 03

Nγe

Nγs

= 20 = 30 = 40

kh


028

041

054

067

08

01 02 03

NγeN

γs

kh

2cB0 = 02cB0 = 0252cB0 = 05


NγeN

γs

kh

033

044

055

066

077

01 02 03

DfB0 = 025DfB0 = 05DfB0 = 1


γs

kh

032

043

054

065

076

01 02 03












3

9

15

21

01 02 03


Nγe

kh





se 0174V2

Ag

kh

A1113890 1113891

minus4

tan αAE (45)




EIf (46)




6 Conclusion




20deg 01 1154 11771 11255 1162 1125 11284 11248 1124702 8714 8524 811 873 81 851 802 801

30deg 01 34681 34854 34535 34942 3453 34591 3453 3452902 24514 24641 24319 2443 24319 24361 2432 24314

40deg 01 13890 13912 13885 13954 13883 13899 1389 1388202 94768 95546 9478 9489 9477 9482 9477 94766


20deg 01 1646 16363 16359 16512 1637 1645 16361 1635902 1253 12325 12581 12524 1233 12812 12324 12321

30deg 01 4691 46579 46942 4676 4659 46751 46575 4657302 33931 33823 3415 3399 3383 3397 33816 33811

40deg 01 17476 17468 174691 17493 17467 17481 17469 1746502 12169 12159 12161 12159 12158 121561 12158 12154











000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000



600500400300200100





Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0



000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





A

Zone I

BC

P

Df q = γDf

Zone IIZone III

V

V1

V1

V2

Vs Vp D

E

B0

(a)

V

V1

V2

( minus + )

( minus )


( minus )

(b)


WminusQv

Qh

AB

E

c

PL

dz

Vs Vp

z

B0








W c

2B20

sin α sin ρcos ϕ

(1)




Vs( ) (2)


Vp( ) (3)


m(z) c

g

1tan α

+1



Qh ckh1

tan α+

1tan ρ

( )intr0 sin ρ


r0 sin ρminus zVs

( )dz


1tan α

+1

tan ρ( ) cos 2π

t

Tminusr0 sin ρ

λ( ) +

r0 sin ρ2πλ

sin 2πt

Tminusr0 sin ρ

λ( )minus sin 2π

t

T [ ]

c

2B20qh

(5)

Qv ckv1

tan α+

1tan ρ

( )intr0 sin ρ


r0 sin ρminus zVp

( )dz

cB2



tan α+

1tan ρ

( ) cos 2πt

Tminusr0 sin ρ

η( ) +

r0 sin ρ2πη

sin 2πt

Tminusr0 sin ρ

η( )minus sin 2π

t

T [ ]

c

2B20qv

(6)

d

q = Dfq = Df

Qh1

W1minusQv1

-

D

B

E

c

C

dz1

c

Vs Vp

z1

Qh2

W2minusQv2




W1 c

2B20e


cosϕ sin ξ (7)


m z1( 1113857 c


1tan(βminus α)

1113896 11138971113890





Vs

1113888 1113889

(9)



Vp

1113888 1113889

(10)



tan(β minus α)1113896 11138971113946


0r0e



Vs

1113888 11138891113890 1113891dz1

cB2



1tan(β minus α)

1113896 11138971113890cos2πt

Tminus


λ1113888 1113889

+r0e


2πλsin2π

t

Tminus


λ1113888 1113889 minus sin2π

t

T1113896 11138971113891

c

2B20qh1

(11)


tan(β minus α)1113896 11138971113946


0r0e



Vp

1113888 11138891113890 1113891dz1

cB2



1tan(β minus α)

1113896 11138971113890cos2πt

Tminus


η1113888 1113889

+r0e


2πηsin2π

t

Tminus


η1113888 1113889 minus sin2π

t

T1113896 11138971113891

c

2B20qv1

(12)


W2 12

1113946β

0r20dθ


4 tanϕ


2 sin 2ϕ

(13)



m(z)i c

g


1113946Hi

0zi(dz)i (14)




Vs

1113888 1113889 (15)


Vp

1113888 1113889 (16)



Qh( )i 2πcdβkh


0zi sinω tminus

Hi minus ziVs

( )(dz)i

cdβkhλB2



Hi2πsin 2π

t

Tminus sin 2π

t

TminusHi

λ( ) minus cos 2π

t

T[ ]

c

2B20qh2 e


(17)

Qv( )i 2πcdβkv


0zi sinω tminus

Hi minus ziVp

( )(dz)i

cdβkvηB2



Hi2πsin 2π

t

Tminus sin 2π

t

TminusHi

η( ) minus cos 2π

t

T[ ]

c

2B20qv2 e


(18)


Qh2 sumn

i1Qh( )i

c

2B20qh2sum

n


2

(19)

Qv2 sumn

i1Qv( )i

c

2B20qv2sum

n


2 (20)



TminusH


t

TminusH

η( ) sin (α minus ϕ)[ ] (21)

WABE c

4B0


V1 (22)

WBCD minusc


cosϕ sin ξV1 (23)



t

TminusH


t

TminusH

η( )( ) sin (β minus α + ϕ) [ ]V1

(24)

WBDE minusc

2intβ


minusc

2B20



(25)

d

Hizi

Qh2


ith slice

B

D

E







(27)

WQh2 c

2B20qh2 sum

n

i1e


idβ tanϕ1113872 1113873

21113946

idβ


c

2B20qh2

V1

(1 + tanϕ)sumn

in


+ eidβ tan ϕ

1113872 111387321113882e




(28)

WQv2 c

2B20qv2sum

ni1 e


idβ tan ϕ1113872 1113873

21113946

idβ


c

2B20qv2

V1

1 + tan2 ϕ( 1113857sumn


+ eidβ tan ϕ

1113872 111387321113882e




(29)





sin(βminus α)V1

(31)

DDE c1113946β

0r0dθVθ cosϕ


2 tanϕV1 (32)




Nc sinα2tanϕ

e2βtanϕ


sin(β minus α)1113890 1113891 (34)

Nq sinαe2βtanϕ


t

Tminus

H


t

Tminus

H

η1113888 11138891113888 1113889sin(β minus α+ϕ)1113896 11138971113890 11138911113890 1113891 (35)

Nc 1113890sin2α







minussumn


1+ tan2ϕ( 11138571113876qh2sum

ni11113882e







(36)



PL c

2B0N (37)


N 1113890sin2 α

cos2 ϕ 1 + 9 tan2 ϕ( 11138573 tanϕ e




+ 2Df

B0

sin αe2β tan ϕ


t

Tminus

H

λ1113874 1113875 cos (β minus α + ϕ)1113882 1113883


Tminus

H





2

1 + tan2 ϕ( 11138571113876qh2sum

ni1e




+ qv2sumni11113882e




cB0

sin α2 tanϕ

e2β tan ϕ


sin (β minus α)1113896 11138971113891

(38)


Qult 05cB0Nce (39)











Orgm

i 05Orgm


j1113872 11138732

1113874 1113875fk + Orgmj1113872 1113873

2minus Orgm

k( 11138572

1113874 1113875fi + Orgmk( 1113857


21113872 1113873fj

Orgmi minusOrg




(40)







(41)




Step 2 Main loop





(43)














FEs = FEs + 1

Start



Main loop





YesNo



Stop




Φ 2ccB0DfB0

025 05 075 1

20deg0 8349 11756 15087 18377

025 11175 14488 17769 2102905 13886 17155 20407 23649

30deg0 30439 40168 49719 59177

025 35903 45497 54975 6439105 41263 50771 60189 69557

40deg0 14424 17837 21178 24489

025 15748 19117 2245 257505 17047 20394 2371 2701









kh 01

Φ 2ccB0


025 05 075 1 025 05 075 1 025 05 075 1

20deg0 5881 8559 11172 13753 5882 8538 11128 13687 5878 851 11079 13614

025 8589 1117 1373 16277 869 11247 13782 16303 8797 11323 13832 1632905 11146 13688 16224 18754 11331 13851 16359 18865 11523 14015 16502 1898

30deg0 2198 29575 37021 44385 22107 29638 37028 44331 22223 29691 3707 44243

025 2685 34306 41675 48996 27128 34529 41834 49081 27415 34739 41979 4916905 3159 38965 46287 53564 32016 39325 46573 53793 32447 39688 46879 54028

40deg0 10141 12717 15255 17772 10232 12786 15302 17789 10317 12849 15341 17811

025 11236 1379 16311 18816 11351 13882 16385 18869 11474 13983 16461 1891605 12312 1485 17367 19858 12463 14979 17465 19942 12615 15103 17572 20021

kh 02


DfB0025 05 075 1 025 05 075 1 025 05 075 1

20deg0 3627 5554 7437 9299 3495 5257 7019 8781 345 5049 6644 824

025 6254 8093 9926 11754 6276 801 9739 11464 6345 794 9535 1113105 8672 10493 12312 14129 8889 10605 12321 14036 9201 10801 12394 13987

30deg0 14856 20524 26077 3156 14451 19881 25193 3044 1388 19024 24045 29002

025 19156 24697 30169 35606 19026 24314 29539 34734 18803 23785 28716 3362105 23312 28779 34201 39615 23416 28634 33811 3898 23505 28422 33304 38169

40deg0 68468 87465 10613 12461 67336 85521 1034 12114 6574 82948 99846 11659

025 77438 96227 11479 13322 76783 94766 11251 13011 75728 92708 10951 126105 86315 10498 12341 14179 86076 10387 12154 13908 8549 10238 11902 13561

kh 03


DfB0025 05 075 1 025 05 075 1 025 05 075 1



30deg0 93 13312 1724 21118 8356 11737 15116 18496 815 10853 13557 16261

025 13109 17 20856 24684 12336 15716 19085 22433 12366 15074 17783 2049205 16736 2057 24389 28195 16285 19633 2296 26286 16541 1925 21958 24667

40deg0 44498 58106 71465 84714 40605 52714 64622 76375 35 45195 55199 65068

025 51859 65283 78574 917 48428 60359 72112 83809 43394 53371 63261 7302105 59 72396 8556 98686 56083 6785 79556 91165 51543 61416 71164 80912




03

046

062

078

01 02 03

Nγe

Nγs

= 20 = 30 = 40

kh


028

041

054

067

08

01 02 03

NγeN

γs

kh

2cB0 = 02cB0 = 0252cB0 = 05


NγeN

γs

kh

033

044

055

066

077

01 02 03

DfB0 = 025DfB0 = 05DfB0 = 1


γs

kh

032

043

054

065

076

01 02 03












3

9

15

21

01 02 03


Nγe

kh





se 0174V2

Ag

kh

A1113890 1113891

minus4

tan αAE (45)




EIf (46)




6 Conclusion




20deg 01 1154 11771 11255 1162 1125 11284 11248 1124702 8714 8524 811 873 81 851 802 801

30deg 01 34681 34854 34535 34942 3453 34591 3453 3452902 24514 24641 24319 2443 24319 24361 2432 24314

40deg 01 13890 13912 13885 13954 13883 13899 1389 1388202 94768 95546 9478 9489 9477 9482 9477 94766


20deg 01 1646 16363 16359 16512 1637 1645 16361 1635902 1253 12325 12581 12524 1233 12812 12324 12321

30deg 01 4691 46579 46942 4676 4659 46751 46575 4657302 33931 33823 3415 3399 3383 3397 33816 33811

40deg 01 17476 17468 174691 17493 17467 17481 17469 1746502 12169 12159 12161 12159 12158 121561 12158 12154











000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000



600500400300200100





Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0



000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







W c

2B20

sin α sin ρcos ϕ

(1)




Vs( ) (2)


Vp( ) (3)


m(z) c

g

1tan α

+1



Qh ckh1

tan α+

1tan ρ

( )intr0 sin ρ


r0 sin ρminus zVs

( )dz


1tan α

+1

tan ρ( ) cos 2π

t

Tminusr0 sin ρ

λ( ) +

r0 sin ρ2πλ

sin 2πt

Tminusr0 sin ρ

λ( )minus sin 2π

t

T [ ]

c

2B20qh

(5)

Qv ckv1

tan α+

1tan ρ

( )intr0 sin ρ


r0 sin ρminus zVp

( )dz

cB2



tan α+

1tan ρ

( ) cos 2πt

Tminusr0 sin ρ

η( ) +

r0 sin ρ2πη

sin 2πt

Tminusr0 sin ρ

η( )minus sin 2π

t

T [ ]

c

2B20qv

(6)

d

q = Dfq = Df

Qh1

W1minusQv1

-

D

B

E

c

C

dz1

c

Vs Vp

z1

Qh2

W2minusQv2




W1 c

2B20e


cosϕ sin ξ (7)


m z1( 1113857 c


1tan(βminus α)

1113896 11138971113890





Vs

1113888 1113889

(9)



Vp

1113888 1113889

(10)



tan(β minus α)1113896 11138971113946


0r0e



Vs

1113888 11138891113890 1113891dz1

cB2



1tan(β minus α)

1113896 11138971113890cos2πt

Tminus


λ1113888 1113889

+r0e


2πλsin2π

t

Tminus


λ1113888 1113889 minus sin2π

t

T1113896 11138971113891

c

2B20qh1

(11)


tan(β minus α)1113896 11138971113946


0r0e



Vp

1113888 11138891113890 1113891dz1

cB2



1tan(β minus α)

1113896 11138971113890cos2πt

Tminus


η1113888 1113889

+r0e


2πηsin2π

t

Tminus


η1113888 1113889 minus sin2π

t

T1113896 11138971113891

c

2B20qv1

(12)


W2 12

1113946β

0r20dθ


4 tanϕ


2 sin 2ϕ

(13)



m(z)i c

g


1113946Hi

0zi(dz)i (14)




Vs

1113888 1113889 (15)


Vp

1113888 1113889 (16)



Qh( )i 2πcdβkh


0zi sinω tminus

Hi minus ziVs

( )(dz)i

cdβkhλB2



Hi2πsin 2π

t

Tminus sin 2π

t

TminusHi

λ( ) minus cos 2π

t

T[ ]

c

2B20qh2 e


(17)

Qv( )i 2πcdβkv


0zi sinω tminus

Hi minus ziVp

( )(dz)i

cdβkvηB2



Hi2πsin 2π

t

Tminus sin 2π

t

TminusHi

η( ) minus cos 2π

t

T[ ]

c

2B20qv2 e


(18)


Qh2 sumn

i1Qh( )i

c

2B20qh2sum

n


2

(19)

Qv2 sumn

i1Qv( )i

c

2B20qv2sum

n


2 (20)



TminusH


t

TminusH

η( ) sin (α minus ϕ)[ ] (21)

WABE c

4B0


V1 (22)

WBCD minusc


cosϕ sin ξV1 (23)



t

TminusH


t

TminusH

η( )( ) sin (β minus α + ϕ) [ ]V1

(24)

WBDE minusc

2intβ


minusc

2B20



(25)

d

Hizi

Qh2


ith slice

B

D

E







(27)

WQh2 c

2B20qh2 sum

n

i1e


idβ tanϕ1113872 1113873

21113946

idβ


c

2B20qh2

V1

(1 + tanϕ)sumn

in


+ eidβ tan ϕ

1113872 111387321113882e




(28)

WQv2 c

2B20qv2sum

ni1 e


idβ tan ϕ1113872 1113873

21113946

idβ


c

2B20qv2

V1

1 + tan2 ϕ( 1113857sumn


+ eidβ tan ϕ

1113872 111387321113882e




(29)





sin(βminus α)V1

(31)

DDE c1113946β

0r0dθVθ cosϕ


2 tanϕV1 (32)




Nc sinα2tanϕ

e2βtanϕ


sin(β minus α)1113890 1113891 (34)

Nq sinαe2βtanϕ


t

Tminus

H


t

Tminus

H

η1113888 11138891113888 1113889sin(β minus α+ϕ)1113896 11138971113890 11138911113890 1113891 (35)

Nc 1113890sin2α







minussumn


1+ tan2ϕ( 11138571113876qh2sum

ni11113882e







(36)



PL c

2B0N (37)


N 1113890sin2 α

cos2 ϕ 1 + 9 tan2 ϕ( 11138573 tanϕ e




+ 2Df

B0

sin αe2β tan ϕ


t

Tminus

H

λ1113874 1113875 cos (β minus α + ϕ)1113882 1113883


Tminus

H





2

1 + tan2 ϕ( 11138571113876qh2sum

ni1e




+ qv2sumni11113882e




cB0

sin α2 tanϕ

e2β tan ϕ


sin (β minus α)1113896 11138971113891

(38)


Qult 05cB0Nce (39)











Orgm

i 05Orgm


j1113872 11138732

1113874 1113875fk + Orgmj1113872 1113873

2minus Orgm

k( 11138572

1113874 1113875fi + Orgmk( 1113857


21113872 1113873fj

Orgmi minusOrg




(40)







(41)




Step 2 Main loop





(43)














FEs = FEs + 1

Start



Main loop





YesNo



Stop




Φ 2ccB0DfB0

025 05 075 1

20deg0 8349 11756 15087 18377

025 11175 14488 17769 2102905 13886 17155 20407 23649

30deg0 30439 40168 49719 59177

025 35903 45497 54975 6439105 41263 50771 60189 69557

40deg0 14424 17837 21178 24489

025 15748 19117 2245 257505 17047 20394 2371 2701









kh 01

Φ 2ccB0


025 05 075 1 025 05 075 1 025 05 075 1

20deg0 5881 8559 11172 13753 5882 8538 11128 13687 5878 851 11079 13614

025 8589 1117 1373 16277 869 11247 13782 16303 8797 11323 13832 1632905 11146 13688 16224 18754 11331 13851 16359 18865 11523 14015 16502 1898

30deg0 2198 29575 37021 44385 22107 29638 37028 44331 22223 29691 3707 44243

025 2685 34306 41675 48996 27128 34529 41834 49081 27415 34739 41979 4916905 3159 38965 46287 53564 32016 39325 46573 53793 32447 39688 46879 54028

40deg0 10141 12717 15255 17772 10232 12786 15302 17789 10317 12849 15341 17811

025 11236 1379 16311 18816 11351 13882 16385 18869 11474 13983 16461 1891605 12312 1485 17367 19858 12463 14979 17465 19942 12615 15103 17572 20021

kh 02


DfB0025 05 075 1 025 05 075 1 025 05 075 1

20deg0 3627 5554 7437 9299 3495 5257 7019 8781 345 5049 6644 824

025 6254 8093 9926 11754 6276 801 9739 11464 6345 794 9535 1113105 8672 10493 12312 14129 8889 10605 12321 14036 9201 10801 12394 13987

30deg0 14856 20524 26077 3156 14451 19881 25193 3044 1388 19024 24045 29002

025 19156 24697 30169 35606 19026 24314 29539 34734 18803 23785 28716 3362105 23312 28779 34201 39615 23416 28634 33811 3898 23505 28422 33304 38169

40deg0 68468 87465 10613 12461 67336 85521 1034 12114 6574 82948 99846 11659

025 77438 96227 11479 13322 76783 94766 11251 13011 75728 92708 10951 126105 86315 10498 12341 14179 86076 10387 12154 13908 8549 10238 11902 13561

kh 03


DfB0025 05 075 1 025 05 075 1 025 05 075 1



30deg0 93 13312 1724 21118 8356 11737 15116 18496 815 10853 13557 16261

025 13109 17 20856 24684 12336 15716 19085 22433 12366 15074 17783 2049205 16736 2057 24389 28195 16285 19633 2296 26286 16541 1925 21958 24667

40deg0 44498 58106 71465 84714 40605 52714 64622 76375 35 45195 55199 65068

025 51859 65283 78574 917 48428 60359 72112 83809 43394 53371 63261 7302105 59 72396 8556 98686 56083 6785 79556 91165 51543 61416 71164 80912




03

046

062

078

01 02 03

Nγe

Nγs

= 20 = 30 = 40

kh


028

041

054

067

08

01 02 03

NγeN

γs

kh

2cB0 = 02cB0 = 0252cB0 = 05


NγeN

γs

kh

033

044

055

066

077

01 02 03

DfB0 = 025DfB0 = 05DfB0 = 1


γs

kh

032

043

054

065

076

01 02 03












3

9

15

21

01 02 03


Nγe

kh





se 0174V2

Ag

kh

A1113890 1113891

minus4

tan αAE (45)




EIf (46)




6 Conclusion




20deg 01 1154 11771 11255 1162 1125 11284 11248 1124702 8714 8524 811 873 81 851 802 801

30deg 01 34681 34854 34535 34942 3453 34591 3453 3452902 24514 24641 24319 2443 24319 24361 2432 24314

40deg 01 13890 13912 13885 13954 13883 13899 1389 1388202 94768 95546 9478 9489 9477 9482 9477 94766


20deg 01 1646 16363 16359 16512 1637 1645 16361 1635902 1253 12325 12581 12524 1233 12812 12324 12321

30deg 01 4691 46579 46942 4676 4659 46751 46575 4657302 33931 33823 3415 3399 3383 3397 33816 33811

40deg 01 17476 17468 174691 17493 17467 17481 17469 1746502 12169 12159 12161 12159 12158 121561 12158 12154











000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000



600500400300200100





Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0



000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



W1 c

2B20e


cosϕ sin ξ (7)


m z1( 1113857 c


1tan(βminus α)

1113896 11138971113890





Vs

1113888 1113889

(9)



Vp

1113888 1113889

(10)



tan(β minus α)1113896 11138971113946


0r0e



Vs

1113888 11138891113890 1113891dz1

cB2



1tan(β minus α)

1113896 11138971113890cos2πt

Tminus


λ1113888 1113889

+r0e


2πλsin2π

t

Tminus


λ1113888 1113889 minus sin2π

t

T1113896 11138971113891

c

2B20qh1

(11)


tan(β minus α)1113896 11138971113946


0r0e



Vp

1113888 11138891113890 1113891dz1

cB2



1tan(β minus α)

1113896 11138971113890cos2πt

Tminus


η1113888 1113889

+r0e


2πηsin2π

t

Tminus


η1113888 1113889 minus sin2π

t

T1113896 11138971113891

c

2B20qv1

(12)


W2 12

1113946β

0r20dθ


4 tanϕ


2 sin 2ϕ

(13)



m(z)i c

g


1113946Hi

0zi(dz)i (14)




Vs

1113888 1113889 (15)


Vp

1113888 1113889 (16)



Qh( )i 2πcdβkh


0zi sinω tminus

Hi minus ziVs

( )(dz)i

cdβkhλB2



Hi2πsin 2π

t

Tminus sin 2π

t

TminusHi

λ( ) minus cos 2π

t

T[ ]

c

2B20qh2 e


(17)

Qv( )i 2πcdβkv


0zi sinω tminus

Hi minus ziVp

( )(dz)i

cdβkvηB2



Hi2πsin 2π

t

Tminus sin 2π

t

TminusHi

η( ) minus cos 2π

t

T[ ]

c

2B20qv2 e


(18)


Qh2 sumn

i1Qh( )i

c

2B20qh2sum

n


2

(19)

Qv2 sumn

i1Qv( )i

c

2B20qv2sum

n


2 (20)



TminusH


t

TminusH

η( ) sin (α minus ϕ)[ ] (21)

WABE c

4B0


V1 (22)

WBCD minusc


cosϕ sin ξV1 (23)



t

TminusH


t

TminusH

η( )( ) sin (β minus α + ϕ) [ ]V1

(24)

WBDE minusc

2intβ


minusc

2B20



(25)

d

Hizi

Qh2


ith slice

B

D

E







(27)

WQh2 c

2B20qh2 sum

n

i1e


idβ tanϕ1113872 1113873

21113946

idβ


c

2B20qh2

V1

(1 + tanϕ)sumn

in


+ eidβ tan ϕ

1113872 111387321113882e




(28)

WQv2 c

2B20qv2sum

ni1 e


idβ tan ϕ1113872 1113873

21113946

idβ


c

2B20qv2

V1

1 + tan2 ϕ( 1113857sumn


+ eidβ tan ϕ

1113872 111387321113882e




(29)





sin(βminus α)V1

(31)

DDE c1113946β

0r0dθVθ cosϕ


2 tanϕV1 (32)




Nc sinα2tanϕ

e2βtanϕ


sin(β minus α)1113890 1113891 (34)

Nq sinαe2βtanϕ


t

Tminus

H


t

Tminus

H

η1113888 11138891113888 1113889sin(β minus α+ϕ)1113896 11138971113890 11138911113890 1113891 (35)

Nc 1113890sin2α







minussumn


1+ tan2ϕ( 11138571113876qh2sum

ni11113882e







(36)



PL c

2B0N (37)


N 1113890sin2 α

cos2 ϕ 1 + 9 tan2 ϕ( 11138573 tanϕ e




+ 2Df

B0

sin αe2β tan ϕ


t

Tminus

H

λ1113874 1113875 cos (β minus α + ϕ)1113882 1113883


Tminus

H





2

1 + tan2 ϕ( 11138571113876qh2sum

ni1e




+ qv2sumni11113882e




cB0

sin α2 tanϕ

e2β tan ϕ


sin (β minus α)1113896 11138971113891

(38)


Qult 05cB0Nce (39)











Orgm

i 05Orgm


j1113872 11138732

1113874 1113875fk + Orgmj1113872 1113873

2minus Orgm

k( 11138572

1113874 1113875fi + Orgmk( 1113857


21113872 1113873fj

Orgmi minusOrg




(40)







(41)




Step 2 Main loop





(43)














FEs = FEs + 1

Start



Main loop





YesNo



Stop




Φ 2ccB0DfB0

025 05 075 1

20deg0 8349 11756 15087 18377

025 11175 14488 17769 2102905 13886 17155 20407 23649

30deg0 30439 40168 49719 59177

025 35903 45497 54975 6439105 41263 50771 60189 69557

40deg0 14424 17837 21178 24489

025 15748 19117 2245 257505 17047 20394 2371 2701









kh 01

Φ 2ccB0


025 05 075 1 025 05 075 1 025 05 075 1

20deg0 5881 8559 11172 13753 5882 8538 11128 13687 5878 851 11079 13614

025 8589 1117 1373 16277 869 11247 13782 16303 8797 11323 13832 1632905 11146 13688 16224 18754 11331 13851 16359 18865 11523 14015 16502 1898

30deg0 2198 29575 37021 44385 22107 29638 37028 44331 22223 29691 3707 44243

025 2685 34306 41675 48996 27128 34529 41834 49081 27415 34739 41979 4916905 3159 38965 46287 53564 32016 39325 46573 53793 32447 39688 46879 54028

40deg0 10141 12717 15255 17772 10232 12786 15302 17789 10317 12849 15341 17811

025 11236 1379 16311 18816 11351 13882 16385 18869 11474 13983 16461 1891605 12312 1485 17367 19858 12463 14979 17465 19942 12615 15103 17572 20021

kh 02


DfB0025 05 075 1 025 05 075 1 025 05 075 1

20deg0 3627 5554 7437 9299 3495 5257 7019 8781 345 5049 6644 824

025 6254 8093 9926 11754 6276 801 9739 11464 6345 794 9535 1113105 8672 10493 12312 14129 8889 10605 12321 14036 9201 10801 12394 13987

30deg0 14856 20524 26077 3156 14451 19881 25193 3044 1388 19024 24045 29002

025 19156 24697 30169 35606 19026 24314 29539 34734 18803 23785 28716 3362105 23312 28779 34201 39615 23416 28634 33811 3898 23505 28422 33304 38169

40deg0 68468 87465 10613 12461 67336 85521 1034 12114 6574 82948 99846 11659

025 77438 96227 11479 13322 76783 94766 11251 13011 75728 92708 10951 126105 86315 10498 12341 14179 86076 10387 12154 13908 8549 10238 11902 13561

kh 03


DfB0025 05 075 1 025 05 075 1 025 05 075 1



30deg0 93 13312 1724 21118 8356 11737 15116 18496 815 10853 13557 16261

025 13109 17 20856 24684 12336 15716 19085 22433 12366 15074 17783 2049205 16736 2057 24389 28195 16285 19633 2296 26286 16541 1925 21958 24667

40deg0 44498 58106 71465 84714 40605 52714 64622 76375 35 45195 55199 65068

025 51859 65283 78574 917 48428 60359 72112 83809 43394 53371 63261 7302105 59 72396 8556 98686 56083 6785 79556 91165 51543 61416 71164 80912




03

046

062

078

01 02 03

Nγe

Nγs

= 20 = 30 = 40

kh


028

041

054

067

08

01 02 03

NγeN

γs

kh

2cB0 = 02cB0 = 0252cB0 = 05


NγeN

γs

kh

033

044

055

066

077

01 02 03

DfB0 = 025DfB0 = 05DfB0 = 1


γs

kh

032

043

054

065

076

01 02 03












3

9

15

21

01 02 03


Nγe

kh





se 0174V2

Ag

kh

A1113890 1113891

minus4

tan αAE (45)




EIf (46)




6 Conclusion




20deg 01 1154 11771 11255 1162 1125 11284 11248 1124702 8714 8524 811 873 81 851 802 801

30deg 01 34681 34854 34535 34942 3453 34591 3453 3452902 24514 24641 24319 2443 24319 24361 2432 24314

40deg 01 13890 13912 13885 13954 13883 13899 1389 1388202 94768 95546 9478 9489 9477 9482 9477 94766


20deg 01 1646 16363 16359 16512 1637 1645 16361 1635902 1253 12325 12581 12524 1233 12812 12324 12321

30deg 01 4691 46579 46942 4676 4659 46751 46575 4657302 33931 33823 3415 3399 3383 3397 33816 33811

40deg 01 17476 17468 174691 17493 17467 17481 17469 1746502 12169 12159 12161 12159 12158 121561 12158 12154











000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000



600500400300200100





Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0



000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Qh( )i 2πcdβkh


0zi sinω tminus

Hi minus ziVs

( )(dz)i

cdβkhλB2



Hi2πsin 2π

t

Tminus sin 2π

t

TminusHi

λ( ) minus cos 2π

t

T[ ]

c

2B20qh2 e


(17)

Qv( )i 2πcdβkv


0zi sinω tminus

Hi minus ziVp

( )(dz)i

cdβkvηB2



Hi2πsin 2π

t

Tminus sin 2π

t

TminusHi

η( ) minus cos 2π

t

T[ ]

c

2B20qv2 e


(18)


Qh2 sumn

i1Qh( )i

c

2B20qh2sum

n


2

(19)

Qv2 sumn

i1Qv( )i

c

2B20qv2sum

n


2 (20)



TminusH


t

TminusH

η( ) sin (α minus ϕ)[ ] (21)

WABE c

4B0


V1 (22)

WBCD minusc


cosϕ sin ξV1 (23)



t

TminusH


t

TminusH

η( )( ) sin (β minus α + ϕ) [ ]V1

(24)

WBDE minusc

2intβ


minusc

2B20



(25)

d

Hizi

Qh2


ith slice

B

D

E







(27)

WQh2 c

2B20qh2 sum

n

i1e


idβ tanϕ1113872 1113873

21113946

idβ


c

2B20qh2

V1

(1 + tanϕ)sumn

in


+ eidβ tan ϕ

1113872 111387321113882e




(28)

WQv2 c

2B20qv2sum

ni1 e


idβ tan ϕ1113872 1113873

21113946

idβ


c

2B20qv2

V1

1 + tan2 ϕ( 1113857sumn


+ eidβ tan ϕ

1113872 111387321113882e




(29)





sin(βminus α)V1

(31)

DDE c1113946β

0r0dθVθ cosϕ


2 tanϕV1 (32)




Nc sinα2tanϕ

e2βtanϕ


sin(β minus α)1113890 1113891 (34)

Nq sinαe2βtanϕ


t

Tminus

H


t

Tminus

H

η1113888 11138891113888 1113889sin(β minus α+ϕ)1113896 11138971113890 11138911113890 1113891 (35)

Nc 1113890sin2α







minussumn


1+ tan2ϕ( 11138571113876qh2sum

ni11113882e







(36)



PL c

2B0N (37)


N 1113890sin2 α

cos2 ϕ 1 + 9 tan2 ϕ( 11138573 tanϕ e




+ 2Df

B0

sin αe2β tan ϕ


t

Tminus

H

λ1113874 1113875 cos (β minus α + ϕ)1113882 1113883


Tminus

H





2

1 + tan2 ϕ( 11138571113876qh2sum

ni1e




+ qv2sumni11113882e




cB0

sin α2 tanϕ

e2β tan ϕ


sin (β minus α)1113896 11138971113891

(38)


Qult 05cB0Nce (39)











Orgm

i 05Orgm


j1113872 11138732

1113874 1113875fk + Orgmj1113872 1113873

2minus Orgm

k( 11138572

1113874 1113875fi + Orgmk( 1113857


21113872 1113873fj

Orgmi minusOrg




(40)







(41)




Step 2 Main loop





(43)














FEs = FEs + 1

Start



Main loop





YesNo



Stop




Φ 2ccB0DfB0

025 05 075 1

20deg0 8349 11756 15087 18377

025 11175 14488 17769 2102905 13886 17155 20407 23649

30deg0 30439 40168 49719 59177

025 35903 45497 54975 6439105 41263 50771 60189 69557

40deg0 14424 17837 21178 24489

025 15748 19117 2245 257505 17047 20394 2371 2701









kh 01

Φ 2ccB0


025 05 075 1 025 05 075 1 025 05 075 1

20deg0 5881 8559 11172 13753 5882 8538 11128 13687 5878 851 11079 13614

025 8589 1117 1373 16277 869 11247 13782 16303 8797 11323 13832 1632905 11146 13688 16224 18754 11331 13851 16359 18865 11523 14015 16502 1898

30deg0 2198 29575 37021 44385 22107 29638 37028 44331 22223 29691 3707 44243

025 2685 34306 41675 48996 27128 34529 41834 49081 27415 34739 41979 4916905 3159 38965 46287 53564 32016 39325 46573 53793 32447 39688 46879 54028

40deg0 10141 12717 15255 17772 10232 12786 15302 17789 10317 12849 15341 17811

025 11236 1379 16311 18816 11351 13882 16385 18869 11474 13983 16461 1891605 12312 1485 17367 19858 12463 14979 17465 19942 12615 15103 17572 20021

kh 02


DfB0025 05 075 1 025 05 075 1 025 05 075 1

20deg0 3627 5554 7437 9299 3495 5257 7019 8781 345 5049 6644 824

025 6254 8093 9926 11754 6276 801 9739 11464 6345 794 9535 1113105 8672 10493 12312 14129 8889 10605 12321 14036 9201 10801 12394 13987

30deg0 14856 20524 26077 3156 14451 19881 25193 3044 1388 19024 24045 29002

025 19156 24697 30169 35606 19026 24314 29539 34734 18803 23785 28716 3362105 23312 28779 34201 39615 23416 28634 33811 3898 23505 28422 33304 38169

40deg0 68468 87465 10613 12461 67336 85521 1034 12114 6574 82948 99846 11659

025 77438 96227 11479 13322 76783 94766 11251 13011 75728 92708 10951 126105 86315 10498 12341 14179 86076 10387 12154 13908 8549 10238 11902 13561

kh 03


DfB0025 05 075 1 025 05 075 1 025 05 075 1



30deg0 93 13312 1724 21118 8356 11737 15116 18496 815 10853 13557 16261

025 13109 17 20856 24684 12336 15716 19085 22433 12366 15074 17783 2049205 16736 2057 24389 28195 16285 19633 2296 26286 16541 1925 21958 24667

40deg0 44498 58106 71465 84714 40605 52714 64622 76375 35 45195 55199 65068

025 51859 65283 78574 917 48428 60359 72112 83809 43394 53371 63261 7302105 59 72396 8556 98686 56083 6785 79556 91165 51543 61416 71164 80912




03

046

062

078

01 02 03

Nγe

Nγs

= 20 = 30 = 40

kh


028

041

054

067

08

01 02 03

NγeN

γs

kh

2cB0 = 02cB0 = 0252cB0 = 05


NγeN

γs

kh

033

044

055

066

077

01 02 03

DfB0 = 025DfB0 = 05DfB0 = 1


γs

kh

032

043

054

065

076

01 02 03












3

9

15

21

01 02 03


Nγe

kh





se 0174V2

Ag

kh

A1113890 1113891

minus4

tan αAE (45)




EIf (46)




6 Conclusion




20deg 01 1154 11771 11255 1162 1125 11284 11248 1124702 8714 8524 811 873 81 851 802 801

30deg 01 34681 34854 34535 34942 3453 34591 3453 3452902 24514 24641 24319 2443 24319 24361 2432 24314

40deg 01 13890 13912 13885 13954 13883 13899 1389 1388202 94768 95546 9478 9489 9477 9482 9477 94766


20deg 01 1646 16363 16359 16512 1637 1645 16361 1635902 1253 12325 12581 12524 1233 12812 12324 12321

30deg 01 4691 46579 46942 4676 4659 46751 46575 4657302 33931 33823 3415 3399 3383 3397 33816 33811

40deg 01 17476 17468 174691 17493 17467 17481 17469 1746502 12169 12159 12161 12159 12158 121561 12158 12154











000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000



600500400300200100





Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0



000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






(27)

WQh2 c

2B20qh2 sum

n

i1e


idβ tanϕ1113872 1113873

21113946

idβ


c

2B20qh2

V1

(1 + tanϕ)sumn

in


+ eidβ tan ϕ

1113872 111387321113882e




(28)

WQv2 c

2B20qv2sum

ni1 e


idβ tan ϕ1113872 1113873

21113946

idβ


c

2B20qv2

V1

1 + tan2 ϕ( 1113857sumn


+ eidβ tan ϕ

1113872 111387321113882e




(29)





sin(βminus α)V1

(31)

DDE c1113946β

0r0dθVθ cosϕ


2 tanϕV1 (32)




Nc sinα2tanϕ

e2βtanϕ


sin(β minus α)1113890 1113891 (34)

Nq sinαe2βtanϕ


t

Tminus

H


t

Tminus

H

η1113888 11138891113888 1113889sin(β minus α+ϕ)1113896 11138971113890 11138911113890 1113891 (35)

Nc 1113890sin2α







minussumn


1+ tan2ϕ( 11138571113876qh2sum

ni11113882e







(36)



PL c

2B0N (37)


N 1113890sin2 α

cos2 ϕ 1 + 9 tan2 ϕ( 11138573 tanϕ e




+ 2Df

B0

sin αe2β tan ϕ


t

Tminus

H

λ1113874 1113875 cos (β minus α + ϕ)1113882 1113883


Tminus

H





2

1 + tan2 ϕ( 11138571113876qh2sum

ni1e




+ qv2sumni11113882e




cB0

sin α2 tanϕ

e2β tan ϕ


sin (β minus α)1113896 11138971113891

(38)


Qult 05cB0Nce (39)











Orgm

i 05Orgm


j1113872 11138732

1113874 1113875fk + Orgmj1113872 1113873

2minus Orgm

k( 11138572

1113874 1113875fi + Orgmk( 1113857


21113872 1113873fj

Orgmi minusOrg




(40)







(41)




Step 2 Main loop





(43)














FEs = FEs + 1

Start



Main loop





YesNo



Stop




Φ 2ccB0DfB0

025 05 075 1

20deg0 8349 11756 15087 18377

025 11175 14488 17769 2102905 13886 17155 20407 23649

30deg0 30439 40168 49719 59177

025 35903 45497 54975 6439105 41263 50771 60189 69557

40deg0 14424 17837 21178 24489

025 15748 19117 2245 257505 17047 20394 2371 2701









kh 01

Φ 2ccB0


025 05 075 1 025 05 075 1 025 05 075 1

20deg0 5881 8559 11172 13753 5882 8538 11128 13687 5878 851 11079 13614

025 8589 1117 1373 16277 869 11247 13782 16303 8797 11323 13832 1632905 11146 13688 16224 18754 11331 13851 16359 18865 11523 14015 16502 1898

30deg0 2198 29575 37021 44385 22107 29638 37028 44331 22223 29691 3707 44243

025 2685 34306 41675 48996 27128 34529 41834 49081 27415 34739 41979 4916905 3159 38965 46287 53564 32016 39325 46573 53793 32447 39688 46879 54028

40deg0 10141 12717 15255 17772 10232 12786 15302 17789 10317 12849 15341 17811

025 11236 1379 16311 18816 11351 13882 16385 18869 11474 13983 16461 1891605 12312 1485 17367 19858 12463 14979 17465 19942 12615 15103 17572 20021

kh 02


DfB0025 05 075 1 025 05 075 1 025 05 075 1

20deg0 3627 5554 7437 9299 3495 5257 7019 8781 345 5049 6644 824

025 6254 8093 9926 11754 6276 801 9739 11464 6345 794 9535 1113105 8672 10493 12312 14129 8889 10605 12321 14036 9201 10801 12394 13987

30deg0 14856 20524 26077 3156 14451 19881 25193 3044 1388 19024 24045 29002

025 19156 24697 30169 35606 19026 24314 29539 34734 18803 23785 28716 3362105 23312 28779 34201 39615 23416 28634 33811 3898 23505 28422 33304 38169

40deg0 68468 87465 10613 12461 67336 85521 1034 12114 6574 82948 99846 11659

025 77438 96227 11479 13322 76783 94766 11251 13011 75728 92708 10951 126105 86315 10498 12341 14179 86076 10387 12154 13908 8549 10238 11902 13561

kh 03


DfB0025 05 075 1 025 05 075 1 025 05 075 1



30deg0 93 13312 1724 21118 8356 11737 15116 18496 815 10853 13557 16261

025 13109 17 20856 24684 12336 15716 19085 22433 12366 15074 17783 2049205 16736 2057 24389 28195 16285 19633 2296 26286 16541 1925 21958 24667

40deg0 44498 58106 71465 84714 40605 52714 64622 76375 35 45195 55199 65068

025 51859 65283 78574 917 48428 60359 72112 83809 43394 53371 63261 7302105 59 72396 8556 98686 56083 6785 79556 91165 51543 61416 71164 80912




03

046

062

078

01 02 03

Nγe

Nγs

= 20 = 30 = 40

kh


028

041

054

067

08

01 02 03

NγeN

γs

kh

2cB0 = 02cB0 = 0252cB0 = 05


NγeN

γs

kh

033

044

055

066

077

01 02 03

DfB0 = 025DfB0 = 05DfB0 = 1


γs

kh

032

043

054

065

076

01 02 03












3

9

15

21

01 02 03


Nγe

kh





se 0174V2

Ag

kh

A1113890 1113891

minus4

tan αAE (45)




EIf (46)




6 Conclusion




20deg 01 1154 11771 11255 1162 1125 11284 11248 1124702 8714 8524 811 873 81 851 802 801

30deg 01 34681 34854 34535 34942 3453 34591 3453 3452902 24514 24641 24319 2443 24319 24361 2432 24314

40deg 01 13890 13912 13885 13954 13883 13899 1389 1388202 94768 95546 9478 9489 9477 9482 9477 94766


20deg 01 1646 16363 16359 16512 1637 1645 16361 1635902 1253 12325 12581 12524 1233 12812 12324 12321

30deg 01 4691 46579 46942 4676 4659 46751 46575 4657302 33931 33823 3415 3399 3383 3397 33816 33811

40deg 01 17476 17468 174691 17493 17467 17481 17469 1746502 12169 12159 12161 12159 12158 121561 12158 12154











000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000



600500400300200100





Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0



000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



PL c

2B0N (37)


N 1113890sin2 α

cos2 ϕ 1 + 9 tan2 ϕ( 11138573 tanϕ e




+ 2Df

B0

sin αe2β tan ϕ


t

Tminus

H

λ1113874 1113875 cos (β minus α + ϕ)1113882 1113883


Tminus

H





2

1 + tan2 ϕ( 11138571113876qh2sum

ni1e




+ qv2sumni11113882e




cB0

sin α2 tanϕ

e2β tan ϕ


sin (β minus α)1113896 11138971113891

(38)


Qult 05cB0Nce (39)











Orgm

i 05Orgm


j1113872 11138732

1113874 1113875fk + Orgmj1113872 1113873

2minus Orgm

k( 11138572

1113874 1113875fi + Orgmk( 1113857


21113872 1113873fj

Orgmi minusOrg




(40)







(41)




Step 2 Main loop





(43)














FEs = FEs + 1

Start



Main loop





YesNo



Stop




Φ 2ccB0DfB0

025 05 075 1

20deg0 8349 11756 15087 18377

025 11175 14488 17769 2102905 13886 17155 20407 23649

30deg0 30439 40168 49719 59177

025 35903 45497 54975 6439105 41263 50771 60189 69557

40deg0 14424 17837 21178 24489

025 15748 19117 2245 257505 17047 20394 2371 2701









kh 01

Φ 2ccB0


025 05 075 1 025 05 075 1 025 05 075 1

20deg0 5881 8559 11172 13753 5882 8538 11128 13687 5878 851 11079 13614

025 8589 1117 1373 16277 869 11247 13782 16303 8797 11323 13832 1632905 11146 13688 16224 18754 11331 13851 16359 18865 11523 14015 16502 1898

30deg0 2198 29575 37021 44385 22107 29638 37028 44331 22223 29691 3707 44243

025 2685 34306 41675 48996 27128 34529 41834 49081 27415 34739 41979 4916905 3159 38965 46287 53564 32016 39325 46573 53793 32447 39688 46879 54028

40deg0 10141 12717 15255 17772 10232 12786 15302 17789 10317 12849 15341 17811

025 11236 1379 16311 18816 11351 13882 16385 18869 11474 13983 16461 1891605 12312 1485 17367 19858 12463 14979 17465 19942 12615 15103 17572 20021

kh 02


DfB0025 05 075 1 025 05 075 1 025 05 075 1

20deg0 3627 5554 7437 9299 3495 5257 7019 8781 345 5049 6644 824

025 6254 8093 9926 11754 6276 801 9739 11464 6345 794 9535 1113105 8672 10493 12312 14129 8889 10605 12321 14036 9201 10801 12394 13987

30deg0 14856 20524 26077 3156 14451 19881 25193 3044 1388 19024 24045 29002

025 19156 24697 30169 35606 19026 24314 29539 34734 18803 23785 28716 3362105 23312 28779 34201 39615 23416 28634 33811 3898 23505 28422 33304 38169

40deg0 68468 87465 10613 12461 67336 85521 1034 12114 6574 82948 99846 11659

025 77438 96227 11479 13322 76783 94766 11251 13011 75728 92708 10951 126105 86315 10498 12341 14179 86076 10387 12154 13908 8549 10238 11902 13561

kh 03


DfB0025 05 075 1 025 05 075 1 025 05 075 1



30deg0 93 13312 1724 21118 8356 11737 15116 18496 815 10853 13557 16261

025 13109 17 20856 24684 12336 15716 19085 22433 12366 15074 17783 2049205 16736 2057 24389 28195 16285 19633 2296 26286 16541 1925 21958 24667

40deg0 44498 58106 71465 84714 40605 52714 64622 76375 35 45195 55199 65068

025 51859 65283 78574 917 48428 60359 72112 83809 43394 53371 63261 7302105 59 72396 8556 98686 56083 6785 79556 91165 51543 61416 71164 80912




03

046

062

078

01 02 03

Nγe

Nγs

= 20 = 30 = 40

kh


028

041

054

067

08

01 02 03

NγeN

γs

kh

2cB0 = 02cB0 = 0252cB0 = 05


NγeN

γs

kh

033

044

055

066

077

01 02 03

DfB0 = 025DfB0 = 05DfB0 = 1


γs

kh

032

043

054

065

076

01 02 03












3

9

15

21

01 02 03


Nγe

kh





se 0174V2

Ag

kh

A1113890 1113891

minus4

tan αAE (45)




EIf (46)




6 Conclusion




20deg 01 1154 11771 11255 1162 1125 11284 11248 1124702 8714 8524 811 873 81 851 802 801

30deg 01 34681 34854 34535 34942 3453 34591 3453 3452902 24514 24641 24319 2443 24319 24361 2432 24314

40deg 01 13890 13912 13885 13954 13883 13899 1389 1388202 94768 95546 9478 9489 9477 9482 9477 94766


20deg 01 1646 16363 16359 16512 1637 1645 16361 1635902 1253 12325 12581 12524 1233 12812 12324 12321

30deg 01 4691 46579 46942 4676 4659 46751 46575 4657302 33931 33823 3415 3399 3383 3397 33816 33811

40deg 01 17476 17468 174691 17493 17467 17481 17469 1746502 12169 12159 12161 12159 12158 121561 12158 12154











000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000



600500400300200100





Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0



000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






Orgm

i 05Orgm


j1113872 11138732

1113874 1113875fk + Orgmj1113872 1113873

2minus Orgm

k( 11138572

1113874 1113875fi + Orgmk( 1113857


21113872 1113873fj

Orgmi minusOrg




(40)







(41)




Step 2 Main loop





(43)














FEs = FEs + 1

Start



Main loop





YesNo



Stop




Φ 2ccB0DfB0

025 05 075 1

20deg0 8349 11756 15087 18377

025 11175 14488 17769 2102905 13886 17155 20407 23649

30deg0 30439 40168 49719 59177

025 35903 45497 54975 6439105 41263 50771 60189 69557

40deg0 14424 17837 21178 24489

025 15748 19117 2245 257505 17047 20394 2371 2701









kh 01

Φ 2ccB0


025 05 075 1 025 05 075 1 025 05 075 1

20deg0 5881 8559 11172 13753 5882 8538 11128 13687 5878 851 11079 13614

025 8589 1117 1373 16277 869 11247 13782 16303 8797 11323 13832 1632905 11146 13688 16224 18754 11331 13851 16359 18865 11523 14015 16502 1898

30deg0 2198 29575 37021 44385 22107 29638 37028 44331 22223 29691 3707 44243

025 2685 34306 41675 48996 27128 34529 41834 49081 27415 34739 41979 4916905 3159 38965 46287 53564 32016 39325 46573 53793 32447 39688 46879 54028

40deg0 10141 12717 15255 17772 10232 12786 15302 17789 10317 12849 15341 17811

025 11236 1379 16311 18816 11351 13882 16385 18869 11474 13983 16461 1891605 12312 1485 17367 19858 12463 14979 17465 19942 12615 15103 17572 20021

kh 02


DfB0025 05 075 1 025 05 075 1 025 05 075 1

20deg0 3627 5554 7437 9299 3495 5257 7019 8781 345 5049 6644 824

025 6254 8093 9926 11754 6276 801 9739 11464 6345 794 9535 1113105 8672 10493 12312 14129 8889 10605 12321 14036 9201 10801 12394 13987

30deg0 14856 20524 26077 3156 14451 19881 25193 3044 1388 19024 24045 29002

025 19156 24697 30169 35606 19026 24314 29539 34734 18803 23785 28716 3362105 23312 28779 34201 39615 23416 28634 33811 3898 23505 28422 33304 38169

40deg0 68468 87465 10613 12461 67336 85521 1034 12114 6574 82948 99846 11659

025 77438 96227 11479 13322 76783 94766 11251 13011 75728 92708 10951 126105 86315 10498 12341 14179 86076 10387 12154 13908 8549 10238 11902 13561

kh 03


DfB0025 05 075 1 025 05 075 1 025 05 075 1



30deg0 93 13312 1724 21118 8356 11737 15116 18496 815 10853 13557 16261

025 13109 17 20856 24684 12336 15716 19085 22433 12366 15074 17783 2049205 16736 2057 24389 28195 16285 19633 2296 26286 16541 1925 21958 24667

40deg0 44498 58106 71465 84714 40605 52714 64622 76375 35 45195 55199 65068

025 51859 65283 78574 917 48428 60359 72112 83809 43394 53371 63261 7302105 59 72396 8556 98686 56083 6785 79556 91165 51543 61416 71164 80912




03

046

062

078

01 02 03

Nγe

Nγs

= 20 = 30 = 40

kh


028

041

054

067

08

01 02 03

NγeN

γs

kh

2cB0 = 02cB0 = 0252cB0 = 05


NγeN

γs

kh

033

044

055

066

077

01 02 03

DfB0 = 025DfB0 = 05DfB0 = 1


γs

kh

032

043

054

065

076

01 02 03












3

9

15

21

01 02 03


Nγe

kh





se 0174V2

Ag

kh

A1113890 1113891

minus4

tan αAE (45)




EIf (46)




6 Conclusion




20deg 01 1154 11771 11255 1162 1125 11284 11248 1124702 8714 8524 811 873 81 851 802 801

30deg 01 34681 34854 34535 34942 3453 34591 3453 3452902 24514 24641 24319 2443 24319 24361 2432 24314

40deg 01 13890 13912 13885 13954 13883 13899 1389 1388202 94768 95546 9478 9489 9477 9482 9477 94766


20deg 01 1646 16363 16359 16512 1637 1645 16361 1635902 1253 12325 12581 12524 1233 12812 12324 12321

30deg 01 4691 46579 46942 4676 4659 46751 46575 4657302 33931 33823 3415 3399 3383 3397 33816 33811

40deg 01 17476 17468 174691 17493 17467 17481 17469 1746502 12169 12159 12161 12159 12158 121561 12158 12154











000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000



600500400300200100





Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0



000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







FEs = FEs + 1

Start



Main loop





YesNo



Stop




Φ 2ccB0DfB0

025 05 075 1

20deg0 8349 11756 15087 18377

025 11175 14488 17769 2102905 13886 17155 20407 23649

30deg0 30439 40168 49719 59177

025 35903 45497 54975 6439105 41263 50771 60189 69557

40deg0 14424 17837 21178 24489

025 15748 19117 2245 257505 17047 20394 2371 2701









kh 01

Φ 2ccB0


025 05 075 1 025 05 075 1 025 05 075 1

20deg0 5881 8559 11172 13753 5882 8538 11128 13687 5878 851 11079 13614

025 8589 1117 1373 16277 869 11247 13782 16303 8797 11323 13832 1632905 11146 13688 16224 18754 11331 13851 16359 18865 11523 14015 16502 1898

30deg0 2198 29575 37021 44385 22107 29638 37028 44331 22223 29691 3707 44243

025 2685 34306 41675 48996 27128 34529 41834 49081 27415 34739 41979 4916905 3159 38965 46287 53564 32016 39325 46573 53793 32447 39688 46879 54028

40deg0 10141 12717 15255 17772 10232 12786 15302 17789 10317 12849 15341 17811

025 11236 1379 16311 18816 11351 13882 16385 18869 11474 13983 16461 1891605 12312 1485 17367 19858 12463 14979 17465 19942 12615 15103 17572 20021

kh 02


DfB0025 05 075 1 025 05 075 1 025 05 075 1

20deg0 3627 5554 7437 9299 3495 5257 7019 8781 345 5049 6644 824

025 6254 8093 9926 11754 6276 801 9739 11464 6345 794 9535 1113105 8672 10493 12312 14129 8889 10605 12321 14036 9201 10801 12394 13987

30deg0 14856 20524 26077 3156 14451 19881 25193 3044 1388 19024 24045 29002

025 19156 24697 30169 35606 19026 24314 29539 34734 18803 23785 28716 3362105 23312 28779 34201 39615 23416 28634 33811 3898 23505 28422 33304 38169

40deg0 68468 87465 10613 12461 67336 85521 1034 12114 6574 82948 99846 11659

025 77438 96227 11479 13322 76783 94766 11251 13011 75728 92708 10951 126105 86315 10498 12341 14179 86076 10387 12154 13908 8549 10238 11902 13561

kh 03


DfB0025 05 075 1 025 05 075 1 025 05 075 1



30deg0 93 13312 1724 21118 8356 11737 15116 18496 815 10853 13557 16261

025 13109 17 20856 24684 12336 15716 19085 22433 12366 15074 17783 2049205 16736 2057 24389 28195 16285 19633 2296 26286 16541 1925 21958 24667

40deg0 44498 58106 71465 84714 40605 52714 64622 76375 35 45195 55199 65068

025 51859 65283 78574 917 48428 60359 72112 83809 43394 53371 63261 7302105 59 72396 8556 98686 56083 6785 79556 91165 51543 61416 71164 80912




03

046

062

078

01 02 03

Nγe

Nγs

= 20 = 30 = 40

kh


028

041

054

067

08

01 02 03

NγeN

γs

kh

2cB0 = 02cB0 = 0252cB0 = 05


NγeN

γs

kh

033

044

055

066

077

01 02 03

DfB0 = 025DfB0 = 05DfB0 = 1


γs

kh

032

043

054

065

076

01 02 03












3

9

15

21

01 02 03


Nγe

kh





se 0174V2

Ag

kh

A1113890 1113891

minus4

tan αAE (45)




EIf (46)




6 Conclusion




20deg 01 1154 11771 11255 1162 1125 11284 11248 1124702 8714 8524 811 873 81 851 802 801

30deg 01 34681 34854 34535 34942 3453 34591 3453 3452902 24514 24641 24319 2443 24319 24361 2432 24314

40deg 01 13890 13912 13885 13954 13883 13899 1389 1388202 94768 95546 9478 9489 9477 9482 9477 94766


20deg 01 1646 16363 16359 16512 1637 1645 16361 1635902 1253 12325 12581 12524 1233 12812 12324 12321

30deg 01 4691 46579 46942 4676 4659 46751 46575 4657302 33931 33823 3415 3399 3383 3397 33816 33811

40deg 01 17476 17468 174691 17493 17467 17481 17469 1746502 12169 12159 12161 12159 12158 121561 12158 12154











000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000



600500400300200100





Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0



000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









kh 01

Φ 2ccB0


025 05 075 1 025 05 075 1 025 05 075 1

20deg0 5881 8559 11172 13753 5882 8538 11128 13687 5878 851 11079 13614

025 8589 1117 1373 16277 869 11247 13782 16303 8797 11323 13832 1632905 11146 13688 16224 18754 11331 13851 16359 18865 11523 14015 16502 1898

30deg0 2198 29575 37021 44385 22107 29638 37028 44331 22223 29691 3707 44243

025 2685 34306 41675 48996 27128 34529 41834 49081 27415 34739 41979 4916905 3159 38965 46287 53564 32016 39325 46573 53793 32447 39688 46879 54028

40deg0 10141 12717 15255 17772 10232 12786 15302 17789 10317 12849 15341 17811

025 11236 1379 16311 18816 11351 13882 16385 18869 11474 13983 16461 1891605 12312 1485 17367 19858 12463 14979 17465 19942 12615 15103 17572 20021

kh 02


DfB0025 05 075 1 025 05 075 1 025 05 075 1

20deg0 3627 5554 7437 9299 3495 5257 7019 8781 345 5049 6644 824

025 6254 8093 9926 11754 6276 801 9739 11464 6345 794 9535 1113105 8672 10493 12312 14129 8889 10605 12321 14036 9201 10801 12394 13987

30deg0 14856 20524 26077 3156 14451 19881 25193 3044 1388 19024 24045 29002

025 19156 24697 30169 35606 19026 24314 29539 34734 18803 23785 28716 3362105 23312 28779 34201 39615 23416 28634 33811 3898 23505 28422 33304 38169

40deg0 68468 87465 10613 12461 67336 85521 1034 12114 6574 82948 99846 11659

025 77438 96227 11479 13322 76783 94766 11251 13011 75728 92708 10951 126105 86315 10498 12341 14179 86076 10387 12154 13908 8549 10238 11902 13561

kh 03


DfB0025 05 075 1 025 05 075 1 025 05 075 1



30deg0 93 13312 1724 21118 8356 11737 15116 18496 815 10853 13557 16261

025 13109 17 20856 24684 12336 15716 19085 22433 12366 15074 17783 2049205 16736 2057 24389 28195 16285 19633 2296 26286 16541 1925 21958 24667

40deg0 44498 58106 71465 84714 40605 52714 64622 76375 35 45195 55199 65068

025 51859 65283 78574 917 48428 60359 72112 83809 43394 53371 63261 7302105 59 72396 8556 98686 56083 6785 79556 91165 51543 61416 71164 80912




03

046

062

078

01 02 03

Nγe

Nγs

= 20 = 30 = 40

kh


028

041

054

067

08

01 02 03

NγeN

γs

kh

2cB0 = 02cB0 = 0252cB0 = 05


NγeN

γs

kh

033

044

055

066

077

01 02 03

DfB0 = 025DfB0 = 05DfB0 = 1


γs

kh

032

043

054

065

076

01 02 03












3

9

15

21

01 02 03


Nγe

kh





se 0174V2

Ag

kh

A1113890 1113891

minus4

tan αAE (45)




EIf (46)




6 Conclusion




20deg 01 1154 11771 11255 1162 1125 11284 11248 1124702 8714 8524 811 873 81 851 802 801

30deg 01 34681 34854 34535 34942 3453 34591 3453 3452902 24514 24641 24319 2443 24319 24361 2432 24314

40deg 01 13890 13912 13885 13954 13883 13899 1389 1388202 94768 95546 9478 9489 9477 9482 9477 94766


20deg 01 1646 16363 16359 16512 1637 1645 16361 1635902 1253 12325 12581 12524 1233 12812 12324 12321

30deg 01 4691 46579 46942 4676 4659 46751 46575 4657302 33931 33823 3415 3399 3383 3397 33816 33811

40deg 01 17476 17468 174691 17493 17467 17481 17469 1746502 12169 12159 12161 12159 12158 121561 12158 12154











000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000



600500400300200100





Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0



000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




03

046

062

078

01 02 03

Nγe

Nγs

= 20 = 30 = 40

kh


028

041

054

067

08

01 02 03

NγeN

γs

kh

2cB0 = 02cB0 = 0252cB0 = 05


NγeN

γs

kh

033

044

055

066

077

01 02 03

DfB0 = 025DfB0 = 05DfB0 = 1


γs

kh

032

043

054

065

076

01 02 03












3

9

15

21

01 02 03


Nγe

kh





se 0174V2

Ag

kh

A1113890 1113891

minus4

tan αAE (45)




EIf (46)




6 Conclusion




20deg 01 1154 11771 11255 1162 1125 11284 11248 1124702 8714 8524 811 873 81 851 802 801

30deg 01 34681 34854 34535 34942 3453 34591 3453 3452902 24514 24641 24319 2443 24319 24361 2432 24314

40deg 01 13890 13912 13885 13954 13883 13899 1389 1388202 94768 95546 9478 9489 9477 9482 9477 94766


20deg 01 1646 16363 16359 16512 1637 1645 16361 1635902 1253 12325 12581 12524 1233 12812 12324 12321

30deg 01 4691 46579 46942 4676 4659 46751 46575 4657302 33931 33823 3415 3399 3383 3397 33816 33811

40deg 01 17476 17468 174691 17493 17467 17481 17469 1746502 12169 12159 12161 12159 12158 121561 12158 12154











000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000



600500400300200100





Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0



000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










3

9

15

21

01 02 03


Nγe

kh





se 0174V2

Ag

kh

A1113890 1113891

minus4

tan αAE (45)




EIf (46)




6 Conclusion




20deg 01 1154 11771 11255 1162 1125 11284 11248 1124702 8714 8524 811 873 81 851 802 801

30deg 01 34681 34854 34535 34942 3453 34591 3453 3452902 24514 24641 24319 2443 24319 24361 2432 24314

40deg 01 13890 13912 13885 13954 13883 13899 1389 1388202 94768 95546 9478 9489 9477 9482 9477 94766


20deg 01 1646 16363 16359 16512 1637 1645 16361 1635902 1253 12325 12581 12524 1233 12812 12324 12321

30deg 01 4691 46579 46942 4676 4659 46751 46575 4657302 33931 33823 3415 3399 3383 3397 33816 33811

40deg 01 17476 17468 174691 17493 17467 17481 17469 1746502 12169 12159 12161 12159 12158 121561 12158 12154











000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000



600500400300200100





Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0



000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




se 0174V2

Ag

kh

A1113890 1113891

minus4

tan αAE (45)




EIf (46)




6 Conclusion




20deg 01 1154 11771 11255 1162 1125 11284 11248 1124702 8714 8524 811 873 81 851 802 801

30deg 01 34681 34854 34535 34942 3453 34591 3453 3452902 24514 24641 24319 2443 24319 24361 2432 24314

40deg 01 13890 13912 13885 13954 13883 13899 1389 1388202 94768 95546 9478 9489 9477 9482 9477 94766


20deg 01 1646 16363 16359 16512 1637 1645 16361 1635902 1253 12325 12581 12524 1233 12812 12324 12321

30deg 01 4691 46579 46942 4676 4659 46751 46575 4657302 33931 33823 3415 3399 3383 3397 33816 33811

40deg 01 17476 17468 174691 17493 17467 17481 17469 1746502 12169 12159 12161 12159 12158 121561 12158 12154











000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000



600500400300200100





Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0



000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




000 1000 2000 3000 4000 5000

A AB B BB B B

6115487

9 10

3

0 X 1

2

6000 7000 8000 9000 10000



600500400300200100





Acce

lera

tion

(cm

s2)

0 6235 1247 18705 2494 31175Time (s)

3741 43645 4988 56115 6235

0



000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


000 1000 2000 3000

3000

2000

1000

000



4000 5000 6000 7000 8000 9000 10000 11000


000 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

2500(10minus3 m)

minus5000

minus12500

minus27500

minus42500

minus20000

minus35000

minus50000

3000

2000

1000

000







Sample 1

0 112 4957 002 127 41 4897025 195 4282 014 1827 53 360105 271 4147 024 1065 68 33071 319 401 028 913 75 3061

Sample 2

0 103 4745 001 255 37 366025 159 4099 01 2558 49 310805 214 3822 018 1827 64 3001 261 3447 025 1023 72 2733


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Nomenclature










References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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