Research ArticleNonlinear Dynamic Analysis of a Timoshenko Beam Resting ona Viscoelastic Foundation and Traveled by a Moving Mass

Ahmad Mamandi1 and Mohammad H Kargarnovin2

1 Department of Mechanical Engineering Parand Branch Islamic Azad University Tehran Iran2Department of Mechanical Engineering Sharif University of Technology Tehran Iran

Correspondence should be addressed to Ahmad Mamandi am 2001hyahoocom

Received 15 March 2013 Accepted 2 September 2013 Published 27 February 2014

Academic Editor Ahmet S Yigit

Copyright copy 2014 A Mamandi and M H Kargarnovin This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The dynamic response of a Timoshenko beam with immovable ends resting on a nonlinear viscoelastic foundation and subjectedto motion of a traveling mass moving with a constant velocity is studied Primarily the beamrsquos nonlinear governing coupled PDEsof motion for the lateral and longitudinal displacements as well as the beamrsquos cross-sectional rotation are derived using Hamiltonrsquosprinciple On deriving these nonlinear coupled PDEs the stretching effect of the beamrsquos neutral axis due to the beamrsquos fixed endconditions in conjunction with the von-Karman strain-displacement relations is considered To obtain the dynamic responses ofthe beam under the act of a moving mass derived nonlinear coupled PDEs of motion are solved by applying Galerkinrsquos methodThen the beamrsquos dynamic responses are obtained using mode summation technique Furthermore after verification of our resultswith other sources in the literature a parametric study on the dynamic response of the beam is conducted by changing the velocityof the moving mass damping coefficient and stiffnesses of the foundation including linear and cubic nonlinear parts respectivelyIt is observed that the inclusion of geometrical and foundation stiffness nonlinearities into the system in presence of the foundationdamping will produce significant effect in the beamrsquos dynamic response

1 Introduction

The topic of vibration study of structural elements such asstrings beams plates and shells under the act of a movingmass is of great interest and importance in the field ofstructural dynamics It should be noted that the review ofnumerous reported studies related to the dynamic behavior ofmechanicalstructural systems discloses that almost linearbehavior of such systems is considered Indeed in reality suchsystems inherently and naturally have nonlinear behavior forexample due to the geometrical nonlinearity orwhen they aresubjected to external loadings comparatively large enoughAswe will see later on in the modeling of the problem thestretching of the beamrsquos neutral axis due to fixed ends condi-tion adds another nonlinearity to the dynamical behavior ofthe system In addition there are some other external distinctmechanical elements having nonlinear behavior attached tosuch structures like shock energy absorbers or viscoelasticfoundationswhichwill add further other nonlinearities in themodel analysis From mechanical point of view any beam

structure can be modeled as a thin or thick beam for whichdifferent theories usually can be implemented In extendingthe issue of the linear analysis of a Timoshenko beam rest-ing on a viscoelastic foundation and under the act of a mov-ing mass certainly including the nonlinear behavior in theanalysis will well provide more reliable and accurate resultsspecifically in cases where the energy damping or any res-traint on the beamrsquos amplitude vibrations plays an importantrole in this filed

Some examples of actual cases can be addressed as abridge when traveled by moving vehicles moving trains onrailway tracks simulation of high axial speedmachining pro-cesses in milling operation and internal fluid flow in pipingsystems resting on a soil foundation Moreover shafts in therotating machinery resting on the elastic supports (journalbearings) and floating in an industrial lubricant can be mod-eled as beams on viscoelastic foundation This approach canbe viewed in modeling of rotating blades in a helicopter or inrotating blades of turbomachineries

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 242090 10 pageshttpdxdoiorg1011552014242090

2 Shock and Vibration

In [1] the linear finite element analysis of an Euler-Bernoulli beam resting on an elastic foundation modeled byspring of variable stiffness subjected to moving point loads isstudied The dynamic response of an infinite Timoshenkobeam on a viscoelastic foundation and under the act of aharmonic moving load has been investigated in [2] In [3]the linear vibration anddynamic buckling of an axially loadedinfinite shear beam-columnon an elastic foundation and sub-jected to traveling loadswith constant amplitude or harmonicamplitude variation with a constant velocity is studied In thispaper formulation in the transformed field domains of timeandmoving space is developed and dynamic responses of thebeam are obtained using Fourier transform method Thedynamic response of a simply supported Timoshenko beamresting on an elastic foundation and traveled by a movingconcentrated load is obtained using method of images in [4]In [5] the wave solution is obtained by the Laplace-transformmethod for a semi-infinite Timoshenko beam traveled bya concentrated moving load with a velocity which may besupersonic intersonic or subsonic with respect to the bend-ing and shear-wave velocities of the beam In [6] using Gale-rkinrsquosmethod the dynamic stability of the transverse vibrationof a simply supported Euler-Bernoulli beam resting on a vis-coelastic foundation and subjected to a continuous series ofequally spaced concentratedmoving loads is analyzed In thispaper the Floquet theory is utilized to study the parametricregions of stability of the beamrsquos response In [7] the deter-ministic and randomdynamic responses of a nonlinear Euler-Bernoulli beam on an elastic foundation and under the act ofa moving load are analyzed using Galerkinrsquos method in con-junction with the dynamic finite element method In thisstudy the effects of longitudinal and transverse deflections areconsidered in the beamrsquos vibration analysis In [8] a newfiniteelement method is extracted to analyze the small amplitudevibration of a Timoshenko beam on variable two-parameterWinkler type of foundation The solution is based on usingcubic polynomial expressions for the total deflection and thebending slope of the beam The linear dynamic analysis oftransverse vibration of a simply supported Timoshenko beamresting on a Winkler type of elastic foundation and traveledby a moving concentrated massforce is investigated in [9] byapplying Galerkinrsquos method using themode summation tech-nique Besides in [10] the linear finite element analysis of aTimoshenko beam under the act of a moving mass is con-ducted The linear dynamic responses of an axially loadeddamped Timoshenko beam resting on a viscoelastic founda-tion are studied in [11] Recently in [12] a new analytical sol-ution to alleviate difficulties related to Fourier analysis andnumerical integration for vibration analysis of an infiniteTimoshenko beam on a nonlinear viscoelastic foundationand subjected to amoving load has been studied A new semi-analytical solution for the Timoshenko beam subjected to thefinite series of distributedmoving loads harmonically varyingin time is considered as a representation of a moving train in[13] In [14] the dynamic response of a Rayleigh beam restingon a nonlinear viscoelastic foundation and traveled by amoving load is developed by employing theAdomian decom-position method in conjunction with coiflet expansion Anapproximate closed form solution has been derived and

condition for the convergence of the decomposition series hasbeen introduced to study the dynamic response of the beamThen a parametric study on vibration analysis is investigatedusing wavelet filters of Coiflet type as a very effective toolto adapt approximating method Transverse vibrations in-duced by a load moving at a constant speed along a finite oran infinite beam resting on a piece-wise homogeneous vis-coelastic foundation are studied in [15] The obtained resultshave direct application on the analysis of railway track vibra-tions induced by high-speed trains crossing regions with sig-nificantly different foundation stiffness In [16] a numericalapproach is developed to the stationary solution of an infiniteEuler-Bernoulli beam on a Winkler foundation and under amoving harmonic load In [17] a boundary element basedmethod is applied to study the nonlinear dynamic analysis ofshear deformable beams resting on tensionless nonlinearPasternak-type viscoelastic foundation under general bound-ary conditions and traversed by moving loads Analyses areperformed to illustrate the accuracy of the developedmethodto investigate the effects of various parameters such as theload velocity load frequency shear deformation foundationnonlinearity damping and axial compression on the beamrsquosdisplacements and stresses In [18] the issue of instabilityphenomenon of a three-mass oscillator uniformly movingalong a continuously viscoelastic supported Timoshenkobeam due to the excited Doppler waves is investigated Theproposed model corresponds to a two-level suspension vehi-cle running on a track and it includes the wheelrail contactnonlinearities which contain the Hertzian contact charac-teristic and the possibility of contact loss The velocities atthe stability limit are calculated by means of the D-decom-position method The characteristic equation has been ob-tained using Greenrsquos function of the differential operatorof the Laplace transformed equations of motion The time-domain analysis of the dynamic behavior due to the stabilityloss has been performed by solving the equations ofmotion invirtue of the convolution theorem Moreover the limit cyclecharacterizing the unstable motion is analyzed

Based on this presumption and on the same line as theother studies this work is initiated In the present study forthe first time three nonlinear governing coupled PDEs ofmotion for a Timoshenko beam resting on a nonlinear vis-coelastic foundation and under the act of a moving mass arederivedThen by applying Galerkinrsquos method these three ob-tained governing nonlinear coupled second order ODEs aresolved numerically using Adams-Bashforth-Moulton integra-tion method via MATLAB solver package It should be notedthat in the present study the effects of changes of differentparameters on the dynamic response of the Timoshenkobeam including linear and nonlinear stiffness parts of foun-dation damping coefficient of viscoelastic foundation andvelocity ofmovingmass alongwith the inclusion of stretchingeffect of the beamrsquos neutral axis are all considered in thedynamic analysis of the beamrsquos deflection

2 Mathematical Modeling

21 Problem Statement In this study the dynamic responsesof a pinned-pinned Timoshenko beam resting on a nonlinear

Shock and Vibration 3

Beam 120588 E G k l A Id

k1 c

z w

Γ

y

x u

m

120577(t)

g

(a)

k1 k3 c

z w

Γ

y

x u

m

120577(t)

g

(b)

k1 k3 c

z w

Γ

y

x u

m

120577(t)

g

(c)

Figure 1 Timoshenko beam under the motion of a concentrated point mass resting on a viscoelastic foundation (a) Timoshenko beam withno stretching effect on simply supported ends resting on a linear viscoelastic foundation M-1 model (b) Timoshenko beamwith no stretchingeffect on simply supported ends resting on a nonlinear viscoelastic foundation M-2 model and (c) Timoshenko beam with stretching effecton pinned-pinned ends resting on a nonlinear viscoelastic foundation M-3 model

viscoelastic foundation and under the act of a moving massare the main subject in our analysis In fact our analysis focu-sed on the study of the effects of the nonlinearities introduceddue to large deflection of the beam (geometric nonlinearity)and the nonlinear stiffness of the foundation on the dynamicbehaviour of above mentioned Timoshenko beam

Consider an isotropic and homogenous Timoshenkobeam with beam density 120588 Youngrsquos modulus 119864 shear modu-lus119866 shear correction factor 119896 span length 119897 constant cross-section 119860 and diametral second cross-sectional moment 119868119889To conduct our analysis three Timoshenko beam modelsillustrated in Figure 1 are considered In the first systemshown in Figure 1(a) a simply supported Timoshenko beamwith no stretching effect resting on a linear viscoelastic founda-tionwith damping coefficient 119888 and linear foundation stiffness1198961 is subjected to the motion of a point mass119898 traveling witha constant velocity V along the beam length From now on wecall this model M-1 In the next system that is Figure 1(b)similar Timoshenko beamwith the same specifications asM-1 but in addition having cubic nonlinear stiffness 1198963 is con-sidered From now on we call this modelM-2 And finally thesystem shown in Figure 1(c) is also similar to the model M-2but with immovable pinned-pinned end boundary conditionsthat is with stretching effect From now on we call this modelM-3 Note that in both M-2 and M-3 models beam is restingon a nonlinear viscoelastic foundation that is in addition to alinear spring with stiffness of 1198961 a cubic nonlinear spring withstiffness of 1198963 is also included It should be mentioned that inour upcoming analysis when the moving mass enters the leftend of the beam at time 119905 = 0 zero initial conditions are

assumed Moreover in our analysis it has been assumed thatthe moving mass during its travel never loses its contact withthe beam

22 Formulation According to von-Karmanrsquos theory thekinematics relations for the beam shown in Figure 1(b) are[19ndash21]

120576119909119909 = 119906119909 +1

21199081199092 120574119909119911 = 119908119909 minus Γ 120581119909 = Γ119909 (1)

Note that 119906 = 119906(119909 119905) and 119908 = 119908(119909 119905) represent the dis-placements of an arbitrary point located on the beamrsquos neutralaxis in the axial and the transverse that is the 119909 and 119911

directions respectively measured from the equilibrium posi-tion when unloaded Moreover Γ = Γ(119909 119905) is the rotation ofany arbitrary section around 119910-axis Also in our notation thesubscripts ( 119905) and ( 119909) stand for the derivativewith respect tothe time (119905) and spatial coordinate (119909) respectively In addi-tion 120576 120574 and 120581 are the longitudinal (or normal) strain shearstrain and curvature at a point located on the neutral axis ofthe beam respectively To obtain the governing differentialequations of motion by applying Hamiltonrsquos principle thekinetic energy 119879 and the strain energy 119880 of the beam underconsideration are [19]

119879 =120588

2int

119897

0

[119860 (1199061199052+ 1199081199052) + 119868119889Γ119905

2] 119889119909

119880 =1

2int

119897

0

(1198641198601205762

119909119909+ 119864119868119889120581119909

2+ 119896119866119860120574119909119911

2) 119889119909

(2)

4 Shock and Vibration

Now we can establish the Lagrangian function of the systemas 119871 = 119879minus(119880minus119882119890) ApplyingHamiltonrsquos principle on 119871 yields[19]

120575int

1199052

1199051

119871 119889119905 = 0 997904rArr 120575int

1199052

1199051

(119880 minus 119879) 119889119905 = int

1199052

1199051

120575119882119890 119889119905 (3)

Note that the total virtual work done 120575119882119890 by the nonlinearviscoelastic foundation and the traveling mass acting on thebeam at the location 119909 = 120577(119905) is [9 20 21]

120575119882119890 = int

119897

0

[ minus 11989811990611990511990512057511990610038161003816100381610038161003816119909=120577(119905)

minus 119898(119892 + 119908119905119905 + 2V119908119909119905 + V2119908119909119909)

times 120575119908|119909=120577(119905) + (1198961119908 + 11989631199083+ 119888119908119905) 120575119908] 119889119909

(4)

After doing some mathematics one would get the nonlineargoverning coupled PDEs of motion (EOMs) as follows [9 2021]

Force relation in 119909 direction

120588119860119906119905119905 minus 119864119860 (119906119909119909 + 119908119909119908119909119909)

= minus1198981199061199051199051003816100381610038161003816119909=120577(119905)

120575 (119909 minus 120577 (119905)) 120594 (119905)

(5)

Force relation in 119911 direction

120588119860119908119905119905 + 119896119866119860 (Γ119909 minus 119908119909119909) minus 119864119860

times (119906119909119909119908119909 + 119906119909119908119909119909 +3

2119908119909119909119908119909

2)

+ 1198961119908 + 11989631199083+ 119888119908119905

= minus119898(119892 + 119908119905119905 + 2V119908119909119905 + V2119908119909119909)10038161003816100381610038161003816119909=120577(119905)

times 120575 (119909 minus 120577 (119905)) 120594 (119905)

(6)

Moment relation about 119910 direction

120588119868119889Γ119905119905 minus 119864119868119889Γ119909119909 + 119896119866119860 (Γ minus 119908119909) = 0 (7)

where 119864119868119889 and 120588119860 are the beamrsquos flexural rigidity and beamrsquosmass per unit length respectively Furthermore 120575(119909minus120577(119905)) isDiracrsquos delta function in which 120577(119905) is the instantaneous posi-tion of the moving mass traveling on the beam In case wherethe mass is traveling with a constant velocity V then its insta-ntaneous position is given by 120577(119905) = V119905 + 1199090 where 1199090represents the initial position of the mass at the start of itsmotion Also the 120594(119905) is the pulse function which is equal to1 while the mass is traveling on the beam and 0 when the tra-veling mass is outside the beam span that is 120594(119905) = 119906(119905) minus

119906(119905 minus 119897V) in which 119906(119905) represents the unit step function

3 Solution Method

In this study Galerkinrsquos method is chosen as a powerful math-ematical tool to analyze the vibrations of a Timoshenko

beam Based on the separation of variables technique theresponse of the Timoshenko beam in terms of the linear free-oscillation modes can be assumed as follows [9 19ndash21]

119906 (119909 119905) =

119899

sum

119895=1

120579119895 (119909) 119903119895 (119905) = Θ119879(119909)R (119905) (8)

119908 (119909 119905) =

119899

sum

119895=1

120601119895 (119909) 119901119895 (119905) = Φ119879(119909)P (119905) (9)

Γ (119909 119905) =

119899

sum

119895=1

120595119895 (119909) 119902119895 (119905) = Ψ119879(119909)Q (119905) (10)

where P(119905) Q(119905) and R(119905) are vectors of order 119899 listing thegeneralized coordinate119901119895(119905) 119902119895(119905) and 119903119895(119905) respectively andΘ(119909)Φ(119909) andΨ(119909) are some vectorial functions collectingthe first 119899mode shapes (eigen-functions) of 120579119895(119909) 120601119895(119909) and120595119895(119909) respectively

In the next step primarily we substitute (8) to (10) into (5)to (7) then on the resulting relations premultiplying bothsides of (5) by Θ119879(119909) (6) by Φ119879(119909) and (7) by Ψ119879(119909) inte-grating over the interval (0 119897) and imposing the orthogonal-ity along with Diracrsquos delta function conditions the resultingnonlinear coupledmodal equations ofmotion inmatrix formare as follows119899

sum

119895=1

[120588119860J119894119895 + 119898O119894119895 (119905) 120594 (t)] 119903119895 (119905) minus 119864119860

119899

sum

119895=1

N119894119895119903119895 (119905)

minus 119864119860

119899

sum

119895=1

119899

sum

119896=1

119901j (119905) L119894119895119896119901119896 (119905) = 0 119894 = 1 2 119899

119899

sum

119895=1

[120588119860M119894119895 + 119898B119894119895 (119905) 120594 (t)] 119895 (119905)

+

119899

sum

119895=1

[2119898V120594(119905)A119894119895 (119905) + 119888M119894119895] 119895 (119905)

+ 119896119866119860

119899

sum

119895=1

F119894119895119902119895 (119905)

+

119899

sum

119895=1

[119898V2C119894119895 (119905) 120594 (t) minus 119896119866119860H119894119895 + 1198961M119894119895] 119901119895 (119905)

+ 1198963

119899

sum

119895=1

119899

sum

119896=1

119899

sum

119897=1

119901119895 (119905) (R2)119894119895119896119897119901119896 (119905) 119901119897 (119905)

minus 119864119860[

[

119899

sum

119895=1

119899

sum

119896=1

119903j (119905)G119894119895119896119901119896 (119905)

+

119899

sum

119895=1

119899

sum

119896=1

119901j (119905) (R1)119894119895119896119903119896 (119905)

+3

2

119899

sum

119895=1

119899

sum

119896=1

119901j (119905) I119894119895119896119901119896(119905)2]

]

Shock and Vibration 5

= minus119898119892120594(119905) a119894 (119905) 119894 = 1 2 119899

120588119868119889

119899

sum

119895=1

S119894119895 119902119895 (119905) minus119899

sum

119895=1

[119864119868119889K119894119895 minus 119896119866119860S119894119895] 119902119895 (119905)

minus 119896119866119860

119899

sum

119895=1

E119894119895119901119895 (119905) = 0 119894 = 1 2 119899

(11)

inwhich dotmark over any parameter indicates the derivativewith respect to the time (119905) All matrices appearing in aboverelations are given in the appendix It is clear that (11) isthree nonlinear coupled second-order ordinary differentialequations (ODEs) The boundary conditions for a pinned-pinned Timoshenko beam are

Essential BCs 119906 (0 119905) = 119906 (119897 119905) = 0

therefore 120579119895 (119909) = 0 at 119909 = 0 119897

119908 (0 119905) = 119908 (119897 119905) = 0

therefore 120601119895 (119909) = 0 at 119909 = 0 119897

Natural BCs 119872119910 (0 119905) = 119872119910 (119897 119905) = 0

therefore 120595119895119909 (119909) = 0 at 119909 = 0 119897

(12)

Moreover initial conditions (ICs) for the Timoshenko beamare

ICs 119906 (119909 0) = 119906119905 (119909 0) = 119908 (119909 0) = 119908119905 (119909 0)

= Γ (119909 0) = Γ119905 (119909 0) = 0

(13)

This is typically a well-known eigen-function eigen-valueproblem out of which the final result for the mode shapes ofthe Timoshenko beam with pinned-pinned ends is expressedas the following [9 19ndash21]

120579119895 (119909) =radic2 sin(

119895120587119909

119897) 120601119895 (119909) =

radic2 sin(119895120587119909

119897)

120595119895 (119909) =radic2 cos(

119895120587119909

119897)

with 119895 = 1 2 3 119899

(14)

Now we use (14) to calculate all matrix quantities given in theappendix In the next step these evaluated matrices will beused in (11) and later the set of equations will be solvednumerically using the Adams-Bashforth-Moulton integrationmethod viaMATLAB solver package to obtain values of119901119895(119905)119902119895(119905) and 119903119895(119905) By back substitution of119901119895(119905) 119902119895(119905) and 119903119895(119905) in(8) to (10) 119906(119909 119905)119908(119909 119905) and Γ(119909 119905) can be obtained respec-tively Subsequently after obtaining values for 119906(119909 119905)119908(119909 119905)and Γ(119909 119905) the dynamic response of a pinned-pinned Timo-shenko beam resting on a nonlinear viscoelastic foundationand subjected to a moving mass traveling with a constantvelocity can be calculated

4 Verification of Results and Case Studies

As mentioned in the introduction at the moment no specificresults exist for the problem under consideration in theliterature Therefore to verify the validity of the resultsobtained in this study we primarily consider some specialcases by which our results can be compared with thoseexisting in the literature

41 Verification of Results for the Linear andNonlinear Analyses

411 Linear Analysis In the first attempt we set (119906119909119909 +119908119909119908119909119909) and (119906119909119909119908119909 + 119906119909119908119909119909 + (32)119908119909119909119908119909

2) in the left-

hand side of (5) and (6) respectively equal to zeroMoreoverwe take 1198963 = 0 and 119888 = 0 This will lead us to a set ofnew relations for 119906(119909 119905) 119908(119909 119905) and Γ(119909 119905) representing thelinear form of governing EOMs of a Timoshenko beam ona linear Winkler type of foundation To verify the validityof the obtained results out of our analysis we consider thedata given in [9] as 119897 = 1m 119864 = 207 times 10

9Nm2 119866 =

776 times 109Nm2 119896 = 09 120588 = 7700 kgm3 119898 = 02120588119860119897 kg

1198961 = 104Nm2 120572 = 011 119892 = 981ms2 and 120573 = 003 in

which120572 and120573 are the velocity ratio andRayleighrsquos slendernesscoefficient respectively Note that 120573 = 1205871199030119897 with 1199030 takenas the beamrsquos radius of gyration and 120572 = VV119888119903 with V119888119903 asthe critical velocity of a concentratedmoving force defined asV119888119903 = 1205961119897120587 [9] with 1205961 as the first natural angular frequencyof transversal vibration of this beam given in general as (120596119894) =(119894120587119897)2radic119864119868119889120588119860(119894 = 1 2 119899) [9 10] In addition it can be

seen that the critical velocity for the first mode (119894 = 1) is givenas V119888119903 = (120587119897)radic119864119868119889120588119860 [9 10]

Based on the above data the computer code was run forthis case and the normalized instantaneous lateral displace-ment (119908119901119908119904) of the instantaneous positions of the movingmass (119909119897) is calculated and the dimensionless outcomeresults are depicted and compared with other existing resultsin Figure 2 The normalization parameter for the lateraldisplacement that is 119908119904 is actually a midpoint deflection ofa simply supported beam under midspan concentrated loadofmg that is 119908119904 = 119898119892119897348119864119868119889 [9 10] A close inspection ofthe curves in Figure 2 indicates perfect agreements betweenthe two outcome results

412 Nonlinear Analysis A simply supported Timoshenkobeam resting on a Pasternak-type viscoelastic and a shearfoundation and traveled by a constant velocity moving forceas shown in Figure 3 is considered For this system (5) and(6) are rewritten as follows

120588119860119906119905119905 minus 119864119860 (119906119909119909 + 119908119909119908119909119909) = 0

120588119860119908119905119905 + 119896119866119860 (Γ119909 minus 119908119909119909)

minus119864119860(119906119909119909119908119909 + 119906119909119908119909119909 +3

2119908119909119909119908119909

2)

minus119896119901119908119909119909 + 1198961119908 + 11989631199083+ 119888119908119905

= minus119875120575 (119909 minus 120577 (119905)) 120594 (119905)

(15)

6 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

xl

wpw

s

Figure 2 Instantaneous normalized lateral displacement 119908119901119908119904under a moving mass of 119898 = 02120588119860119897 kg 1198961 = 10

4Nm2 120572 = 011and 120573 = 003 (mdash) linear analysis present study () linear analysis[9]

Beam 120588 E G k l A Id

kpk3k1 c

P

z w

Γ

y

x u

120577(t)

Figure 3 A simply supported Timoshenko beam resting on anonlinear Pasternak viscoelastic foundation and subjected to amoving force 119875 (see [17])

in which 119896119901 and119875 are Pasternak or shear foundationmodulusand the value of moving force respectively and (7) remainsunchanged

To check on the validity of our solutions based on (7)and (15) we consider the reported results on [17] in whichanalyses shear deformable beams on tensionless nonlinearPasternak-type viscoelastic foundations and subjected tomoving forces The following data that is 119897 = 10m 119868119889 =395times10

minus6m4119860 = 8613times10minus4m2119864 = 207times109Nm2119866 =

7961 times 109Nm2 119896 = 085 120588 = 7820 kgm3 1198961 = 20 times

106Nm2 1198963 = 0Nm

4 119896119901 = 69times103N 119888 = 138times103Nsdotsm2

119875 = 144 times 103N and V = 167ms are adapted from [17]

Based on above mentioned equations a computer programwas developed and the outcome nonlinear results are de-picted and compared with [17] in Figure 4 A close inspectionof the curves reveals very close agreements between the twosolutions

42 Results and Discussions After being satisfied with thevalidity of solution technique the beamrsquos instantaneousdynamic lateral deflection is calculated in the next step In

0 1 2 3 4 5 6 7 8 9 10minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05times10minus3

x (m)

w(x03

) (m

)

Figure 4 Lateral displacement 119908(119909 03) at instant 119905 = 03 s alonglength of the beam (mdash) nonlinear analysis present study (- - - - -)nonlinear analysis [17]

obtaining these results the following data [9] are taken intoconsideration unless otherwise specified

119897 = 1m 119864 = 207 times 109Nm2 119866 = 776 times 10

9Nm2

119896 = 09 120588 = 7700 kgm3 119868119889 = 1045 times 10minus3m4

119860 = 01146m2 119898 = 3120588119860119897 kg 1198961 = 104Nm2

1198963 = 1019Nm4 119888 = 10

3N sdot sm2 120573 = 03

(16)

It should be mentioned that all deflection variationsversus moving mass instantaneous positions are given in anondimensional form that is 119908119901119908119904 Moreover it has to bepointed out that based on the conducted convergence studyrelated to the linear and nonlinear analyses ten modes ofvibration are taken into account for the steady state answersfor 119906(119909 119905) 119908(119909 119905) and Γ(119909 119905) that is 119899 = 10

The effect of variation of nonlinear part of foundationstiffness 1198963 and geometrical nonlinearity due to the beamrsquosend boundary conditions on the dynamic responses of a Tim-oshenko beam traversed by a moving mass are depicted inFigure 5 In this analysis 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2 119888 =103Nsdotsm2 and 120573 = 03 and different speeds for the traveling

mass (120572 = 011 025 and 05) are considered It should benoted that under adopted data the results for M-2 and M-3models are similar and hence the variations are overlaidThismeans that as the beamgets thicker it resistsmore againstthe stretching of midplane surfaceTherefore stretching doesnot play an important role in the beamrsquos dynamical responseespecially In addition in this figure the differences betweendynamical behavior of the beam betweenM-2 orM-3modelsand M-1 model are compared with each other Out of thisfigure and as it is expected it can be concluded that byincreasing the value of 1198963 the difference between M-2 or M-3models and M-1 model becomes more pronounced In otherwords by including geometrical and foundation nonlineari-ties into the M-1 model as the reference beam as the size ofnonlinearity increases the difference of the values of 119908119901119908119904

Shock and Vibration 7

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

16

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019

k3 = 1020

xl

wpw

s

(a)

0 01 02 03 04 05 06 07 08 09 1

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019k3 = 1020

xl

wpw

s

0

05

1

15

2

(b)

0 01 02 03 04 05 06 07 08 09 1

0

02

04

06

08

1

12

14

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019 k3 = 1020

xl

wpw

s

(c)

Figure 5 Variation of dimensionless instantaneous dynamicresponse119908119901119908119904 versus normalized instantaneous position of mov-ing mass 119909119897 of the Timoshenko beam with different 1198963 valuestraveled at different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2119888 = 10

3Nsdotsm2 and 120573 = 03 (a) 120572 = 011 (b) 120572 = 025 (c) 120572 = 05(- - - - -) M-1 model (mdash) M-2 model or M-3 model

in the nonlinear models (M-2 or M-3) compared to the M-1model increases This difference becomes more pronouncedfor 1198963 = 10

23Nm4 Furthermore when the mass velocityincreases the maximum value of the beamrsquos dynamicresponse increases up to 120572 = 025 and decreases afterwardIn addition it can be seen that for the 1198963 values greater than1021Nm4 by increasing the value of 120572 the fluctuation ofdynamical response of the beam reduces accordingly andfinally for 1198963 = 10

23Nm4 and 120572 = 011 this fluctuationreaches its highest value

Figure 6 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position of moving mass 119909119897 for three different Tim-oshenko beams of M-1 M-2 and M-3 traversed by a moving

0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2 120572 = 025120572 = 05

120572 = 11120572 = 075

120572 = 011

xl

wpw

s

Figure 6 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam traveled at differentspeeds with 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2 119888 = 103Nsdotsm2 and

120573 = 03 (- - - - -) M-1 model (mdash) M-2 or M-3 model with1198963 = 10

21 Nm4

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

16

k1 = 1012

k1 = 1011

k1 = 0

k1 = 1010k1 = 109

xl

wpw

s

(a)

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

k1 = 1012

k1 = 1011

k1 = 0

k1 = 1010

k1 = 109

xl

wpw

s

(b)

Figure 7 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 1198961 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 119888 = 103Nsdotsm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 1198963 = 10

22Nm4

mass The following data are considered 119898 = 3120588119860119897 kg1198961 = 10

4Nm2 1198963 = 1021Nm4 119888 = 103Nsdotsm2 and 120573 = 03

Moreover different speeds for the traveling mass are consid-ered (120572 = 011 025 05 075 and 11) It can be seen from thisfigure that the 119908119901119908119904 results for the M-2 or M-3 models arealways lower than the one obtained from the linear beammodel that is M-1 model However the119908119901119908119904 results for M-2 and M-3 are almost the same and independent of the value

8 Shock and Vibration

0010203040506070809

c = 0c = 106

c = 107

c = 108

c = 109

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0

02

04

06

08

1

12

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

c = 0

c = 106 c = 107

c = 108 c = 109

(b)

Figure 8 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 119888 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

10 Nm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 119896

3= 1022 Nm4

of velocity ratio The maximum difference between M-1 andM-2 orM-3models always happens at120572 = 025 Furthermoreit can be seen that in this case the maximum lateral dynamicdeflection 119908119901119908119904 increases up to 120572 = 025 and a reversingtrend prevails afterward nomatterwhat types of beammodelsare used

Using linear (M-1 model) and nonlinear (M-2 and M-3models) Timoshenko beammodels Figure 7 shows the varia-tion of normalized lateral dynamic displacement119908119901119908119904 ver-sus normalized instantaneous position of moving mass 119909119897As it was explained before these beams are resting on a vis-coelastic foundation For further analysis different values of1198961 are considered for these beam models The following dataare chosen 1198963 = 10

22Nm4 119888 = 103Nsdotsm2 and 120573 = 03

with mass moving at two different speeds (120572 = 011 and 05)It can be seen from this figure that the results of 119908119901119908119904obtained from theM-2 andM-3models with 1198963 = 10

22Nm4are the same and always lower than the one obtained fromthe M-1 model where the maximum difference between M-1andM-2 orM-3models always happens at 1198961 = 0 and is inde-pendent of the value of mass velocity Also when the valueof 1198961increases the difference between M-1 and M-2 or M-3models decreases accordingly For example when 1198961 =

1011Nm2 this difference becomes negligible Furthermore

by comparing Figure 7(a) with Figure 7(b) it can be observedthat the beamrsquos nonlinear dynamic response forM-2 andM-3

0

05

1

15

2 120573 = 03

120573 = 0015

120573 = 015

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0020406081

121416182

0 01 02 03 04 05 06 07 08 09 1

xlwpw

s

120573 = 03

120573 = 0015

120573 = 015

(b)

Figure 9 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with 119898 = 5120588119860119897 kgand 120572 = 025 and different 120573 values (a) 1198961 = 1198963 = 119888 = 0 (b)1198961 = 10

4 Nm2 1198963 = 104Nm4 and 119888 = 10

3Nsdotsm2 (- - - - -) M-1model and (mdash) M-2 or M-3 model

models at 120572 = 05 is greater than the one related to 120572 = 011Nevertheless a reversing trend is seen for the beamrsquos M-1model dynamic response

Figure 8 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams ofM-1 M-2 and M-3 traversed by a moving mass In this casedifferent values of 119888 are considered The following data arechosen 119898 = 3120588119860119897 kg 1198961 = 10

10Nm2 1198963 = 1022Nm4 and

120573 = 03 with mass moving at two different speeds (120572 =

011 and 05) It can be seen from this figure that the beamsdynamic responses of 119908119901119908119904 obtained from the M-2 or M-3 models are always lower than the one obtained from theM-1 model where the maximum difference between M-1 and M-2 or M-3 models always happens at 119888 = 0 nomatter what values of mass velocity would be Also when thevalue of 119888 increases the beamrsquos dynamical response differencebetween M-1 and M-2 or M-3 models decreases accordinglywhere for 119888 = 109Nsdotsm2 this difference becomes negligibleFurthermore by comparing Figure 8(a) with Figure 8(b) itcan be concluded that for 119888 lt 108Nsdotsm2 the obtained beamrsquosdynamic response at 120572 = 05 is almost greater than the oneobtained at 120572 = 011 no matter what type of beam models isused

Figure 9 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams of

Shock and Vibration 9

M-1 M-2 and M-3 The following data are assumed 119898 =

5120588119860119897 kg and mass traveling speed is 120572 = 025 The resultsare presented for three different slenderness ratios (120573) Fur-thermore it should be noted that in Figure 9(a) 1198961 = 1198963 = 119888 =0 and in Figure 9(b) 1198961 = 10

4Nm2 1198963 = 104Nm4 and 119888 =

103Nsdotsm2 are considered It can be seen from this figure that

the maximum value of dynamical response119908119901119908119904 increasesas 120573 increases Moreover the position of maximum value of119908119901119908119904 moves to the right end of the beam as 120573 increases Itcan be concluded from this figure that inclusion of nonlinearfoundation parameters will decrease the difference of linearand nonlinear analyses It should be further mentioned thatthe results for linear and nonlinear analyses are the same for120573 = 015 and 03

5 Conclusions

The longitudinal and lateral deflections as well as the rotationof warped cross section of three different Timoshenko beamsof M-1 M-2 and M-3 models resting on viscoelastic founda-tion and subjected to a moving mass with constant velocityare all considered by including the nonlinear nature in thebeamrsquos geometry and the stiffness of viscoelastic foundationThe outcome results are as the following

(1) It is concluded that by including geometrical andfoundation nonlinearities into the reference M-1model as the size of nonlinearity increases the differ-ence of the beamrsquos dynamic response in the nonlinearmodels (M-2 or M-3) and M-1 model increases Thisdifference becomesmore pronounced for the compar-atively large enough values of nonlinear stiffness partof viscoelastic foundation

(2) It is observed that the beamrsquos dynamic response resultsfor the fully nonlinear beam model that is M-3model are always lower than the one obtained fromthe linear beam model that is M-1 model

(3) It is concluded that when the value of linear stiffness-part of foundation increases the difference betweenM-1 and M-2 or M-3 models decreases accordinglywhere forthe large values of linear stiffness part offoundation this difference becomes negligible More-over when the value of foundation damping increasesthe beamrsquos dynamical response difference betweenM-1 and M-2 or M-3 models decreases accordingly

(4) Because both stretching effect and nonlinear stiffnesspart of viscoelastic foundation in conjunction withmoving mass condition and the other values of thefoundationrsquos parameters are significant factors in thenonlinear dynamic behavior of the beam with chang-ing the beamrsquos slenderness ratio 120573 it appears that for120573 le 0015 nonlinear behavior predominantly can beseen However for120573 gt 0015 no specific values for thecritical value of the traveling mass 119898 and foundationnonlinear stiffness 1198963 were obtained which could bean idea for further investigation

Appendix

The definition of different matrices used in calculation of thenonlinear coupled ODEs of modal relations (11) is as

(M)119894119895 = int119897

0

120601119894 (119909) 120601119895 (119909) 119889119909 (H)119894119895 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 119889119909

(F)119894119895 = int119897

0

120601119894 (119909) 1205951015840

119895(119909) 119889119909 (S)119894119895 = int

119897

0

120595119894 (119909) 120595119895 (119909) 119889119909

(K)119894119895 = int119897

0

120595119894 (119909) 12059510158401015840

119895(119909) 119889119909 (E)119894119895 = int

119897

0

120595119894 (119909) 1206011015840

119895(119909) 119889119909

(J)119894119895 = int119897

0

120579119894 (119909) 120579119895 (119909) 119889119909 (N)119894119895 = int119897

0

120579119894 (119909) 12057910158401015840

119895(119909) 119889119909

(G)119894119895119896 = int119897

0

120601119894 (119909) 12057910158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(I)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 12060110158402

119896(119909) 119889119909

(L)119894119895119896 = int119897

0

120579119894 (119909) 12060110158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(R1)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 1205791015840

119896(119909) 119889119909

(R2)119894119895119896119897 = int119897

0

120601119894 (119909) 120601119895 (119909) 120601119896 (119909) 120601119897 (119909) 119889119909

(A)119894119895 = 120601119894 (119909 = 120577 (119905)) 1206011015840

119895(119909 = 120577 (119905))

(B)119894119895 = 120601119894 (119909 = 120577 (119905)) 120601119895 (119909 = 120577 (119905))

(C)119894119895 = 120601119894 (119909 = 120577 (119905)) 12060110158401015840

119895(119909 = 120577 (119905))

(O)119894119895 = 120579119894 (119909 = 120577 (119905)) 120579119895 (119909 = 120577 (119905)) (A1)

where 119894 119895 119896 119897 = 1 2 3 119899 in which prime mark over anyparameter indicates the derivativewith respect to the position(119909) Furthermore the vector of a in the form of 119899 times 1 columnis given by

120601119894 (119909 = 120577 (119905)) = (a)119894 (A2)

Conflict of Interests

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

Acknowledgment

This paper is dedicated to Ahmad Mamandirsquos dearest Pro-fessor Dr Mohammad Hossein Kargarnovin passed awayon Monday November 4 2013 He was a fine kind decentsincere good tempered and well-liked professor who will besadlymissed by all who knew himMay his soul rest in peace

10 Shock and Vibration

References

[1] DThambiratnam andY Zhuge ldquoDynamic analysis of beams onan elastic foundation subjected to moving loadsrdquo Journal ofSound and Vibration vol 198 no 2 pp 149ndash169 1996

[2] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of an infi-nite tomoshenko beam on a viscoelastic foundation to a har-monic moving loadrdquo Journal of Sound and Vibration vol 241no 5 pp 809ndash824 2001

[3] S-M Kim and Y-H Cho ldquoVibration and dynamic buckling ofshear beam-columns on elastic foundation under moving har-monic loadsrdquo International Journal of Solids and Structures vol43 no 3-4 pp 393ndash412 2006

[4] C R Steele ldquoThe finite beam with a moving loadrdquo Journal ofApplied Mechanics vol 34 pp 111ndash118 1967

[5] A L Florence ldquoTraveling force on aTimoshenkobeamrdquo Journalof Applied Mechanics vol 32 pp 351ndash358 1965

[6] H D Nelson and R A Conover ldquoDynamic stability of a beamcarrying moving massesrdquo Journal of Applied Mechanics vol 38no 4 pp 1003ndash1006 1971

[7] T-P Chang and Y-N Liu ldquoDynamic finite element analysis of anonlinear beam subjected to a moving loadrdquo International Jour-nal of Solids and Structures vol 33 no 12 pp 1673ndash1688 1996

[8] Y C Hou C H Tseng and S F Ling ldquoA new high-ordernon-uniformTimoshenko beamfinite element on variable two-parameter foundations for vibration analysisrdquo Journal of Soundand Vibration vol 191 no 1 pp 91ndash106 1996

[9] H P Lee ldquoDynamic response of a Timoshenko beam on awinkler foundation subjected to amovingmassrdquoAppliedAcous-tics vol 55 no 3 pp 203ndash215 1998

[10] P Lou G-L Dai and Q-Y Zeng ldquoFinite-element analysis for aTimoshenko beam subjected to a moving massrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal of Mech-anical Engineering Science vol 220 no 5 pp 669ndash678 2006

[11] Y-H Chen and J-T Sheu ldquoAxially-loaded damped Timo-shenko beam on viscoelastic foundationrdquo International Journalfor Numerical Methods in Engineering vol 36 no 6 pp 1013ndash1027 1993

[12] Z Hryniewicz and P Koziol ldquoWavelet-based solution for vibra-tions of a beam on a nonlinear viscoelastic foundation due tomoving loadrdquo Journal ofTheoretical and AppliedMechanics vol51 no 1 pp 215ndash224 2013

[13] P Koziol and ZHryniewicz ldquoDynamic response of a beam rest-ing on a nonlinear foundation to a moving load coiflet-basedsolutionrdquo Shock and Vibration vol 19 pp 995ndash1007 2012

[14] Z Hryniewicz ldquoDynamics of Rayleigh beam on nonlinearfoundation due to moving load using Adomian decompositionand coiflet expansionrdquo Soil Dynamics and Earthquake Engineer-ing vol 31 no 8 pp 1123ndash1131 2011

[15] Z Dimitrovova ldquoA general procedure for the dynamic analysisof finite and infinite beams on piece-wise homogeneous foun-dationundermoving loadsrdquo Journal of Sound andVibration vol329 no 13 pp 2635ndash2653 2010

[16] V-H Nguyen and D Duhamel ldquoFinite element procedures fornonlinear structures in moving coordinates Part II infinitebeam under moving harmonic loadsrdquo Computers and Struc-tures vol 86 no 21-22 pp 2056ndash2063 2008

[17] E J Sapountzakis and A E Kampitsis ldquoNonlinear response ofshear deformable beams on tensionless nonlinear viscoelasticfoundation under moving loadsrdquo Journal of Sound and Vibra-tion vol 330 no 22 pp 5410ndash5426 2011

[18] T Mazilu M Dumitriu and C Tudorache ldquoInstability of anoscillator moving along a Timoshenko beam on viscoelasticfoundationrdquo Nonlinear Dynamics vol 67 no 2 pp 1273ndash12932012

[19] L Meirovitch Principles and Techniques of Vibrations Prentice-Hall Prentice NJ USA 1997

[20] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010

[21] A Mamandi and M H Kargarnovin ldquoNonlinear dynamicanalysis of an inclined Timoshenko beam subjected to amovingmassforce with beamrsquos weight includedrdquo Shock and Vibrationvol 18 no 6 pp 875ndash891 2011

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2 Shock and Vibration

In [1] the linear finite element analysis of an Euler-Bernoulli beam resting on an elastic foundation modeled byspring of variable stiffness subjected to moving point loads isstudied The dynamic response of an infinite Timoshenkobeam on a viscoelastic foundation and under the act of aharmonic moving load has been investigated in [2] In [3]the linear vibration anddynamic buckling of an axially loadedinfinite shear beam-columnon an elastic foundation and sub-jected to traveling loadswith constant amplitude or harmonicamplitude variation with a constant velocity is studied In thispaper formulation in the transformed field domains of timeandmoving space is developed and dynamic responses of thebeam are obtained using Fourier transform method Thedynamic response of a simply supported Timoshenko beamresting on an elastic foundation and traveled by a movingconcentrated load is obtained using method of images in [4]In [5] the wave solution is obtained by the Laplace-transformmethod for a semi-infinite Timoshenko beam traveled bya concentrated moving load with a velocity which may besupersonic intersonic or subsonic with respect to the bend-ing and shear-wave velocities of the beam In [6] using Gale-rkinrsquosmethod the dynamic stability of the transverse vibrationof a simply supported Euler-Bernoulli beam resting on a vis-coelastic foundation and subjected to a continuous series ofequally spaced concentratedmoving loads is analyzed In thispaper the Floquet theory is utilized to study the parametricregions of stability of the beamrsquos response In [7] the deter-ministic and randomdynamic responses of a nonlinear Euler-Bernoulli beam on an elastic foundation and under the act ofa moving load are analyzed using Galerkinrsquos method in con-junction with the dynamic finite element method In thisstudy the effects of longitudinal and transverse deflections areconsidered in the beamrsquos vibration analysis In [8] a newfiniteelement method is extracted to analyze the small amplitudevibration of a Timoshenko beam on variable two-parameterWinkler type of foundation The solution is based on usingcubic polynomial expressions for the total deflection and thebending slope of the beam The linear dynamic analysis oftransverse vibration of a simply supported Timoshenko beamresting on a Winkler type of elastic foundation and traveledby a moving concentrated massforce is investigated in [9] byapplying Galerkinrsquos method using themode summation tech-nique Besides in [10] the linear finite element analysis of aTimoshenko beam under the act of a moving mass is con-ducted The linear dynamic responses of an axially loadeddamped Timoshenko beam resting on a viscoelastic founda-tion are studied in [11] Recently in [12] a new analytical sol-ution to alleviate difficulties related to Fourier analysis andnumerical integration for vibration analysis of an infiniteTimoshenko beam on a nonlinear viscoelastic foundationand subjected to amoving load has been studied A new semi-analytical solution for the Timoshenko beam subjected to thefinite series of distributedmoving loads harmonically varyingin time is considered as a representation of a moving train in[13] In [14] the dynamic response of a Rayleigh beam restingon a nonlinear viscoelastic foundation and traveled by amoving load is developed by employing theAdomian decom-position method in conjunction with coiflet expansion Anapproximate closed form solution has been derived and

condition for the convergence of the decomposition series hasbeen introduced to study the dynamic response of the beamThen a parametric study on vibration analysis is investigatedusing wavelet filters of Coiflet type as a very effective toolto adapt approximating method Transverse vibrations in-duced by a load moving at a constant speed along a finite oran infinite beam resting on a piece-wise homogeneous vis-coelastic foundation are studied in [15] The obtained resultshave direct application on the analysis of railway track vibra-tions induced by high-speed trains crossing regions with sig-nificantly different foundation stiffness In [16] a numericalapproach is developed to the stationary solution of an infiniteEuler-Bernoulli beam on a Winkler foundation and under amoving harmonic load In [17] a boundary element basedmethod is applied to study the nonlinear dynamic analysis ofshear deformable beams resting on tensionless nonlinearPasternak-type viscoelastic foundation under general bound-ary conditions and traversed by moving loads Analyses areperformed to illustrate the accuracy of the developedmethodto investigate the effects of various parameters such as theload velocity load frequency shear deformation foundationnonlinearity damping and axial compression on the beamrsquosdisplacements and stresses In [18] the issue of instabilityphenomenon of a three-mass oscillator uniformly movingalong a continuously viscoelastic supported Timoshenkobeam due to the excited Doppler waves is investigated Theproposed model corresponds to a two-level suspension vehi-cle running on a track and it includes the wheelrail contactnonlinearities which contain the Hertzian contact charac-teristic and the possibility of contact loss The velocities atthe stability limit are calculated by means of the D-decom-position method The characteristic equation has been ob-tained using Greenrsquos function of the differential operatorof the Laplace transformed equations of motion The time-domain analysis of the dynamic behavior due to the stabilityloss has been performed by solving the equations ofmotion invirtue of the convolution theorem Moreover the limit cyclecharacterizing the unstable motion is analyzed

Based on this presumption and on the same line as theother studies this work is initiated In the present study forthe first time three nonlinear governing coupled PDEs ofmotion for a Timoshenko beam resting on a nonlinear vis-coelastic foundation and under the act of a moving mass arederivedThen by applying Galerkinrsquos method these three ob-tained governing nonlinear coupled second order ODEs aresolved numerically using Adams-Bashforth-Moulton integra-tion method via MATLAB solver package It should be notedthat in the present study the effects of changes of differentparameters on the dynamic response of the Timoshenkobeam including linear and nonlinear stiffness parts of foun-dation damping coefficient of viscoelastic foundation andvelocity ofmovingmass alongwith the inclusion of stretchingeffect of the beamrsquos neutral axis are all considered in thedynamic analysis of the beamrsquos deflection

2 Mathematical Modeling

21 Problem Statement In this study the dynamic responsesof a pinned-pinned Timoshenko beam resting on a nonlinear

Shock and Vibration 3

Beam 120588 E G k l A Id

k1 c

z w

Γ

y

x u

m

120577(t)

g

(a)

k1 k3 c

z w

Γ

y

x u

m

120577(t)

g

(b)

k1 k3 c

z w

Γ

y

x u

m

120577(t)

g

(c)

Figure 1 Timoshenko beam under the motion of a concentrated point mass resting on a viscoelastic foundation (a) Timoshenko beam withno stretching effect on simply supported ends resting on a linear viscoelastic foundation M-1 model (b) Timoshenko beamwith no stretchingeffect on simply supported ends resting on a nonlinear viscoelastic foundation M-2 model and (c) Timoshenko beam with stretching effecton pinned-pinned ends resting on a nonlinear viscoelastic foundation M-3 model

viscoelastic foundation and under the act of a moving massare the main subject in our analysis In fact our analysis focu-sed on the study of the effects of the nonlinearities introduceddue to large deflection of the beam (geometric nonlinearity)and the nonlinear stiffness of the foundation on the dynamicbehaviour of above mentioned Timoshenko beam

Consider an isotropic and homogenous Timoshenkobeam with beam density 120588 Youngrsquos modulus 119864 shear modu-lus119866 shear correction factor 119896 span length 119897 constant cross-section 119860 and diametral second cross-sectional moment 119868119889To conduct our analysis three Timoshenko beam modelsillustrated in Figure 1 are considered In the first systemshown in Figure 1(a) a simply supported Timoshenko beamwith no stretching effect resting on a linear viscoelastic founda-tionwith damping coefficient 119888 and linear foundation stiffness1198961 is subjected to the motion of a point mass119898 traveling witha constant velocity V along the beam length From now on wecall this model M-1 In the next system that is Figure 1(b)similar Timoshenko beamwith the same specifications asM-1 but in addition having cubic nonlinear stiffness 1198963 is con-sidered From now on we call this modelM-2 And finally thesystem shown in Figure 1(c) is also similar to the model M-2but with immovable pinned-pinned end boundary conditionsthat is with stretching effect From now on we call this modelM-3 Note that in both M-2 and M-3 models beam is restingon a nonlinear viscoelastic foundation that is in addition to alinear spring with stiffness of 1198961 a cubic nonlinear spring withstiffness of 1198963 is also included It should be mentioned that inour upcoming analysis when the moving mass enters the leftend of the beam at time 119905 = 0 zero initial conditions are

assumed Moreover in our analysis it has been assumed thatthe moving mass during its travel never loses its contact withthe beam

22 Formulation According to von-Karmanrsquos theory thekinematics relations for the beam shown in Figure 1(b) are[19ndash21]

120576119909119909 = 119906119909 +1

21199081199092 120574119909119911 = 119908119909 minus Γ 120581119909 = Γ119909 (1)

Note that 119906 = 119906(119909 119905) and 119908 = 119908(119909 119905) represent the dis-placements of an arbitrary point located on the beamrsquos neutralaxis in the axial and the transverse that is the 119909 and 119911

directions respectively measured from the equilibrium posi-tion when unloaded Moreover Γ = Γ(119909 119905) is the rotation ofany arbitrary section around 119910-axis Also in our notation thesubscripts ( 119905) and ( 119909) stand for the derivativewith respect tothe time (119905) and spatial coordinate (119909) respectively In addi-tion 120576 120574 and 120581 are the longitudinal (or normal) strain shearstrain and curvature at a point located on the neutral axis ofthe beam respectively To obtain the governing differentialequations of motion by applying Hamiltonrsquos principle thekinetic energy 119879 and the strain energy 119880 of the beam underconsideration are [19]

119879 =120588

2int

119897

0

[119860 (1199061199052+ 1199081199052) + 119868119889Γ119905

2] 119889119909

119880 =1

2int

119897

0

(1198641198601205762

119909119909+ 119864119868119889120581119909

2+ 119896119866119860120574119909119911

2) 119889119909

(2)

4 Shock and Vibration

Now we can establish the Lagrangian function of the systemas 119871 = 119879minus(119880minus119882119890) ApplyingHamiltonrsquos principle on 119871 yields[19]

120575int

1199052

1199051

119871 119889119905 = 0 997904rArr 120575int

1199052

1199051

(119880 minus 119879) 119889119905 = int

1199052

1199051

120575119882119890 119889119905 (3)

Note that the total virtual work done 120575119882119890 by the nonlinearviscoelastic foundation and the traveling mass acting on thebeam at the location 119909 = 120577(119905) is [9 20 21]

120575119882119890 = int

119897

0

[ minus 11989811990611990511990512057511990610038161003816100381610038161003816119909=120577(119905)

minus 119898(119892 + 119908119905119905 + 2V119908119909119905 + V2119908119909119909)

times 120575119908|119909=120577(119905) + (1198961119908 + 11989631199083+ 119888119908119905) 120575119908] 119889119909

(4)

After doing some mathematics one would get the nonlineargoverning coupled PDEs of motion (EOMs) as follows [9 2021]

Force relation in 119909 direction

120588119860119906119905119905 minus 119864119860 (119906119909119909 + 119908119909119908119909119909)

= minus1198981199061199051199051003816100381610038161003816119909=120577(119905)

120575 (119909 minus 120577 (119905)) 120594 (119905)

(5)

Force relation in 119911 direction

120588119860119908119905119905 + 119896119866119860 (Γ119909 minus 119908119909119909) minus 119864119860

times (119906119909119909119908119909 + 119906119909119908119909119909 +3

2119908119909119909119908119909

2)

+ 1198961119908 + 11989631199083+ 119888119908119905

= minus119898(119892 + 119908119905119905 + 2V119908119909119905 + V2119908119909119909)10038161003816100381610038161003816119909=120577(119905)

times 120575 (119909 minus 120577 (119905)) 120594 (119905)

(6)

Moment relation about 119910 direction

120588119868119889Γ119905119905 minus 119864119868119889Γ119909119909 + 119896119866119860 (Γ minus 119908119909) = 0 (7)

where 119864119868119889 and 120588119860 are the beamrsquos flexural rigidity and beamrsquosmass per unit length respectively Furthermore 120575(119909minus120577(119905)) isDiracrsquos delta function in which 120577(119905) is the instantaneous posi-tion of the moving mass traveling on the beam In case wherethe mass is traveling with a constant velocity V then its insta-ntaneous position is given by 120577(119905) = V119905 + 1199090 where 1199090represents the initial position of the mass at the start of itsmotion Also the 120594(119905) is the pulse function which is equal to1 while the mass is traveling on the beam and 0 when the tra-veling mass is outside the beam span that is 120594(119905) = 119906(119905) minus

119906(119905 minus 119897V) in which 119906(119905) represents the unit step function

3 Solution Method

In this study Galerkinrsquos method is chosen as a powerful math-ematical tool to analyze the vibrations of a Timoshenko

beam Based on the separation of variables technique theresponse of the Timoshenko beam in terms of the linear free-oscillation modes can be assumed as follows [9 19ndash21]

119906 (119909 119905) =

119899

sum

119895=1

120579119895 (119909) 119903119895 (119905) = Θ119879(119909)R (119905) (8)

119908 (119909 119905) =

119899

sum

119895=1

120601119895 (119909) 119901119895 (119905) = Φ119879(119909)P (119905) (9)

Γ (119909 119905) =

119899

sum

119895=1

120595119895 (119909) 119902119895 (119905) = Ψ119879(119909)Q (119905) (10)

where P(119905) Q(119905) and R(119905) are vectors of order 119899 listing thegeneralized coordinate119901119895(119905) 119902119895(119905) and 119903119895(119905) respectively andΘ(119909)Φ(119909) andΨ(119909) are some vectorial functions collectingthe first 119899mode shapes (eigen-functions) of 120579119895(119909) 120601119895(119909) and120595119895(119909) respectively

In the next step primarily we substitute (8) to (10) into (5)to (7) then on the resulting relations premultiplying bothsides of (5) by Θ119879(119909) (6) by Φ119879(119909) and (7) by Ψ119879(119909) inte-grating over the interval (0 119897) and imposing the orthogonal-ity along with Diracrsquos delta function conditions the resultingnonlinear coupledmodal equations ofmotion inmatrix formare as follows119899

sum

119895=1

[120588119860J119894119895 + 119898O119894119895 (119905) 120594 (t)] 119903119895 (119905) minus 119864119860

119899

sum

119895=1

N119894119895119903119895 (119905)

minus 119864119860

119899

sum

119895=1

119899

sum

119896=1

119901j (119905) L119894119895119896119901119896 (119905) = 0 119894 = 1 2 119899

119899

sum

119895=1

[120588119860M119894119895 + 119898B119894119895 (119905) 120594 (t)] 119895 (119905)

+

119899

sum

119895=1

[2119898V120594(119905)A119894119895 (119905) + 119888M119894119895] 119895 (119905)

+ 119896119866119860

119899

sum

119895=1

F119894119895119902119895 (119905)

+

119899

sum

119895=1

[119898V2C119894119895 (119905) 120594 (t) minus 119896119866119860H119894119895 + 1198961M119894119895] 119901119895 (119905)

+ 1198963

119899

sum

119895=1

119899

sum

119896=1

119899

sum

119897=1

119901119895 (119905) (R2)119894119895119896119897119901119896 (119905) 119901119897 (119905)

minus 119864119860[

[

119899

sum

119895=1

119899

sum

119896=1

119903j (119905)G119894119895119896119901119896 (119905)

+

119899

sum

119895=1

119899

sum

119896=1

119901j (119905) (R1)119894119895119896119903119896 (119905)

+3

2

119899

sum

119895=1

119899

sum

119896=1

119901j (119905) I119894119895119896119901119896(119905)2]

]

Shock and Vibration 5

= minus119898119892120594(119905) a119894 (119905) 119894 = 1 2 119899

120588119868119889

119899

sum

119895=1

S119894119895 119902119895 (119905) minus119899

sum

119895=1

[119864119868119889K119894119895 minus 119896119866119860S119894119895] 119902119895 (119905)

minus 119896119866119860

119899

sum

119895=1

E119894119895119901119895 (119905) = 0 119894 = 1 2 119899

(11)

inwhich dotmark over any parameter indicates the derivativewith respect to the time (119905) All matrices appearing in aboverelations are given in the appendix It is clear that (11) isthree nonlinear coupled second-order ordinary differentialequations (ODEs) The boundary conditions for a pinned-pinned Timoshenko beam are

Essential BCs 119906 (0 119905) = 119906 (119897 119905) = 0

therefore 120579119895 (119909) = 0 at 119909 = 0 119897

119908 (0 119905) = 119908 (119897 119905) = 0

therefore 120601119895 (119909) = 0 at 119909 = 0 119897

Natural BCs 119872119910 (0 119905) = 119872119910 (119897 119905) = 0

therefore 120595119895119909 (119909) = 0 at 119909 = 0 119897

(12)

Moreover initial conditions (ICs) for the Timoshenko beamare

ICs 119906 (119909 0) = 119906119905 (119909 0) = 119908 (119909 0) = 119908119905 (119909 0)

= Γ (119909 0) = Γ119905 (119909 0) = 0

(13)

This is typically a well-known eigen-function eigen-valueproblem out of which the final result for the mode shapes ofthe Timoshenko beam with pinned-pinned ends is expressedas the following [9 19ndash21]

120579119895 (119909) =radic2 sin(

119895120587119909

119897) 120601119895 (119909) =

radic2 sin(119895120587119909

119897)

120595119895 (119909) =radic2 cos(

119895120587119909

119897)

with 119895 = 1 2 3 119899

(14)

Now we use (14) to calculate all matrix quantities given in theappendix In the next step these evaluated matrices will beused in (11) and later the set of equations will be solvednumerically using the Adams-Bashforth-Moulton integrationmethod viaMATLAB solver package to obtain values of119901119895(119905)119902119895(119905) and 119903119895(119905) By back substitution of119901119895(119905) 119902119895(119905) and 119903119895(119905) in(8) to (10) 119906(119909 119905)119908(119909 119905) and Γ(119909 119905) can be obtained respec-tively Subsequently after obtaining values for 119906(119909 119905)119908(119909 119905)and Γ(119909 119905) the dynamic response of a pinned-pinned Timo-shenko beam resting on a nonlinear viscoelastic foundationand subjected to a moving mass traveling with a constantvelocity can be calculated

4 Verification of Results and Case Studies

As mentioned in the introduction at the moment no specificresults exist for the problem under consideration in theliterature Therefore to verify the validity of the resultsobtained in this study we primarily consider some specialcases by which our results can be compared with thoseexisting in the literature

41 Verification of Results for the Linear andNonlinear Analyses

411 Linear Analysis In the first attempt we set (119906119909119909 +119908119909119908119909119909) and (119906119909119909119908119909 + 119906119909119908119909119909 + (32)119908119909119909119908119909

2) in the left-

hand side of (5) and (6) respectively equal to zeroMoreoverwe take 1198963 = 0 and 119888 = 0 This will lead us to a set ofnew relations for 119906(119909 119905) 119908(119909 119905) and Γ(119909 119905) representing thelinear form of governing EOMs of a Timoshenko beam ona linear Winkler type of foundation To verify the validityof the obtained results out of our analysis we consider thedata given in [9] as 119897 = 1m 119864 = 207 times 10

9Nm2 119866 =

776 times 109Nm2 119896 = 09 120588 = 7700 kgm3 119898 = 02120588119860119897 kg

1198961 = 104Nm2 120572 = 011 119892 = 981ms2 and 120573 = 003 in

which120572 and120573 are the velocity ratio andRayleighrsquos slendernesscoefficient respectively Note that 120573 = 1205871199030119897 with 1199030 takenas the beamrsquos radius of gyration and 120572 = VV119888119903 with V119888119903 asthe critical velocity of a concentratedmoving force defined asV119888119903 = 1205961119897120587 [9] with 1205961 as the first natural angular frequencyof transversal vibration of this beam given in general as (120596119894) =(119894120587119897)2radic119864119868119889120588119860(119894 = 1 2 119899) [9 10] In addition it can be

seen that the critical velocity for the first mode (119894 = 1) is givenas V119888119903 = (120587119897)radic119864119868119889120588119860 [9 10]

Based on the above data the computer code was run forthis case and the normalized instantaneous lateral displace-ment (119908119901119908119904) of the instantaneous positions of the movingmass (119909119897) is calculated and the dimensionless outcomeresults are depicted and compared with other existing resultsin Figure 2 The normalization parameter for the lateraldisplacement that is 119908119904 is actually a midpoint deflection ofa simply supported beam under midspan concentrated loadofmg that is 119908119904 = 119898119892119897348119864119868119889 [9 10] A close inspection ofthe curves in Figure 2 indicates perfect agreements betweenthe two outcome results

412 Nonlinear Analysis A simply supported Timoshenkobeam resting on a Pasternak-type viscoelastic and a shearfoundation and traveled by a constant velocity moving forceas shown in Figure 3 is considered For this system (5) and(6) are rewritten as follows

120588119860119906119905119905 minus 119864119860 (119906119909119909 + 119908119909119908119909119909) = 0

120588119860119908119905119905 + 119896119866119860 (Γ119909 minus 119908119909119909)

minus119864119860(119906119909119909119908119909 + 119906119909119908119909119909 +3

2119908119909119909119908119909

2)

minus119896119901119908119909119909 + 1198961119908 + 11989631199083+ 119888119908119905

= minus119875120575 (119909 minus 120577 (119905)) 120594 (119905)

(15)

6 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

xl

wpw

s

Figure 2 Instantaneous normalized lateral displacement 119908119901119908119904under a moving mass of 119898 = 02120588119860119897 kg 1198961 = 10

4Nm2 120572 = 011and 120573 = 003 (mdash) linear analysis present study () linear analysis[9]

Beam 120588 E G k l A Id

kpk3k1 c

P

z w

Γ

y

x u

120577(t)

Figure 3 A simply supported Timoshenko beam resting on anonlinear Pasternak viscoelastic foundation and subjected to amoving force 119875 (see [17])

in which 119896119901 and119875 are Pasternak or shear foundationmodulusand the value of moving force respectively and (7) remainsunchanged

To check on the validity of our solutions based on (7)and (15) we consider the reported results on [17] in whichanalyses shear deformable beams on tensionless nonlinearPasternak-type viscoelastic foundations and subjected tomoving forces The following data that is 119897 = 10m 119868119889 =395times10

minus6m4119860 = 8613times10minus4m2119864 = 207times109Nm2119866 =

7961 times 109Nm2 119896 = 085 120588 = 7820 kgm3 1198961 = 20 times

106Nm2 1198963 = 0Nm

4 119896119901 = 69times103N 119888 = 138times103Nsdotsm2

119875 = 144 times 103N and V = 167ms are adapted from [17]

Based on above mentioned equations a computer programwas developed and the outcome nonlinear results are de-picted and compared with [17] in Figure 4 A close inspectionof the curves reveals very close agreements between the twosolutions

42 Results and Discussions After being satisfied with thevalidity of solution technique the beamrsquos instantaneousdynamic lateral deflection is calculated in the next step In

0 1 2 3 4 5 6 7 8 9 10minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05times10minus3

x (m)

w(x03

) (m

)

Figure 4 Lateral displacement 119908(119909 03) at instant 119905 = 03 s alonglength of the beam (mdash) nonlinear analysis present study (- - - - -)nonlinear analysis [17]

obtaining these results the following data [9] are taken intoconsideration unless otherwise specified

119897 = 1m 119864 = 207 times 109Nm2 119866 = 776 times 10

9Nm2

119896 = 09 120588 = 7700 kgm3 119868119889 = 1045 times 10minus3m4

119860 = 01146m2 119898 = 3120588119860119897 kg 1198961 = 104Nm2

1198963 = 1019Nm4 119888 = 10

3N sdot sm2 120573 = 03

(16)

It should be mentioned that all deflection variationsversus moving mass instantaneous positions are given in anondimensional form that is 119908119901119908119904 Moreover it has to bepointed out that based on the conducted convergence studyrelated to the linear and nonlinear analyses ten modes ofvibration are taken into account for the steady state answersfor 119906(119909 119905) 119908(119909 119905) and Γ(119909 119905) that is 119899 = 10

The effect of variation of nonlinear part of foundationstiffness 1198963 and geometrical nonlinearity due to the beamrsquosend boundary conditions on the dynamic responses of a Tim-oshenko beam traversed by a moving mass are depicted inFigure 5 In this analysis 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2 119888 =103Nsdotsm2 and 120573 = 03 and different speeds for the traveling

mass (120572 = 011 025 and 05) are considered It should benoted that under adopted data the results for M-2 and M-3models are similar and hence the variations are overlaidThismeans that as the beamgets thicker it resistsmore againstthe stretching of midplane surfaceTherefore stretching doesnot play an important role in the beamrsquos dynamical responseespecially In addition in this figure the differences betweendynamical behavior of the beam betweenM-2 orM-3modelsand M-1 model are compared with each other Out of thisfigure and as it is expected it can be concluded that byincreasing the value of 1198963 the difference between M-2 or M-3models and M-1 model becomes more pronounced In otherwords by including geometrical and foundation nonlineari-ties into the M-1 model as the reference beam as the size ofnonlinearity increases the difference of the values of 119908119901119908119904

Shock and Vibration 7

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

16

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019

k3 = 1020

xl

wpw

s

(a)

0 01 02 03 04 05 06 07 08 09 1

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019k3 = 1020

xl

wpw

s

0

05

1

15

2

(b)

0 01 02 03 04 05 06 07 08 09 1

0

02

04

06

08

1

12

14

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019 k3 = 1020

xl

wpw

s

(c)

Figure 5 Variation of dimensionless instantaneous dynamicresponse119908119901119908119904 versus normalized instantaneous position of mov-ing mass 119909119897 of the Timoshenko beam with different 1198963 valuestraveled at different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2119888 = 10

3Nsdotsm2 and 120573 = 03 (a) 120572 = 011 (b) 120572 = 025 (c) 120572 = 05(- - - - -) M-1 model (mdash) M-2 model or M-3 model

in the nonlinear models (M-2 or M-3) compared to the M-1model increases This difference becomes more pronouncedfor 1198963 = 10

23Nm4 Furthermore when the mass velocityincreases the maximum value of the beamrsquos dynamicresponse increases up to 120572 = 025 and decreases afterwardIn addition it can be seen that for the 1198963 values greater than1021Nm4 by increasing the value of 120572 the fluctuation ofdynamical response of the beam reduces accordingly andfinally for 1198963 = 10

23Nm4 and 120572 = 011 this fluctuationreaches its highest value

Figure 6 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position of moving mass 119909119897 for three different Tim-oshenko beams of M-1 M-2 and M-3 traversed by a moving

0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2 120572 = 025120572 = 05

120572 = 11120572 = 075

120572 = 011

xl

wpw

s

Figure 6 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam traveled at differentspeeds with 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2 119888 = 103Nsdotsm2 and

120573 = 03 (- - - - -) M-1 model (mdash) M-2 or M-3 model with1198963 = 10

21 Nm4

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

16

k1 = 1012

k1 = 1011

k1 = 0

k1 = 1010k1 = 109

xl

wpw

s

(a)

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

k1 = 1012

k1 = 1011

k1 = 0

k1 = 1010

k1 = 109

xl

wpw

s

(b)

Figure 7 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 1198961 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 119888 = 103Nsdotsm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 1198963 = 10

22Nm4

mass The following data are considered 119898 = 3120588119860119897 kg1198961 = 10

4Nm2 1198963 = 1021Nm4 119888 = 103Nsdotsm2 and 120573 = 03

Moreover different speeds for the traveling mass are consid-ered (120572 = 011 025 05 075 and 11) It can be seen from thisfigure that the 119908119901119908119904 results for the M-2 or M-3 models arealways lower than the one obtained from the linear beammodel that is M-1 model However the119908119901119908119904 results for M-2 and M-3 are almost the same and independent of the value

8 Shock and Vibration

0010203040506070809

c = 0c = 106

c = 107

c = 108

c = 109

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0

02

04

06

08

1

12

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

c = 0

c = 106 c = 107

c = 108 c = 109

(b)

Figure 8 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 119888 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

10 Nm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 119896

3= 1022 Nm4

of velocity ratio The maximum difference between M-1 andM-2 orM-3models always happens at120572 = 025 Furthermoreit can be seen that in this case the maximum lateral dynamicdeflection 119908119901119908119904 increases up to 120572 = 025 and a reversingtrend prevails afterward nomatterwhat types of beammodelsare used

Using linear (M-1 model) and nonlinear (M-2 and M-3models) Timoshenko beammodels Figure 7 shows the varia-tion of normalized lateral dynamic displacement119908119901119908119904 ver-sus normalized instantaneous position of moving mass 119909119897As it was explained before these beams are resting on a vis-coelastic foundation For further analysis different values of1198961 are considered for these beam models The following dataare chosen 1198963 = 10

22Nm4 119888 = 103Nsdotsm2 and 120573 = 03

with mass moving at two different speeds (120572 = 011 and 05)It can be seen from this figure that the results of 119908119901119908119904obtained from theM-2 andM-3models with 1198963 = 10

22Nm4are the same and always lower than the one obtained fromthe M-1 model where the maximum difference between M-1andM-2 orM-3models always happens at 1198961 = 0 and is inde-pendent of the value of mass velocity Also when the valueof 1198961increases the difference between M-1 and M-2 or M-3models decreases accordingly For example when 1198961 =

1011Nm2 this difference becomes negligible Furthermore

by comparing Figure 7(a) with Figure 7(b) it can be observedthat the beamrsquos nonlinear dynamic response forM-2 andM-3

0

05

1

15

2 120573 = 03

120573 = 0015

120573 = 015

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0020406081

121416182

0 01 02 03 04 05 06 07 08 09 1

xlwpw

s

120573 = 03

120573 = 0015

120573 = 015

(b)

Figure 9 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with 119898 = 5120588119860119897 kgand 120572 = 025 and different 120573 values (a) 1198961 = 1198963 = 119888 = 0 (b)1198961 = 10

4 Nm2 1198963 = 104Nm4 and 119888 = 10

3Nsdotsm2 (- - - - -) M-1model and (mdash) M-2 or M-3 model

models at 120572 = 05 is greater than the one related to 120572 = 011Nevertheless a reversing trend is seen for the beamrsquos M-1model dynamic response

Figure 8 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams ofM-1 M-2 and M-3 traversed by a moving mass In this casedifferent values of 119888 are considered The following data arechosen 119898 = 3120588119860119897 kg 1198961 = 10

10Nm2 1198963 = 1022Nm4 and

120573 = 03 with mass moving at two different speeds (120572 =

011 and 05) It can be seen from this figure that the beamsdynamic responses of 119908119901119908119904 obtained from the M-2 or M-3 models are always lower than the one obtained from theM-1 model where the maximum difference between M-1 and M-2 or M-3 models always happens at 119888 = 0 nomatter what values of mass velocity would be Also when thevalue of 119888 increases the beamrsquos dynamical response differencebetween M-1 and M-2 or M-3 models decreases accordinglywhere for 119888 = 109Nsdotsm2 this difference becomes negligibleFurthermore by comparing Figure 8(a) with Figure 8(b) itcan be concluded that for 119888 lt 108Nsdotsm2 the obtained beamrsquosdynamic response at 120572 = 05 is almost greater than the oneobtained at 120572 = 011 no matter what type of beam models isused

Figure 9 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams of

Shock and Vibration 9

M-1 M-2 and M-3 The following data are assumed 119898 =

5120588119860119897 kg and mass traveling speed is 120572 = 025 The resultsare presented for three different slenderness ratios (120573) Fur-thermore it should be noted that in Figure 9(a) 1198961 = 1198963 = 119888 =0 and in Figure 9(b) 1198961 = 10

4Nm2 1198963 = 104Nm4 and 119888 =

103Nsdotsm2 are considered It can be seen from this figure that

the maximum value of dynamical response119908119901119908119904 increasesas 120573 increases Moreover the position of maximum value of119908119901119908119904 moves to the right end of the beam as 120573 increases Itcan be concluded from this figure that inclusion of nonlinearfoundation parameters will decrease the difference of linearand nonlinear analyses It should be further mentioned thatthe results for linear and nonlinear analyses are the same for120573 = 015 and 03

5 Conclusions

The longitudinal and lateral deflections as well as the rotationof warped cross section of three different Timoshenko beamsof M-1 M-2 and M-3 models resting on viscoelastic founda-tion and subjected to a moving mass with constant velocityare all considered by including the nonlinear nature in thebeamrsquos geometry and the stiffness of viscoelastic foundationThe outcome results are as the following

(1) It is concluded that by including geometrical andfoundation nonlinearities into the reference M-1model as the size of nonlinearity increases the differ-ence of the beamrsquos dynamic response in the nonlinearmodels (M-2 or M-3) and M-1 model increases Thisdifference becomesmore pronounced for the compar-atively large enough values of nonlinear stiffness partof viscoelastic foundation

(2) It is observed that the beamrsquos dynamic response resultsfor the fully nonlinear beam model that is M-3model are always lower than the one obtained fromthe linear beam model that is M-1 model

(3) It is concluded that when the value of linear stiffness-part of foundation increases the difference betweenM-1 and M-2 or M-3 models decreases accordinglywhere forthe large values of linear stiffness part offoundation this difference becomes negligible More-over when the value of foundation damping increasesthe beamrsquos dynamical response difference betweenM-1 and M-2 or M-3 models decreases accordingly

(4) Because both stretching effect and nonlinear stiffnesspart of viscoelastic foundation in conjunction withmoving mass condition and the other values of thefoundationrsquos parameters are significant factors in thenonlinear dynamic behavior of the beam with chang-ing the beamrsquos slenderness ratio 120573 it appears that for120573 le 0015 nonlinear behavior predominantly can beseen However for120573 gt 0015 no specific values for thecritical value of the traveling mass 119898 and foundationnonlinear stiffness 1198963 were obtained which could bean idea for further investigation

Appendix

The definition of different matrices used in calculation of thenonlinear coupled ODEs of modal relations (11) is as

(M)119894119895 = int119897

0

120601119894 (119909) 120601119895 (119909) 119889119909 (H)119894119895 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 119889119909

(F)119894119895 = int119897

0

120601119894 (119909) 1205951015840

119895(119909) 119889119909 (S)119894119895 = int

119897

0

120595119894 (119909) 120595119895 (119909) 119889119909

(K)119894119895 = int119897

0

120595119894 (119909) 12059510158401015840

119895(119909) 119889119909 (E)119894119895 = int

119897

0

120595119894 (119909) 1206011015840

119895(119909) 119889119909

(J)119894119895 = int119897

0

120579119894 (119909) 120579119895 (119909) 119889119909 (N)119894119895 = int119897

0

120579119894 (119909) 12057910158401015840

119895(119909) 119889119909

(G)119894119895119896 = int119897

0

120601119894 (119909) 12057910158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(I)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 12060110158402

119896(119909) 119889119909

(L)119894119895119896 = int119897

0

120579119894 (119909) 12060110158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(R1)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 1205791015840

119896(119909) 119889119909

(R2)119894119895119896119897 = int119897

0

120601119894 (119909) 120601119895 (119909) 120601119896 (119909) 120601119897 (119909) 119889119909

(A)119894119895 = 120601119894 (119909 = 120577 (119905)) 1206011015840

119895(119909 = 120577 (119905))

(B)119894119895 = 120601119894 (119909 = 120577 (119905)) 120601119895 (119909 = 120577 (119905))

(C)119894119895 = 120601119894 (119909 = 120577 (119905)) 12060110158401015840

119895(119909 = 120577 (119905))

(O)119894119895 = 120579119894 (119909 = 120577 (119905)) 120579119895 (119909 = 120577 (119905)) (A1)

where 119894 119895 119896 119897 = 1 2 3 119899 in which prime mark over anyparameter indicates the derivativewith respect to the position(119909) Furthermore the vector of a in the form of 119899 times 1 columnis given by

120601119894 (119909 = 120577 (119905)) = (a)119894 (A2)

Conflict of Interests

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

Acknowledgment

This paper is dedicated to Ahmad Mamandirsquos dearest Pro-fessor Dr Mohammad Hossein Kargarnovin passed awayon Monday November 4 2013 He was a fine kind decentsincere good tempered and well-liked professor who will besadlymissed by all who knew himMay his soul rest in peace

10 Shock and Vibration

References

[1] DThambiratnam andY Zhuge ldquoDynamic analysis of beams onan elastic foundation subjected to moving loadsrdquo Journal ofSound and Vibration vol 198 no 2 pp 149ndash169 1996

[2] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of an infi-nite tomoshenko beam on a viscoelastic foundation to a har-monic moving loadrdquo Journal of Sound and Vibration vol 241no 5 pp 809ndash824 2001

[3] S-M Kim and Y-H Cho ldquoVibration and dynamic buckling ofshear beam-columns on elastic foundation under moving har-monic loadsrdquo International Journal of Solids and Structures vol43 no 3-4 pp 393ndash412 2006

[4] C R Steele ldquoThe finite beam with a moving loadrdquo Journal ofApplied Mechanics vol 34 pp 111ndash118 1967

[5] A L Florence ldquoTraveling force on aTimoshenkobeamrdquo Journalof Applied Mechanics vol 32 pp 351ndash358 1965

[6] H D Nelson and R A Conover ldquoDynamic stability of a beamcarrying moving massesrdquo Journal of Applied Mechanics vol 38no 4 pp 1003ndash1006 1971

[7] T-P Chang and Y-N Liu ldquoDynamic finite element analysis of anonlinear beam subjected to a moving loadrdquo International Jour-nal of Solids and Structures vol 33 no 12 pp 1673ndash1688 1996

[8] Y C Hou C H Tseng and S F Ling ldquoA new high-ordernon-uniformTimoshenko beamfinite element on variable two-parameter foundations for vibration analysisrdquo Journal of Soundand Vibration vol 191 no 1 pp 91ndash106 1996

[9] H P Lee ldquoDynamic response of a Timoshenko beam on awinkler foundation subjected to amovingmassrdquoAppliedAcous-tics vol 55 no 3 pp 203ndash215 1998

[10] P Lou G-L Dai and Q-Y Zeng ldquoFinite-element analysis for aTimoshenko beam subjected to a moving massrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal of Mech-anical Engineering Science vol 220 no 5 pp 669ndash678 2006

[11] Y-H Chen and J-T Sheu ldquoAxially-loaded damped Timo-shenko beam on viscoelastic foundationrdquo International Journalfor Numerical Methods in Engineering vol 36 no 6 pp 1013ndash1027 1993

[12] Z Hryniewicz and P Koziol ldquoWavelet-based solution for vibra-tions of a beam on a nonlinear viscoelastic foundation due tomoving loadrdquo Journal ofTheoretical and AppliedMechanics vol51 no 1 pp 215ndash224 2013

[13] P Koziol and ZHryniewicz ldquoDynamic response of a beam rest-ing on a nonlinear foundation to a moving load coiflet-basedsolutionrdquo Shock and Vibration vol 19 pp 995ndash1007 2012

[14] Z Hryniewicz ldquoDynamics of Rayleigh beam on nonlinearfoundation due to moving load using Adomian decompositionand coiflet expansionrdquo Soil Dynamics and Earthquake Engineer-ing vol 31 no 8 pp 1123ndash1131 2011

[15] Z Dimitrovova ldquoA general procedure for the dynamic analysisof finite and infinite beams on piece-wise homogeneous foun-dationundermoving loadsrdquo Journal of Sound andVibration vol329 no 13 pp 2635ndash2653 2010

[16] V-H Nguyen and D Duhamel ldquoFinite element procedures fornonlinear structures in moving coordinates Part II infinitebeam under moving harmonic loadsrdquo Computers and Struc-tures vol 86 no 21-22 pp 2056ndash2063 2008

[17] E J Sapountzakis and A E Kampitsis ldquoNonlinear response ofshear deformable beams on tensionless nonlinear viscoelasticfoundation under moving loadsrdquo Journal of Sound and Vibra-tion vol 330 no 22 pp 5410ndash5426 2011

[18] T Mazilu M Dumitriu and C Tudorache ldquoInstability of anoscillator moving along a Timoshenko beam on viscoelasticfoundationrdquo Nonlinear Dynamics vol 67 no 2 pp 1273ndash12932012

[19] L Meirovitch Principles and Techniques of Vibrations Prentice-Hall Prentice NJ USA 1997

[20] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010

[21] A Mamandi and M H Kargarnovin ldquoNonlinear dynamicanalysis of an inclined Timoshenko beam subjected to amovingmassforce with beamrsquos weight includedrdquo Shock and Vibrationvol 18 no 6 pp 875ndash891 2011

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International Journal of

Shock and Vibration 3

Beam 120588 E G k l A Id

k1 c

z w

Γ

y

x u

m

120577(t)

g

(a)

k1 k3 c

z w

Γ

y

x u

m

120577(t)

g

(b)

k1 k3 c

z w

Γ

y

x u

m

120577(t)

g

(c)

Figure 1 Timoshenko beam under the motion of a concentrated point mass resting on a viscoelastic foundation (a) Timoshenko beam withno stretching effect on simply supported ends resting on a linear viscoelastic foundation M-1 model (b) Timoshenko beamwith no stretchingeffect on simply supported ends resting on a nonlinear viscoelastic foundation M-2 model and (c) Timoshenko beam with stretching effecton pinned-pinned ends resting on a nonlinear viscoelastic foundation M-3 model

viscoelastic foundation and under the act of a moving massare the main subject in our analysis In fact our analysis focu-sed on the study of the effects of the nonlinearities introduceddue to large deflection of the beam (geometric nonlinearity)and the nonlinear stiffness of the foundation on the dynamicbehaviour of above mentioned Timoshenko beam

Consider an isotropic and homogenous Timoshenkobeam with beam density 120588 Youngrsquos modulus 119864 shear modu-lus119866 shear correction factor 119896 span length 119897 constant cross-section 119860 and diametral second cross-sectional moment 119868119889To conduct our analysis three Timoshenko beam modelsillustrated in Figure 1 are considered In the first systemshown in Figure 1(a) a simply supported Timoshenko beamwith no stretching effect resting on a linear viscoelastic founda-tionwith damping coefficient 119888 and linear foundation stiffness1198961 is subjected to the motion of a point mass119898 traveling witha constant velocity V along the beam length From now on wecall this model M-1 In the next system that is Figure 1(b)similar Timoshenko beamwith the same specifications asM-1 but in addition having cubic nonlinear stiffness 1198963 is con-sidered From now on we call this modelM-2 And finally thesystem shown in Figure 1(c) is also similar to the model M-2but with immovable pinned-pinned end boundary conditionsthat is with stretching effect From now on we call this modelM-3 Note that in both M-2 and M-3 models beam is restingon a nonlinear viscoelastic foundation that is in addition to alinear spring with stiffness of 1198961 a cubic nonlinear spring withstiffness of 1198963 is also included It should be mentioned that inour upcoming analysis when the moving mass enters the leftend of the beam at time 119905 = 0 zero initial conditions are

assumed Moreover in our analysis it has been assumed thatthe moving mass during its travel never loses its contact withthe beam

22 Formulation According to von-Karmanrsquos theory thekinematics relations for the beam shown in Figure 1(b) are[19ndash21]

120576119909119909 = 119906119909 +1

21199081199092 120574119909119911 = 119908119909 minus Γ 120581119909 = Γ119909 (1)

Note that 119906 = 119906(119909 119905) and 119908 = 119908(119909 119905) represent the dis-placements of an arbitrary point located on the beamrsquos neutralaxis in the axial and the transverse that is the 119909 and 119911

directions respectively measured from the equilibrium posi-tion when unloaded Moreover Γ = Γ(119909 119905) is the rotation ofany arbitrary section around 119910-axis Also in our notation thesubscripts ( 119905) and ( 119909) stand for the derivativewith respect tothe time (119905) and spatial coordinate (119909) respectively In addi-tion 120576 120574 and 120581 are the longitudinal (or normal) strain shearstrain and curvature at a point located on the neutral axis ofthe beam respectively To obtain the governing differentialequations of motion by applying Hamiltonrsquos principle thekinetic energy 119879 and the strain energy 119880 of the beam underconsideration are [19]

119879 =120588

2int

119897

0

[119860 (1199061199052+ 1199081199052) + 119868119889Γ119905

2] 119889119909

119880 =1

2int

119897

0

(1198641198601205762

119909119909+ 119864119868119889120581119909

2+ 119896119866119860120574119909119911

2) 119889119909

(2)

4 Shock and Vibration

Now we can establish the Lagrangian function of the systemas 119871 = 119879minus(119880minus119882119890) ApplyingHamiltonrsquos principle on 119871 yields[19]

120575int

1199052

1199051

119871 119889119905 = 0 997904rArr 120575int

1199052

1199051

(119880 minus 119879) 119889119905 = int

1199052

1199051

120575119882119890 119889119905 (3)

Note that the total virtual work done 120575119882119890 by the nonlinearviscoelastic foundation and the traveling mass acting on thebeam at the location 119909 = 120577(119905) is [9 20 21]

120575119882119890 = int

119897

0

[ minus 11989811990611990511990512057511990610038161003816100381610038161003816119909=120577(119905)

minus 119898(119892 + 119908119905119905 + 2V119908119909119905 + V2119908119909119909)

times 120575119908|119909=120577(119905) + (1198961119908 + 11989631199083+ 119888119908119905) 120575119908] 119889119909

(4)

After doing some mathematics one would get the nonlineargoverning coupled PDEs of motion (EOMs) as follows [9 2021]

Force relation in 119909 direction

120588119860119906119905119905 minus 119864119860 (119906119909119909 + 119908119909119908119909119909)

= minus1198981199061199051199051003816100381610038161003816119909=120577(119905)

120575 (119909 minus 120577 (119905)) 120594 (119905)

(5)

Force relation in 119911 direction

120588119860119908119905119905 + 119896119866119860 (Γ119909 minus 119908119909119909) minus 119864119860

times (119906119909119909119908119909 + 119906119909119908119909119909 +3

2119908119909119909119908119909

2)

+ 1198961119908 + 11989631199083+ 119888119908119905

= minus119898(119892 + 119908119905119905 + 2V119908119909119905 + V2119908119909119909)10038161003816100381610038161003816119909=120577(119905)

times 120575 (119909 minus 120577 (119905)) 120594 (119905)

(6)

Moment relation about 119910 direction

120588119868119889Γ119905119905 minus 119864119868119889Γ119909119909 + 119896119866119860 (Γ minus 119908119909) = 0 (7)

where 119864119868119889 and 120588119860 are the beamrsquos flexural rigidity and beamrsquosmass per unit length respectively Furthermore 120575(119909minus120577(119905)) isDiracrsquos delta function in which 120577(119905) is the instantaneous posi-tion of the moving mass traveling on the beam In case wherethe mass is traveling with a constant velocity V then its insta-ntaneous position is given by 120577(119905) = V119905 + 1199090 where 1199090represents the initial position of the mass at the start of itsmotion Also the 120594(119905) is the pulse function which is equal to1 while the mass is traveling on the beam and 0 when the tra-veling mass is outside the beam span that is 120594(119905) = 119906(119905) minus

119906(119905 minus 119897V) in which 119906(119905) represents the unit step function

3 Solution Method

In this study Galerkinrsquos method is chosen as a powerful math-ematical tool to analyze the vibrations of a Timoshenko

beam Based on the separation of variables technique theresponse of the Timoshenko beam in terms of the linear free-oscillation modes can be assumed as follows [9 19ndash21]

119906 (119909 119905) =

119899

sum

119895=1

120579119895 (119909) 119903119895 (119905) = Θ119879(119909)R (119905) (8)

119908 (119909 119905) =

119899

sum

119895=1

120601119895 (119909) 119901119895 (119905) = Φ119879(119909)P (119905) (9)

Γ (119909 119905) =

119899

sum

119895=1

120595119895 (119909) 119902119895 (119905) = Ψ119879(119909)Q (119905) (10)

where P(119905) Q(119905) and R(119905) are vectors of order 119899 listing thegeneralized coordinate119901119895(119905) 119902119895(119905) and 119903119895(119905) respectively andΘ(119909)Φ(119909) andΨ(119909) are some vectorial functions collectingthe first 119899mode shapes (eigen-functions) of 120579119895(119909) 120601119895(119909) and120595119895(119909) respectively

In the next step primarily we substitute (8) to (10) into (5)to (7) then on the resulting relations premultiplying bothsides of (5) by Θ119879(119909) (6) by Φ119879(119909) and (7) by Ψ119879(119909) inte-grating over the interval (0 119897) and imposing the orthogonal-ity along with Diracrsquos delta function conditions the resultingnonlinear coupledmodal equations ofmotion inmatrix formare as follows119899

sum

119895=1

[120588119860J119894119895 + 119898O119894119895 (119905) 120594 (t)] 119903119895 (119905) minus 119864119860

119899

sum

119895=1

N119894119895119903119895 (119905)

minus 119864119860

119899

sum

119895=1

119899

sum

119896=1

119901j (119905) L119894119895119896119901119896 (119905) = 0 119894 = 1 2 119899

119899

sum

119895=1

[120588119860M119894119895 + 119898B119894119895 (119905) 120594 (t)] 119895 (119905)

+

119899

sum

119895=1

[2119898V120594(119905)A119894119895 (119905) + 119888M119894119895] 119895 (119905)

+ 119896119866119860

119899

sum

119895=1

F119894119895119902119895 (119905)

+

119899

sum

119895=1

[119898V2C119894119895 (119905) 120594 (t) minus 119896119866119860H119894119895 + 1198961M119894119895] 119901119895 (119905)

+ 1198963

119899

sum

119895=1

119899

sum

119896=1

119899

sum

119897=1

119901119895 (119905) (R2)119894119895119896119897119901119896 (119905) 119901119897 (119905)

minus 119864119860[

[

119899

sum

119895=1

119899

sum

119896=1

119903j (119905)G119894119895119896119901119896 (119905)

+

119899

sum

119895=1

119899

sum

119896=1

119901j (119905) (R1)119894119895119896119903119896 (119905)

+3

2

119899

sum

119895=1

119899

sum

119896=1

119901j (119905) I119894119895119896119901119896(119905)2]

]

Shock and Vibration 5

= minus119898119892120594(119905) a119894 (119905) 119894 = 1 2 119899

120588119868119889

119899

sum

119895=1

S119894119895 119902119895 (119905) minus119899

sum

119895=1

[119864119868119889K119894119895 minus 119896119866119860S119894119895] 119902119895 (119905)

minus 119896119866119860

119899

sum

119895=1

E119894119895119901119895 (119905) = 0 119894 = 1 2 119899

(11)

inwhich dotmark over any parameter indicates the derivativewith respect to the time (119905) All matrices appearing in aboverelations are given in the appendix It is clear that (11) isthree nonlinear coupled second-order ordinary differentialequations (ODEs) The boundary conditions for a pinned-pinned Timoshenko beam are

Essential BCs 119906 (0 119905) = 119906 (119897 119905) = 0

therefore 120579119895 (119909) = 0 at 119909 = 0 119897

119908 (0 119905) = 119908 (119897 119905) = 0

therefore 120601119895 (119909) = 0 at 119909 = 0 119897

Natural BCs 119872119910 (0 119905) = 119872119910 (119897 119905) = 0

therefore 120595119895119909 (119909) = 0 at 119909 = 0 119897

(12)

Moreover initial conditions (ICs) for the Timoshenko beamare

ICs 119906 (119909 0) = 119906119905 (119909 0) = 119908 (119909 0) = 119908119905 (119909 0)

= Γ (119909 0) = Γ119905 (119909 0) = 0

(13)

This is typically a well-known eigen-function eigen-valueproblem out of which the final result for the mode shapes ofthe Timoshenko beam with pinned-pinned ends is expressedas the following [9 19ndash21]

120579119895 (119909) =radic2 sin(

119895120587119909

119897) 120601119895 (119909) =

radic2 sin(119895120587119909

119897)

120595119895 (119909) =radic2 cos(

119895120587119909

119897)

with 119895 = 1 2 3 119899

(14)

Now we use (14) to calculate all matrix quantities given in theappendix In the next step these evaluated matrices will beused in (11) and later the set of equations will be solvednumerically using the Adams-Bashforth-Moulton integrationmethod viaMATLAB solver package to obtain values of119901119895(119905)119902119895(119905) and 119903119895(119905) By back substitution of119901119895(119905) 119902119895(119905) and 119903119895(119905) in(8) to (10) 119906(119909 119905)119908(119909 119905) and Γ(119909 119905) can be obtained respec-tively Subsequently after obtaining values for 119906(119909 119905)119908(119909 119905)and Γ(119909 119905) the dynamic response of a pinned-pinned Timo-shenko beam resting on a nonlinear viscoelastic foundationand subjected to a moving mass traveling with a constantvelocity can be calculated

4 Verification of Results and Case Studies

As mentioned in the introduction at the moment no specificresults exist for the problem under consideration in theliterature Therefore to verify the validity of the resultsobtained in this study we primarily consider some specialcases by which our results can be compared with thoseexisting in the literature

41 Verification of Results for the Linear andNonlinear Analyses

411 Linear Analysis In the first attempt we set (119906119909119909 +119908119909119908119909119909) and (119906119909119909119908119909 + 119906119909119908119909119909 + (32)119908119909119909119908119909

2) in the left-

hand side of (5) and (6) respectively equal to zeroMoreoverwe take 1198963 = 0 and 119888 = 0 This will lead us to a set ofnew relations for 119906(119909 119905) 119908(119909 119905) and Γ(119909 119905) representing thelinear form of governing EOMs of a Timoshenko beam ona linear Winkler type of foundation To verify the validityof the obtained results out of our analysis we consider thedata given in [9] as 119897 = 1m 119864 = 207 times 10

9Nm2 119866 =

776 times 109Nm2 119896 = 09 120588 = 7700 kgm3 119898 = 02120588119860119897 kg

1198961 = 104Nm2 120572 = 011 119892 = 981ms2 and 120573 = 003 in

which120572 and120573 are the velocity ratio andRayleighrsquos slendernesscoefficient respectively Note that 120573 = 1205871199030119897 with 1199030 takenas the beamrsquos radius of gyration and 120572 = VV119888119903 with V119888119903 asthe critical velocity of a concentratedmoving force defined asV119888119903 = 1205961119897120587 [9] with 1205961 as the first natural angular frequencyof transversal vibration of this beam given in general as (120596119894) =(119894120587119897)2radic119864119868119889120588119860(119894 = 1 2 119899) [9 10] In addition it can be

seen that the critical velocity for the first mode (119894 = 1) is givenas V119888119903 = (120587119897)radic119864119868119889120588119860 [9 10]

Based on the above data the computer code was run forthis case and the normalized instantaneous lateral displace-ment (119908119901119908119904) of the instantaneous positions of the movingmass (119909119897) is calculated and the dimensionless outcomeresults are depicted and compared with other existing resultsin Figure 2 The normalization parameter for the lateraldisplacement that is 119908119904 is actually a midpoint deflection ofa simply supported beam under midspan concentrated loadofmg that is 119908119904 = 119898119892119897348119864119868119889 [9 10] A close inspection ofthe curves in Figure 2 indicates perfect agreements betweenthe two outcome results

412 Nonlinear Analysis A simply supported Timoshenkobeam resting on a Pasternak-type viscoelastic and a shearfoundation and traveled by a constant velocity moving forceas shown in Figure 3 is considered For this system (5) and(6) are rewritten as follows

120588119860119906119905119905 minus 119864119860 (119906119909119909 + 119908119909119908119909119909) = 0

120588119860119908119905119905 + 119896119866119860 (Γ119909 minus 119908119909119909)

minus119864119860(119906119909119909119908119909 + 119906119909119908119909119909 +3

2119908119909119909119908119909

2)

minus119896119901119908119909119909 + 1198961119908 + 11989631199083+ 119888119908119905

= minus119875120575 (119909 minus 120577 (119905)) 120594 (119905)

(15)

6 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

xl

wpw

s

Figure 2 Instantaneous normalized lateral displacement 119908119901119908119904under a moving mass of 119898 = 02120588119860119897 kg 1198961 = 10

4Nm2 120572 = 011and 120573 = 003 (mdash) linear analysis present study () linear analysis[9]

Beam 120588 E G k l A Id

kpk3k1 c

P

z w

Γ

y

x u

120577(t)

Figure 3 A simply supported Timoshenko beam resting on anonlinear Pasternak viscoelastic foundation and subjected to amoving force 119875 (see [17])

in which 119896119901 and119875 are Pasternak or shear foundationmodulusand the value of moving force respectively and (7) remainsunchanged

To check on the validity of our solutions based on (7)and (15) we consider the reported results on [17] in whichanalyses shear deformable beams on tensionless nonlinearPasternak-type viscoelastic foundations and subjected tomoving forces The following data that is 119897 = 10m 119868119889 =395times10

minus6m4119860 = 8613times10minus4m2119864 = 207times109Nm2119866 =

7961 times 109Nm2 119896 = 085 120588 = 7820 kgm3 1198961 = 20 times

106Nm2 1198963 = 0Nm

4 119896119901 = 69times103N 119888 = 138times103Nsdotsm2

119875 = 144 times 103N and V = 167ms are adapted from [17]

Based on above mentioned equations a computer programwas developed and the outcome nonlinear results are de-picted and compared with [17] in Figure 4 A close inspectionof the curves reveals very close agreements between the twosolutions

42 Results and Discussions After being satisfied with thevalidity of solution technique the beamrsquos instantaneousdynamic lateral deflection is calculated in the next step In

0 1 2 3 4 5 6 7 8 9 10minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05times10minus3

x (m)

w(x03

) (m

)

Figure 4 Lateral displacement 119908(119909 03) at instant 119905 = 03 s alonglength of the beam (mdash) nonlinear analysis present study (- - - - -)nonlinear analysis [17]

obtaining these results the following data [9] are taken intoconsideration unless otherwise specified

119897 = 1m 119864 = 207 times 109Nm2 119866 = 776 times 10

9Nm2

119896 = 09 120588 = 7700 kgm3 119868119889 = 1045 times 10minus3m4

119860 = 01146m2 119898 = 3120588119860119897 kg 1198961 = 104Nm2

1198963 = 1019Nm4 119888 = 10

3N sdot sm2 120573 = 03

(16)

It should be mentioned that all deflection variationsversus moving mass instantaneous positions are given in anondimensional form that is 119908119901119908119904 Moreover it has to bepointed out that based on the conducted convergence studyrelated to the linear and nonlinear analyses ten modes ofvibration are taken into account for the steady state answersfor 119906(119909 119905) 119908(119909 119905) and Γ(119909 119905) that is 119899 = 10

The effect of variation of nonlinear part of foundationstiffness 1198963 and geometrical nonlinearity due to the beamrsquosend boundary conditions on the dynamic responses of a Tim-oshenko beam traversed by a moving mass are depicted inFigure 5 In this analysis 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2 119888 =103Nsdotsm2 and 120573 = 03 and different speeds for the traveling

mass (120572 = 011 025 and 05) are considered It should benoted that under adopted data the results for M-2 and M-3models are similar and hence the variations are overlaidThismeans that as the beamgets thicker it resistsmore againstthe stretching of midplane surfaceTherefore stretching doesnot play an important role in the beamrsquos dynamical responseespecially In addition in this figure the differences betweendynamical behavior of the beam betweenM-2 orM-3modelsand M-1 model are compared with each other Out of thisfigure and as it is expected it can be concluded that byincreasing the value of 1198963 the difference between M-2 or M-3models and M-1 model becomes more pronounced In otherwords by including geometrical and foundation nonlineari-ties into the M-1 model as the reference beam as the size ofnonlinearity increases the difference of the values of 119908119901119908119904

Shock and Vibration 7

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

16

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019

k3 = 1020

xl

wpw

s

(a)

0 01 02 03 04 05 06 07 08 09 1

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019k3 = 1020

xl

wpw

s

0

05

1

15

2

(b)

0 01 02 03 04 05 06 07 08 09 1

0

02

04

06

08

1

12

14

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019 k3 = 1020

xl

wpw

s

(c)

Figure 5 Variation of dimensionless instantaneous dynamicresponse119908119901119908119904 versus normalized instantaneous position of mov-ing mass 119909119897 of the Timoshenko beam with different 1198963 valuestraveled at different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2119888 = 10

3Nsdotsm2 and 120573 = 03 (a) 120572 = 011 (b) 120572 = 025 (c) 120572 = 05(- - - - -) M-1 model (mdash) M-2 model or M-3 model

in the nonlinear models (M-2 or M-3) compared to the M-1model increases This difference becomes more pronouncedfor 1198963 = 10

23Nm4 Furthermore when the mass velocityincreases the maximum value of the beamrsquos dynamicresponse increases up to 120572 = 025 and decreases afterwardIn addition it can be seen that for the 1198963 values greater than1021Nm4 by increasing the value of 120572 the fluctuation ofdynamical response of the beam reduces accordingly andfinally for 1198963 = 10

23Nm4 and 120572 = 011 this fluctuationreaches its highest value

Figure 6 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position of moving mass 119909119897 for three different Tim-oshenko beams of M-1 M-2 and M-3 traversed by a moving

0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2 120572 = 025120572 = 05

120572 = 11120572 = 075

120572 = 011

xl

wpw

s

Figure 6 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam traveled at differentspeeds with 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2 119888 = 103Nsdotsm2 and

120573 = 03 (- - - - -) M-1 model (mdash) M-2 or M-3 model with1198963 = 10

21 Nm4

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

16

k1 = 1012

k1 = 1011

k1 = 0

k1 = 1010k1 = 109

xl

wpw

s

(a)

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

k1 = 1012

k1 = 1011

k1 = 0

k1 = 1010

k1 = 109

xl

wpw

s

(b)

Figure 7 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 1198961 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 119888 = 103Nsdotsm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 1198963 = 10

22Nm4

mass The following data are considered 119898 = 3120588119860119897 kg1198961 = 10

4Nm2 1198963 = 1021Nm4 119888 = 103Nsdotsm2 and 120573 = 03

Moreover different speeds for the traveling mass are consid-ered (120572 = 011 025 05 075 and 11) It can be seen from thisfigure that the 119908119901119908119904 results for the M-2 or M-3 models arealways lower than the one obtained from the linear beammodel that is M-1 model However the119908119901119908119904 results for M-2 and M-3 are almost the same and independent of the value

8 Shock and Vibration

0010203040506070809

c = 0c = 106

c = 107

c = 108

c = 109

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0

02

04

06

08

1

12

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

c = 0

c = 106 c = 107

c = 108 c = 109

(b)

Figure 8 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 119888 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

10 Nm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 119896

3= 1022 Nm4

of velocity ratio The maximum difference between M-1 andM-2 orM-3models always happens at120572 = 025 Furthermoreit can be seen that in this case the maximum lateral dynamicdeflection 119908119901119908119904 increases up to 120572 = 025 and a reversingtrend prevails afterward nomatterwhat types of beammodelsare used

Using linear (M-1 model) and nonlinear (M-2 and M-3models) Timoshenko beammodels Figure 7 shows the varia-tion of normalized lateral dynamic displacement119908119901119908119904 ver-sus normalized instantaneous position of moving mass 119909119897As it was explained before these beams are resting on a vis-coelastic foundation For further analysis different values of1198961 are considered for these beam models The following dataare chosen 1198963 = 10

22Nm4 119888 = 103Nsdotsm2 and 120573 = 03

with mass moving at two different speeds (120572 = 011 and 05)It can be seen from this figure that the results of 119908119901119908119904obtained from theM-2 andM-3models with 1198963 = 10

22Nm4are the same and always lower than the one obtained fromthe M-1 model where the maximum difference between M-1andM-2 orM-3models always happens at 1198961 = 0 and is inde-pendent of the value of mass velocity Also when the valueof 1198961increases the difference between M-1 and M-2 or M-3models decreases accordingly For example when 1198961 =

1011Nm2 this difference becomes negligible Furthermore

by comparing Figure 7(a) with Figure 7(b) it can be observedthat the beamrsquos nonlinear dynamic response forM-2 andM-3

0

05

1

15

2 120573 = 03

120573 = 0015

120573 = 015

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0020406081

121416182

0 01 02 03 04 05 06 07 08 09 1

xlwpw

s

120573 = 03

120573 = 0015

120573 = 015

(b)

Figure 9 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with 119898 = 5120588119860119897 kgand 120572 = 025 and different 120573 values (a) 1198961 = 1198963 = 119888 = 0 (b)1198961 = 10

4 Nm2 1198963 = 104Nm4 and 119888 = 10

3Nsdotsm2 (- - - - -) M-1model and (mdash) M-2 or M-3 model

models at 120572 = 05 is greater than the one related to 120572 = 011Nevertheless a reversing trend is seen for the beamrsquos M-1model dynamic response

Figure 8 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams ofM-1 M-2 and M-3 traversed by a moving mass In this casedifferent values of 119888 are considered The following data arechosen 119898 = 3120588119860119897 kg 1198961 = 10

10Nm2 1198963 = 1022Nm4 and

120573 = 03 with mass moving at two different speeds (120572 =

011 and 05) It can be seen from this figure that the beamsdynamic responses of 119908119901119908119904 obtained from the M-2 or M-3 models are always lower than the one obtained from theM-1 model where the maximum difference between M-1 and M-2 or M-3 models always happens at 119888 = 0 nomatter what values of mass velocity would be Also when thevalue of 119888 increases the beamrsquos dynamical response differencebetween M-1 and M-2 or M-3 models decreases accordinglywhere for 119888 = 109Nsdotsm2 this difference becomes negligibleFurthermore by comparing Figure 8(a) with Figure 8(b) itcan be concluded that for 119888 lt 108Nsdotsm2 the obtained beamrsquosdynamic response at 120572 = 05 is almost greater than the oneobtained at 120572 = 011 no matter what type of beam models isused

Figure 9 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams of

Shock and Vibration 9

M-1 M-2 and M-3 The following data are assumed 119898 =

5120588119860119897 kg and mass traveling speed is 120572 = 025 The resultsare presented for three different slenderness ratios (120573) Fur-thermore it should be noted that in Figure 9(a) 1198961 = 1198963 = 119888 =0 and in Figure 9(b) 1198961 = 10

4Nm2 1198963 = 104Nm4 and 119888 =

103Nsdotsm2 are considered It can be seen from this figure that

the maximum value of dynamical response119908119901119908119904 increasesas 120573 increases Moreover the position of maximum value of119908119901119908119904 moves to the right end of the beam as 120573 increases Itcan be concluded from this figure that inclusion of nonlinearfoundation parameters will decrease the difference of linearand nonlinear analyses It should be further mentioned thatthe results for linear and nonlinear analyses are the same for120573 = 015 and 03

5 Conclusions

The longitudinal and lateral deflections as well as the rotationof warped cross section of three different Timoshenko beamsof M-1 M-2 and M-3 models resting on viscoelastic founda-tion and subjected to a moving mass with constant velocityare all considered by including the nonlinear nature in thebeamrsquos geometry and the stiffness of viscoelastic foundationThe outcome results are as the following

(1) It is concluded that by including geometrical andfoundation nonlinearities into the reference M-1model as the size of nonlinearity increases the differ-ence of the beamrsquos dynamic response in the nonlinearmodels (M-2 or M-3) and M-1 model increases Thisdifference becomesmore pronounced for the compar-atively large enough values of nonlinear stiffness partof viscoelastic foundation

(2) It is observed that the beamrsquos dynamic response resultsfor the fully nonlinear beam model that is M-3model are always lower than the one obtained fromthe linear beam model that is M-1 model

(3) It is concluded that when the value of linear stiffness-part of foundation increases the difference betweenM-1 and M-2 or M-3 models decreases accordinglywhere forthe large values of linear stiffness part offoundation this difference becomes negligible More-over when the value of foundation damping increasesthe beamrsquos dynamical response difference betweenM-1 and M-2 or M-3 models decreases accordingly

(4) Because both stretching effect and nonlinear stiffnesspart of viscoelastic foundation in conjunction withmoving mass condition and the other values of thefoundationrsquos parameters are significant factors in thenonlinear dynamic behavior of the beam with chang-ing the beamrsquos slenderness ratio 120573 it appears that for120573 le 0015 nonlinear behavior predominantly can beseen However for120573 gt 0015 no specific values for thecritical value of the traveling mass 119898 and foundationnonlinear stiffness 1198963 were obtained which could bean idea for further investigation

Appendix

The definition of different matrices used in calculation of thenonlinear coupled ODEs of modal relations (11) is as

(M)119894119895 = int119897

0

120601119894 (119909) 120601119895 (119909) 119889119909 (H)119894119895 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 119889119909

(F)119894119895 = int119897

0

120601119894 (119909) 1205951015840

119895(119909) 119889119909 (S)119894119895 = int

119897

0

120595119894 (119909) 120595119895 (119909) 119889119909

(K)119894119895 = int119897

0

120595119894 (119909) 12059510158401015840

119895(119909) 119889119909 (E)119894119895 = int

119897

0

120595119894 (119909) 1206011015840

119895(119909) 119889119909

(J)119894119895 = int119897

0

120579119894 (119909) 120579119895 (119909) 119889119909 (N)119894119895 = int119897

0

120579119894 (119909) 12057910158401015840

119895(119909) 119889119909

(G)119894119895119896 = int119897

0

120601119894 (119909) 12057910158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(I)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 12060110158402

119896(119909) 119889119909

(L)119894119895119896 = int119897

0

120579119894 (119909) 12060110158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(R1)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 1205791015840

119896(119909) 119889119909

(R2)119894119895119896119897 = int119897

0

120601119894 (119909) 120601119895 (119909) 120601119896 (119909) 120601119897 (119909) 119889119909

(A)119894119895 = 120601119894 (119909 = 120577 (119905)) 1206011015840

119895(119909 = 120577 (119905))

(B)119894119895 = 120601119894 (119909 = 120577 (119905)) 120601119895 (119909 = 120577 (119905))

(C)119894119895 = 120601119894 (119909 = 120577 (119905)) 12060110158401015840

119895(119909 = 120577 (119905))

(O)119894119895 = 120579119894 (119909 = 120577 (119905)) 120579119895 (119909 = 120577 (119905)) (A1)

where 119894 119895 119896 119897 = 1 2 3 119899 in which prime mark over anyparameter indicates the derivativewith respect to the position(119909) Furthermore the vector of a in the form of 119899 times 1 columnis given by

120601119894 (119909 = 120577 (119905)) = (a)119894 (A2)

Conflict of Interests

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

Acknowledgment

This paper is dedicated to Ahmad Mamandirsquos dearest Pro-fessor Dr Mohammad Hossein Kargarnovin passed awayon Monday November 4 2013 He was a fine kind decentsincere good tempered and well-liked professor who will besadlymissed by all who knew himMay his soul rest in peace

10 Shock and Vibration

References

[1] DThambiratnam andY Zhuge ldquoDynamic analysis of beams onan elastic foundation subjected to moving loadsrdquo Journal ofSound and Vibration vol 198 no 2 pp 149ndash169 1996

[2] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of an infi-nite tomoshenko beam on a viscoelastic foundation to a har-monic moving loadrdquo Journal of Sound and Vibration vol 241no 5 pp 809ndash824 2001

[3] S-M Kim and Y-H Cho ldquoVibration and dynamic buckling ofshear beam-columns on elastic foundation under moving har-monic loadsrdquo International Journal of Solids and Structures vol43 no 3-4 pp 393ndash412 2006

[4] C R Steele ldquoThe finite beam with a moving loadrdquo Journal ofApplied Mechanics vol 34 pp 111ndash118 1967

[5] A L Florence ldquoTraveling force on aTimoshenkobeamrdquo Journalof Applied Mechanics vol 32 pp 351ndash358 1965

[6] H D Nelson and R A Conover ldquoDynamic stability of a beamcarrying moving massesrdquo Journal of Applied Mechanics vol 38no 4 pp 1003ndash1006 1971

[7] T-P Chang and Y-N Liu ldquoDynamic finite element analysis of anonlinear beam subjected to a moving loadrdquo International Jour-nal of Solids and Structures vol 33 no 12 pp 1673ndash1688 1996

[8] Y C Hou C H Tseng and S F Ling ldquoA new high-ordernon-uniformTimoshenko beamfinite element on variable two-parameter foundations for vibration analysisrdquo Journal of Soundand Vibration vol 191 no 1 pp 91ndash106 1996

[9] H P Lee ldquoDynamic response of a Timoshenko beam on awinkler foundation subjected to amovingmassrdquoAppliedAcous-tics vol 55 no 3 pp 203ndash215 1998

[10] P Lou G-L Dai and Q-Y Zeng ldquoFinite-element analysis for aTimoshenko beam subjected to a moving massrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal of Mech-anical Engineering Science vol 220 no 5 pp 669ndash678 2006

[11] Y-H Chen and J-T Sheu ldquoAxially-loaded damped Timo-shenko beam on viscoelastic foundationrdquo International Journalfor Numerical Methods in Engineering vol 36 no 6 pp 1013ndash1027 1993

[12] Z Hryniewicz and P Koziol ldquoWavelet-based solution for vibra-tions of a beam on a nonlinear viscoelastic foundation due tomoving loadrdquo Journal ofTheoretical and AppliedMechanics vol51 no 1 pp 215ndash224 2013

[13] P Koziol and ZHryniewicz ldquoDynamic response of a beam rest-ing on a nonlinear foundation to a moving load coiflet-basedsolutionrdquo Shock and Vibration vol 19 pp 995ndash1007 2012

[14] Z Hryniewicz ldquoDynamics of Rayleigh beam on nonlinearfoundation due to moving load using Adomian decompositionand coiflet expansionrdquo Soil Dynamics and Earthquake Engineer-ing vol 31 no 8 pp 1123ndash1131 2011

[15] Z Dimitrovova ldquoA general procedure for the dynamic analysisof finite and infinite beams on piece-wise homogeneous foun-dationundermoving loadsrdquo Journal of Sound andVibration vol329 no 13 pp 2635ndash2653 2010

[16] V-H Nguyen and D Duhamel ldquoFinite element procedures fornonlinear structures in moving coordinates Part II infinitebeam under moving harmonic loadsrdquo Computers and Struc-tures vol 86 no 21-22 pp 2056ndash2063 2008

[17] E J Sapountzakis and A E Kampitsis ldquoNonlinear response ofshear deformable beams on tensionless nonlinear viscoelasticfoundation under moving loadsrdquo Journal of Sound and Vibra-tion vol 330 no 22 pp 5410ndash5426 2011

[18] T Mazilu M Dumitriu and C Tudorache ldquoInstability of anoscillator moving along a Timoshenko beam on viscoelasticfoundationrdquo Nonlinear Dynamics vol 67 no 2 pp 1273ndash12932012

[19] L Meirovitch Principles and Techniques of Vibrations Prentice-Hall Prentice NJ USA 1997

[20] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010

[21] A Mamandi and M H Kargarnovin ldquoNonlinear dynamicanalysis of an inclined Timoshenko beam subjected to amovingmassforce with beamrsquos weight includedrdquo Shock and Vibrationvol 18 no 6 pp 875ndash891 2011

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Shock and Vibration

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4 Shock and Vibration

Now we can establish the Lagrangian function of the systemas 119871 = 119879minus(119880minus119882119890) ApplyingHamiltonrsquos principle on 119871 yields[19]

120575int

1199052

1199051

119871 119889119905 = 0 997904rArr 120575int

1199052

1199051

(119880 minus 119879) 119889119905 = int

1199052

1199051

120575119882119890 119889119905 (3)

Note that the total virtual work done 120575119882119890 by the nonlinearviscoelastic foundation and the traveling mass acting on thebeam at the location 119909 = 120577(119905) is [9 20 21]

120575119882119890 = int

119897

0

[ minus 11989811990611990511990512057511990610038161003816100381610038161003816119909=120577(119905)

minus 119898(119892 + 119908119905119905 + 2V119908119909119905 + V2119908119909119909)

times 120575119908|119909=120577(119905) + (1198961119908 + 11989631199083+ 119888119908119905) 120575119908] 119889119909

(4)

After doing some mathematics one would get the nonlineargoverning coupled PDEs of motion (EOMs) as follows [9 2021]

Force relation in 119909 direction

120588119860119906119905119905 minus 119864119860 (119906119909119909 + 119908119909119908119909119909)

= minus1198981199061199051199051003816100381610038161003816119909=120577(119905)

120575 (119909 minus 120577 (119905)) 120594 (119905)

(5)

Force relation in 119911 direction

120588119860119908119905119905 + 119896119866119860 (Γ119909 minus 119908119909119909) minus 119864119860

times (119906119909119909119908119909 + 119906119909119908119909119909 +3

2119908119909119909119908119909

2)

+ 1198961119908 + 11989631199083+ 119888119908119905

= minus119898(119892 + 119908119905119905 + 2V119908119909119905 + V2119908119909119909)10038161003816100381610038161003816119909=120577(119905)

times 120575 (119909 minus 120577 (119905)) 120594 (119905)

(6)

Moment relation about 119910 direction

120588119868119889Γ119905119905 minus 119864119868119889Γ119909119909 + 119896119866119860 (Γ minus 119908119909) = 0 (7)

where 119864119868119889 and 120588119860 are the beamrsquos flexural rigidity and beamrsquosmass per unit length respectively Furthermore 120575(119909minus120577(119905)) isDiracrsquos delta function in which 120577(119905) is the instantaneous posi-tion of the moving mass traveling on the beam In case wherethe mass is traveling with a constant velocity V then its insta-ntaneous position is given by 120577(119905) = V119905 + 1199090 where 1199090represents the initial position of the mass at the start of itsmotion Also the 120594(119905) is the pulse function which is equal to1 while the mass is traveling on the beam and 0 when the tra-veling mass is outside the beam span that is 120594(119905) = 119906(119905) minus

119906(119905 minus 119897V) in which 119906(119905) represents the unit step function

3 Solution Method

In this study Galerkinrsquos method is chosen as a powerful math-ematical tool to analyze the vibrations of a Timoshenko

beam Based on the separation of variables technique theresponse of the Timoshenko beam in terms of the linear free-oscillation modes can be assumed as follows [9 19ndash21]

119906 (119909 119905) =

119899

sum

119895=1

120579119895 (119909) 119903119895 (119905) = Θ119879(119909)R (119905) (8)

119908 (119909 119905) =

119899

sum

119895=1

120601119895 (119909) 119901119895 (119905) = Φ119879(119909)P (119905) (9)

Γ (119909 119905) =

119899

sum

119895=1

120595119895 (119909) 119902119895 (119905) = Ψ119879(119909)Q (119905) (10)

where P(119905) Q(119905) and R(119905) are vectors of order 119899 listing thegeneralized coordinate119901119895(119905) 119902119895(119905) and 119903119895(119905) respectively andΘ(119909)Φ(119909) andΨ(119909) are some vectorial functions collectingthe first 119899mode shapes (eigen-functions) of 120579119895(119909) 120601119895(119909) and120595119895(119909) respectively

In the next step primarily we substitute (8) to (10) into (5)to (7) then on the resulting relations premultiplying bothsides of (5) by Θ119879(119909) (6) by Φ119879(119909) and (7) by Ψ119879(119909) inte-grating over the interval (0 119897) and imposing the orthogonal-ity along with Diracrsquos delta function conditions the resultingnonlinear coupledmodal equations ofmotion inmatrix formare as follows119899

sum

119895=1

[120588119860J119894119895 + 119898O119894119895 (119905) 120594 (t)] 119903119895 (119905) minus 119864119860

119899

sum

119895=1

N119894119895119903119895 (119905)

minus 119864119860

119899

sum

119895=1

119899

sum

119896=1

119901j (119905) L119894119895119896119901119896 (119905) = 0 119894 = 1 2 119899

119899

sum

119895=1

[120588119860M119894119895 + 119898B119894119895 (119905) 120594 (t)] 119895 (119905)

+

119899

sum

119895=1

[2119898V120594(119905)A119894119895 (119905) + 119888M119894119895] 119895 (119905)

+ 119896119866119860

119899

sum

119895=1

F119894119895119902119895 (119905)

+

119899

sum

119895=1

[119898V2C119894119895 (119905) 120594 (t) minus 119896119866119860H119894119895 + 1198961M119894119895] 119901119895 (119905)

+ 1198963

119899

sum

119895=1

119899

sum

119896=1

119899

sum

119897=1

119901119895 (119905) (R2)119894119895119896119897119901119896 (119905) 119901119897 (119905)

minus 119864119860[

[

119899

sum

119895=1

119899

sum

119896=1

119903j (119905)G119894119895119896119901119896 (119905)

+

119899

sum

119895=1

119899

sum

119896=1

119901j (119905) (R1)119894119895119896119903119896 (119905)

+3

2

119899

sum

119895=1

119899

sum

119896=1

119901j (119905) I119894119895119896119901119896(119905)2]

]

Shock and Vibration 5

= minus119898119892120594(119905) a119894 (119905) 119894 = 1 2 119899

120588119868119889

119899

sum

119895=1

S119894119895 119902119895 (119905) minus119899

sum

119895=1

[119864119868119889K119894119895 minus 119896119866119860S119894119895] 119902119895 (119905)

minus 119896119866119860

119899

sum

119895=1

E119894119895119901119895 (119905) = 0 119894 = 1 2 119899

(11)

inwhich dotmark over any parameter indicates the derivativewith respect to the time (119905) All matrices appearing in aboverelations are given in the appendix It is clear that (11) isthree nonlinear coupled second-order ordinary differentialequations (ODEs) The boundary conditions for a pinned-pinned Timoshenko beam are

Essential BCs 119906 (0 119905) = 119906 (119897 119905) = 0

therefore 120579119895 (119909) = 0 at 119909 = 0 119897

119908 (0 119905) = 119908 (119897 119905) = 0

therefore 120601119895 (119909) = 0 at 119909 = 0 119897

Natural BCs 119872119910 (0 119905) = 119872119910 (119897 119905) = 0

therefore 120595119895119909 (119909) = 0 at 119909 = 0 119897

(12)

Moreover initial conditions (ICs) for the Timoshenko beamare

ICs 119906 (119909 0) = 119906119905 (119909 0) = 119908 (119909 0) = 119908119905 (119909 0)

= Γ (119909 0) = Γ119905 (119909 0) = 0

(13)

This is typically a well-known eigen-function eigen-valueproblem out of which the final result for the mode shapes ofthe Timoshenko beam with pinned-pinned ends is expressedas the following [9 19ndash21]

120579119895 (119909) =radic2 sin(

119895120587119909

119897) 120601119895 (119909) =

radic2 sin(119895120587119909

119897)

120595119895 (119909) =radic2 cos(

119895120587119909

119897)

with 119895 = 1 2 3 119899

(14)

Now we use (14) to calculate all matrix quantities given in theappendix In the next step these evaluated matrices will beused in (11) and later the set of equations will be solvednumerically using the Adams-Bashforth-Moulton integrationmethod viaMATLAB solver package to obtain values of119901119895(119905)119902119895(119905) and 119903119895(119905) By back substitution of119901119895(119905) 119902119895(119905) and 119903119895(119905) in(8) to (10) 119906(119909 119905)119908(119909 119905) and Γ(119909 119905) can be obtained respec-tively Subsequently after obtaining values for 119906(119909 119905)119908(119909 119905)and Γ(119909 119905) the dynamic response of a pinned-pinned Timo-shenko beam resting on a nonlinear viscoelastic foundationand subjected to a moving mass traveling with a constantvelocity can be calculated

4 Verification of Results and Case Studies

As mentioned in the introduction at the moment no specificresults exist for the problem under consideration in theliterature Therefore to verify the validity of the resultsobtained in this study we primarily consider some specialcases by which our results can be compared with thoseexisting in the literature

41 Verification of Results for the Linear andNonlinear Analyses

411 Linear Analysis In the first attempt we set (119906119909119909 +119908119909119908119909119909) and (119906119909119909119908119909 + 119906119909119908119909119909 + (32)119908119909119909119908119909

2) in the left-

hand side of (5) and (6) respectively equal to zeroMoreoverwe take 1198963 = 0 and 119888 = 0 This will lead us to a set ofnew relations for 119906(119909 119905) 119908(119909 119905) and Γ(119909 119905) representing thelinear form of governing EOMs of a Timoshenko beam ona linear Winkler type of foundation To verify the validityof the obtained results out of our analysis we consider thedata given in [9] as 119897 = 1m 119864 = 207 times 10

9Nm2 119866 =

776 times 109Nm2 119896 = 09 120588 = 7700 kgm3 119898 = 02120588119860119897 kg

1198961 = 104Nm2 120572 = 011 119892 = 981ms2 and 120573 = 003 in

which120572 and120573 are the velocity ratio andRayleighrsquos slendernesscoefficient respectively Note that 120573 = 1205871199030119897 with 1199030 takenas the beamrsquos radius of gyration and 120572 = VV119888119903 with V119888119903 asthe critical velocity of a concentratedmoving force defined asV119888119903 = 1205961119897120587 [9] with 1205961 as the first natural angular frequencyof transversal vibration of this beam given in general as (120596119894) =(119894120587119897)2radic119864119868119889120588119860(119894 = 1 2 119899) [9 10] In addition it can be

seen that the critical velocity for the first mode (119894 = 1) is givenas V119888119903 = (120587119897)radic119864119868119889120588119860 [9 10]

Based on the above data the computer code was run forthis case and the normalized instantaneous lateral displace-ment (119908119901119908119904) of the instantaneous positions of the movingmass (119909119897) is calculated and the dimensionless outcomeresults are depicted and compared with other existing resultsin Figure 2 The normalization parameter for the lateraldisplacement that is 119908119904 is actually a midpoint deflection ofa simply supported beam under midspan concentrated loadofmg that is 119908119904 = 119898119892119897348119864119868119889 [9 10] A close inspection ofthe curves in Figure 2 indicates perfect agreements betweenthe two outcome results

412 Nonlinear Analysis A simply supported Timoshenkobeam resting on a Pasternak-type viscoelastic and a shearfoundation and traveled by a constant velocity moving forceas shown in Figure 3 is considered For this system (5) and(6) are rewritten as follows

120588119860119906119905119905 minus 119864119860 (119906119909119909 + 119908119909119908119909119909) = 0

120588119860119908119905119905 + 119896119866119860 (Γ119909 minus 119908119909119909)

minus119864119860(119906119909119909119908119909 + 119906119909119908119909119909 +3

2119908119909119909119908119909

2)

minus119896119901119908119909119909 + 1198961119908 + 11989631199083+ 119888119908119905

= minus119875120575 (119909 minus 120577 (119905)) 120594 (119905)

(15)

6 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

xl

wpw

s

Figure 2 Instantaneous normalized lateral displacement 119908119901119908119904under a moving mass of 119898 = 02120588119860119897 kg 1198961 = 10

4Nm2 120572 = 011and 120573 = 003 (mdash) linear analysis present study () linear analysis[9]

Beam 120588 E G k l A Id

kpk3k1 c

P

z w

Γ

y

x u

120577(t)

Figure 3 A simply supported Timoshenko beam resting on anonlinear Pasternak viscoelastic foundation and subjected to amoving force 119875 (see [17])

in which 119896119901 and119875 are Pasternak or shear foundationmodulusand the value of moving force respectively and (7) remainsunchanged

To check on the validity of our solutions based on (7)and (15) we consider the reported results on [17] in whichanalyses shear deformable beams on tensionless nonlinearPasternak-type viscoelastic foundations and subjected tomoving forces The following data that is 119897 = 10m 119868119889 =395times10

minus6m4119860 = 8613times10minus4m2119864 = 207times109Nm2119866 =

7961 times 109Nm2 119896 = 085 120588 = 7820 kgm3 1198961 = 20 times

106Nm2 1198963 = 0Nm

4 119896119901 = 69times103N 119888 = 138times103Nsdotsm2

119875 = 144 times 103N and V = 167ms are adapted from [17]

Based on above mentioned equations a computer programwas developed and the outcome nonlinear results are de-picted and compared with [17] in Figure 4 A close inspectionof the curves reveals very close agreements between the twosolutions

42 Results and Discussions After being satisfied with thevalidity of solution technique the beamrsquos instantaneousdynamic lateral deflection is calculated in the next step In

0 1 2 3 4 5 6 7 8 9 10minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05times10minus3

x (m)

w(x03

) (m

)

Figure 4 Lateral displacement 119908(119909 03) at instant 119905 = 03 s alonglength of the beam (mdash) nonlinear analysis present study (- - - - -)nonlinear analysis [17]

obtaining these results the following data [9] are taken intoconsideration unless otherwise specified

119897 = 1m 119864 = 207 times 109Nm2 119866 = 776 times 10

9Nm2

119896 = 09 120588 = 7700 kgm3 119868119889 = 1045 times 10minus3m4

119860 = 01146m2 119898 = 3120588119860119897 kg 1198961 = 104Nm2

1198963 = 1019Nm4 119888 = 10

3N sdot sm2 120573 = 03

(16)

It should be mentioned that all deflection variationsversus moving mass instantaneous positions are given in anondimensional form that is 119908119901119908119904 Moreover it has to bepointed out that based on the conducted convergence studyrelated to the linear and nonlinear analyses ten modes ofvibration are taken into account for the steady state answersfor 119906(119909 119905) 119908(119909 119905) and Γ(119909 119905) that is 119899 = 10

The effect of variation of nonlinear part of foundationstiffness 1198963 and geometrical nonlinearity due to the beamrsquosend boundary conditions on the dynamic responses of a Tim-oshenko beam traversed by a moving mass are depicted inFigure 5 In this analysis 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2 119888 =103Nsdotsm2 and 120573 = 03 and different speeds for the traveling

mass (120572 = 011 025 and 05) are considered It should benoted that under adopted data the results for M-2 and M-3models are similar and hence the variations are overlaidThismeans that as the beamgets thicker it resistsmore againstthe stretching of midplane surfaceTherefore stretching doesnot play an important role in the beamrsquos dynamical responseespecially In addition in this figure the differences betweendynamical behavior of the beam betweenM-2 orM-3modelsand M-1 model are compared with each other Out of thisfigure and as it is expected it can be concluded that byincreasing the value of 1198963 the difference between M-2 or M-3models and M-1 model becomes more pronounced In otherwords by including geometrical and foundation nonlineari-ties into the M-1 model as the reference beam as the size ofnonlinearity increases the difference of the values of 119908119901119908119904

Shock and Vibration 7

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

16

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019

k3 = 1020

xl

wpw

s

(a)

0 01 02 03 04 05 06 07 08 09 1

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019k3 = 1020

xl

wpw

s

0

05

1

15

2

(b)

0 01 02 03 04 05 06 07 08 09 1

0

02

04

06

08

1

12

14

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019 k3 = 1020

xl

wpw

s

(c)

Figure 5 Variation of dimensionless instantaneous dynamicresponse119908119901119908119904 versus normalized instantaneous position of mov-ing mass 119909119897 of the Timoshenko beam with different 1198963 valuestraveled at different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2119888 = 10

3Nsdotsm2 and 120573 = 03 (a) 120572 = 011 (b) 120572 = 025 (c) 120572 = 05(- - - - -) M-1 model (mdash) M-2 model or M-3 model

in the nonlinear models (M-2 or M-3) compared to the M-1model increases This difference becomes more pronouncedfor 1198963 = 10

23Nm4 Furthermore when the mass velocityincreases the maximum value of the beamrsquos dynamicresponse increases up to 120572 = 025 and decreases afterwardIn addition it can be seen that for the 1198963 values greater than1021Nm4 by increasing the value of 120572 the fluctuation ofdynamical response of the beam reduces accordingly andfinally for 1198963 = 10

23Nm4 and 120572 = 011 this fluctuationreaches its highest value

Figure 6 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position of moving mass 119909119897 for three different Tim-oshenko beams of M-1 M-2 and M-3 traversed by a moving

0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2 120572 = 025120572 = 05

120572 = 11120572 = 075

120572 = 011

xl

wpw

s

Figure 6 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam traveled at differentspeeds with 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2 119888 = 103Nsdotsm2 and

120573 = 03 (- - - - -) M-1 model (mdash) M-2 or M-3 model with1198963 = 10

21 Nm4

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

16

k1 = 1012

k1 = 1011

k1 = 0

k1 = 1010k1 = 109

xl

wpw

s

(a)

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

k1 = 1012

k1 = 1011

k1 = 0

k1 = 1010

k1 = 109

xl

wpw

s

(b)

Figure 7 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 1198961 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 119888 = 103Nsdotsm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 1198963 = 10

22Nm4

mass The following data are considered 119898 = 3120588119860119897 kg1198961 = 10

4Nm2 1198963 = 1021Nm4 119888 = 103Nsdotsm2 and 120573 = 03

Moreover different speeds for the traveling mass are consid-ered (120572 = 011 025 05 075 and 11) It can be seen from thisfigure that the 119908119901119908119904 results for the M-2 or M-3 models arealways lower than the one obtained from the linear beammodel that is M-1 model However the119908119901119908119904 results for M-2 and M-3 are almost the same and independent of the value

8 Shock and Vibration

0010203040506070809

c = 0c = 106

c = 107

c = 108

c = 109

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0

02

04

06

08

1

12

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

c = 0

c = 106 c = 107

c = 108 c = 109

(b)

Figure 8 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 119888 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

10 Nm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 119896

3= 1022 Nm4

of velocity ratio The maximum difference between M-1 andM-2 orM-3models always happens at120572 = 025 Furthermoreit can be seen that in this case the maximum lateral dynamicdeflection 119908119901119908119904 increases up to 120572 = 025 and a reversingtrend prevails afterward nomatterwhat types of beammodelsare used

Using linear (M-1 model) and nonlinear (M-2 and M-3models) Timoshenko beammodels Figure 7 shows the varia-tion of normalized lateral dynamic displacement119908119901119908119904 ver-sus normalized instantaneous position of moving mass 119909119897As it was explained before these beams are resting on a vis-coelastic foundation For further analysis different values of1198961 are considered for these beam models The following dataare chosen 1198963 = 10

22Nm4 119888 = 103Nsdotsm2 and 120573 = 03

with mass moving at two different speeds (120572 = 011 and 05)It can be seen from this figure that the results of 119908119901119908119904obtained from theM-2 andM-3models with 1198963 = 10

22Nm4are the same and always lower than the one obtained fromthe M-1 model where the maximum difference between M-1andM-2 orM-3models always happens at 1198961 = 0 and is inde-pendent of the value of mass velocity Also when the valueof 1198961increases the difference between M-1 and M-2 or M-3models decreases accordingly For example when 1198961 =

1011Nm2 this difference becomes negligible Furthermore

by comparing Figure 7(a) with Figure 7(b) it can be observedthat the beamrsquos nonlinear dynamic response forM-2 andM-3

0

05

1

15

2 120573 = 03

120573 = 0015

120573 = 015

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0020406081

121416182

0 01 02 03 04 05 06 07 08 09 1

xlwpw

s

120573 = 03

120573 = 0015

120573 = 015

(b)

Figure 9 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with 119898 = 5120588119860119897 kgand 120572 = 025 and different 120573 values (a) 1198961 = 1198963 = 119888 = 0 (b)1198961 = 10

4 Nm2 1198963 = 104Nm4 and 119888 = 10

3Nsdotsm2 (- - - - -) M-1model and (mdash) M-2 or M-3 model

models at 120572 = 05 is greater than the one related to 120572 = 011Nevertheless a reversing trend is seen for the beamrsquos M-1model dynamic response

Figure 8 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams ofM-1 M-2 and M-3 traversed by a moving mass In this casedifferent values of 119888 are considered The following data arechosen 119898 = 3120588119860119897 kg 1198961 = 10

10Nm2 1198963 = 1022Nm4 and

120573 = 03 with mass moving at two different speeds (120572 =

011 and 05) It can be seen from this figure that the beamsdynamic responses of 119908119901119908119904 obtained from the M-2 or M-3 models are always lower than the one obtained from theM-1 model where the maximum difference between M-1 and M-2 or M-3 models always happens at 119888 = 0 nomatter what values of mass velocity would be Also when thevalue of 119888 increases the beamrsquos dynamical response differencebetween M-1 and M-2 or M-3 models decreases accordinglywhere for 119888 = 109Nsdotsm2 this difference becomes negligibleFurthermore by comparing Figure 8(a) with Figure 8(b) itcan be concluded that for 119888 lt 108Nsdotsm2 the obtained beamrsquosdynamic response at 120572 = 05 is almost greater than the oneobtained at 120572 = 011 no matter what type of beam models isused

Figure 9 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams of

Shock and Vibration 9

M-1 M-2 and M-3 The following data are assumed 119898 =

5120588119860119897 kg and mass traveling speed is 120572 = 025 The resultsare presented for three different slenderness ratios (120573) Fur-thermore it should be noted that in Figure 9(a) 1198961 = 1198963 = 119888 =0 and in Figure 9(b) 1198961 = 10

4Nm2 1198963 = 104Nm4 and 119888 =

103Nsdotsm2 are considered It can be seen from this figure that

the maximum value of dynamical response119908119901119908119904 increasesas 120573 increases Moreover the position of maximum value of119908119901119908119904 moves to the right end of the beam as 120573 increases Itcan be concluded from this figure that inclusion of nonlinearfoundation parameters will decrease the difference of linearand nonlinear analyses It should be further mentioned thatthe results for linear and nonlinear analyses are the same for120573 = 015 and 03

5 Conclusions

The longitudinal and lateral deflections as well as the rotationof warped cross section of three different Timoshenko beamsof M-1 M-2 and M-3 models resting on viscoelastic founda-tion and subjected to a moving mass with constant velocityare all considered by including the nonlinear nature in thebeamrsquos geometry and the stiffness of viscoelastic foundationThe outcome results are as the following

(1) It is concluded that by including geometrical andfoundation nonlinearities into the reference M-1model as the size of nonlinearity increases the differ-ence of the beamrsquos dynamic response in the nonlinearmodels (M-2 or M-3) and M-1 model increases Thisdifference becomesmore pronounced for the compar-atively large enough values of nonlinear stiffness partof viscoelastic foundation

(2) It is observed that the beamrsquos dynamic response resultsfor the fully nonlinear beam model that is M-3model are always lower than the one obtained fromthe linear beam model that is M-1 model

(3) It is concluded that when the value of linear stiffness-part of foundation increases the difference betweenM-1 and M-2 or M-3 models decreases accordinglywhere forthe large values of linear stiffness part offoundation this difference becomes negligible More-over when the value of foundation damping increasesthe beamrsquos dynamical response difference betweenM-1 and M-2 or M-3 models decreases accordingly

(4) Because both stretching effect and nonlinear stiffnesspart of viscoelastic foundation in conjunction withmoving mass condition and the other values of thefoundationrsquos parameters are significant factors in thenonlinear dynamic behavior of the beam with chang-ing the beamrsquos slenderness ratio 120573 it appears that for120573 le 0015 nonlinear behavior predominantly can beseen However for120573 gt 0015 no specific values for thecritical value of the traveling mass 119898 and foundationnonlinear stiffness 1198963 were obtained which could bean idea for further investigation

Appendix

The definition of different matrices used in calculation of thenonlinear coupled ODEs of modal relations (11) is as

(M)119894119895 = int119897

0

120601119894 (119909) 120601119895 (119909) 119889119909 (H)119894119895 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 119889119909

(F)119894119895 = int119897

0

120601119894 (119909) 1205951015840

119895(119909) 119889119909 (S)119894119895 = int

119897

0

120595119894 (119909) 120595119895 (119909) 119889119909

(K)119894119895 = int119897

0

120595119894 (119909) 12059510158401015840

119895(119909) 119889119909 (E)119894119895 = int

119897

0

120595119894 (119909) 1206011015840

119895(119909) 119889119909

(J)119894119895 = int119897

0

120579119894 (119909) 120579119895 (119909) 119889119909 (N)119894119895 = int119897

0

120579119894 (119909) 12057910158401015840

119895(119909) 119889119909

(G)119894119895119896 = int119897

0

120601119894 (119909) 12057910158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(I)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 12060110158402

119896(119909) 119889119909

(L)119894119895119896 = int119897

0

120579119894 (119909) 12060110158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(R1)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 1205791015840

119896(119909) 119889119909

(R2)119894119895119896119897 = int119897

0

120601119894 (119909) 120601119895 (119909) 120601119896 (119909) 120601119897 (119909) 119889119909

(A)119894119895 = 120601119894 (119909 = 120577 (119905)) 1206011015840

119895(119909 = 120577 (119905))

(B)119894119895 = 120601119894 (119909 = 120577 (119905)) 120601119895 (119909 = 120577 (119905))

(C)119894119895 = 120601119894 (119909 = 120577 (119905)) 12060110158401015840

119895(119909 = 120577 (119905))

(O)119894119895 = 120579119894 (119909 = 120577 (119905)) 120579119895 (119909 = 120577 (119905)) (A1)

where 119894 119895 119896 119897 = 1 2 3 119899 in which prime mark over anyparameter indicates the derivativewith respect to the position(119909) Furthermore the vector of a in the form of 119899 times 1 columnis given by

120601119894 (119909 = 120577 (119905)) = (a)119894 (A2)

Conflict of Interests

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

Acknowledgment

This paper is dedicated to Ahmad Mamandirsquos dearest Pro-fessor Dr Mohammad Hossein Kargarnovin passed awayon Monday November 4 2013 He was a fine kind decentsincere good tempered and well-liked professor who will besadlymissed by all who knew himMay his soul rest in peace

10 Shock and Vibration

References

[1] DThambiratnam andY Zhuge ldquoDynamic analysis of beams onan elastic foundation subjected to moving loadsrdquo Journal ofSound and Vibration vol 198 no 2 pp 149ndash169 1996

[2] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of an infi-nite tomoshenko beam on a viscoelastic foundation to a har-monic moving loadrdquo Journal of Sound and Vibration vol 241no 5 pp 809ndash824 2001

[3] S-M Kim and Y-H Cho ldquoVibration and dynamic buckling ofshear beam-columns on elastic foundation under moving har-monic loadsrdquo International Journal of Solids and Structures vol43 no 3-4 pp 393ndash412 2006

[4] C R Steele ldquoThe finite beam with a moving loadrdquo Journal ofApplied Mechanics vol 34 pp 111ndash118 1967

[5] A L Florence ldquoTraveling force on aTimoshenkobeamrdquo Journalof Applied Mechanics vol 32 pp 351ndash358 1965

[6] H D Nelson and R A Conover ldquoDynamic stability of a beamcarrying moving massesrdquo Journal of Applied Mechanics vol 38no 4 pp 1003ndash1006 1971

[7] T-P Chang and Y-N Liu ldquoDynamic finite element analysis of anonlinear beam subjected to a moving loadrdquo International Jour-nal of Solids and Structures vol 33 no 12 pp 1673ndash1688 1996

[8] Y C Hou C H Tseng and S F Ling ldquoA new high-ordernon-uniformTimoshenko beamfinite element on variable two-parameter foundations for vibration analysisrdquo Journal of Soundand Vibration vol 191 no 1 pp 91ndash106 1996

[9] H P Lee ldquoDynamic response of a Timoshenko beam on awinkler foundation subjected to amovingmassrdquoAppliedAcous-tics vol 55 no 3 pp 203ndash215 1998

[10] P Lou G-L Dai and Q-Y Zeng ldquoFinite-element analysis for aTimoshenko beam subjected to a moving massrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal of Mech-anical Engineering Science vol 220 no 5 pp 669ndash678 2006

[11] Y-H Chen and J-T Sheu ldquoAxially-loaded damped Timo-shenko beam on viscoelastic foundationrdquo International Journalfor Numerical Methods in Engineering vol 36 no 6 pp 1013ndash1027 1993

[12] Z Hryniewicz and P Koziol ldquoWavelet-based solution for vibra-tions of a beam on a nonlinear viscoelastic foundation due tomoving loadrdquo Journal ofTheoretical and AppliedMechanics vol51 no 1 pp 215ndash224 2013

[13] P Koziol and ZHryniewicz ldquoDynamic response of a beam rest-ing on a nonlinear foundation to a moving load coiflet-basedsolutionrdquo Shock and Vibration vol 19 pp 995ndash1007 2012

[14] Z Hryniewicz ldquoDynamics of Rayleigh beam on nonlinearfoundation due to moving load using Adomian decompositionand coiflet expansionrdquo Soil Dynamics and Earthquake Engineer-ing vol 31 no 8 pp 1123ndash1131 2011

[15] Z Dimitrovova ldquoA general procedure for the dynamic analysisof finite and infinite beams on piece-wise homogeneous foun-dationundermoving loadsrdquo Journal of Sound andVibration vol329 no 13 pp 2635ndash2653 2010

[16] V-H Nguyen and D Duhamel ldquoFinite element procedures fornonlinear structures in moving coordinates Part II infinitebeam under moving harmonic loadsrdquo Computers and Struc-tures vol 86 no 21-22 pp 2056ndash2063 2008

[17] E J Sapountzakis and A E Kampitsis ldquoNonlinear response ofshear deformable beams on tensionless nonlinear viscoelasticfoundation under moving loadsrdquo Journal of Sound and Vibra-tion vol 330 no 22 pp 5410ndash5426 2011

[18] T Mazilu M Dumitriu and C Tudorache ldquoInstability of anoscillator moving along a Timoshenko beam on viscoelasticfoundationrdquo Nonlinear Dynamics vol 67 no 2 pp 1273ndash12932012

[19] L Meirovitch Principles and Techniques of Vibrations Prentice-Hall Prentice NJ USA 1997

[20] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010

[21] A Mamandi and M H Kargarnovin ldquoNonlinear dynamicanalysis of an inclined Timoshenko beam subjected to amovingmassforce with beamrsquos weight includedrdquo Shock and Vibrationvol 18 no 6 pp 875ndash891 2011

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Shock and Vibration 5

= minus119898119892120594(119905) a119894 (119905) 119894 = 1 2 119899

120588119868119889

119899

sum

119895=1

S119894119895 119902119895 (119905) minus119899

sum

119895=1

[119864119868119889K119894119895 minus 119896119866119860S119894119895] 119902119895 (119905)

minus 119896119866119860

119899

sum

119895=1

E119894119895119901119895 (119905) = 0 119894 = 1 2 119899

(11)

inwhich dotmark over any parameter indicates the derivativewith respect to the time (119905) All matrices appearing in aboverelations are given in the appendix It is clear that (11) isthree nonlinear coupled second-order ordinary differentialequations (ODEs) The boundary conditions for a pinned-pinned Timoshenko beam are

Essential BCs 119906 (0 119905) = 119906 (119897 119905) = 0

therefore 120579119895 (119909) = 0 at 119909 = 0 119897

119908 (0 119905) = 119908 (119897 119905) = 0

therefore 120601119895 (119909) = 0 at 119909 = 0 119897

Natural BCs 119872119910 (0 119905) = 119872119910 (119897 119905) = 0

therefore 120595119895119909 (119909) = 0 at 119909 = 0 119897

(12)

Moreover initial conditions (ICs) for the Timoshenko beamare

ICs 119906 (119909 0) = 119906119905 (119909 0) = 119908 (119909 0) = 119908119905 (119909 0)

= Γ (119909 0) = Γ119905 (119909 0) = 0

(13)

This is typically a well-known eigen-function eigen-valueproblem out of which the final result for the mode shapes ofthe Timoshenko beam with pinned-pinned ends is expressedas the following [9 19ndash21]

120579119895 (119909) =radic2 sin(

119895120587119909

119897) 120601119895 (119909) =

radic2 sin(119895120587119909

119897)

120595119895 (119909) =radic2 cos(

119895120587119909

119897)

with 119895 = 1 2 3 119899

(14)

Now we use (14) to calculate all matrix quantities given in theappendix In the next step these evaluated matrices will beused in (11) and later the set of equations will be solvednumerically using the Adams-Bashforth-Moulton integrationmethod viaMATLAB solver package to obtain values of119901119895(119905)119902119895(119905) and 119903119895(119905) By back substitution of119901119895(119905) 119902119895(119905) and 119903119895(119905) in(8) to (10) 119906(119909 119905)119908(119909 119905) and Γ(119909 119905) can be obtained respec-tively Subsequently after obtaining values for 119906(119909 119905)119908(119909 119905)and Γ(119909 119905) the dynamic response of a pinned-pinned Timo-shenko beam resting on a nonlinear viscoelastic foundationand subjected to a moving mass traveling with a constantvelocity can be calculated

4 Verification of Results and Case Studies

As mentioned in the introduction at the moment no specificresults exist for the problem under consideration in theliterature Therefore to verify the validity of the resultsobtained in this study we primarily consider some specialcases by which our results can be compared with thoseexisting in the literature

41 Verification of Results for the Linear andNonlinear Analyses

411 Linear Analysis In the first attempt we set (119906119909119909 +119908119909119908119909119909) and (119906119909119909119908119909 + 119906119909119908119909119909 + (32)119908119909119909119908119909

2) in the left-

hand side of (5) and (6) respectively equal to zeroMoreoverwe take 1198963 = 0 and 119888 = 0 This will lead us to a set ofnew relations for 119906(119909 119905) 119908(119909 119905) and Γ(119909 119905) representing thelinear form of governing EOMs of a Timoshenko beam ona linear Winkler type of foundation To verify the validityof the obtained results out of our analysis we consider thedata given in [9] as 119897 = 1m 119864 = 207 times 10

9Nm2 119866 =

776 times 109Nm2 119896 = 09 120588 = 7700 kgm3 119898 = 02120588119860119897 kg

1198961 = 104Nm2 120572 = 011 119892 = 981ms2 and 120573 = 003 in

which120572 and120573 are the velocity ratio andRayleighrsquos slendernesscoefficient respectively Note that 120573 = 1205871199030119897 with 1199030 takenas the beamrsquos radius of gyration and 120572 = VV119888119903 with V119888119903 asthe critical velocity of a concentratedmoving force defined asV119888119903 = 1205961119897120587 [9] with 1205961 as the first natural angular frequencyof transversal vibration of this beam given in general as (120596119894) =(119894120587119897)2radic119864119868119889120588119860(119894 = 1 2 119899) [9 10] In addition it can be

seen that the critical velocity for the first mode (119894 = 1) is givenas V119888119903 = (120587119897)radic119864119868119889120588119860 [9 10]

Based on the above data the computer code was run forthis case and the normalized instantaneous lateral displace-ment (119908119901119908119904) of the instantaneous positions of the movingmass (119909119897) is calculated and the dimensionless outcomeresults are depicted and compared with other existing resultsin Figure 2 The normalization parameter for the lateraldisplacement that is 119908119904 is actually a midpoint deflection ofa simply supported beam under midspan concentrated loadofmg that is 119908119904 = 119898119892119897348119864119868119889 [9 10] A close inspection ofthe curves in Figure 2 indicates perfect agreements betweenthe two outcome results

412 Nonlinear Analysis A simply supported Timoshenkobeam resting on a Pasternak-type viscoelastic and a shearfoundation and traveled by a constant velocity moving forceas shown in Figure 3 is considered For this system (5) and(6) are rewritten as follows

120588119860119906119905119905 minus 119864119860 (119906119909119909 + 119908119909119908119909119909) = 0

120588119860119908119905119905 + 119896119866119860 (Γ119909 minus 119908119909119909)

minus119864119860(119906119909119909119908119909 + 119906119909119908119909119909 +3

2119908119909119909119908119909

2)

minus119896119901119908119909119909 + 1198961119908 + 11989631199083+ 119888119908119905

= minus119875120575 (119909 minus 120577 (119905)) 120594 (119905)

(15)

6 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

xl

wpw

s

Figure 2 Instantaneous normalized lateral displacement 119908119901119908119904under a moving mass of 119898 = 02120588119860119897 kg 1198961 = 10

4Nm2 120572 = 011and 120573 = 003 (mdash) linear analysis present study () linear analysis[9]

Beam 120588 E G k l A Id

kpk3k1 c

P

z w

Γ

y

x u

120577(t)

Figure 3 A simply supported Timoshenko beam resting on anonlinear Pasternak viscoelastic foundation and subjected to amoving force 119875 (see [17])

in which 119896119901 and119875 are Pasternak or shear foundationmodulusand the value of moving force respectively and (7) remainsunchanged

To check on the validity of our solutions based on (7)and (15) we consider the reported results on [17] in whichanalyses shear deformable beams on tensionless nonlinearPasternak-type viscoelastic foundations and subjected tomoving forces The following data that is 119897 = 10m 119868119889 =395times10

minus6m4119860 = 8613times10minus4m2119864 = 207times109Nm2119866 =

7961 times 109Nm2 119896 = 085 120588 = 7820 kgm3 1198961 = 20 times

106Nm2 1198963 = 0Nm

4 119896119901 = 69times103N 119888 = 138times103Nsdotsm2

119875 = 144 times 103N and V = 167ms are adapted from [17]

Based on above mentioned equations a computer programwas developed and the outcome nonlinear results are de-picted and compared with [17] in Figure 4 A close inspectionof the curves reveals very close agreements between the twosolutions

42 Results and Discussions After being satisfied with thevalidity of solution technique the beamrsquos instantaneousdynamic lateral deflection is calculated in the next step In

0 1 2 3 4 5 6 7 8 9 10minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05times10minus3

x (m)

w(x03

) (m

)

Figure 4 Lateral displacement 119908(119909 03) at instant 119905 = 03 s alonglength of the beam (mdash) nonlinear analysis present study (- - - - -)nonlinear analysis [17]

obtaining these results the following data [9] are taken intoconsideration unless otherwise specified

119897 = 1m 119864 = 207 times 109Nm2 119866 = 776 times 10

9Nm2

119896 = 09 120588 = 7700 kgm3 119868119889 = 1045 times 10minus3m4

119860 = 01146m2 119898 = 3120588119860119897 kg 1198961 = 104Nm2

1198963 = 1019Nm4 119888 = 10

3N sdot sm2 120573 = 03

(16)

It should be mentioned that all deflection variationsversus moving mass instantaneous positions are given in anondimensional form that is 119908119901119908119904 Moreover it has to bepointed out that based on the conducted convergence studyrelated to the linear and nonlinear analyses ten modes ofvibration are taken into account for the steady state answersfor 119906(119909 119905) 119908(119909 119905) and Γ(119909 119905) that is 119899 = 10

The effect of variation of nonlinear part of foundationstiffness 1198963 and geometrical nonlinearity due to the beamrsquosend boundary conditions on the dynamic responses of a Tim-oshenko beam traversed by a moving mass are depicted inFigure 5 In this analysis 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2 119888 =103Nsdotsm2 and 120573 = 03 and different speeds for the traveling

mass (120572 = 011 025 and 05) are considered It should benoted that under adopted data the results for M-2 and M-3models are similar and hence the variations are overlaidThismeans that as the beamgets thicker it resistsmore againstthe stretching of midplane surfaceTherefore stretching doesnot play an important role in the beamrsquos dynamical responseespecially In addition in this figure the differences betweendynamical behavior of the beam betweenM-2 orM-3modelsand M-1 model are compared with each other Out of thisfigure and as it is expected it can be concluded that byincreasing the value of 1198963 the difference between M-2 or M-3models and M-1 model becomes more pronounced In otherwords by including geometrical and foundation nonlineari-ties into the M-1 model as the reference beam as the size ofnonlinearity increases the difference of the values of 119908119901119908119904

Shock and Vibration 7

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

16

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019

k3 = 1020

xl

wpw

s

(a)

0 01 02 03 04 05 06 07 08 09 1

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019k3 = 1020

xl

wpw

s

0

05

1

15

2

(b)

0 01 02 03 04 05 06 07 08 09 1

0

02

04

06

08

1

12

14

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019 k3 = 1020

xl

wpw

s

(c)

Figure 5 Variation of dimensionless instantaneous dynamicresponse119908119901119908119904 versus normalized instantaneous position of mov-ing mass 119909119897 of the Timoshenko beam with different 1198963 valuestraveled at different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2119888 = 10

3Nsdotsm2 and 120573 = 03 (a) 120572 = 011 (b) 120572 = 025 (c) 120572 = 05(- - - - -) M-1 model (mdash) M-2 model or M-3 model

in the nonlinear models (M-2 or M-3) compared to the M-1model increases This difference becomes more pronouncedfor 1198963 = 10

23Nm4 Furthermore when the mass velocityincreases the maximum value of the beamrsquos dynamicresponse increases up to 120572 = 025 and decreases afterwardIn addition it can be seen that for the 1198963 values greater than1021Nm4 by increasing the value of 120572 the fluctuation ofdynamical response of the beam reduces accordingly andfinally for 1198963 = 10

23Nm4 and 120572 = 011 this fluctuationreaches its highest value

Figure 6 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position of moving mass 119909119897 for three different Tim-oshenko beams of M-1 M-2 and M-3 traversed by a moving

0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2 120572 = 025120572 = 05

120572 = 11120572 = 075

120572 = 011

xl

wpw

s

Figure 6 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam traveled at differentspeeds with 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2 119888 = 103Nsdotsm2 and

120573 = 03 (- - - - -) M-1 model (mdash) M-2 or M-3 model with1198963 = 10

21 Nm4

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

16

k1 = 1012

k1 = 1011

k1 = 0

k1 = 1010k1 = 109

xl

wpw

s

(a)

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

k1 = 1012

k1 = 1011

k1 = 0

k1 = 1010

k1 = 109

xl

wpw

s

(b)

Figure 7 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 1198961 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 119888 = 103Nsdotsm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 1198963 = 10

22Nm4

mass The following data are considered 119898 = 3120588119860119897 kg1198961 = 10

4Nm2 1198963 = 1021Nm4 119888 = 103Nsdotsm2 and 120573 = 03

Moreover different speeds for the traveling mass are consid-ered (120572 = 011 025 05 075 and 11) It can be seen from thisfigure that the 119908119901119908119904 results for the M-2 or M-3 models arealways lower than the one obtained from the linear beammodel that is M-1 model However the119908119901119908119904 results for M-2 and M-3 are almost the same and independent of the value

8 Shock and Vibration

0010203040506070809

c = 0c = 106

c = 107

c = 108

c = 109

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0

02

04

06

08

1

12

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

c = 0

c = 106 c = 107

c = 108 c = 109

(b)

Figure 8 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 119888 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

10 Nm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 119896

3= 1022 Nm4

of velocity ratio The maximum difference between M-1 andM-2 orM-3models always happens at120572 = 025 Furthermoreit can be seen that in this case the maximum lateral dynamicdeflection 119908119901119908119904 increases up to 120572 = 025 and a reversingtrend prevails afterward nomatterwhat types of beammodelsare used

Using linear (M-1 model) and nonlinear (M-2 and M-3models) Timoshenko beammodels Figure 7 shows the varia-tion of normalized lateral dynamic displacement119908119901119908119904 ver-sus normalized instantaneous position of moving mass 119909119897As it was explained before these beams are resting on a vis-coelastic foundation For further analysis different values of1198961 are considered for these beam models The following dataare chosen 1198963 = 10

22Nm4 119888 = 103Nsdotsm2 and 120573 = 03

with mass moving at two different speeds (120572 = 011 and 05)It can be seen from this figure that the results of 119908119901119908119904obtained from theM-2 andM-3models with 1198963 = 10

22Nm4are the same and always lower than the one obtained fromthe M-1 model where the maximum difference between M-1andM-2 orM-3models always happens at 1198961 = 0 and is inde-pendent of the value of mass velocity Also when the valueof 1198961increases the difference between M-1 and M-2 or M-3models decreases accordingly For example when 1198961 =

1011Nm2 this difference becomes negligible Furthermore

by comparing Figure 7(a) with Figure 7(b) it can be observedthat the beamrsquos nonlinear dynamic response forM-2 andM-3

0

05

1

15

2 120573 = 03

120573 = 0015

120573 = 015

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0020406081

121416182

0 01 02 03 04 05 06 07 08 09 1

xlwpw

s

120573 = 03

120573 = 0015

120573 = 015

(b)

Figure 9 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with 119898 = 5120588119860119897 kgand 120572 = 025 and different 120573 values (a) 1198961 = 1198963 = 119888 = 0 (b)1198961 = 10

4 Nm2 1198963 = 104Nm4 and 119888 = 10

3Nsdotsm2 (- - - - -) M-1model and (mdash) M-2 or M-3 model

models at 120572 = 05 is greater than the one related to 120572 = 011Nevertheless a reversing trend is seen for the beamrsquos M-1model dynamic response

Figure 8 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams ofM-1 M-2 and M-3 traversed by a moving mass In this casedifferent values of 119888 are considered The following data arechosen 119898 = 3120588119860119897 kg 1198961 = 10

10Nm2 1198963 = 1022Nm4 and

120573 = 03 with mass moving at two different speeds (120572 =

011 and 05) It can be seen from this figure that the beamsdynamic responses of 119908119901119908119904 obtained from the M-2 or M-3 models are always lower than the one obtained from theM-1 model where the maximum difference between M-1 and M-2 or M-3 models always happens at 119888 = 0 nomatter what values of mass velocity would be Also when thevalue of 119888 increases the beamrsquos dynamical response differencebetween M-1 and M-2 or M-3 models decreases accordinglywhere for 119888 = 109Nsdotsm2 this difference becomes negligibleFurthermore by comparing Figure 8(a) with Figure 8(b) itcan be concluded that for 119888 lt 108Nsdotsm2 the obtained beamrsquosdynamic response at 120572 = 05 is almost greater than the oneobtained at 120572 = 011 no matter what type of beam models isused

Figure 9 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams of

Shock and Vibration 9

M-1 M-2 and M-3 The following data are assumed 119898 =

5120588119860119897 kg and mass traveling speed is 120572 = 025 The resultsare presented for three different slenderness ratios (120573) Fur-thermore it should be noted that in Figure 9(a) 1198961 = 1198963 = 119888 =0 and in Figure 9(b) 1198961 = 10

4Nm2 1198963 = 104Nm4 and 119888 =

103Nsdotsm2 are considered It can be seen from this figure that

the maximum value of dynamical response119908119901119908119904 increasesas 120573 increases Moreover the position of maximum value of119908119901119908119904 moves to the right end of the beam as 120573 increases Itcan be concluded from this figure that inclusion of nonlinearfoundation parameters will decrease the difference of linearand nonlinear analyses It should be further mentioned thatthe results for linear and nonlinear analyses are the same for120573 = 015 and 03

5 Conclusions

The longitudinal and lateral deflections as well as the rotationof warped cross section of three different Timoshenko beamsof M-1 M-2 and M-3 models resting on viscoelastic founda-tion and subjected to a moving mass with constant velocityare all considered by including the nonlinear nature in thebeamrsquos geometry and the stiffness of viscoelastic foundationThe outcome results are as the following

(1) It is concluded that by including geometrical andfoundation nonlinearities into the reference M-1model as the size of nonlinearity increases the differ-ence of the beamrsquos dynamic response in the nonlinearmodels (M-2 or M-3) and M-1 model increases Thisdifference becomesmore pronounced for the compar-atively large enough values of nonlinear stiffness partof viscoelastic foundation

(2) It is observed that the beamrsquos dynamic response resultsfor the fully nonlinear beam model that is M-3model are always lower than the one obtained fromthe linear beam model that is M-1 model

(3) It is concluded that when the value of linear stiffness-part of foundation increases the difference betweenM-1 and M-2 or M-3 models decreases accordinglywhere forthe large values of linear stiffness part offoundation this difference becomes negligible More-over when the value of foundation damping increasesthe beamrsquos dynamical response difference betweenM-1 and M-2 or M-3 models decreases accordingly

(4) Because both stretching effect and nonlinear stiffnesspart of viscoelastic foundation in conjunction withmoving mass condition and the other values of thefoundationrsquos parameters are significant factors in thenonlinear dynamic behavior of the beam with chang-ing the beamrsquos slenderness ratio 120573 it appears that for120573 le 0015 nonlinear behavior predominantly can beseen However for120573 gt 0015 no specific values for thecritical value of the traveling mass 119898 and foundationnonlinear stiffness 1198963 were obtained which could bean idea for further investigation

Appendix

The definition of different matrices used in calculation of thenonlinear coupled ODEs of modal relations (11) is as

(M)119894119895 = int119897

0

120601119894 (119909) 120601119895 (119909) 119889119909 (H)119894119895 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 119889119909

(F)119894119895 = int119897

0

120601119894 (119909) 1205951015840

119895(119909) 119889119909 (S)119894119895 = int

119897

0

120595119894 (119909) 120595119895 (119909) 119889119909

(K)119894119895 = int119897

0

120595119894 (119909) 12059510158401015840

119895(119909) 119889119909 (E)119894119895 = int

119897

0

120595119894 (119909) 1206011015840

119895(119909) 119889119909

(J)119894119895 = int119897

0

120579119894 (119909) 120579119895 (119909) 119889119909 (N)119894119895 = int119897

0

120579119894 (119909) 12057910158401015840

119895(119909) 119889119909

(G)119894119895119896 = int119897

0

120601119894 (119909) 12057910158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(I)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 12060110158402

119896(119909) 119889119909

(L)119894119895119896 = int119897

0

120579119894 (119909) 12060110158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(R1)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 1205791015840

119896(119909) 119889119909

(R2)119894119895119896119897 = int119897

0

120601119894 (119909) 120601119895 (119909) 120601119896 (119909) 120601119897 (119909) 119889119909

(A)119894119895 = 120601119894 (119909 = 120577 (119905)) 1206011015840

119895(119909 = 120577 (119905))

(B)119894119895 = 120601119894 (119909 = 120577 (119905)) 120601119895 (119909 = 120577 (119905))

(C)119894119895 = 120601119894 (119909 = 120577 (119905)) 12060110158401015840

119895(119909 = 120577 (119905))

(O)119894119895 = 120579119894 (119909 = 120577 (119905)) 120579119895 (119909 = 120577 (119905)) (A1)

where 119894 119895 119896 119897 = 1 2 3 119899 in which prime mark over anyparameter indicates the derivativewith respect to the position(119909) Furthermore the vector of a in the form of 119899 times 1 columnis given by

120601119894 (119909 = 120577 (119905)) = (a)119894 (A2)

Conflict of Interests

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

Acknowledgment

This paper is dedicated to Ahmad Mamandirsquos dearest Pro-fessor Dr Mohammad Hossein Kargarnovin passed awayon Monday November 4 2013 He was a fine kind decentsincere good tempered and well-liked professor who will besadlymissed by all who knew himMay his soul rest in peace

10 Shock and Vibration

References

[1] DThambiratnam andY Zhuge ldquoDynamic analysis of beams onan elastic foundation subjected to moving loadsrdquo Journal ofSound and Vibration vol 198 no 2 pp 149ndash169 1996

[2] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of an infi-nite tomoshenko beam on a viscoelastic foundation to a har-monic moving loadrdquo Journal of Sound and Vibration vol 241no 5 pp 809ndash824 2001

[3] S-M Kim and Y-H Cho ldquoVibration and dynamic buckling ofshear beam-columns on elastic foundation under moving har-monic loadsrdquo International Journal of Solids and Structures vol43 no 3-4 pp 393ndash412 2006

[4] C R Steele ldquoThe finite beam with a moving loadrdquo Journal ofApplied Mechanics vol 34 pp 111ndash118 1967

[5] A L Florence ldquoTraveling force on aTimoshenkobeamrdquo Journalof Applied Mechanics vol 32 pp 351ndash358 1965

[6] H D Nelson and R A Conover ldquoDynamic stability of a beamcarrying moving massesrdquo Journal of Applied Mechanics vol 38no 4 pp 1003ndash1006 1971

[7] T-P Chang and Y-N Liu ldquoDynamic finite element analysis of anonlinear beam subjected to a moving loadrdquo International Jour-nal of Solids and Structures vol 33 no 12 pp 1673ndash1688 1996

[8] Y C Hou C H Tseng and S F Ling ldquoA new high-ordernon-uniformTimoshenko beamfinite element on variable two-parameter foundations for vibration analysisrdquo Journal of Soundand Vibration vol 191 no 1 pp 91ndash106 1996

[9] H P Lee ldquoDynamic response of a Timoshenko beam on awinkler foundation subjected to amovingmassrdquoAppliedAcous-tics vol 55 no 3 pp 203ndash215 1998

[10] P Lou G-L Dai and Q-Y Zeng ldquoFinite-element analysis for aTimoshenko beam subjected to a moving massrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal of Mech-anical Engineering Science vol 220 no 5 pp 669ndash678 2006

[11] Y-H Chen and J-T Sheu ldquoAxially-loaded damped Timo-shenko beam on viscoelastic foundationrdquo International Journalfor Numerical Methods in Engineering vol 36 no 6 pp 1013ndash1027 1993

[12] Z Hryniewicz and P Koziol ldquoWavelet-based solution for vibra-tions of a beam on a nonlinear viscoelastic foundation due tomoving loadrdquo Journal ofTheoretical and AppliedMechanics vol51 no 1 pp 215ndash224 2013

[13] P Koziol and ZHryniewicz ldquoDynamic response of a beam rest-ing on a nonlinear foundation to a moving load coiflet-basedsolutionrdquo Shock and Vibration vol 19 pp 995ndash1007 2012

[14] Z Hryniewicz ldquoDynamics of Rayleigh beam on nonlinearfoundation due to moving load using Adomian decompositionand coiflet expansionrdquo Soil Dynamics and Earthquake Engineer-ing vol 31 no 8 pp 1123ndash1131 2011

[15] Z Dimitrovova ldquoA general procedure for the dynamic analysisof finite and infinite beams on piece-wise homogeneous foun-dationundermoving loadsrdquo Journal of Sound andVibration vol329 no 13 pp 2635ndash2653 2010

[16] V-H Nguyen and D Duhamel ldquoFinite element procedures fornonlinear structures in moving coordinates Part II infinitebeam under moving harmonic loadsrdquo Computers and Struc-tures vol 86 no 21-22 pp 2056ndash2063 2008

[17] E J Sapountzakis and A E Kampitsis ldquoNonlinear response ofshear deformable beams on tensionless nonlinear viscoelasticfoundation under moving loadsrdquo Journal of Sound and Vibra-tion vol 330 no 22 pp 5410ndash5426 2011

[18] T Mazilu M Dumitriu and C Tudorache ldquoInstability of anoscillator moving along a Timoshenko beam on viscoelasticfoundationrdquo Nonlinear Dynamics vol 67 no 2 pp 1273ndash12932012

[19] L Meirovitch Principles and Techniques of Vibrations Prentice-Hall Prentice NJ USA 1997

[20] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010

[21] A Mamandi and M H Kargarnovin ldquoNonlinear dynamicanalysis of an inclined Timoshenko beam subjected to amovingmassforce with beamrsquos weight includedrdquo Shock and Vibrationvol 18 no 6 pp 875ndash891 2011

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Shock and Vibration

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DistributedSensor Networks

International Journal of

6 Shock and Vibration

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

xl

wpw

s

Figure 2 Instantaneous normalized lateral displacement 119908119901119908119904under a moving mass of 119898 = 02120588119860119897 kg 1198961 = 10

4Nm2 120572 = 011and 120573 = 003 (mdash) linear analysis present study () linear analysis[9]

Beam 120588 E G k l A Id

kpk3k1 c

P

z w

Γ

y

x u

120577(t)

Figure 3 A simply supported Timoshenko beam resting on anonlinear Pasternak viscoelastic foundation and subjected to amoving force 119875 (see [17])

in which 119896119901 and119875 are Pasternak or shear foundationmodulusand the value of moving force respectively and (7) remainsunchanged

To check on the validity of our solutions based on (7)and (15) we consider the reported results on [17] in whichanalyses shear deformable beams on tensionless nonlinearPasternak-type viscoelastic foundations and subjected tomoving forces The following data that is 119897 = 10m 119868119889 =395times10

minus6m4119860 = 8613times10minus4m2119864 = 207times109Nm2119866 =

7961 times 109Nm2 119896 = 085 120588 = 7820 kgm3 1198961 = 20 times

106Nm2 1198963 = 0Nm

4 119896119901 = 69times103N 119888 = 138times103Nsdotsm2

119875 = 144 times 103N and V = 167ms are adapted from [17]

Based on above mentioned equations a computer programwas developed and the outcome nonlinear results are de-picted and compared with [17] in Figure 4 A close inspectionof the curves reveals very close agreements between the twosolutions

42 Results and Discussions After being satisfied with thevalidity of solution technique the beamrsquos instantaneousdynamic lateral deflection is calculated in the next step In

0 1 2 3 4 5 6 7 8 9 10minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05times10minus3

x (m)

w(x03

) (m

)

Figure 4 Lateral displacement 119908(119909 03) at instant 119905 = 03 s alonglength of the beam (mdash) nonlinear analysis present study (- - - - -)nonlinear analysis [17]

obtaining these results the following data [9] are taken intoconsideration unless otherwise specified

119897 = 1m 119864 = 207 times 109Nm2 119866 = 776 times 10

9Nm2

119896 = 09 120588 = 7700 kgm3 119868119889 = 1045 times 10minus3m4

119860 = 01146m2 119898 = 3120588119860119897 kg 1198961 = 104Nm2

1198963 = 1019Nm4 119888 = 10

3N sdot sm2 120573 = 03

(16)

It should be mentioned that all deflection variationsversus moving mass instantaneous positions are given in anondimensional form that is 119908119901119908119904 Moreover it has to bepointed out that based on the conducted convergence studyrelated to the linear and nonlinear analyses ten modes ofvibration are taken into account for the steady state answersfor 119906(119909 119905) 119908(119909 119905) and Γ(119909 119905) that is 119899 = 10

The effect of variation of nonlinear part of foundationstiffness 1198963 and geometrical nonlinearity due to the beamrsquosend boundary conditions on the dynamic responses of a Tim-oshenko beam traversed by a moving mass are depicted inFigure 5 In this analysis 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2 119888 =103Nsdotsm2 and 120573 = 03 and different speeds for the traveling

mass (120572 = 011 025 and 05) are considered It should benoted that under adopted data the results for M-2 and M-3models are similar and hence the variations are overlaidThismeans that as the beamgets thicker it resistsmore againstthe stretching of midplane surfaceTherefore stretching doesnot play an important role in the beamrsquos dynamical responseespecially In addition in this figure the differences betweendynamical behavior of the beam betweenM-2 orM-3modelsand M-1 model are compared with each other Out of thisfigure and as it is expected it can be concluded that byincreasing the value of 1198963 the difference between M-2 or M-3models and M-1 model becomes more pronounced In otherwords by including geometrical and foundation nonlineari-ties into the M-1 model as the reference beam as the size ofnonlinearity increases the difference of the values of 119908119901119908119904

Shock and Vibration 7

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

16

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019

k3 = 1020

xl

wpw

s

(a)

0 01 02 03 04 05 06 07 08 09 1

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019k3 = 1020

xl

wpw

s

0

05

1

15

2

(b)

0 01 02 03 04 05 06 07 08 09 1

0

02

04

06

08

1

12

14

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019 k3 = 1020

xl

wpw

s

(c)

Figure 5 Variation of dimensionless instantaneous dynamicresponse119908119901119908119904 versus normalized instantaneous position of mov-ing mass 119909119897 of the Timoshenko beam with different 1198963 valuestraveled at different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2119888 = 10

3Nsdotsm2 and 120573 = 03 (a) 120572 = 011 (b) 120572 = 025 (c) 120572 = 05(- - - - -) M-1 model (mdash) M-2 model or M-3 model

in the nonlinear models (M-2 or M-3) compared to the M-1model increases This difference becomes more pronouncedfor 1198963 = 10

23Nm4 Furthermore when the mass velocityincreases the maximum value of the beamrsquos dynamicresponse increases up to 120572 = 025 and decreases afterwardIn addition it can be seen that for the 1198963 values greater than1021Nm4 by increasing the value of 120572 the fluctuation ofdynamical response of the beam reduces accordingly andfinally for 1198963 = 10

23Nm4 and 120572 = 011 this fluctuationreaches its highest value

Figure 6 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position of moving mass 119909119897 for three different Tim-oshenko beams of M-1 M-2 and M-3 traversed by a moving

0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2 120572 = 025120572 = 05

120572 = 11120572 = 075

120572 = 011

xl

wpw

s

Figure 6 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam traveled at differentspeeds with 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2 119888 = 103Nsdotsm2 and

120573 = 03 (- - - - -) M-1 model (mdash) M-2 or M-3 model with1198963 = 10

21 Nm4

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

16

k1 = 1012

k1 = 1011

k1 = 0

k1 = 1010k1 = 109

xl

wpw

s

(a)

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

k1 = 1012

k1 = 1011

k1 = 0

k1 = 1010

k1 = 109

xl

wpw

s

(b)

Figure 7 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 1198961 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 119888 = 103Nsdotsm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 1198963 = 10

22Nm4

mass The following data are considered 119898 = 3120588119860119897 kg1198961 = 10

4Nm2 1198963 = 1021Nm4 119888 = 103Nsdotsm2 and 120573 = 03

Moreover different speeds for the traveling mass are consid-ered (120572 = 011 025 05 075 and 11) It can be seen from thisfigure that the 119908119901119908119904 results for the M-2 or M-3 models arealways lower than the one obtained from the linear beammodel that is M-1 model However the119908119901119908119904 results for M-2 and M-3 are almost the same and independent of the value

8 Shock and Vibration

0010203040506070809

c = 0c = 106

c = 107

c = 108

c = 109

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0

02

04

06

08

1

12

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

c = 0

c = 106 c = 107

c = 108 c = 109

(b)

Figure 8 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 119888 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

10 Nm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 119896

3= 1022 Nm4

of velocity ratio The maximum difference between M-1 andM-2 orM-3models always happens at120572 = 025 Furthermoreit can be seen that in this case the maximum lateral dynamicdeflection 119908119901119908119904 increases up to 120572 = 025 and a reversingtrend prevails afterward nomatterwhat types of beammodelsare used

Using linear (M-1 model) and nonlinear (M-2 and M-3models) Timoshenko beammodels Figure 7 shows the varia-tion of normalized lateral dynamic displacement119908119901119908119904 ver-sus normalized instantaneous position of moving mass 119909119897As it was explained before these beams are resting on a vis-coelastic foundation For further analysis different values of1198961 are considered for these beam models The following dataare chosen 1198963 = 10

22Nm4 119888 = 103Nsdotsm2 and 120573 = 03

with mass moving at two different speeds (120572 = 011 and 05)It can be seen from this figure that the results of 119908119901119908119904obtained from theM-2 andM-3models with 1198963 = 10

22Nm4are the same and always lower than the one obtained fromthe M-1 model where the maximum difference between M-1andM-2 orM-3models always happens at 1198961 = 0 and is inde-pendent of the value of mass velocity Also when the valueof 1198961increases the difference between M-1 and M-2 or M-3models decreases accordingly For example when 1198961 =

1011Nm2 this difference becomes negligible Furthermore

by comparing Figure 7(a) with Figure 7(b) it can be observedthat the beamrsquos nonlinear dynamic response forM-2 andM-3

0

05

1

15

2 120573 = 03

120573 = 0015

120573 = 015

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0020406081

121416182

0 01 02 03 04 05 06 07 08 09 1

xlwpw

s

120573 = 03

120573 = 0015

120573 = 015

(b)

Figure 9 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with 119898 = 5120588119860119897 kgand 120572 = 025 and different 120573 values (a) 1198961 = 1198963 = 119888 = 0 (b)1198961 = 10

4 Nm2 1198963 = 104Nm4 and 119888 = 10

3Nsdotsm2 (- - - - -) M-1model and (mdash) M-2 or M-3 model

models at 120572 = 05 is greater than the one related to 120572 = 011Nevertheless a reversing trend is seen for the beamrsquos M-1model dynamic response

Figure 8 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams ofM-1 M-2 and M-3 traversed by a moving mass In this casedifferent values of 119888 are considered The following data arechosen 119898 = 3120588119860119897 kg 1198961 = 10

10Nm2 1198963 = 1022Nm4 and

120573 = 03 with mass moving at two different speeds (120572 =

011 and 05) It can be seen from this figure that the beamsdynamic responses of 119908119901119908119904 obtained from the M-2 or M-3 models are always lower than the one obtained from theM-1 model where the maximum difference between M-1 and M-2 or M-3 models always happens at 119888 = 0 nomatter what values of mass velocity would be Also when thevalue of 119888 increases the beamrsquos dynamical response differencebetween M-1 and M-2 or M-3 models decreases accordinglywhere for 119888 = 109Nsdotsm2 this difference becomes negligibleFurthermore by comparing Figure 8(a) with Figure 8(b) itcan be concluded that for 119888 lt 108Nsdotsm2 the obtained beamrsquosdynamic response at 120572 = 05 is almost greater than the oneobtained at 120572 = 011 no matter what type of beam models isused

Figure 9 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams of

Shock and Vibration 9

M-1 M-2 and M-3 The following data are assumed 119898 =

5120588119860119897 kg and mass traveling speed is 120572 = 025 The resultsare presented for three different slenderness ratios (120573) Fur-thermore it should be noted that in Figure 9(a) 1198961 = 1198963 = 119888 =0 and in Figure 9(b) 1198961 = 10

4Nm2 1198963 = 104Nm4 and 119888 =

103Nsdotsm2 are considered It can be seen from this figure that

the maximum value of dynamical response119908119901119908119904 increasesas 120573 increases Moreover the position of maximum value of119908119901119908119904 moves to the right end of the beam as 120573 increases Itcan be concluded from this figure that inclusion of nonlinearfoundation parameters will decrease the difference of linearand nonlinear analyses It should be further mentioned thatthe results for linear and nonlinear analyses are the same for120573 = 015 and 03

5 Conclusions

The longitudinal and lateral deflections as well as the rotationof warped cross section of three different Timoshenko beamsof M-1 M-2 and M-3 models resting on viscoelastic founda-tion and subjected to a moving mass with constant velocityare all considered by including the nonlinear nature in thebeamrsquos geometry and the stiffness of viscoelastic foundationThe outcome results are as the following

(1) It is concluded that by including geometrical andfoundation nonlinearities into the reference M-1model as the size of nonlinearity increases the differ-ence of the beamrsquos dynamic response in the nonlinearmodels (M-2 or M-3) and M-1 model increases Thisdifference becomesmore pronounced for the compar-atively large enough values of nonlinear stiffness partof viscoelastic foundation

(2) It is observed that the beamrsquos dynamic response resultsfor the fully nonlinear beam model that is M-3model are always lower than the one obtained fromthe linear beam model that is M-1 model

(3) It is concluded that when the value of linear stiffness-part of foundation increases the difference betweenM-1 and M-2 or M-3 models decreases accordinglywhere forthe large values of linear stiffness part offoundation this difference becomes negligible More-over when the value of foundation damping increasesthe beamrsquos dynamical response difference betweenM-1 and M-2 or M-3 models decreases accordingly

(4) Because both stretching effect and nonlinear stiffnesspart of viscoelastic foundation in conjunction withmoving mass condition and the other values of thefoundationrsquos parameters are significant factors in thenonlinear dynamic behavior of the beam with chang-ing the beamrsquos slenderness ratio 120573 it appears that for120573 le 0015 nonlinear behavior predominantly can beseen However for120573 gt 0015 no specific values for thecritical value of the traveling mass 119898 and foundationnonlinear stiffness 1198963 were obtained which could bean idea for further investigation

Appendix

The definition of different matrices used in calculation of thenonlinear coupled ODEs of modal relations (11) is as

(M)119894119895 = int119897

0

120601119894 (119909) 120601119895 (119909) 119889119909 (H)119894119895 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 119889119909

(F)119894119895 = int119897

0

120601119894 (119909) 1205951015840

119895(119909) 119889119909 (S)119894119895 = int

119897

0

120595119894 (119909) 120595119895 (119909) 119889119909

(K)119894119895 = int119897

0

120595119894 (119909) 12059510158401015840

119895(119909) 119889119909 (E)119894119895 = int

119897

0

120595119894 (119909) 1206011015840

119895(119909) 119889119909

(J)119894119895 = int119897

0

120579119894 (119909) 120579119895 (119909) 119889119909 (N)119894119895 = int119897

0

120579119894 (119909) 12057910158401015840

119895(119909) 119889119909

(G)119894119895119896 = int119897

0

120601119894 (119909) 12057910158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(I)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 12060110158402

119896(119909) 119889119909

(L)119894119895119896 = int119897

0

120579119894 (119909) 12060110158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(R1)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 1205791015840

119896(119909) 119889119909

(R2)119894119895119896119897 = int119897

0

120601119894 (119909) 120601119895 (119909) 120601119896 (119909) 120601119897 (119909) 119889119909

(A)119894119895 = 120601119894 (119909 = 120577 (119905)) 1206011015840

119895(119909 = 120577 (119905))

(B)119894119895 = 120601119894 (119909 = 120577 (119905)) 120601119895 (119909 = 120577 (119905))

(C)119894119895 = 120601119894 (119909 = 120577 (119905)) 12060110158401015840

119895(119909 = 120577 (119905))

(O)119894119895 = 120579119894 (119909 = 120577 (119905)) 120579119895 (119909 = 120577 (119905)) (A1)

where 119894 119895 119896 119897 = 1 2 3 119899 in which prime mark over anyparameter indicates the derivativewith respect to the position(119909) Furthermore the vector of a in the form of 119899 times 1 columnis given by

120601119894 (119909 = 120577 (119905)) = (a)119894 (A2)

Conflict of Interests

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

Acknowledgment

This paper is dedicated to Ahmad Mamandirsquos dearest Pro-fessor Dr Mohammad Hossein Kargarnovin passed awayon Monday November 4 2013 He was a fine kind decentsincere good tempered and well-liked professor who will besadlymissed by all who knew himMay his soul rest in peace

10 Shock and Vibration

References

[1] DThambiratnam andY Zhuge ldquoDynamic analysis of beams onan elastic foundation subjected to moving loadsrdquo Journal ofSound and Vibration vol 198 no 2 pp 149ndash169 1996

[2] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of an infi-nite tomoshenko beam on a viscoelastic foundation to a har-monic moving loadrdquo Journal of Sound and Vibration vol 241no 5 pp 809ndash824 2001

[3] S-M Kim and Y-H Cho ldquoVibration and dynamic buckling ofshear beam-columns on elastic foundation under moving har-monic loadsrdquo International Journal of Solids and Structures vol43 no 3-4 pp 393ndash412 2006

[4] C R Steele ldquoThe finite beam with a moving loadrdquo Journal ofApplied Mechanics vol 34 pp 111ndash118 1967

[5] A L Florence ldquoTraveling force on aTimoshenkobeamrdquo Journalof Applied Mechanics vol 32 pp 351ndash358 1965

[6] H D Nelson and R A Conover ldquoDynamic stability of a beamcarrying moving massesrdquo Journal of Applied Mechanics vol 38no 4 pp 1003ndash1006 1971

[7] T-P Chang and Y-N Liu ldquoDynamic finite element analysis of anonlinear beam subjected to a moving loadrdquo International Jour-nal of Solids and Structures vol 33 no 12 pp 1673ndash1688 1996

[8] Y C Hou C H Tseng and S F Ling ldquoA new high-ordernon-uniformTimoshenko beamfinite element on variable two-parameter foundations for vibration analysisrdquo Journal of Soundand Vibration vol 191 no 1 pp 91ndash106 1996

[9] H P Lee ldquoDynamic response of a Timoshenko beam on awinkler foundation subjected to amovingmassrdquoAppliedAcous-tics vol 55 no 3 pp 203ndash215 1998

[10] P Lou G-L Dai and Q-Y Zeng ldquoFinite-element analysis for aTimoshenko beam subjected to a moving massrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal of Mech-anical Engineering Science vol 220 no 5 pp 669ndash678 2006

[11] Y-H Chen and J-T Sheu ldquoAxially-loaded damped Timo-shenko beam on viscoelastic foundationrdquo International Journalfor Numerical Methods in Engineering vol 36 no 6 pp 1013ndash1027 1993

[12] Z Hryniewicz and P Koziol ldquoWavelet-based solution for vibra-tions of a beam on a nonlinear viscoelastic foundation due tomoving loadrdquo Journal ofTheoretical and AppliedMechanics vol51 no 1 pp 215ndash224 2013

[13] P Koziol and ZHryniewicz ldquoDynamic response of a beam rest-ing on a nonlinear foundation to a moving load coiflet-basedsolutionrdquo Shock and Vibration vol 19 pp 995ndash1007 2012

[14] Z Hryniewicz ldquoDynamics of Rayleigh beam on nonlinearfoundation due to moving load using Adomian decompositionand coiflet expansionrdquo Soil Dynamics and Earthquake Engineer-ing vol 31 no 8 pp 1123ndash1131 2011

[15] Z Dimitrovova ldquoA general procedure for the dynamic analysisof finite and infinite beams on piece-wise homogeneous foun-dationundermoving loadsrdquo Journal of Sound andVibration vol329 no 13 pp 2635ndash2653 2010

[16] V-H Nguyen and D Duhamel ldquoFinite element procedures fornonlinear structures in moving coordinates Part II infinitebeam under moving harmonic loadsrdquo Computers and Struc-tures vol 86 no 21-22 pp 2056ndash2063 2008

[17] E J Sapountzakis and A E Kampitsis ldquoNonlinear response ofshear deformable beams on tensionless nonlinear viscoelasticfoundation under moving loadsrdquo Journal of Sound and Vibra-tion vol 330 no 22 pp 5410ndash5426 2011

[18] T Mazilu M Dumitriu and C Tudorache ldquoInstability of anoscillator moving along a Timoshenko beam on viscoelasticfoundationrdquo Nonlinear Dynamics vol 67 no 2 pp 1273ndash12932012

[19] L Meirovitch Principles and Techniques of Vibrations Prentice-Hall Prentice NJ USA 1997

[20] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010

[21] A Mamandi and M H Kargarnovin ldquoNonlinear dynamicanalysis of an inclined Timoshenko beam subjected to amovingmassforce with beamrsquos weight includedrdquo Shock and Vibrationvol 18 no 6 pp 875ndash891 2011

International Journal of

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Shock and Vibration

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DistributedSensor Networks

International Journal of

Shock and Vibration 7

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

16

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019

k3 = 1020

xl

wpw

s

(a)

0 01 02 03 04 05 06 07 08 09 1

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019k3 = 1020

xl

wpw

s

0

05

1

15

2

(b)

0 01 02 03 04 05 06 07 08 09 1

0

02

04

06

08

1

12

14

k3 = 1021

k3 = 1023

k3 = 1022

k3 = 0

k3 = 1019 k3 = 1020

xl

wpw

s

(c)

Figure 5 Variation of dimensionless instantaneous dynamicresponse119908119901119908119904 versus normalized instantaneous position of mov-ing mass 119909119897 of the Timoshenko beam with different 1198963 valuestraveled at different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2119888 = 10

3Nsdotsm2 and 120573 = 03 (a) 120572 = 011 (b) 120572 = 025 (c) 120572 = 05(- - - - -) M-1 model (mdash) M-2 model or M-3 model

in the nonlinear models (M-2 or M-3) compared to the M-1model increases This difference becomes more pronouncedfor 1198963 = 10

23Nm4 Furthermore when the mass velocityincreases the maximum value of the beamrsquos dynamicresponse increases up to 120572 = 025 and decreases afterwardIn addition it can be seen that for the 1198963 values greater than1021Nm4 by increasing the value of 120572 the fluctuation ofdynamical response of the beam reduces accordingly andfinally for 1198963 = 10

23Nm4 and 120572 = 011 this fluctuationreaches its highest value

Figure 6 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position of moving mass 119909119897 for three different Tim-oshenko beams of M-1 M-2 and M-3 traversed by a moving

0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2 120572 = 025120572 = 05

120572 = 11120572 = 075

120572 = 011

xl

wpw

s

Figure 6 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam traveled at differentspeeds with 119898 = 3120588119860119897 kg 1198961 = 10

4Nm2 119888 = 103Nsdotsm2 and

120573 = 03 (- - - - -) M-1 model (mdash) M-2 or M-3 model with1198963 = 10

21 Nm4

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

16

k1 = 1012

k1 = 1011

k1 = 0

k1 = 1010k1 = 109

xl

wpw

s

(a)

0 01 02 03 04 05 06 07 08 09 10

02

04

06

08

1

12

14

k1 = 1012

k1 = 1011

k1 = 0

k1 = 1010

k1 = 109

xl

wpw

s

(b)

Figure 7 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 1198961 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 119888 = 103Nsdotsm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 1198963 = 10

22Nm4

mass The following data are considered 119898 = 3120588119860119897 kg1198961 = 10

4Nm2 1198963 = 1021Nm4 119888 = 103Nsdotsm2 and 120573 = 03

Moreover different speeds for the traveling mass are consid-ered (120572 = 011 025 05 075 and 11) It can be seen from thisfigure that the 119908119901119908119904 results for the M-2 or M-3 models arealways lower than the one obtained from the linear beammodel that is M-1 model However the119908119901119908119904 results for M-2 and M-3 are almost the same and independent of the value

8 Shock and Vibration

0010203040506070809

c = 0c = 106

c = 107

c = 108

c = 109

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0

02

04

06

08

1

12

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

c = 0

c = 106 c = 107

c = 108 c = 109

(b)

Figure 8 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 119888 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

10 Nm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 119896

3= 1022 Nm4

of velocity ratio The maximum difference between M-1 andM-2 orM-3models always happens at120572 = 025 Furthermoreit can be seen that in this case the maximum lateral dynamicdeflection 119908119901119908119904 increases up to 120572 = 025 and a reversingtrend prevails afterward nomatterwhat types of beammodelsare used

Using linear (M-1 model) and nonlinear (M-2 and M-3models) Timoshenko beammodels Figure 7 shows the varia-tion of normalized lateral dynamic displacement119908119901119908119904 ver-sus normalized instantaneous position of moving mass 119909119897As it was explained before these beams are resting on a vis-coelastic foundation For further analysis different values of1198961 are considered for these beam models The following dataare chosen 1198963 = 10

22Nm4 119888 = 103Nsdotsm2 and 120573 = 03

with mass moving at two different speeds (120572 = 011 and 05)It can be seen from this figure that the results of 119908119901119908119904obtained from theM-2 andM-3models with 1198963 = 10

22Nm4are the same and always lower than the one obtained fromthe M-1 model where the maximum difference between M-1andM-2 orM-3models always happens at 1198961 = 0 and is inde-pendent of the value of mass velocity Also when the valueof 1198961increases the difference between M-1 and M-2 or M-3models decreases accordingly For example when 1198961 =

1011Nm2 this difference becomes negligible Furthermore

by comparing Figure 7(a) with Figure 7(b) it can be observedthat the beamrsquos nonlinear dynamic response forM-2 andM-3

0

05

1

15

2 120573 = 03

120573 = 0015

120573 = 015

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0020406081

121416182

0 01 02 03 04 05 06 07 08 09 1

xlwpw

s

120573 = 03

120573 = 0015

120573 = 015

(b)

Figure 9 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with 119898 = 5120588119860119897 kgand 120572 = 025 and different 120573 values (a) 1198961 = 1198963 = 119888 = 0 (b)1198961 = 10

4 Nm2 1198963 = 104Nm4 and 119888 = 10

3Nsdotsm2 (- - - - -) M-1model and (mdash) M-2 or M-3 model

models at 120572 = 05 is greater than the one related to 120572 = 011Nevertheless a reversing trend is seen for the beamrsquos M-1model dynamic response

Figure 8 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams ofM-1 M-2 and M-3 traversed by a moving mass In this casedifferent values of 119888 are considered The following data arechosen 119898 = 3120588119860119897 kg 1198961 = 10

10Nm2 1198963 = 1022Nm4 and

120573 = 03 with mass moving at two different speeds (120572 =

011 and 05) It can be seen from this figure that the beamsdynamic responses of 119908119901119908119904 obtained from the M-2 or M-3 models are always lower than the one obtained from theM-1 model where the maximum difference between M-1 and M-2 or M-3 models always happens at 119888 = 0 nomatter what values of mass velocity would be Also when thevalue of 119888 increases the beamrsquos dynamical response differencebetween M-1 and M-2 or M-3 models decreases accordinglywhere for 119888 = 109Nsdotsm2 this difference becomes negligibleFurthermore by comparing Figure 8(a) with Figure 8(b) itcan be concluded that for 119888 lt 108Nsdotsm2 the obtained beamrsquosdynamic response at 120572 = 05 is almost greater than the oneobtained at 120572 = 011 no matter what type of beam models isused

Figure 9 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams of

Shock and Vibration 9

M-1 M-2 and M-3 The following data are assumed 119898 =

5120588119860119897 kg and mass traveling speed is 120572 = 025 The resultsare presented for three different slenderness ratios (120573) Fur-thermore it should be noted that in Figure 9(a) 1198961 = 1198963 = 119888 =0 and in Figure 9(b) 1198961 = 10

4Nm2 1198963 = 104Nm4 and 119888 =

103Nsdotsm2 are considered It can be seen from this figure that

the maximum value of dynamical response119908119901119908119904 increasesas 120573 increases Moreover the position of maximum value of119908119901119908119904 moves to the right end of the beam as 120573 increases Itcan be concluded from this figure that inclusion of nonlinearfoundation parameters will decrease the difference of linearand nonlinear analyses It should be further mentioned thatthe results for linear and nonlinear analyses are the same for120573 = 015 and 03

5 Conclusions

The longitudinal and lateral deflections as well as the rotationof warped cross section of three different Timoshenko beamsof M-1 M-2 and M-3 models resting on viscoelastic founda-tion and subjected to a moving mass with constant velocityare all considered by including the nonlinear nature in thebeamrsquos geometry and the stiffness of viscoelastic foundationThe outcome results are as the following

(1) It is concluded that by including geometrical andfoundation nonlinearities into the reference M-1model as the size of nonlinearity increases the differ-ence of the beamrsquos dynamic response in the nonlinearmodels (M-2 or M-3) and M-1 model increases Thisdifference becomesmore pronounced for the compar-atively large enough values of nonlinear stiffness partof viscoelastic foundation

(2) It is observed that the beamrsquos dynamic response resultsfor the fully nonlinear beam model that is M-3model are always lower than the one obtained fromthe linear beam model that is M-1 model

(3) It is concluded that when the value of linear stiffness-part of foundation increases the difference betweenM-1 and M-2 or M-3 models decreases accordinglywhere forthe large values of linear stiffness part offoundation this difference becomes negligible More-over when the value of foundation damping increasesthe beamrsquos dynamical response difference betweenM-1 and M-2 or M-3 models decreases accordingly

(4) Because both stretching effect and nonlinear stiffnesspart of viscoelastic foundation in conjunction withmoving mass condition and the other values of thefoundationrsquos parameters are significant factors in thenonlinear dynamic behavior of the beam with chang-ing the beamrsquos slenderness ratio 120573 it appears that for120573 le 0015 nonlinear behavior predominantly can beseen However for120573 gt 0015 no specific values for thecritical value of the traveling mass 119898 and foundationnonlinear stiffness 1198963 were obtained which could bean idea for further investigation

Appendix

The definition of different matrices used in calculation of thenonlinear coupled ODEs of modal relations (11) is as

(M)119894119895 = int119897

0

120601119894 (119909) 120601119895 (119909) 119889119909 (H)119894119895 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 119889119909

(F)119894119895 = int119897

0

120601119894 (119909) 1205951015840

119895(119909) 119889119909 (S)119894119895 = int

119897

0

120595119894 (119909) 120595119895 (119909) 119889119909

(K)119894119895 = int119897

0

120595119894 (119909) 12059510158401015840

119895(119909) 119889119909 (E)119894119895 = int

119897

0

120595119894 (119909) 1206011015840

119895(119909) 119889119909

(J)119894119895 = int119897

0

120579119894 (119909) 120579119895 (119909) 119889119909 (N)119894119895 = int119897

0

120579119894 (119909) 12057910158401015840

119895(119909) 119889119909

(G)119894119895119896 = int119897

0

120601119894 (119909) 12057910158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(I)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 12060110158402

119896(119909) 119889119909

(L)119894119895119896 = int119897

0

120579119894 (119909) 12060110158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(R1)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 1205791015840

119896(119909) 119889119909

(R2)119894119895119896119897 = int119897

0

120601119894 (119909) 120601119895 (119909) 120601119896 (119909) 120601119897 (119909) 119889119909

(A)119894119895 = 120601119894 (119909 = 120577 (119905)) 1206011015840

119895(119909 = 120577 (119905))

(B)119894119895 = 120601119894 (119909 = 120577 (119905)) 120601119895 (119909 = 120577 (119905))

(C)119894119895 = 120601119894 (119909 = 120577 (119905)) 12060110158401015840

119895(119909 = 120577 (119905))

(O)119894119895 = 120579119894 (119909 = 120577 (119905)) 120579119895 (119909 = 120577 (119905)) (A1)

where 119894 119895 119896 119897 = 1 2 3 119899 in which prime mark over anyparameter indicates the derivativewith respect to the position(119909) Furthermore the vector of a in the form of 119899 times 1 columnis given by

120601119894 (119909 = 120577 (119905)) = (a)119894 (A2)

Conflict of Interests

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

Acknowledgment

This paper is dedicated to Ahmad Mamandirsquos dearest Pro-fessor Dr Mohammad Hossein Kargarnovin passed awayon Monday November 4 2013 He was a fine kind decentsincere good tempered and well-liked professor who will besadlymissed by all who knew himMay his soul rest in peace

10 Shock and Vibration

References

[1] DThambiratnam andY Zhuge ldquoDynamic analysis of beams onan elastic foundation subjected to moving loadsrdquo Journal ofSound and Vibration vol 198 no 2 pp 149ndash169 1996

[2] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of an infi-nite tomoshenko beam on a viscoelastic foundation to a har-monic moving loadrdquo Journal of Sound and Vibration vol 241no 5 pp 809ndash824 2001

[3] S-M Kim and Y-H Cho ldquoVibration and dynamic buckling ofshear beam-columns on elastic foundation under moving har-monic loadsrdquo International Journal of Solids and Structures vol43 no 3-4 pp 393ndash412 2006

[4] C R Steele ldquoThe finite beam with a moving loadrdquo Journal ofApplied Mechanics vol 34 pp 111ndash118 1967

[5] A L Florence ldquoTraveling force on aTimoshenkobeamrdquo Journalof Applied Mechanics vol 32 pp 351ndash358 1965

[6] H D Nelson and R A Conover ldquoDynamic stability of a beamcarrying moving massesrdquo Journal of Applied Mechanics vol 38no 4 pp 1003ndash1006 1971

[7] T-P Chang and Y-N Liu ldquoDynamic finite element analysis of anonlinear beam subjected to a moving loadrdquo International Jour-nal of Solids and Structures vol 33 no 12 pp 1673ndash1688 1996

[8] Y C Hou C H Tseng and S F Ling ldquoA new high-ordernon-uniformTimoshenko beamfinite element on variable two-parameter foundations for vibration analysisrdquo Journal of Soundand Vibration vol 191 no 1 pp 91ndash106 1996

[9] H P Lee ldquoDynamic response of a Timoshenko beam on awinkler foundation subjected to amovingmassrdquoAppliedAcous-tics vol 55 no 3 pp 203ndash215 1998

[10] P Lou G-L Dai and Q-Y Zeng ldquoFinite-element analysis for aTimoshenko beam subjected to a moving massrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal of Mech-anical Engineering Science vol 220 no 5 pp 669ndash678 2006

[11] Y-H Chen and J-T Sheu ldquoAxially-loaded damped Timo-shenko beam on viscoelastic foundationrdquo International Journalfor Numerical Methods in Engineering vol 36 no 6 pp 1013ndash1027 1993

[12] Z Hryniewicz and P Koziol ldquoWavelet-based solution for vibra-tions of a beam on a nonlinear viscoelastic foundation due tomoving loadrdquo Journal ofTheoretical and AppliedMechanics vol51 no 1 pp 215ndash224 2013

[13] P Koziol and ZHryniewicz ldquoDynamic response of a beam rest-ing on a nonlinear foundation to a moving load coiflet-basedsolutionrdquo Shock and Vibration vol 19 pp 995ndash1007 2012

[14] Z Hryniewicz ldquoDynamics of Rayleigh beam on nonlinearfoundation due to moving load using Adomian decompositionand coiflet expansionrdquo Soil Dynamics and Earthquake Engineer-ing vol 31 no 8 pp 1123ndash1131 2011

[15] Z Dimitrovova ldquoA general procedure for the dynamic analysisof finite and infinite beams on piece-wise homogeneous foun-dationundermoving loadsrdquo Journal of Sound andVibration vol329 no 13 pp 2635ndash2653 2010

[16] V-H Nguyen and D Duhamel ldquoFinite element procedures fornonlinear structures in moving coordinates Part II infinitebeam under moving harmonic loadsrdquo Computers and Struc-tures vol 86 no 21-22 pp 2056ndash2063 2008

[17] E J Sapountzakis and A E Kampitsis ldquoNonlinear response ofshear deformable beams on tensionless nonlinear viscoelasticfoundation under moving loadsrdquo Journal of Sound and Vibra-tion vol 330 no 22 pp 5410ndash5426 2011

[18] T Mazilu M Dumitriu and C Tudorache ldquoInstability of anoscillator moving along a Timoshenko beam on viscoelasticfoundationrdquo Nonlinear Dynamics vol 67 no 2 pp 1273ndash12932012

[19] L Meirovitch Principles and Techniques of Vibrations Prentice-Hall Prentice NJ USA 1997

[20] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010

[21] A Mamandi and M H Kargarnovin ldquoNonlinear dynamicanalysis of an inclined Timoshenko beam subjected to amovingmassforce with beamrsquos weight includedrdquo Shock and Vibrationvol 18 no 6 pp 875ndash891 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

0010203040506070809

c = 0c = 106

c = 107

c = 108

c = 109

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0

02

04

06

08

1

12

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

c = 0

c = 106 c = 107

c = 108 c = 109

(b)

Figure 8 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with different 119888 valuestraveled at two different speeds with 119898 = 3120588119860119897 kg 1198961 = 10

10 Nm2and 120573 = 03 (a) 120572 = 011 (b) 120572 = 05 (- - - - -) M-1 model and (mdash)M-2 or M-3 model with 119896

3= 1022 Nm4

of velocity ratio The maximum difference between M-1 andM-2 orM-3models always happens at120572 = 025 Furthermoreit can be seen that in this case the maximum lateral dynamicdeflection 119908119901119908119904 increases up to 120572 = 025 and a reversingtrend prevails afterward nomatterwhat types of beammodelsare used

Using linear (M-1 model) and nonlinear (M-2 and M-3models) Timoshenko beammodels Figure 7 shows the varia-tion of normalized lateral dynamic displacement119908119901119908119904 ver-sus normalized instantaneous position of moving mass 119909119897As it was explained before these beams are resting on a vis-coelastic foundation For further analysis different values of1198961 are considered for these beam models The following dataare chosen 1198963 = 10

22Nm4 119888 = 103Nsdotsm2 and 120573 = 03

with mass moving at two different speeds (120572 = 011 and 05)It can be seen from this figure that the results of 119908119901119908119904obtained from theM-2 andM-3models with 1198963 = 10

22Nm4are the same and always lower than the one obtained fromthe M-1 model where the maximum difference between M-1andM-2 orM-3models always happens at 1198961 = 0 and is inde-pendent of the value of mass velocity Also when the valueof 1198961increases the difference between M-1 and M-2 or M-3models decreases accordingly For example when 1198961 =

1011Nm2 this difference becomes negligible Furthermore

by comparing Figure 7(a) with Figure 7(b) it can be observedthat the beamrsquos nonlinear dynamic response forM-2 andM-3

0

05

1

15

2 120573 = 03

120573 = 0015

120573 = 015

0 01 02 03 04 05 06 07 08 09 1

xl

wpw

s

(a)

0020406081

121416182

0 01 02 03 04 05 06 07 08 09 1

xlwpw

s

120573 = 03

120573 = 0015

120573 = 015

(b)

Figure 9 Variation of dimensionless instantaneous dynamic lateraldeflection 119908119901119908119904 versus normalized instantaneous position ofmoving mass 119909119897 of the Timoshenko beam with 119898 = 5120588119860119897 kgand 120572 = 025 and different 120573 values (a) 1198961 = 1198963 = 119888 = 0 (b)1198961 = 10

4 Nm2 1198963 = 104Nm4 and 119888 = 10

3Nsdotsm2 (- - - - -) M-1model and (mdash) M-2 or M-3 model

models at 120572 = 05 is greater than the one related to 120572 = 011Nevertheless a reversing trend is seen for the beamrsquos M-1model dynamic response

Figure 8 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams ofM-1 M-2 and M-3 traversed by a moving mass In this casedifferent values of 119888 are considered The following data arechosen 119898 = 3120588119860119897 kg 1198961 = 10

10Nm2 1198963 = 1022Nm4 and

120573 = 03 with mass moving at two different speeds (120572 =

011 and 05) It can be seen from this figure that the beamsdynamic responses of 119908119901119908119904 obtained from the M-2 or M-3 models are always lower than the one obtained from theM-1 model where the maximum difference between M-1 and M-2 or M-3 models always happens at 119888 = 0 nomatter what values of mass velocity would be Also when thevalue of 119888 increases the beamrsquos dynamical response differencebetween M-1 and M-2 or M-3 models decreases accordinglywhere for 119888 = 109Nsdotsm2 this difference becomes negligibleFurthermore by comparing Figure 8(a) with Figure 8(b) itcan be concluded that for 119888 lt 108Nsdotsm2 the obtained beamrsquosdynamic response at 120572 = 05 is almost greater than the oneobtained at 120572 = 011 no matter what type of beam models isused

Figure 9 shows the variation of normalized lateraldynamic displacement 119908119901119908119904 versus normalized instanta-neous position 119909119897 for three different Timoshenko beams of

Shock and Vibration 9

M-1 M-2 and M-3 The following data are assumed 119898 =

5120588119860119897 kg and mass traveling speed is 120572 = 025 The resultsare presented for three different slenderness ratios (120573) Fur-thermore it should be noted that in Figure 9(a) 1198961 = 1198963 = 119888 =0 and in Figure 9(b) 1198961 = 10

4Nm2 1198963 = 104Nm4 and 119888 =

103Nsdotsm2 are considered It can be seen from this figure that

the maximum value of dynamical response119908119901119908119904 increasesas 120573 increases Moreover the position of maximum value of119908119901119908119904 moves to the right end of the beam as 120573 increases Itcan be concluded from this figure that inclusion of nonlinearfoundation parameters will decrease the difference of linearand nonlinear analyses It should be further mentioned thatthe results for linear and nonlinear analyses are the same for120573 = 015 and 03

5 Conclusions

The longitudinal and lateral deflections as well as the rotationof warped cross section of three different Timoshenko beamsof M-1 M-2 and M-3 models resting on viscoelastic founda-tion and subjected to a moving mass with constant velocityare all considered by including the nonlinear nature in thebeamrsquos geometry and the stiffness of viscoelastic foundationThe outcome results are as the following

(1) It is concluded that by including geometrical andfoundation nonlinearities into the reference M-1model as the size of nonlinearity increases the differ-ence of the beamrsquos dynamic response in the nonlinearmodels (M-2 or M-3) and M-1 model increases Thisdifference becomesmore pronounced for the compar-atively large enough values of nonlinear stiffness partof viscoelastic foundation

(2) It is observed that the beamrsquos dynamic response resultsfor the fully nonlinear beam model that is M-3model are always lower than the one obtained fromthe linear beam model that is M-1 model

(3) It is concluded that when the value of linear stiffness-part of foundation increases the difference betweenM-1 and M-2 or M-3 models decreases accordinglywhere forthe large values of linear stiffness part offoundation this difference becomes negligible More-over when the value of foundation damping increasesthe beamrsquos dynamical response difference betweenM-1 and M-2 or M-3 models decreases accordingly

(4) Because both stretching effect and nonlinear stiffnesspart of viscoelastic foundation in conjunction withmoving mass condition and the other values of thefoundationrsquos parameters are significant factors in thenonlinear dynamic behavior of the beam with chang-ing the beamrsquos slenderness ratio 120573 it appears that for120573 le 0015 nonlinear behavior predominantly can beseen However for120573 gt 0015 no specific values for thecritical value of the traveling mass 119898 and foundationnonlinear stiffness 1198963 were obtained which could bean idea for further investigation

Appendix

The definition of different matrices used in calculation of thenonlinear coupled ODEs of modal relations (11) is as

(M)119894119895 = int119897

0

120601119894 (119909) 120601119895 (119909) 119889119909 (H)119894119895 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 119889119909

(F)119894119895 = int119897

0

120601119894 (119909) 1205951015840

119895(119909) 119889119909 (S)119894119895 = int

119897

0

120595119894 (119909) 120595119895 (119909) 119889119909

(K)119894119895 = int119897

0

120595119894 (119909) 12059510158401015840

119895(119909) 119889119909 (E)119894119895 = int

119897

0

120595119894 (119909) 1206011015840

119895(119909) 119889119909

(J)119894119895 = int119897

0

120579119894 (119909) 120579119895 (119909) 119889119909 (N)119894119895 = int119897

0

120579119894 (119909) 12057910158401015840

119895(119909) 119889119909

(G)119894119895119896 = int119897

0

120601119894 (119909) 12057910158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(I)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 12060110158402

119896(119909) 119889119909

(L)119894119895119896 = int119897

0

120579119894 (119909) 12060110158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(R1)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 1205791015840

119896(119909) 119889119909

(R2)119894119895119896119897 = int119897

0

120601119894 (119909) 120601119895 (119909) 120601119896 (119909) 120601119897 (119909) 119889119909

(A)119894119895 = 120601119894 (119909 = 120577 (119905)) 1206011015840

119895(119909 = 120577 (119905))

(B)119894119895 = 120601119894 (119909 = 120577 (119905)) 120601119895 (119909 = 120577 (119905))

(C)119894119895 = 120601119894 (119909 = 120577 (119905)) 12060110158401015840

119895(119909 = 120577 (119905))

(O)119894119895 = 120579119894 (119909 = 120577 (119905)) 120579119895 (119909 = 120577 (119905)) (A1)

where 119894 119895 119896 119897 = 1 2 3 119899 in which prime mark over anyparameter indicates the derivativewith respect to the position(119909) Furthermore the vector of a in the form of 119899 times 1 columnis given by

120601119894 (119909 = 120577 (119905)) = (a)119894 (A2)

Conflict of Interests

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

Acknowledgment

This paper is dedicated to Ahmad Mamandirsquos dearest Pro-fessor Dr Mohammad Hossein Kargarnovin passed awayon Monday November 4 2013 He was a fine kind decentsincere good tempered and well-liked professor who will besadlymissed by all who knew himMay his soul rest in peace

10 Shock and Vibration

References

[1] DThambiratnam andY Zhuge ldquoDynamic analysis of beams onan elastic foundation subjected to moving loadsrdquo Journal ofSound and Vibration vol 198 no 2 pp 149ndash169 1996

[2] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of an infi-nite tomoshenko beam on a viscoelastic foundation to a har-monic moving loadrdquo Journal of Sound and Vibration vol 241no 5 pp 809ndash824 2001

[3] S-M Kim and Y-H Cho ldquoVibration and dynamic buckling ofshear beam-columns on elastic foundation under moving har-monic loadsrdquo International Journal of Solids and Structures vol43 no 3-4 pp 393ndash412 2006

[4] C R Steele ldquoThe finite beam with a moving loadrdquo Journal ofApplied Mechanics vol 34 pp 111ndash118 1967

[5] A L Florence ldquoTraveling force on aTimoshenkobeamrdquo Journalof Applied Mechanics vol 32 pp 351ndash358 1965

[6] H D Nelson and R A Conover ldquoDynamic stability of a beamcarrying moving massesrdquo Journal of Applied Mechanics vol 38no 4 pp 1003ndash1006 1971

[7] T-P Chang and Y-N Liu ldquoDynamic finite element analysis of anonlinear beam subjected to a moving loadrdquo International Jour-nal of Solids and Structures vol 33 no 12 pp 1673ndash1688 1996

[8] Y C Hou C H Tseng and S F Ling ldquoA new high-ordernon-uniformTimoshenko beamfinite element on variable two-parameter foundations for vibration analysisrdquo Journal of Soundand Vibration vol 191 no 1 pp 91ndash106 1996

[9] H P Lee ldquoDynamic response of a Timoshenko beam on awinkler foundation subjected to amovingmassrdquoAppliedAcous-tics vol 55 no 3 pp 203ndash215 1998

[10] P Lou G-L Dai and Q-Y Zeng ldquoFinite-element analysis for aTimoshenko beam subjected to a moving massrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal of Mech-anical Engineering Science vol 220 no 5 pp 669ndash678 2006

[11] Y-H Chen and J-T Sheu ldquoAxially-loaded damped Timo-shenko beam on viscoelastic foundationrdquo International Journalfor Numerical Methods in Engineering vol 36 no 6 pp 1013ndash1027 1993

[12] Z Hryniewicz and P Koziol ldquoWavelet-based solution for vibra-tions of a beam on a nonlinear viscoelastic foundation due tomoving loadrdquo Journal ofTheoretical and AppliedMechanics vol51 no 1 pp 215ndash224 2013

[13] P Koziol and ZHryniewicz ldquoDynamic response of a beam rest-ing on a nonlinear foundation to a moving load coiflet-basedsolutionrdquo Shock and Vibration vol 19 pp 995ndash1007 2012

[14] Z Hryniewicz ldquoDynamics of Rayleigh beam on nonlinearfoundation due to moving load using Adomian decompositionand coiflet expansionrdquo Soil Dynamics and Earthquake Engineer-ing vol 31 no 8 pp 1123ndash1131 2011

[15] Z Dimitrovova ldquoA general procedure for the dynamic analysisof finite and infinite beams on piece-wise homogeneous foun-dationundermoving loadsrdquo Journal of Sound andVibration vol329 no 13 pp 2635ndash2653 2010

[16] V-H Nguyen and D Duhamel ldquoFinite element procedures fornonlinear structures in moving coordinates Part II infinitebeam under moving harmonic loadsrdquo Computers and Struc-tures vol 86 no 21-22 pp 2056ndash2063 2008

[17] E J Sapountzakis and A E Kampitsis ldquoNonlinear response ofshear deformable beams on tensionless nonlinear viscoelasticfoundation under moving loadsrdquo Journal of Sound and Vibra-tion vol 330 no 22 pp 5410ndash5426 2011

[18] T Mazilu M Dumitriu and C Tudorache ldquoInstability of anoscillator moving along a Timoshenko beam on viscoelasticfoundationrdquo Nonlinear Dynamics vol 67 no 2 pp 1273ndash12932012

[19] L Meirovitch Principles and Techniques of Vibrations Prentice-Hall Prentice NJ USA 1997

[20] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010

[21] A Mamandi and M H Kargarnovin ldquoNonlinear dynamicanalysis of an inclined Timoshenko beam subjected to amovingmassforce with beamrsquos weight includedrdquo Shock and Vibrationvol 18 no 6 pp 875ndash891 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

M-1 M-2 and M-3 The following data are assumed 119898 =

5120588119860119897 kg and mass traveling speed is 120572 = 025 The resultsare presented for three different slenderness ratios (120573) Fur-thermore it should be noted that in Figure 9(a) 1198961 = 1198963 = 119888 =0 and in Figure 9(b) 1198961 = 10

4Nm2 1198963 = 104Nm4 and 119888 =

103Nsdotsm2 are considered It can be seen from this figure that

the maximum value of dynamical response119908119901119908119904 increasesas 120573 increases Moreover the position of maximum value of119908119901119908119904 moves to the right end of the beam as 120573 increases Itcan be concluded from this figure that inclusion of nonlinearfoundation parameters will decrease the difference of linearand nonlinear analyses It should be further mentioned thatthe results for linear and nonlinear analyses are the same for120573 = 015 and 03

5 Conclusions

The longitudinal and lateral deflections as well as the rotationof warped cross section of three different Timoshenko beamsof M-1 M-2 and M-3 models resting on viscoelastic founda-tion and subjected to a moving mass with constant velocityare all considered by including the nonlinear nature in thebeamrsquos geometry and the stiffness of viscoelastic foundationThe outcome results are as the following

(1) It is concluded that by including geometrical andfoundation nonlinearities into the reference M-1model as the size of nonlinearity increases the differ-ence of the beamrsquos dynamic response in the nonlinearmodels (M-2 or M-3) and M-1 model increases Thisdifference becomesmore pronounced for the compar-atively large enough values of nonlinear stiffness partof viscoelastic foundation

(2) It is observed that the beamrsquos dynamic response resultsfor the fully nonlinear beam model that is M-3model are always lower than the one obtained fromthe linear beam model that is M-1 model

(3) It is concluded that when the value of linear stiffness-part of foundation increases the difference betweenM-1 and M-2 or M-3 models decreases accordinglywhere forthe large values of linear stiffness part offoundation this difference becomes negligible More-over when the value of foundation damping increasesthe beamrsquos dynamical response difference betweenM-1 and M-2 or M-3 models decreases accordingly

(4) Because both stretching effect and nonlinear stiffnesspart of viscoelastic foundation in conjunction withmoving mass condition and the other values of thefoundationrsquos parameters are significant factors in thenonlinear dynamic behavior of the beam with chang-ing the beamrsquos slenderness ratio 120573 it appears that for120573 le 0015 nonlinear behavior predominantly can beseen However for120573 gt 0015 no specific values for thecritical value of the traveling mass 119898 and foundationnonlinear stiffness 1198963 were obtained which could bean idea for further investigation

Appendix

The definition of different matrices used in calculation of thenonlinear coupled ODEs of modal relations (11) is as

(M)119894119895 = int119897

0

120601119894 (119909) 120601119895 (119909) 119889119909 (H)119894119895 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 119889119909

(F)119894119895 = int119897

0

120601119894 (119909) 1205951015840

119895(119909) 119889119909 (S)119894119895 = int

119897

0

120595119894 (119909) 120595119895 (119909) 119889119909

(K)119894119895 = int119897

0

120595119894 (119909) 12059510158401015840

119895(119909) 119889119909 (E)119894119895 = int

119897

0

120595119894 (119909) 1206011015840

119895(119909) 119889119909

(J)119894119895 = int119897

0

120579119894 (119909) 120579119895 (119909) 119889119909 (N)119894119895 = int119897

0

120579119894 (119909) 12057910158401015840

119895(119909) 119889119909

(G)119894119895119896 = int119897

0

120601119894 (119909) 12057910158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(I)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 12060110158402

119896(119909) 119889119909

(L)119894119895119896 = int119897

0

120579119894 (119909) 12060110158401015840

119895(119909) 1206011015840

119896(119909) 119889119909

(R1)119894119895119896 = int119897

0

120601119894 (119909) 12060110158401015840

119895(119909) 1205791015840

119896(119909) 119889119909

(R2)119894119895119896119897 = int119897

0

120601119894 (119909) 120601119895 (119909) 120601119896 (119909) 120601119897 (119909) 119889119909

(A)119894119895 = 120601119894 (119909 = 120577 (119905)) 1206011015840

119895(119909 = 120577 (119905))

(B)119894119895 = 120601119894 (119909 = 120577 (119905)) 120601119895 (119909 = 120577 (119905))

(C)119894119895 = 120601119894 (119909 = 120577 (119905)) 12060110158401015840

119895(119909 = 120577 (119905))

(O)119894119895 = 120579119894 (119909 = 120577 (119905)) 120579119895 (119909 = 120577 (119905)) (A1)

where 119894 119895 119896 119897 = 1 2 3 119899 in which prime mark over anyparameter indicates the derivativewith respect to the position(119909) Furthermore the vector of a in the form of 119899 times 1 columnis given by

120601119894 (119909 = 120577 (119905)) = (a)119894 (A2)

Conflict of Interests

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

Acknowledgment

This paper is dedicated to Ahmad Mamandirsquos dearest Pro-fessor Dr Mohammad Hossein Kargarnovin passed awayon Monday November 4 2013 He was a fine kind decentsincere good tempered and well-liked professor who will besadlymissed by all who knew himMay his soul rest in peace

10 Shock and Vibration

References

[1] DThambiratnam andY Zhuge ldquoDynamic analysis of beams onan elastic foundation subjected to moving loadsrdquo Journal ofSound and Vibration vol 198 no 2 pp 149ndash169 1996

[2] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of an infi-nite tomoshenko beam on a viscoelastic foundation to a har-monic moving loadrdquo Journal of Sound and Vibration vol 241no 5 pp 809ndash824 2001

[3] S-M Kim and Y-H Cho ldquoVibration and dynamic buckling ofshear beam-columns on elastic foundation under moving har-monic loadsrdquo International Journal of Solids and Structures vol43 no 3-4 pp 393ndash412 2006

[4] C R Steele ldquoThe finite beam with a moving loadrdquo Journal ofApplied Mechanics vol 34 pp 111ndash118 1967

[5] A L Florence ldquoTraveling force on aTimoshenkobeamrdquo Journalof Applied Mechanics vol 32 pp 351ndash358 1965

[6] H D Nelson and R A Conover ldquoDynamic stability of a beamcarrying moving massesrdquo Journal of Applied Mechanics vol 38no 4 pp 1003ndash1006 1971

[7] T-P Chang and Y-N Liu ldquoDynamic finite element analysis of anonlinear beam subjected to a moving loadrdquo International Jour-nal of Solids and Structures vol 33 no 12 pp 1673ndash1688 1996

[8] Y C Hou C H Tseng and S F Ling ldquoA new high-ordernon-uniformTimoshenko beamfinite element on variable two-parameter foundations for vibration analysisrdquo Journal of Soundand Vibration vol 191 no 1 pp 91ndash106 1996

[9] H P Lee ldquoDynamic response of a Timoshenko beam on awinkler foundation subjected to amovingmassrdquoAppliedAcous-tics vol 55 no 3 pp 203ndash215 1998

[10] P Lou G-L Dai and Q-Y Zeng ldquoFinite-element analysis for aTimoshenko beam subjected to a moving massrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal of Mech-anical Engineering Science vol 220 no 5 pp 669ndash678 2006

[11] Y-H Chen and J-T Sheu ldquoAxially-loaded damped Timo-shenko beam on viscoelastic foundationrdquo International Journalfor Numerical Methods in Engineering vol 36 no 6 pp 1013ndash1027 1993

[12] Z Hryniewicz and P Koziol ldquoWavelet-based solution for vibra-tions of a beam on a nonlinear viscoelastic foundation due tomoving loadrdquo Journal ofTheoretical and AppliedMechanics vol51 no 1 pp 215ndash224 2013

[13] P Koziol and ZHryniewicz ldquoDynamic response of a beam rest-ing on a nonlinear foundation to a moving load coiflet-basedsolutionrdquo Shock and Vibration vol 19 pp 995ndash1007 2012

[14] Z Hryniewicz ldquoDynamics of Rayleigh beam on nonlinearfoundation due to moving load using Adomian decompositionand coiflet expansionrdquo Soil Dynamics and Earthquake Engineer-ing vol 31 no 8 pp 1123ndash1131 2011

[15] Z Dimitrovova ldquoA general procedure for the dynamic analysisof finite and infinite beams on piece-wise homogeneous foun-dationundermoving loadsrdquo Journal of Sound andVibration vol329 no 13 pp 2635ndash2653 2010

[16] V-H Nguyen and D Duhamel ldquoFinite element procedures fornonlinear structures in moving coordinates Part II infinitebeam under moving harmonic loadsrdquo Computers and Struc-tures vol 86 no 21-22 pp 2056ndash2063 2008

[17] E J Sapountzakis and A E Kampitsis ldquoNonlinear response ofshear deformable beams on tensionless nonlinear viscoelasticfoundation under moving loadsrdquo Journal of Sound and Vibra-tion vol 330 no 22 pp 5410ndash5426 2011

[18] T Mazilu M Dumitriu and C Tudorache ldquoInstability of anoscillator moving along a Timoshenko beam on viscoelasticfoundationrdquo Nonlinear Dynamics vol 67 no 2 pp 1273ndash12932012

[19] L Meirovitch Principles and Techniques of Vibrations Prentice-Hall Prentice NJ USA 1997

[20] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010

[21] A Mamandi and M H Kargarnovin ldquoNonlinear dynamicanalysis of an inclined Timoshenko beam subjected to amovingmassforce with beamrsquos weight includedrdquo Shock and Vibrationvol 18 no 6 pp 875ndash891 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

References

[1] DThambiratnam andY Zhuge ldquoDynamic analysis of beams onan elastic foundation subjected to moving loadsrdquo Journal ofSound and Vibration vol 198 no 2 pp 149ndash169 1996

[2] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of an infi-nite tomoshenko beam on a viscoelastic foundation to a har-monic moving loadrdquo Journal of Sound and Vibration vol 241no 5 pp 809ndash824 2001

[3] S-M Kim and Y-H Cho ldquoVibration and dynamic buckling ofshear beam-columns on elastic foundation under moving har-monic loadsrdquo International Journal of Solids and Structures vol43 no 3-4 pp 393ndash412 2006

[4] C R Steele ldquoThe finite beam with a moving loadrdquo Journal ofApplied Mechanics vol 34 pp 111ndash118 1967

[5] A L Florence ldquoTraveling force on aTimoshenkobeamrdquo Journalof Applied Mechanics vol 32 pp 351ndash358 1965

[6] H D Nelson and R A Conover ldquoDynamic stability of a beamcarrying moving massesrdquo Journal of Applied Mechanics vol 38no 4 pp 1003ndash1006 1971

[7] T-P Chang and Y-N Liu ldquoDynamic finite element analysis of anonlinear beam subjected to a moving loadrdquo International Jour-nal of Solids and Structures vol 33 no 12 pp 1673ndash1688 1996

[8] Y C Hou C H Tseng and S F Ling ldquoA new high-ordernon-uniformTimoshenko beamfinite element on variable two-parameter foundations for vibration analysisrdquo Journal of Soundand Vibration vol 191 no 1 pp 91ndash106 1996

[9] H P Lee ldquoDynamic response of a Timoshenko beam on awinkler foundation subjected to amovingmassrdquoAppliedAcous-tics vol 55 no 3 pp 203ndash215 1998

[10] P Lou G-L Dai and Q-Y Zeng ldquoFinite-element analysis for aTimoshenko beam subjected to a moving massrdquo Proceedings ofthe Institution of Mechanical Engineers Part C Journal of Mech-anical Engineering Science vol 220 no 5 pp 669ndash678 2006

[11] Y-H Chen and J-T Sheu ldquoAxially-loaded damped Timo-shenko beam on viscoelastic foundationrdquo International Journalfor Numerical Methods in Engineering vol 36 no 6 pp 1013ndash1027 1993

[12] Z Hryniewicz and P Koziol ldquoWavelet-based solution for vibra-tions of a beam on a nonlinear viscoelastic foundation due tomoving loadrdquo Journal ofTheoretical and AppliedMechanics vol51 no 1 pp 215ndash224 2013

[13] P Koziol and ZHryniewicz ldquoDynamic response of a beam rest-ing on a nonlinear foundation to a moving load coiflet-basedsolutionrdquo Shock and Vibration vol 19 pp 995ndash1007 2012

[14] Z Hryniewicz ldquoDynamics of Rayleigh beam on nonlinearfoundation due to moving load using Adomian decompositionand coiflet expansionrdquo Soil Dynamics and Earthquake Engineer-ing vol 31 no 8 pp 1123ndash1131 2011

[15] Z Dimitrovova ldquoA general procedure for the dynamic analysisof finite and infinite beams on piece-wise homogeneous foun-dationundermoving loadsrdquo Journal of Sound andVibration vol329 no 13 pp 2635ndash2653 2010

[16] V-H Nguyen and D Duhamel ldquoFinite element procedures fornonlinear structures in moving coordinates Part II infinitebeam under moving harmonic loadsrdquo Computers and Struc-tures vol 86 no 21-22 pp 2056ndash2063 2008

[17] E J Sapountzakis and A E Kampitsis ldquoNonlinear response ofshear deformable beams on tensionless nonlinear viscoelasticfoundation under moving loadsrdquo Journal of Sound and Vibra-tion vol 330 no 22 pp 5410ndash5426 2011

[18] T Mazilu M Dumitriu and C Tudorache ldquoInstability of anoscillator moving along a Timoshenko beam on viscoelasticfoundationrdquo Nonlinear Dynamics vol 67 no 2 pp 1273ndash12932012

[19] L Meirovitch Principles and Techniques of Vibrations Prentice-Hall Prentice NJ USA 1997

[20] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010

[21] A Mamandi and M H Kargarnovin ldquoNonlinear dynamicanalysis of an inclined Timoshenko beam subjected to amovingmassforce with beamrsquos weight includedrdquo Shock and Vibrationvol 18 no 6 pp 875ndash891 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of