Research ArticleAn Analytical Study on Dynamic Response of Multiple SimplySupported Beam System Subjected to Moving Loads

Zhipeng Lai 1 Lizhong Jiang12 andWangbao Zhou 12

1School of Civil Engineering Central South University Changsha 410075 China2National Engineering Laboratory for High Speed Railway Construction Changsha 410075 China

Correspondence should be addressed to Wangbao Zhou zhouwangbao163com

Received 22 March 2018 Revised 13 June 2018 Accepted 26 June 2018 Published 9 July 2018

Academic Editor Sundararajan Natarajan

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

Based on EulerndashBernoulli beam theory first partial differential equations were established for the vibration of multiple simplysupported beams subjected tomoving loads Then integral transforms were conducted on the spatial displacement coordinate andtime in the partial differential equations and the frequency-domain response of multiple simply supported beams subjected tomoving loads was obtained Next by conducting the corresponding inverse transforms on the displacement frequency-domainresponses of multiple simply supported beams the spatial displacement time-domain responses were obtained Finally to validatethe analytical method reported in this paper the dynamic response of a typical double simply supported rail-bridge beam system ofhigh-speed railway in China subjected to a moving load was carried out The results show that the analytical solution proposed inthis paper is consistent with the results obtained from a finite element analysis validating and rationalizing the analytical solutionMoreover the system parameters were analyzed for the dynamic response of double simply supported rail-bridge beam system inhigh-speed railway subjected to a moving load with different speeds the conclusions can be beneficial for engineering practice

1 Introduction

Moving-load dynamic problems are very common in engi-neering and daily life [1] and have attracted much attentionfor decades and recently raised many engineering concernssuch as the traffic comfort and safety of high-speed railwaylines and new materials for mechanical and aerospace engi-neering The problem of dynamic response of a structuresubjected to moving loads has always been the focus inengineering practice

For decades the dynamic response of a single beam hasbeen extensively investigated Many previous studies haveprovided better solutions to this problem [2ndash6] and there arealsomany papers on beamvibration excited by amovingmass[7ndash10] In the recent decades much effort has been devotedto the analysis of dynamic response of a double-beam systemand the results are significant [11ndash16]

The free and forced vibration analyses of a double-beamsystem have been extensively investigated ONISZCZUK [1718] reported the free vibration analysis of two parallel simplysupported beams continuously joined by a Winkler elastic

layer Zhang [19] investigated the vibration and buckling ofa double-beam system with identical elastic modulus undercompressive axial loading on the basis of EulerndashBernoullibeam theory The properties of free transverse vibration ofthe system significantly depended on the axial compressionsSimilarly Stojanovic [20] studied the forced vibration andbuckling of a Rayleigh and Timoshenko double-beam systemcontinuously joined by a Winkler elastic layer under com-pressive axial loading while the elastic modulus of doublebeam was also the same Palmeri [21] proposed a Galerkin-type state-space approach for the transverse vibrations ofslender double-beam systems joined with a viscoelastic innerlayer and conducted numerical applications onmodal shapesmodal frequencies and forced vibrations Zhang [22] inves-tigated the transverse vibration of two parallel Timoshenkobeams connected by discrete springs and coupled withdiscontinuities by considering both free and forced vibrationLi and Hu [23] developed a semianalytical method to inves-tigate the natural frequencies and mode shapes of a double-beam system interconnected by a viscoelastic layer whilethe modal-expansion iterated method was further applied

HindawiShock and VibrationVolume 2018 Article ID 2149251 14 pageshttpsdoiorg10115520182149251

2 Shock and Vibration

to determine the forced vibration responses of the double-beam system Abu-Hilal studied the dynamic response of adouble-beam system traversed by a constant moving load[24] Although the dynamic deflections of both beams wereprovided in analytical closed forms the two simply supportedprismatic beams should be identical Unlike Abu-HilalrsquosresearchWu [25] investigated the dynamic response of a sim-ply supported viscously damped double-beam system undermoving harmonic loads the double-beam system consistedof two finite elastic homogeneous isotropic beams whichwere also identical parallel and connected continuously bya layer of elastic springs with viscous damping Meanwhilebased on the double-beam theory the dynamic characteristicsof a railway system have been investigated [26ndash29]

The dynamic responses of a triple-beam system have alsointrigued many scholarsrsquo attention Stojanovic studied thestochastic stability of three elastically connected Euler beamson an elastic foundation using a perturbation approach[30] Li developed a dynamic stiffness method for a gen-eral elastically connected three-beam system based on bothEulerndashBernoulli beam theory [31] and Timoshenko beamtheory [32] A dynamic stiffness matrix was formulated fromthe analytical closed-form solutions of differential equationsof motion of the three-beam element in free transversevibration and thenused alongwith the automatedMuller rootsearch algorithm to calculate the free vibration characteristicsof three-beam system

Increasing efforts have also been devoted to analyze thedynamic responses of a multiple-beam system Rao [33]investigated the natural vibrations of elastically connectedmulti-Timoshenko beams Mao [34] used the Adomianmod-ified decomposition method (AMDM) to investigate the freevibrations of N elastically connected parallel EulerndashBernoullibeams continuously joined by a Winkler-type elastic layerand a general solution for the free vibration problem of elas-tically connected beams under general boundary conditionswas obtained Stojanovic developed a general procedure forthe determination of natural frequencies and buckling loadfor a set of beam system under compressive axial load usingTimoshenko and high-order shear deformation theory [35]The set of beams were simply supported and continuouslyjoined by a Winkler elastic layer Kargarnovin [36] proposeda closed-form solution to study the dynamics of a compositebeam with a single delamination under the action of amoving constant force the delaminated beam was dividedinto four interconnected beams using the delamination limitsas their boundaries Ariaei [37] investigated the transversedynamic response of an elastically connected multiple beambased on Timoshenko beam theory involving a change invariables and modal analysis to decouple and solve thegoverning differential equations respectively However theparallel beams were assumed to be identical and connectedby a finite number of springs

These studies indicate that although the moving loadsproblem on beam-like structure has been investigated pro-foundly in a way studies on the dynamic response of a beammainly focused on single-beam and double-beam modelsand many studies have some limitations For example thematerials and geometric properties of each beam are required

to be identical the analytic expression forms are too complexto be extended to multiple beams and studies are lackingon the dynamic response of multiple beams with differentproperties subjected tomoving loadsTherefore the dynamicresponses of a multiple simply supported EulerndashBernoullibeam system with elastic connection were evaluated inthis study combined with finite sin-Fourier transform andnumerical Laplace transform based on Durbin transformthe spatial displacement time-domain response of a mul-tiple simply supported beam system was obtained Thenan analytical method developed in this study was used tocalculate the dynamic response of a rail-bridge double simplysupported beam system of a high-speed railway system inChina subjected to a moving load validating and rational-izing the analytical solution reported in this paper drawingsome conclusionsmeaningful to engineering design and thuslaying foundations for further applications of the multiplesimply supported beam system in engineering

2 Model of a Multiple Simply Supported BeamSystem Subjected to Moving Loads

The N-layer simply supported beam system is shown inFigure 1 The length of each beam is L the beams areconnected vertically through springs evenly distributed alongthe beam length Based on EulerndashBernoulli beam theoryaccording to the dynamic equilibrium relationship of themultiple simply supported beam system subjected to movingloads the corresponding partial differential equations for thevibration of the system can be established as follows

1198981 12059721199061 (119909 119905)1205971199052 + 11986411198681 120597

41199061 (119909 119905)1205971199094+ 1198961 (1199061 minus 1199062) = 120575 (119909 minus V119905) 1198751 (119905)

(1)

119898119903 1205972119906119903 (119909 119905)1205971199052 + 119864119903119868119903 120597

4119906119903 (119909 119905)1205971199094minus 119896119903minus1 [119906119903minus1 (119909 119905) minus 119906119903 (119909 119905)]+ 119896119903 [119906119903 (119909 119905) minus 119906119903+1 (119909 119905)] = 120575 (119909 minus V119905) 119875119903 (119905)

(2)

1198981198731205972119906119873 (119909 119905)1205971199052 + 119864119873119868119873120597

4119906119873 (119909 119905)1205971199094minus 119896119873minus1 [119906119873minus1 (119909 119905) minus 119906119873 (119909 119905)] = 120575 (119909 minus V119905) 119875119873 (119905)

(3)

where m119903 is the mass per unit length of the 119903th simplysupported beam E119903 and I119903 are the elasticity modulus andhorizontal inertia moment of the 119903th simply supported beamrespectively and u119903(119909 119905) is the vertical displacement responseof the 119903th simply supported beam P1(119905)simP119873(119905) are theexternal loads on themultiple simply supported beam systemk119903 is the spring stiffness between the 119903th and 119903 + 1th simplysupported beams V is themoving speed of external loads and120575 is Dirac function

To solve the partial differential equations (1)-(3) of thesystem integral transform methods were used in this paperFirst finite sin-Fourier Transform was conducted on spatial

Shock and Vibration 3

11

2

r

n-1

1

2

Ｌ

Ｈ-1

n

v

Figure 1 Model of a multiple simply supported beam system

coordinate 119909 Regarding 0 le 119909 le L the transform can bedefined as follows [38]

119865 [119906 (119909)] = (120585119896) = int119871

0

119906 (119909) sin (120585119896119909) 119889119909 (4)

119865minus1 [ (120585119896)] = 119906 (119909) = 2119871infinsum119896=1

(120585119896) sin (120585119896119909) (5)

where

120585119896 = 119896120587119871 119896 = 1 2 3 (6)

Based on (4) finite sin-Fourier transform was conductedon 119906(4)(119909) the quartic derivative of displacement functionwith respect to 119909 and the following expression was obtained

119865d4119906 (119909)d1199092 = minus12058511989611990610158401015840 (119871) (minus1)119896 + 12058511989611990610158401015840 (0) + 1205854119896 (120585119896)

minus 1205853119896119906 (0) + (minus1)119896 1205853119896119906 (119871)(7)

Considering a small deformation the boundary condi-tion for the simply supported beam can be described asfollows

119906 (119909 119905)|119909=0 = 011986411986811990610158401015840 (119909 119905)|119909=0 = 0119906 (119909 119905)|119909=119871 = 0

11986411986811990610158401015840 (119909 119905)|119909=119871 = 0(8)

According to the boundary conditions in (8) (7) can besimplified as follows

1198651205974119906 (119909 119905)1205971199094 = 1205854119896 (120585119896 t) (9)

Finite sin-Fourier transform with regard to spatial coor-dinate 119909 was conducted on the left and right sides of (1)-(3)and the following equations can be obtained

1198981 12059721 (120585119896 119905)1205971199052 + 1198641119868112058541198961 (120585119896 119905)+ 1198961 [1 (120585119896 119905) minus 2 (120585119896 119905)] = 1198751 (119905) sin (120585119896V119905)

(10)

119898119903 1205972119903 (120585119896 119905)1205971199052 + 1198641199031198681199031205854119896119903 (120585119896 119905)minus 119896119903minus1 [119903minus1 (120585119896 119905) minus 119903 (120585119896 119905)]+ 119896119903 [119903 (120585119896 119905) minus 119903+1 (120585119896 119905)] = 119875119903 (119905) sin (120585119896V119905)

(11)

1198981198731205972119873 (120585119896 119905)1205971199052 + 1198641198731198681198731205854119896119873 (120585119896 119905)minus [119896119873minus1 [119873minus1 (120585119896 119905) minus 119873 (120585119896 119905)]]= 119875119873 (119905) sin (120585119896V119905)

(12)

According to the derivative theorem of Laplace trans-form

119871d2119906 (119905)d1199052 = 1199042119906lowast (119904) minus 1199041199061015840 (0+) minus 119906 (0+) (13)

where 119904 is the transformed variable corresponding to time 119905The initial conditions of multiple simply supported beam

system can be defined as follows

119906119894 (119909 0) = 0119894 (119909 0) = 0

119894 = 1 r 119873(14)

From (13) and (14) Laplace transformation on the secondderivative of displacement responses of simply supportedbeam system with respect to time can be obtained as follows

1198711205972119906119894 (119909 119905)1205971199052 = 1199042119906119894lowast (119909 119904) 119894 = 1 r 119873 (15)

4 Shock and Vibration

Laplace transformation with respect to time 119905 was con-ducted on the left and right sides of (10)ndash(12) and thefollowing equations can be obtained after reorganizing andcombining similar terms

(11989811199042 + 1198961 + 119864111986811205854119896) 1lowast (120585119896 119904) minus 11989612lowast (120585119896 119904)= 1198621 (120585119896 119904)

(16)

minus 119896119903minus1119903minus1lowast (120585119896 119904)+ [1198981199031199042 + 1198641199031198681199031205854119896 + 119896119903minus1 + 119896119903] 119903lowast (120585119896 119904)minus 119896119903119903+1lowast (120585119896 119904) = 119862119903 (120585119896 119904)

(17)

minus 119896119873minus1119873minus1lowast (120585119896 119904)+ (1198981198731199042 + 1198641198731198681198731205854119896 + 119896119873minus1) 119873lowast (120585119896 119904)= 119862119873 (120585119896 119904)

(18)

where C119894(120585119896 119904) = 119871[119875119894(119905) sin(120585119896V119905)] is the external load offrequency domain on the 119894th simply supported beam Thevector quantities for the frequency-domain responses andexternal loads of frequency domain on each simply supportedbeam can be denoted as follows

ulowast (120585119896 119904)= lowast1 (120585119896 119904) lowast119903 (120585119896 119904) lowast119873 (120585119896 119904)T

(19)

C (120585119896 119904)= 1198621 (120585119896 119904) 119862119903 (120585119896 119904) 119862119873 (120585119896 119904)T

(20)

Therefore (16)ndash(18) can be expressed as follows

[D] ulowast (120585119896 119904) = C (120585119896 119904) (21)

where D is the matrix with respect to 119873 times 119873 and only thefollowing elements are nonzero

11986311 = 11989811199042 + 119864111986811205854119896 + 1198961119863119903minus1119903 = 119863119903119903minus1 (22)

119863119903119903 = 1198981199031199042 + 1198641199031198681199031205854119896 + 119896119903minus1 + 119896119903119863119903119903minus1 = minus119896119903minus1119863119903119903+1 = minus119896119903

(23)

119863119873119873 = 1198981198731199042 + 1198641198731198681198731205854119896 + 119896119873minus1119863119903+1119903 = 119863119903119903+1 (24)

The frequency-domain responses ulowast(120585119896 119904) for eachbeam in the multiple simply supported beam system canbe obtained by solving the linear algebraic equation (21)Finite sin-Fourier inverse transformation was conducted onthe frequency-domain responses ulowast(120585119896 119904) for the multiplesimply supported beam system and the spatial displacementresponses for the multiple simply supported beam system

with respect to the time-frequency domains can be obtainedas follows

119906119894 (119909 119904) =119872sum119896=1

119894lowast (120585119896 119904) sin (120585119896119909) 119889119909119894 = 1 r 119873

(25)

Laplace inverse transform (LIT) with respect to time-frequency domain s was conducted on the left and rightsides of (25) and the spatial displacement time-domainresponses for the multiple simply supported beam system canbe obtained as follows

119906119894 (119909 119905) = 119871minus1 [119906119894 (119909 119904)] 119894 = 1 r 119873 (26)

The solving process in this paper shows that the dis-placement frequency-domain response function 119906119894(119909 119904) foreach beam contains the product of power function and sinefunction However with the increase in the number of layersof simply supported beam the analytical expression of (26)becomes complex and lengthy and it is not easy to conductan analytical expression on it Therefore to effectively solvethe LIT for (26) according to arbitrary numbers of beamsfast LIT (FLIT) [39] proposed by Durbin was introduced inthis paper to obtain the solution This method of numericalinverse transform can be used to solve the transform resultsrelatively accurately It was verified that the error boundaryof this method can be set to be any arbitrarily small valueso that better computing results can be obtained even forcomplex expressions The corresponding solving programsare compiled in MATLAB to solve (21)ndash(26) and the spa-tial displacement time-domain response of different simplysupported beam systems with arbitrary numbers of beamssubjected to moving loads can be obtained

3 Analysis of Numerical Examples

31 Validation of the Analytical Model In the first placeas an application example of a double simply supportedbeam system subjected to moving loads a typical rail-bridgesystem of high-speed railway system in China was selectedfor analysis In this system the first simply supported beamis a rail of ballastless tracks and the second simply supportedbeam is a typical simply supported bridge with a span of 32mThemoving load acts on the first simply supported beam andthe materials of the double simply supported beam systemand geometric parameters are shown as follows

m1 = 60KgmE1 = 21 times 105MpaI1 = 3217 times 10minus5m4m2 = 183 times 103KgmE2 = 345 times 104Mpa

Shock and Vibration 5

(a) (b)

Figure 2 Deflections of double simply supported rail-bridge system with different positions of the moving load (a) first beam (b) secondbeam

I2 = 1095m4L1 = L2 = 32mk1 = 60MNm2p1 = 85KN

(27)

Based on the solving programs compiled inMATLAB thevertical deflection of rail and bridge with different positionsof themoving load in the double simply supported rail-bridgesystem when the moving speed of external load v = 100 mswere plotted as shown in Figure 2

The results in the diagram show that the dynamic verticaldeflection values for the rail and bridge in the double simplysupported system subjected to moving loads reached themaximum near the mid-span (middle point of the system)The displacement time-history responses of the mid-span offirst and second simply supported beams effectively reflectthe maximum displacement responses of the double simplysupported beam system of rail-bridge in a high-speed railwaysystem Hence for the following computational analysisthe mid-span displacement responses for the double simplysupported beam system were selected for analysis

To validate the theoretical solution in this paper ANSYSfinite element software was used to conduct numericalsimulation on the dynamic response of the double simplysupported rail-bridge system subjected to moving loads andthe time-history responses for the mid-span displacements ofthe double simply supported rail-bridge system subjected toexternal load with different moving speeds (32 ms and 64ms) were calculated Regarding the FE model the springswere modeled by Combin14 element in ANSYS and thespacing between the Combin14 elements was set as 01mwhich is a relatively small spacing to model the continuoussupport between both the beams Then the results werecompared with the theoretically calculated results and thecomparison results are shown in Figures 3 and 4 The resultsshow that the theoretically calculated values are consistentwith the results obtained from the ANSYS finite elementnumerical calculation

Next as another validation of the theoretical solution inthis paper the dynamic responses of a quadruple beam sys-tem subjected to moving loads were calculated by theoreticalanalysis model and compared with the calculation results ofANSYS finite element software The material and geometricparameters of the quadruple simply supported beam systemare as follows

m1 = 60KgmE1 = 21 times 105MpaI1 = 3217 times 10minus5m4m2 = 1275KgmE2 = 355 times 104MpaI2 = 17 times 10minus3m4m3 = 140125KgmE3 = 30 times 104MpaI3 = 1686 times 10minus3m4m4 = 18300KgmE4 = 345 times 104MpaI4 = 1095m4L1 = L2 = L3 = L4 = 32mk1 = 60MNm2k2 = 900MNm2k3 = 1375MNm2p1 = 85KN

(28)

Similar to the double-beam systemrsquos analysis the time-history responses for the mid-span displacements of the

6 Shock and Vibration

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

02 04 06 08 1000t (s)

(a)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 3Mid-span deflection time-history for double simply supported rail-bridge systemwith loadmoving speed v = 32ms (a) first beam(b) second beam

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

ANSYSMATLAB

(a)

minus020

minus016

minus012

minus008

minus004

000

004D

eflec

tion

(mm

)

01 02 03 04 0500t (s)

ANSYSMATLAB

(b)

Figure 4 Mid-span deflection time-history for double simply supported rail-bridge system with load moving speed v = 64 ms (a) firstbeam (b) second beam

quadruple simply supported beam system subjected to exter-nal load with different moving speeds (32 ms and 64 ms)were calculated The comparisons between the theoreticallycalculated results and the ANSYS finite element numericalcalculation are shown in Figures 5 and 6 The results alsoshow that the theoretical results are consistent with theresults obtained from the ANSYS finite element numericalcalculation thus validating and rationalizing this theoreticalanalysis model in an effective manner

32 Analysis of Maximum Deflection-Loading Speed for Mid-Span of Double Simply Supported Beam Figure 7 shows themaximum dynamic deflection for the mid-span of doublesimply supported rail-bridge system at different load movingspeeds With the increase in load moving speeds the effectof speed on the dynamic response of the double simply

supported rail-bridge beam system shows a nonlinear rela-tionship a sine curve was observed instead of increasingwith the increase of speed linearly The maximum mid-span deflection for the first simply supported beam showsthe variation of sine curve around the amplitude of 102mm Besides with the increase of speed the period andamplitude of sine curve also increased The maximum mid-span deflection for the second simply supported beam alsoshows the variation of sine curve but unlike the first beamthe mid-span deflection for this beam shows an overallincreasing trend with the increase of speeds

Due to the very low moment of inertia of the firstbeam compared to the second beam the first beam showshierarchical oscillations while the secondary beam does notshow any local oscillations and the mid-span deflectiondynamic response of the first beam is insensitive to the

Shock and Vibration 7

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02 04 06 08 1000t (s)

(a)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus028

minus021

minus014

minus007

000

007

Defl

ectio

n (m

m)

(b)

ANSYSMATLAB

minus024

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

02 04 06 08 1000t (s)

(c)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(d)

Figure 5 Mid-span deflection time-history for quadruple simply supported beam system with load moving speed v = 32 ms (a) first beam(b) second beam (c) third beam (d) fourth beam

speed Relationship between maximum deflection and load-ing speed for mid-span of double simply supported beamwith a higher I1 value is shown in Figures 8-9 The resultsshow that the local oscillation of the first beamrsquos mid-spanmaximumdeflection reduced significantly when themomentof inertia I1 is revalued as 100 I1 and there are barely no localoscillations of the first beamrsquos mid-span maximum deflectionwhen the moment of inertia I1 is taken as 105 I1

Moreover the speeds atwhich themid-spandeflection forthe double beams reaches the peak value are the same withthe appearance of multiple points of resonance (wave crest)and points of cancellation (wave trough) as shown in Table 1

Figures 10 and 11 show the contrast ofmid-span deflectionresponses of the double simply supported rail-bridge beamsystem under different resonances and cancellation speeds

Table 1 and Figures 10-11 show that with the increasein resonance speeds the time-history curves of mid-spandeflection dynamic response of the first beam are consistentwhile the local oscillation of mid-span deflection time-history curves of the second beam significantly reduced thevibration period becomes longer and the mid-span dynamic

response reaches the amplitude when the moving loads actnear the mid-span of double rail-bridge system With theincrease in cancellation speeds the time-history curves ofmid-span deflection dynamic response of the first beamare consistent and when the mid-span deflection dynamicresponse of the second beam reaches the amplitude thedistance between the moving-load position and mid-spanincreases When the load moving speed is 881 ms the mid-spandynamic deflection response of the second beam reachesthe amplitude while the moving loads are on the 13 and 23positions of the beam

33 Analysis of Parameter Influence of Double Simply Sup-ported Rail-Bridge System Based on the analysis in theabove section further study was conducted regarding theeffect of interlayer spring stiffness beam masses and flexuralrigidities on the dynamic responses of the double simplysupported rail-bridge system in this section The movingspeed of loads v = 100 ms was taken as the example to studythe effect of variation of three parameters

8 Shock and Vibration

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

01 02 03 04 0500t (s)

(a)

ANSYSMATLAB

01 02 03 04 0500t (s)

minus028

minus021

minus014

minus007

000

007

Defl

ectio

n (m

m)

(b)

ANSYSMATLAB

minus024

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

(c)

ANSYSMATLAB

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

(d)

Figure 6 Mid-span deflection time-history for quadruple simply supported beam system with load moving speed v = 64 ms (a) first beam(b) second beam (c) third beam (d) fourth beam

First Beam

40 80 120 160 2000v (ms)

096

100

104

108

112

116

120

Defl

ectio

n (m

m)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 7 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam (a) first beam (b)second beam

Shock and Vibration 9

First Beam038

040

042

044

046

048

050

052D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 8 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 100I1 (a) first beam (b) second beam

First Beam

40 80 120 160 2000v (ms)

0060

0063

0066

0069

0072

0075

0078

Defl

ectio

n (m

m)

(a)

Second Beam0040

0044

0048

0052

0056

0060D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(b)

Figure 9 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 105I1 (a) first beam (b) second beam

Table 1 Relationship between resonance (cancellation) speed and amplitude

Speed at the wavecrest (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)

Speed at the wavetrough (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)189 1024 0161 204 1009 0159232 1025 0162 253 1015 0160299 1027 0165 334 1011 0162411 1039 0169 487 1004 0163687 1058 0180 881 0996 0164

Figure 12 show that interlayer stiffness variation has agreater effect on the dynamic response of the first sim-ply supported beam and the mid-span dynamic deflectionamplitude of the first simply supported beam decreasedwith the increase in interlayer stiffness while the interlayer

stiffness has a negligible effect on the mid-span dynamictime-history of the second simply supported beam Thereason is that in the double simply supported rail-bridgesystem the differences in the stiffness of the first and secondbeams are large the flexural rigidity of the first beam is

10 Shock and Vibration

v=189msv=232msv=299ms

v=411msv=687ms

4 8 12 16 20 24 28 320Load position (m)

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

(a)

v=189msv=232msv=299ms

v=411msv=687ms

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(b)

Figure 10 Mid-span deflection time-history for double simply supported rail-bridge system under resonance condition (a) first beam (b)second beam

v=204msv=253msv=334ms

v=487msv=881ms

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(a)

0 4 8 12 16 20 24 28 32

v=204msv=253msv=334ms

v=487msv=881ms

Load position (m)

minus020

minus016

minus012

minus008

minus004

000

Defl

ectio

n (m

m)

(b)

Figure 11 Mid-span deflection time-history for the double simply supported rail-bridge system under cancellation condition (a) first beam(b) second beam

much smaller than that of the second beam and the bendingresistance of the second beam plays a leading role in thebending resistance of the entire system

Figures 13 and 14 show the effects of mass variationper unit length of the first and second beams on mid-span deflection time-history for double simply supportedrail-bridge system respectively Figure 13 shows that underdifferent masses of the first beam the time-history curvesof mid-span deflection dynamic response for the two simplysupported beams are consistent Therefore the effect of massvariation of the first simply supported beam on the mid-spandeflection dynamic response of the double-beam system canbe neglected Figure 14 shows that the mass variation of thesecond simply supported beam has a significant effect on the

mid-span deflection dynamic response of both beams whenthemass per unit length of the second simply supported beamincreases to four times of the original mass the amplitudes ofmid-span deflection dynamic response of the first and secondsimply supported beams both increase significantly with thegrowth rate reaching up to 10 and 45 respectively

Figures 15 and 16 show the effect of flexural rigidityvariations of the first and second simply supported beamson the mid-span deflection time-history for double simplysupported rail-bridge system Figure 15 shows that the mid-span dynamic deflection amplitude of the first simply sup-ported beam significantly decreased with the increase in thestiffness of this beam while the stiffness of the first beamhas a smaller effect on the mid-span dynamic deflection

Shock and Vibration 11

025kr050kr100kr

200kr400kr

minus28

minus24

minus20

minus16

minus12

minus08

minus04

00

04

Defl

ectio

n (m

m)

008 016 024 032000t (s)

(a)

025kr050kr100kr

200kr400kr

008 016 024 032000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 12 Comparison diagram of mid-span deflection time-history for the double simply supported rail-bridge system under differentinterlayer stiffness (a) first beam (b) second beam

100

200400

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025Ｇ1050Ｇ1

Ｇ1

Ｇ1Ｇ1

(a)

025050100

200400

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

Ｇ1Ｇ1

Ｇ1Ｇ1

Ｇ1

(b)

Figure 13 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof first beam (a) first beam (b) second beam

amplitude of the second simply supported beam Underdifferent stiffness of the first beam the mid-span deflectiondynamic time-history curves for the second simply supportedbeam are consistent Figure 16 shows that the flexural rigidityvariations of the second simply supported beam have a sig-nificant effect on the mid-span deflection dynamic responseof double simply supported rail-bridge system In additionthe mid-span deflection dynamic response amplitudes ofboth beams significantly increased with the decrease in theflexural stiffness of second beam In practical applicationsthe dynamic response of the rail-bridge system in high-speedrailways can be decreased by moderately decreasing the massof the second beam and increasing the flexural stiffness ofboth simply supported beams

4 Conclusions

In this study a dynamic analysis model was established for amultiple simply supported beam system subjected to movingloads in combination of finite sin-Fourier transform andnumerical Laplace transform based on Durbin transforman analytical calculation method was developed for thespatial displacement time-domain response of a multiplesimply supported beam systemThe following conclusions aredrawn by comparing the examples and ANSYS finite elementnumerical computation method(1) A simply supported rail-bridge system of high-speedrailways with a span of 32 m in China was taken as theexample and the theoretically calculated results were com-pared with the results obtained from ANSYS finite element

12 Shock and Vibration

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

100

200400

025Ｇ2050Ｇ2

Ｇ2

Ｇ2Ｇ2

(a)

minus030

minus024

minus018

minus012

minus006

000

006

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025050100

200400Ｇ2

Ｇ2

Ｇ2Ｇ2

Ｇ2

(b)

Figure 14 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof second beam (a) first beam (b) second beam

008 016 024 032000t (s)

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02510501

10012001

(a)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

02510501

10012001

(b)

Figure 15 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof first beam (a) first beam (b) second beam

numerical calculation validating the analytical calculationmethod reported in this paper(2) The effect of load moving speed on the dynamicresponse of a double simply supported rail-bridge system ofhigh-speed railways subjected tomoving loads was evaluatedThe results indicate that the double simply supported rail-bridge system has multiple points of resonance and cancel-lation which should be paid attention in engineering(3) For a typical double simply supported rail-bridgesystem of high-speed railway in China because of a largedifference in the flexural rigidity of the first and secondbeams the variation in interlayer stiffness mainly affectsthe dynamic response of the first simply supported beamexhibiting a positive correlation When the mass and flexuralstiffness parameters of first simply supported beam vary they

mainly affect the dynamic response of this layer of simplysupported beam but slightly affect the dynamic responseof second simply supported beam The variation in theparameters of second simply supported beam has a greatereffect on the double simply supported beam system and adecrease in the mass of this layer of simply supported beamand an increase in the flexural rigidity of this beam canboth decrease the dynamic response of the double simplysupported rail-bridge system

Data Availability

The data used to support the findings of this study areincluded within the article

Shock and Vibration 13

minus14

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

05021002

20024002

(a)

008 016 024 032000t (s)

minus048

minus040

minus032

minus024

minus016

minus008

000

008

Defl

ectio

n (m

m)

05021002

20024002

(b)

Figure 16 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof second beam (a) first beam (b) second beam

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51708630)

References

[1] H Ouyang ldquoMoving-load dynamic problems a tutorial (with abrief overview)rdquoMechanical Systems and Signal Processing vol25 no 6 pp 2039ndash2060 2011

[2] D G Duffy ldquoThe response of an infinite railroad track to amoving vibrating massrdquo Journal of Applied Mechanics vol 57no 1 pp 66ndash73 1990

[3] L Fryba Vibration of solids and structures under moving loadsSpringer Science amp Business Media 2013

[4] G Michaltsos D Sophianopoulos and A N Kounadis ldquoTheeffect of a moving mass and other parameters on the dynamicresponse of a simply supported beamrdquo Journal of Sound andVibration vol 191 no 3 pp 357ndash362 1996

[5] P Museros E Moliner and M D Martınez-Rodrigo ldquoFreevibrations of simply-supported beam bridges under movingloads Maximum resonance cancellation and resonant verticalaccelerationrdquo Journal of Sound and Vibration vol 332 no 2 pp326ndash345 2013

[6] P Sniady ldquoDynamic response of a Timoshenko beam to amoving forcerdquo Journal of Applied Mechanics vol 75 no 2Article ID 024503 2008

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

[8] K Y Lee and A A Renshaw ldquoSolution of the moving massproblem using complex eigenfunction expansionsrdquo Journal ofApplied Mechanics vol 67 no 4 pp 823ndash827 2000

[9] U Lee ldquoSeparation between the flexible structure and themoving mass sliding on itrdquo Journal of Sound and Vibration vol209 no 5 pp 867ndash877 1998

[10] S A Eftekhari and A A Jafari ldquoCoupling ritz method andtriangular quadrature rule for moving mass problemrdquo Journalof Applied Mechanics vol 79 no 2 2012

[11] M Dublin andH R Friedrich ldquoForced responses of two elasticbeams interconnected by spring-damper systemsrdquo Journal of theAeronautical Sciences vol 23 pp 824ndash829 1956

[12] J M Seelig and W H Hoppmann ldquoImpact on an elasticallyconnected double-beam systemrdquo Journal of Applied Mechanicsvol 31 pp 621ndash626 1964

[13] P G Kessel ldquoResonances Excited in an Elastically ConnectedDouble-Beam System by a Cyclic Moving LoadrdquoThe Journal ofthe Acoustical Society of America vol 40 no 3 pp 684ndash6871966

[14] S Chonan ldquoDynamical behaviours of elastically connecteddouble-beam systems subjected to an impulsive loadrdquo Bulletinof JSME vol 19 no 132 pp 595ndash603 1976

[15] H V Vu A M Ordonez and B H Karnopp ldquoVibration of adouble-beam systemrdquo Journal of Sound and Vibration vol 229no 4 pp 807ndash822 2000

[16] H J Xu Z H Ding Z R Lu and J K Liu ldquoStructural DamageDetection Using a Modified Artificial Bee Colony AlgorithmrdquoCMES-Computer Modeling in Engineering amp Sciences vol 111pp 335ndash355 2016

[17] Z Oniszczuk ldquoFree transverse vibrations of elastically con-nected simply supported double-beam complex systemrdquo Jour-nal of Sound and Vibration vol 232 no 2 pp 387ndash403 2000

[18] Z Oniszczuk ldquoForced transverse vibrations of an elasticallyconnected complex simply supported double-beam systemrdquoJournal of Sound and Vibration vol 264 no 2 pp 273ndash2862003

[19] Y Q Zhang Y Lu S L Wang and X Liu ldquoVibration andbuckling of a double-beam system under compressive axialloadingrdquo Journal of Sound and Vibration vol 318 no 1-2 pp341ndash352 2008

[20] V Stojanovic and P Kozic ldquoForced transverse vibration ofRayleigh and Timoshenko double-beam system with effect of

14 Shock and Vibration

compressive axial loadrdquo International Journal of MechanicalSciences vol 60 no 1 pp 59ndash71 2012

[21] A Palmeri and S Adhikari ldquoA Galerkin-type state-spaceapproach for transverse vibrations of slender double-beamsystems with viscoelastic inner layerrdquo Journal of Sound andVibration vol 330 no 26 pp 6372ndash6386 2011

[22] Z Zhang X Huang Z Zhang and H Hua ldquoOn the transversevibration of Timoshenko double-beam systems coupled withvarious discontinuitiesrdquo International Journal of MechanicalSciences vol 89 pp 222ndash241 2014

[23] Y X Li Z J Hu and L Z Sun ldquoDynamical behavior of adouble-beam system interconnected by a viscoelastic layerrdquoInternational Journal of Mechanical Sciences vol 105 pp 291ndash303 2016

[24] M Abu-Hilal ldquoDynamic response of a double EulerndashBernoullibeam due to a moving constant loadrdquo Journal of Sound andVibration vol 297 no 3-5 pp 477ndash491 2006

[25] Y Wu and Y Gao ldquoAnalytical Solutions for Simply SupportedViscously Damped Double-Beam System under Moving Har-monic Loadsrdquo Journal of Engineering Mechanics vol 141 no 7p 04015004 2015

[26] Y-H Chen and Z-M Shiu ldquoResonant curves of an elevatedrailway to harmonic moving loadsrdquo International Journal ofStructural Stability and Dynamics vol 4 no 2 pp 237ndash2572004

[27] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of aninfinite tomoshenko beam on a viscoelastic foundation to aharmonic moving loadrdquo Journal of Sound and Vibration vol241 no 5 pp 809ndash824 2001

[28] M FMHussein andH EMHunt ldquoModelling of floating-slabtracks with continuous slabs under oscillating moving loadsrdquoJournal of Sound andVibration vol 297 no 1-2 pp 37ndash54 2006

[29] H P Wang J Li and K Zhang ldquoVibration analysis of themaglev guideway with the moving loadrdquo Journal of Sound andVibration vol 305 no 4-5 pp 621ndash640 2007

[30] V Stojanovic and M Petkovic ldquoMoment Lyapunov exponentsand stochastic stability of a three-dimensional system on elasticfoundation using a perturbation approachrdquo Journal of AppliedMechanics vol 80 no 5 2013

[31] L Jun and H Hongxing ldquoDynamic stiffness vibration analysisof an elastically connected three-beam systemrdquo Applied Acous-tics vol 69 no 7 pp 591ndash600 2008

[32] J Li Y Chen and H Hua ldquoExact dynamic stiffness matrixof a Timoshenko three-beam systemrdquo International Journal ofMechanical Sciences vol 50 no 6 pp 1023ndash1034 2008

[33] S S Rao ldquoNatural vibrations of systems of elastically connectedTimoshenko beams rdquo The Journal of the Acoustical Society ofAmerica vol 55 no 6 pp 1232ndash1237 1974

[34] Q Mao ldquoFree vibration analysis of elastically connectedmultiple-beams by using the Adomianmodified decompositionmethodrdquo Journal of Sound and Vibration vol 331 no 11 pp2532ndash2542 2012

[35] V Stojanovic P Kozic and G Janevski ldquoExact closed-formsolutions for the natural frequencies and stability of elasticallyconnected multiple beam system using Timoshenko and high-order shear deformation theoryrdquo Journal of Sound and Vibra-tion vol 332 no 3 pp 563ndash576 2013

[36] M H Kargarnovin M T Ahmadian and R-A Jafari-Talookolaei ldquoAnalytical solution for the dynamic analysis of adelaminated composite beam traversed by a moving constantforcerdquo Journal of Vibration and Control vol 19 no 10 pp 1524ndash1537 2013

[37] A Ariaei S Ziaei-Rad and M Ghayour ldquoTransverse vibrationof a multiple-Timoshenko beam system with intermediateelastic connections due to a moving loadrdquo Archive of AppliedMechanics vol 81 no 3 pp 263ndash281 2011

[38] Y Povstenko and J Klekot ldquoThe Dirichlet problem for thetime-fractional advection-diffusion equation in a line segmentrdquoBoundary Value Problems Paper No 89 8 pages 2016

[39] F Durbin ldquoNumerical inversion of Laplace transforms anefficient improvement to Dubner and Abatersquos methodrdquo TheComputer Journal vol 17 pp 371ndash376 1974

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Shock and Vibration 3

11

2

r

n-1

1

2

Ｌ

Ｈ-1

n

v

Figure 1 Model of a multiple simply supported beam system

coordinate 119909 Regarding 0 le 119909 le L the transform can bedefined as follows [38]

119865 [119906 (119909)] = (120585119896) = int119871

0

119906 (119909) sin (120585119896119909) 119889119909 (4)

119865minus1 [ (120585119896)] = 119906 (119909) = 2119871infinsum119896=1

(120585119896) sin (120585119896119909) (5)

where

120585119896 = 119896120587119871 119896 = 1 2 3 (6)

Based on (4) finite sin-Fourier transform was conductedon 119906(4)(119909) the quartic derivative of displacement functionwith respect to 119909 and the following expression was obtained

119865d4119906 (119909)d1199092 = minus12058511989611990610158401015840 (119871) (minus1)119896 + 12058511989611990610158401015840 (0) + 1205854119896 (120585119896)

minus 1205853119896119906 (0) + (minus1)119896 1205853119896119906 (119871)(7)

Considering a small deformation the boundary condi-tion for the simply supported beam can be described asfollows

119906 (119909 119905)|119909=0 = 011986411986811990610158401015840 (119909 119905)|119909=0 = 0119906 (119909 119905)|119909=119871 = 0

11986411986811990610158401015840 (119909 119905)|119909=119871 = 0(8)

According to the boundary conditions in (8) (7) can besimplified as follows

1198651205974119906 (119909 119905)1205971199094 = 1205854119896 (120585119896 t) (9)

Finite sin-Fourier transform with regard to spatial coor-dinate 119909 was conducted on the left and right sides of (1)-(3)and the following equations can be obtained

1198981 12059721 (120585119896 119905)1205971199052 + 1198641119868112058541198961 (120585119896 119905)+ 1198961 [1 (120585119896 119905) minus 2 (120585119896 119905)] = 1198751 (119905) sin (120585119896V119905)

(10)

119898119903 1205972119903 (120585119896 119905)1205971199052 + 1198641199031198681199031205854119896119903 (120585119896 119905)minus 119896119903minus1 [119903minus1 (120585119896 119905) minus 119903 (120585119896 119905)]+ 119896119903 [119903 (120585119896 119905) minus 119903+1 (120585119896 119905)] = 119875119903 (119905) sin (120585119896V119905)

(11)

1198981198731205972119873 (120585119896 119905)1205971199052 + 1198641198731198681198731205854119896119873 (120585119896 119905)minus [119896119873minus1 [119873minus1 (120585119896 119905) minus 119873 (120585119896 119905)]]= 119875119873 (119905) sin (120585119896V119905)

(12)

According to the derivative theorem of Laplace trans-form

119871d2119906 (119905)d1199052 = 1199042119906lowast (119904) minus 1199041199061015840 (0+) minus 119906 (0+) (13)

where 119904 is the transformed variable corresponding to time 119905The initial conditions of multiple simply supported beam

system can be defined as follows

119906119894 (119909 0) = 0119894 (119909 0) = 0

119894 = 1 r 119873(14)

From (13) and (14) Laplace transformation on the secondderivative of displacement responses of simply supportedbeam system with respect to time can be obtained as follows

1198711205972119906119894 (119909 119905)1205971199052 = 1199042119906119894lowast (119909 119904) 119894 = 1 r 119873 (15)

4 Shock and Vibration

Laplace transformation with respect to time 119905 was con-ducted on the left and right sides of (10)ndash(12) and thefollowing equations can be obtained after reorganizing andcombining similar terms

(11989811199042 + 1198961 + 119864111986811205854119896) 1lowast (120585119896 119904) minus 11989612lowast (120585119896 119904)= 1198621 (120585119896 119904)

(16)

minus 119896119903minus1119903minus1lowast (120585119896 119904)+ [1198981199031199042 + 1198641199031198681199031205854119896 + 119896119903minus1 + 119896119903] 119903lowast (120585119896 119904)minus 119896119903119903+1lowast (120585119896 119904) = 119862119903 (120585119896 119904)

(17)

minus 119896119873minus1119873minus1lowast (120585119896 119904)+ (1198981198731199042 + 1198641198731198681198731205854119896 + 119896119873minus1) 119873lowast (120585119896 119904)= 119862119873 (120585119896 119904)

(18)

where C119894(120585119896 119904) = 119871[119875119894(119905) sin(120585119896V119905)] is the external load offrequency domain on the 119894th simply supported beam Thevector quantities for the frequency-domain responses andexternal loads of frequency domain on each simply supportedbeam can be denoted as follows

ulowast (120585119896 119904)= lowast1 (120585119896 119904) lowast119903 (120585119896 119904) lowast119873 (120585119896 119904)T

(19)

C (120585119896 119904)= 1198621 (120585119896 119904) 119862119903 (120585119896 119904) 119862119873 (120585119896 119904)T

(20)

Therefore (16)ndash(18) can be expressed as follows

[D] ulowast (120585119896 119904) = C (120585119896 119904) (21)

where D is the matrix with respect to 119873 times 119873 and only thefollowing elements are nonzero

11986311 = 11989811199042 + 119864111986811205854119896 + 1198961119863119903minus1119903 = 119863119903119903minus1 (22)

119863119903119903 = 1198981199031199042 + 1198641199031198681199031205854119896 + 119896119903minus1 + 119896119903119863119903119903minus1 = minus119896119903minus1119863119903119903+1 = minus119896119903

(23)

119863119873119873 = 1198981198731199042 + 1198641198731198681198731205854119896 + 119896119873minus1119863119903+1119903 = 119863119903119903+1 (24)

The frequency-domain responses ulowast(120585119896 119904) for eachbeam in the multiple simply supported beam system canbe obtained by solving the linear algebraic equation (21)Finite sin-Fourier inverse transformation was conducted onthe frequency-domain responses ulowast(120585119896 119904) for the multiplesimply supported beam system and the spatial displacementresponses for the multiple simply supported beam system

with respect to the time-frequency domains can be obtainedas follows

119906119894 (119909 119904) =119872sum119896=1

119894lowast (120585119896 119904) sin (120585119896119909) 119889119909119894 = 1 r 119873

(25)

Laplace inverse transform (LIT) with respect to time-frequency domain s was conducted on the left and rightsides of (25) and the spatial displacement time-domainresponses for the multiple simply supported beam system canbe obtained as follows

119906119894 (119909 119905) = 119871minus1 [119906119894 (119909 119904)] 119894 = 1 r 119873 (26)

The solving process in this paper shows that the dis-placement frequency-domain response function 119906119894(119909 119904) foreach beam contains the product of power function and sinefunction However with the increase in the number of layersof simply supported beam the analytical expression of (26)becomes complex and lengthy and it is not easy to conductan analytical expression on it Therefore to effectively solvethe LIT for (26) according to arbitrary numbers of beamsfast LIT (FLIT) [39] proposed by Durbin was introduced inthis paper to obtain the solution This method of numericalinverse transform can be used to solve the transform resultsrelatively accurately It was verified that the error boundaryof this method can be set to be any arbitrarily small valueso that better computing results can be obtained even forcomplex expressions The corresponding solving programsare compiled in MATLAB to solve (21)ndash(26) and the spa-tial displacement time-domain response of different simplysupported beam systems with arbitrary numbers of beamssubjected to moving loads can be obtained

3 Analysis of Numerical Examples

31 Validation of the Analytical Model In the first placeas an application example of a double simply supportedbeam system subjected to moving loads a typical rail-bridgesystem of high-speed railway system in China was selectedfor analysis In this system the first simply supported beamis a rail of ballastless tracks and the second simply supportedbeam is a typical simply supported bridge with a span of 32mThemoving load acts on the first simply supported beam andthe materials of the double simply supported beam systemand geometric parameters are shown as follows

m1 = 60KgmE1 = 21 times 105MpaI1 = 3217 times 10minus5m4m2 = 183 times 103KgmE2 = 345 times 104Mpa

Shock and Vibration 5

(a) (b)

Figure 2 Deflections of double simply supported rail-bridge system with different positions of the moving load (a) first beam (b) secondbeam

I2 = 1095m4L1 = L2 = 32mk1 = 60MNm2p1 = 85KN

(27)

Based on the solving programs compiled inMATLAB thevertical deflection of rail and bridge with different positionsof themoving load in the double simply supported rail-bridgesystem when the moving speed of external load v = 100 mswere plotted as shown in Figure 2

The results in the diagram show that the dynamic verticaldeflection values for the rail and bridge in the double simplysupported system subjected to moving loads reached themaximum near the mid-span (middle point of the system)The displacement time-history responses of the mid-span offirst and second simply supported beams effectively reflectthe maximum displacement responses of the double simplysupported beam system of rail-bridge in a high-speed railwaysystem Hence for the following computational analysisthe mid-span displacement responses for the double simplysupported beam system were selected for analysis

To validate the theoretical solution in this paper ANSYSfinite element software was used to conduct numericalsimulation on the dynamic response of the double simplysupported rail-bridge system subjected to moving loads andthe time-history responses for the mid-span displacements ofthe double simply supported rail-bridge system subjected toexternal load with different moving speeds (32 ms and 64ms) were calculated Regarding the FE model the springswere modeled by Combin14 element in ANSYS and thespacing between the Combin14 elements was set as 01mwhich is a relatively small spacing to model the continuoussupport between both the beams Then the results werecompared with the theoretically calculated results and thecomparison results are shown in Figures 3 and 4 The resultsshow that the theoretically calculated values are consistentwith the results obtained from the ANSYS finite elementnumerical calculation

Next as another validation of the theoretical solution inthis paper the dynamic responses of a quadruple beam sys-tem subjected to moving loads were calculated by theoreticalanalysis model and compared with the calculation results ofANSYS finite element software The material and geometricparameters of the quadruple simply supported beam systemare as follows

m1 = 60KgmE1 = 21 times 105MpaI1 = 3217 times 10minus5m4m2 = 1275KgmE2 = 355 times 104MpaI2 = 17 times 10minus3m4m3 = 140125KgmE3 = 30 times 104MpaI3 = 1686 times 10minus3m4m4 = 18300KgmE4 = 345 times 104MpaI4 = 1095m4L1 = L2 = L3 = L4 = 32mk1 = 60MNm2k2 = 900MNm2k3 = 1375MNm2p1 = 85KN

(28)

Similar to the double-beam systemrsquos analysis the time-history responses for the mid-span displacements of the

6 Shock and Vibration

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

02 04 06 08 1000t (s)

(a)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 3Mid-span deflection time-history for double simply supported rail-bridge systemwith loadmoving speed v = 32ms (a) first beam(b) second beam

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

ANSYSMATLAB

(a)

minus020

minus016

minus012

minus008

minus004

000

004D

eflec

tion

(mm

)

01 02 03 04 0500t (s)

ANSYSMATLAB

(b)

Figure 4 Mid-span deflection time-history for double simply supported rail-bridge system with load moving speed v = 64 ms (a) firstbeam (b) second beam

quadruple simply supported beam system subjected to exter-nal load with different moving speeds (32 ms and 64 ms)were calculated The comparisons between the theoreticallycalculated results and the ANSYS finite element numericalcalculation are shown in Figures 5 and 6 The results alsoshow that the theoretical results are consistent with theresults obtained from the ANSYS finite element numericalcalculation thus validating and rationalizing this theoreticalanalysis model in an effective manner

32 Analysis of Maximum Deflection-Loading Speed for Mid-Span of Double Simply Supported Beam Figure 7 shows themaximum dynamic deflection for the mid-span of doublesimply supported rail-bridge system at different load movingspeeds With the increase in load moving speeds the effectof speed on the dynamic response of the double simply

supported rail-bridge beam system shows a nonlinear rela-tionship a sine curve was observed instead of increasingwith the increase of speed linearly The maximum mid-span deflection for the first simply supported beam showsthe variation of sine curve around the amplitude of 102mm Besides with the increase of speed the period andamplitude of sine curve also increased The maximum mid-span deflection for the second simply supported beam alsoshows the variation of sine curve but unlike the first beamthe mid-span deflection for this beam shows an overallincreasing trend with the increase of speeds

Due to the very low moment of inertia of the firstbeam compared to the second beam the first beam showshierarchical oscillations while the secondary beam does notshow any local oscillations and the mid-span deflectiondynamic response of the first beam is insensitive to the

Shock and Vibration 7

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02 04 06 08 1000t (s)

(a)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus028

minus021

minus014

minus007

000

007

Defl

ectio

n (m

m)

(b)

ANSYSMATLAB

minus024

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

02 04 06 08 1000t (s)

(c)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(d)

Figure 5 Mid-span deflection time-history for quadruple simply supported beam system with load moving speed v = 32 ms (a) first beam(b) second beam (c) third beam (d) fourth beam

speed Relationship between maximum deflection and load-ing speed for mid-span of double simply supported beamwith a higher I1 value is shown in Figures 8-9 The resultsshow that the local oscillation of the first beamrsquos mid-spanmaximumdeflection reduced significantly when themomentof inertia I1 is revalued as 100 I1 and there are barely no localoscillations of the first beamrsquos mid-span maximum deflectionwhen the moment of inertia I1 is taken as 105 I1

Moreover the speeds atwhich themid-spandeflection forthe double beams reaches the peak value are the same withthe appearance of multiple points of resonance (wave crest)and points of cancellation (wave trough) as shown in Table 1

Figures 10 and 11 show the contrast ofmid-span deflectionresponses of the double simply supported rail-bridge beamsystem under different resonances and cancellation speeds

Table 1 and Figures 10-11 show that with the increasein resonance speeds the time-history curves of mid-spandeflection dynamic response of the first beam are consistentwhile the local oscillation of mid-span deflection time-history curves of the second beam significantly reduced thevibration period becomes longer and the mid-span dynamic

response reaches the amplitude when the moving loads actnear the mid-span of double rail-bridge system With theincrease in cancellation speeds the time-history curves ofmid-span deflection dynamic response of the first beamare consistent and when the mid-span deflection dynamicresponse of the second beam reaches the amplitude thedistance between the moving-load position and mid-spanincreases When the load moving speed is 881 ms the mid-spandynamic deflection response of the second beam reachesthe amplitude while the moving loads are on the 13 and 23positions of the beam

33 Analysis of Parameter Influence of Double Simply Sup-ported Rail-Bridge System Based on the analysis in theabove section further study was conducted regarding theeffect of interlayer spring stiffness beam masses and flexuralrigidities on the dynamic responses of the double simplysupported rail-bridge system in this section The movingspeed of loads v = 100 ms was taken as the example to studythe effect of variation of three parameters

8 Shock and Vibration

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

01 02 03 04 0500t (s)

(a)

ANSYSMATLAB

01 02 03 04 0500t (s)

minus028

minus021

minus014

minus007

000

007

Defl

ectio

n (m

m)

(b)

ANSYSMATLAB

minus024

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

(c)

ANSYSMATLAB

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

(d)

Figure 6 Mid-span deflection time-history for quadruple simply supported beam system with load moving speed v = 64 ms (a) first beam(b) second beam (c) third beam (d) fourth beam

First Beam

40 80 120 160 2000v (ms)

096

100

104

108

112

116

120

Defl

ectio

n (m

m)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 7 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam (a) first beam (b)second beam

Shock and Vibration 9

First Beam038

040

042

044

046

048

050

052D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 8 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 100I1 (a) first beam (b) second beam

First Beam

40 80 120 160 2000v (ms)

0060

0063

0066

0069

0072

0075

0078

Defl

ectio

n (m

m)

(a)

Second Beam0040

0044

0048

0052

0056

0060D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(b)

Figure 9 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 105I1 (a) first beam (b) second beam

Table 1 Relationship between resonance (cancellation) speed and amplitude

Speed at the wavecrest (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)

Speed at the wavetrough (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)189 1024 0161 204 1009 0159232 1025 0162 253 1015 0160299 1027 0165 334 1011 0162411 1039 0169 487 1004 0163687 1058 0180 881 0996 0164

Figure 12 show that interlayer stiffness variation has agreater effect on the dynamic response of the first sim-ply supported beam and the mid-span dynamic deflectionamplitude of the first simply supported beam decreasedwith the increase in interlayer stiffness while the interlayer

stiffness has a negligible effect on the mid-span dynamictime-history of the second simply supported beam Thereason is that in the double simply supported rail-bridgesystem the differences in the stiffness of the first and secondbeams are large the flexural rigidity of the first beam is

10 Shock and Vibration

v=189msv=232msv=299ms

v=411msv=687ms

4 8 12 16 20 24 28 320Load position (m)

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

(a)

v=189msv=232msv=299ms

v=411msv=687ms

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(b)

Figure 10 Mid-span deflection time-history for double simply supported rail-bridge system under resonance condition (a) first beam (b)second beam

v=204msv=253msv=334ms

v=487msv=881ms

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(a)

0 4 8 12 16 20 24 28 32

v=204msv=253msv=334ms

v=487msv=881ms

Load position (m)

minus020

minus016

minus012

minus008

minus004

000

Defl

ectio

n (m

m)

(b)

Figure 11 Mid-span deflection time-history for the double simply supported rail-bridge system under cancellation condition (a) first beam(b) second beam

much smaller than that of the second beam and the bendingresistance of the second beam plays a leading role in thebending resistance of the entire system

Figures 13 and 14 show the effects of mass variationper unit length of the first and second beams on mid-span deflection time-history for double simply supportedrail-bridge system respectively Figure 13 shows that underdifferent masses of the first beam the time-history curvesof mid-span deflection dynamic response for the two simplysupported beams are consistent Therefore the effect of massvariation of the first simply supported beam on the mid-spandeflection dynamic response of the double-beam system canbe neglected Figure 14 shows that the mass variation of thesecond simply supported beam has a significant effect on the

mid-span deflection dynamic response of both beams whenthemass per unit length of the second simply supported beamincreases to four times of the original mass the amplitudes ofmid-span deflection dynamic response of the first and secondsimply supported beams both increase significantly with thegrowth rate reaching up to 10 and 45 respectively

Figures 15 and 16 show the effect of flexural rigidityvariations of the first and second simply supported beamson the mid-span deflection time-history for double simplysupported rail-bridge system Figure 15 shows that the mid-span dynamic deflection amplitude of the first simply sup-ported beam significantly decreased with the increase in thestiffness of this beam while the stiffness of the first beamhas a smaller effect on the mid-span dynamic deflection

Shock and Vibration 11

025kr050kr100kr

200kr400kr

minus28

minus24

minus20

minus16

minus12

minus08

minus04

00

04

Defl

ectio

n (m

m)

008 016 024 032000t (s)

(a)

025kr050kr100kr

200kr400kr

008 016 024 032000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 12 Comparison diagram of mid-span deflection time-history for the double simply supported rail-bridge system under differentinterlayer stiffness (a) first beam (b) second beam

100

200400

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025Ｇ1050Ｇ1

Ｇ1

Ｇ1Ｇ1

(a)

025050100

200400

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

Ｇ1Ｇ1

Ｇ1Ｇ1

Ｇ1

(b)

Figure 13 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof first beam (a) first beam (b) second beam

amplitude of the second simply supported beam Underdifferent stiffness of the first beam the mid-span deflectiondynamic time-history curves for the second simply supportedbeam are consistent Figure 16 shows that the flexural rigidityvariations of the second simply supported beam have a sig-nificant effect on the mid-span deflection dynamic responseof double simply supported rail-bridge system In additionthe mid-span deflection dynamic response amplitudes ofboth beams significantly increased with the decrease in theflexural stiffness of second beam In practical applicationsthe dynamic response of the rail-bridge system in high-speedrailways can be decreased by moderately decreasing the massof the second beam and increasing the flexural stiffness ofboth simply supported beams

4 Conclusions

In this study a dynamic analysis model was established for amultiple simply supported beam system subjected to movingloads in combination of finite sin-Fourier transform andnumerical Laplace transform based on Durbin transforman analytical calculation method was developed for thespatial displacement time-domain response of a multiplesimply supported beam systemThe following conclusions aredrawn by comparing the examples and ANSYS finite elementnumerical computation method(1) A simply supported rail-bridge system of high-speedrailways with a span of 32 m in China was taken as theexample and the theoretically calculated results were com-pared with the results obtained from ANSYS finite element

12 Shock and Vibration

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

100

200400

025Ｇ2050Ｇ2

Ｇ2

Ｇ2Ｇ2

(a)

minus030

minus024

minus018

minus012

minus006

000

006

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025050100

200400Ｇ2

Ｇ2

Ｇ2Ｇ2

Ｇ2

(b)

Figure 14 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof second beam (a) first beam (b) second beam

008 016 024 032000t (s)

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02510501

10012001

(a)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

02510501

10012001

(b)

Figure 15 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof first beam (a) first beam (b) second beam

numerical calculation validating the analytical calculationmethod reported in this paper(2) The effect of load moving speed on the dynamicresponse of a double simply supported rail-bridge system ofhigh-speed railways subjected tomoving loads was evaluatedThe results indicate that the double simply supported rail-bridge system has multiple points of resonance and cancel-lation which should be paid attention in engineering(3) For a typical double simply supported rail-bridgesystem of high-speed railway in China because of a largedifference in the flexural rigidity of the first and secondbeams the variation in interlayer stiffness mainly affectsthe dynamic response of the first simply supported beamexhibiting a positive correlation When the mass and flexuralstiffness parameters of first simply supported beam vary they

mainly affect the dynamic response of this layer of simplysupported beam but slightly affect the dynamic responseof second simply supported beam The variation in theparameters of second simply supported beam has a greatereffect on the double simply supported beam system and adecrease in the mass of this layer of simply supported beamand an increase in the flexural rigidity of this beam canboth decrease the dynamic response of the double simplysupported rail-bridge system

Data Availability

The data used to support the findings of this study areincluded within the article

Shock and Vibration 13

minus14

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

05021002

20024002

(a)

008 016 024 032000t (s)

minus048

minus040

minus032

minus024

minus016

minus008

000

008

Defl

ectio

n (m

m)

05021002

20024002

(b)

Figure 16 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof second beam (a) first beam (b) second beam

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51708630)

References

[1] H Ouyang ldquoMoving-load dynamic problems a tutorial (with abrief overview)rdquoMechanical Systems and Signal Processing vol25 no 6 pp 2039ndash2060 2011

[2] D G Duffy ldquoThe response of an infinite railroad track to amoving vibrating massrdquo Journal of Applied Mechanics vol 57no 1 pp 66ndash73 1990

[3] L Fryba Vibration of solids and structures under moving loadsSpringer Science amp Business Media 2013

[4] G Michaltsos D Sophianopoulos and A N Kounadis ldquoTheeffect of a moving mass and other parameters on the dynamicresponse of a simply supported beamrdquo Journal of Sound andVibration vol 191 no 3 pp 357ndash362 1996

[5] P Museros E Moliner and M D Martınez-Rodrigo ldquoFreevibrations of simply-supported beam bridges under movingloads Maximum resonance cancellation and resonant verticalaccelerationrdquo Journal of Sound and Vibration vol 332 no 2 pp326ndash345 2013

[6] P Sniady ldquoDynamic response of a Timoshenko beam to amoving forcerdquo Journal of Applied Mechanics vol 75 no 2Article ID 024503 2008

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

[8] K Y Lee and A A Renshaw ldquoSolution of the moving massproblem using complex eigenfunction expansionsrdquo Journal ofApplied Mechanics vol 67 no 4 pp 823ndash827 2000

[9] U Lee ldquoSeparation between the flexible structure and themoving mass sliding on itrdquo Journal of Sound and Vibration vol209 no 5 pp 867ndash877 1998

[10] S A Eftekhari and A A Jafari ldquoCoupling ritz method andtriangular quadrature rule for moving mass problemrdquo Journalof Applied Mechanics vol 79 no 2 2012

[11] M Dublin andH R Friedrich ldquoForced responses of two elasticbeams interconnected by spring-damper systemsrdquo Journal of theAeronautical Sciences vol 23 pp 824ndash829 1956

[12] J M Seelig and W H Hoppmann ldquoImpact on an elasticallyconnected double-beam systemrdquo Journal of Applied Mechanicsvol 31 pp 621ndash626 1964

[13] P G Kessel ldquoResonances Excited in an Elastically ConnectedDouble-Beam System by a Cyclic Moving LoadrdquoThe Journal ofthe Acoustical Society of America vol 40 no 3 pp 684ndash6871966

[14] S Chonan ldquoDynamical behaviours of elastically connecteddouble-beam systems subjected to an impulsive loadrdquo Bulletinof JSME vol 19 no 132 pp 595ndash603 1976

[15] H V Vu A M Ordonez and B H Karnopp ldquoVibration of adouble-beam systemrdquo Journal of Sound and Vibration vol 229no 4 pp 807ndash822 2000

[16] H J Xu Z H Ding Z R Lu and J K Liu ldquoStructural DamageDetection Using a Modified Artificial Bee Colony AlgorithmrdquoCMES-Computer Modeling in Engineering amp Sciences vol 111pp 335ndash355 2016

[17] Z Oniszczuk ldquoFree transverse vibrations of elastically con-nected simply supported double-beam complex systemrdquo Jour-nal of Sound and Vibration vol 232 no 2 pp 387ndash403 2000

[18] Z Oniszczuk ldquoForced transverse vibrations of an elasticallyconnected complex simply supported double-beam systemrdquoJournal of Sound and Vibration vol 264 no 2 pp 273ndash2862003

[19] Y Q Zhang Y Lu S L Wang and X Liu ldquoVibration andbuckling of a double-beam system under compressive axialloadingrdquo Journal of Sound and Vibration vol 318 no 1-2 pp341ndash352 2008

[20] V Stojanovic and P Kozic ldquoForced transverse vibration ofRayleigh and Timoshenko double-beam system with effect of

14 Shock and Vibration

compressive axial loadrdquo International Journal of MechanicalSciences vol 60 no 1 pp 59ndash71 2012

[21] A Palmeri and S Adhikari ldquoA Galerkin-type state-spaceapproach for transverse vibrations of slender double-beamsystems with viscoelastic inner layerrdquo Journal of Sound andVibration vol 330 no 26 pp 6372ndash6386 2011

[22] Z Zhang X Huang Z Zhang and H Hua ldquoOn the transversevibration of Timoshenko double-beam systems coupled withvarious discontinuitiesrdquo International Journal of MechanicalSciences vol 89 pp 222ndash241 2014

[23] Y X Li Z J Hu and L Z Sun ldquoDynamical behavior of adouble-beam system interconnected by a viscoelastic layerrdquoInternational Journal of Mechanical Sciences vol 105 pp 291ndash303 2016

[24] M Abu-Hilal ldquoDynamic response of a double EulerndashBernoullibeam due to a moving constant loadrdquo Journal of Sound andVibration vol 297 no 3-5 pp 477ndash491 2006

[25] Y Wu and Y Gao ldquoAnalytical Solutions for Simply SupportedViscously Damped Double-Beam System under Moving Har-monic Loadsrdquo Journal of Engineering Mechanics vol 141 no 7p 04015004 2015

[26] Y-H Chen and Z-M Shiu ldquoResonant curves of an elevatedrailway to harmonic moving loadsrdquo International Journal ofStructural Stability and Dynamics vol 4 no 2 pp 237ndash2572004

[27] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of aninfinite tomoshenko beam on a viscoelastic foundation to aharmonic moving loadrdquo Journal of Sound and Vibration vol241 no 5 pp 809ndash824 2001

[28] M FMHussein andH EMHunt ldquoModelling of floating-slabtracks with continuous slabs under oscillating moving loadsrdquoJournal of Sound andVibration vol 297 no 1-2 pp 37ndash54 2006

[29] H P Wang J Li and K Zhang ldquoVibration analysis of themaglev guideway with the moving loadrdquo Journal of Sound andVibration vol 305 no 4-5 pp 621ndash640 2007

[30] V Stojanovic and M Petkovic ldquoMoment Lyapunov exponentsand stochastic stability of a three-dimensional system on elasticfoundation using a perturbation approachrdquo Journal of AppliedMechanics vol 80 no 5 2013

[31] L Jun and H Hongxing ldquoDynamic stiffness vibration analysisof an elastically connected three-beam systemrdquo Applied Acous-tics vol 69 no 7 pp 591ndash600 2008

[32] J Li Y Chen and H Hua ldquoExact dynamic stiffness matrixof a Timoshenko three-beam systemrdquo International Journal ofMechanical Sciences vol 50 no 6 pp 1023ndash1034 2008

[33] S S Rao ldquoNatural vibrations of systems of elastically connectedTimoshenko beams rdquo The Journal of the Acoustical Society ofAmerica vol 55 no 6 pp 1232ndash1237 1974

[34] Q Mao ldquoFree vibration analysis of elastically connectedmultiple-beams by using the Adomianmodified decompositionmethodrdquo Journal of Sound and Vibration vol 331 no 11 pp2532ndash2542 2012

[35] V Stojanovic P Kozic and G Janevski ldquoExact closed-formsolutions for the natural frequencies and stability of elasticallyconnected multiple beam system using Timoshenko and high-order shear deformation theoryrdquo Journal of Sound and Vibra-tion vol 332 no 3 pp 563ndash576 2013

[36] M H Kargarnovin M T Ahmadian and R-A Jafari-Talookolaei ldquoAnalytical solution for the dynamic analysis of adelaminated composite beam traversed by a moving constantforcerdquo Journal of Vibration and Control vol 19 no 10 pp 1524ndash1537 2013

[37] A Ariaei S Ziaei-Rad and M Ghayour ldquoTransverse vibrationof a multiple-Timoshenko beam system with intermediateelastic connections due to a moving loadrdquo Archive of AppliedMechanics vol 81 no 3 pp 263ndash281 2011

[38] Y Povstenko and J Klekot ldquoThe Dirichlet problem for thetime-fractional advection-diffusion equation in a line segmentrdquoBoundary Value Problems Paper No 89 8 pages 2016

[39] F Durbin ldquoNumerical inversion of Laplace transforms anefficient improvement to Dubner and Abatersquos methodrdquo TheComputer Journal vol 17 pp 371ndash376 1974

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4 Shock and Vibration

Laplace transformation with respect to time 119905 was con-ducted on the left and right sides of (10)ndash(12) and thefollowing equations can be obtained after reorganizing andcombining similar terms

(11989811199042 + 1198961 + 119864111986811205854119896) 1lowast (120585119896 119904) minus 11989612lowast (120585119896 119904)= 1198621 (120585119896 119904)

(16)

minus 119896119903minus1119903minus1lowast (120585119896 119904)+ [1198981199031199042 + 1198641199031198681199031205854119896 + 119896119903minus1 + 119896119903] 119903lowast (120585119896 119904)minus 119896119903119903+1lowast (120585119896 119904) = 119862119903 (120585119896 119904)

(17)

minus 119896119873minus1119873minus1lowast (120585119896 119904)+ (1198981198731199042 + 1198641198731198681198731205854119896 + 119896119873minus1) 119873lowast (120585119896 119904)= 119862119873 (120585119896 119904)

(18)

where C119894(120585119896 119904) = 119871[119875119894(119905) sin(120585119896V119905)] is the external load offrequency domain on the 119894th simply supported beam Thevector quantities for the frequency-domain responses andexternal loads of frequency domain on each simply supportedbeam can be denoted as follows

ulowast (120585119896 119904)= lowast1 (120585119896 119904) lowast119903 (120585119896 119904) lowast119873 (120585119896 119904)T

(19)

C (120585119896 119904)= 1198621 (120585119896 119904) 119862119903 (120585119896 119904) 119862119873 (120585119896 119904)T

(20)

Therefore (16)ndash(18) can be expressed as follows

[D] ulowast (120585119896 119904) = C (120585119896 119904) (21)

where D is the matrix with respect to 119873 times 119873 and only thefollowing elements are nonzero

11986311 = 11989811199042 + 119864111986811205854119896 + 1198961119863119903minus1119903 = 119863119903119903minus1 (22)

119863119903119903 = 1198981199031199042 + 1198641199031198681199031205854119896 + 119896119903minus1 + 119896119903119863119903119903minus1 = minus119896119903minus1119863119903119903+1 = minus119896119903

(23)

119863119873119873 = 1198981198731199042 + 1198641198731198681198731205854119896 + 119896119873minus1119863119903+1119903 = 119863119903119903+1 (24)

The frequency-domain responses ulowast(120585119896 119904) for eachbeam in the multiple simply supported beam system canbe obtained by solving the linear algebraic equation (21)Finite sin-Fourier inverse transformation was conducted onthe frequency-domain responses ulowast(120585119896 119904) for the multiplesimply supported beam system and the spatial displacementresponses for the multiple simply supported beam system

with respect to the time-frequency domains can be obtainedas follows

119906119894 (119909 119904) =119872sum119896=1

119894lowast (120585119896 119904) sin (120585119896119909) 119889119909119894 = 1 r 119873

(25)

Laplace inverse transform (LIT) with respect to time-frequency domain s was conducted on the left and rightsides of (25) and the spatial displacement time-domainresponses for the multiple simply supported beam system canbe obtained as follows

119906119894 (119909 119905) = 119871minus1 [119906119894 (119909 119904)] 119894 = 1 r 119873 (26)

The solving process in this paper shows that the dis-placement frequency-domain response function 119906119894(119909 119904) foreach beam contains the product of power function and sinefunction However with the increase in the number of layersof simply supported beam the analytical expression of (26)becomes complex and lengthy and it is not easy to conductan analytical expression on it Therefore to effectively solvethe LIT for (26) according to arbitrary numbers of beamsfast LIT (FLIT) [39] proposed by Durbin was introduced inthis paper to obtain the solution This method of numericalinverse transform can be used to solve the transform resultsrelatively accurately It was verified that the error boundaryof this method can be set to be any arbitrarily small valueso that better computing results can be obtained even forcomplex expressions The corresponding solving programsare compiled in MATLAB to solve (21)ndash(26) and the spa-tial displacement time-domain response of different simplysupported beam systems with arbitrary numbers of beamssubjected to moving loads can be obtained

3 Analysis of Numerical Examples

31 Validation of the Analytical Model In the first placeas an application example of a double simply supportedbeam system subjected to moving loads a typical rail-bridgesystem of high-speed railway system in China was selectedfor analysis In this system the first simply supported beamis a rail of ballastless tracks and the second simply supportedbeam is a typical simply supported bridge with a span of 32mThemoving load acts on the first simply supported beam andthe materials of the double simply supported beam systemand geometric parameters are shown as follows

m1 = 60KgmE1 = 21 times 105MpaI1 = 3217 times 10minus5m4m2 = 183 times 103KgmE2 = 345 times 104Mpa

Shock and Vibration 5

(a) (b)

Figure 2 Deflections of double simply supported rail-bridge system with different positions of the moving load (a) first beam (b) secondbeam

I2 = 1095m4L1 = L2 = 32mk1 = 60MNm2p1 = 85KN

(27)

Based on the solving programs compiled inMATLAB thevertical deflection of rail and bridge with different positionsof themoving load in the double simply supported rail-bridgesystem when the moving speed of external load v = 100 mswere plotted as shown in Figure 2

The results in the diagram show that the dynamic verticaldeflection values for the rail and bridge in the double simplysupported system subjected to moving loads reached themaximum near the mid-span (middle point of the system)The displacement time-history responses of the mid-span offirst and second simply supported beams effectively reflectthe maximum displacement responses of the double simplysupported beam system of rail-bridge in a high-speed railwaysystem Hence for the following computational analysisthe mid-span displacement responses for the double simplysupported beam system were selected for analysis

To validate the theoretical solution in this paper ANSYSfinite element software was used to conduct numericalsimulation on the dynamic response of the double simplysupported rail-bridge system subjected to moving loads andthe time-history responses for the mid-span displacements ofthe double simply supported rail-bridge system subjected toexternal load with different moving speeds (32 ms and 64ms) were calculated Regarding the FE model the springswere modeled by Combin14 element in ANSYS and thespacing between the Combin14 elements was set as 01mwhich is a relatively small spacing to model the continuoussupport between both the beams Then the results werecompared with the theoretically calculated results and thecomparison results are shown in Figures 3 and 4 The resultsshow that the theoretically calculated values are consistentwith the results obtained from the ANSYS finite elementnumerical calculation

Next as another validation of the theoretical solution inthis paper the dynamic responses of a quadruple beam sys-tem subjected to moving loads were calculated by theoreticalanalysis model and compared with the calculation results ofANSYS finite element software The material and geometricparameters of the quadruple simply supported beam systemare as follows

m1 = 60KgmE1 = 21 times 105MpaI1 = 3217 times 10minus5m4m2 = 1275KgmE2 = 355 times 104MpaI2 = 17 times 10minus3m4m3 = 140125KgmE3 = 30 times 104MpaI3 = 1686 times 10minus3m4m4 = 18300KgmE4 = 345 times 104MpaI4 = 1095m4L1 = L2 = L3 = L4 = 32mk1 = 60MNm2k2 = 900MNm2k3 = 1375MNm2p1 = 85KN

(28)

Similar to the double-beam systemrsquos analysis the time-history responses for the mid-span displacements of the

6 Shock and Vibration

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

02 04 06 08 1000t (s)

(a)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 3Mid-span deflection time-history for double simply supported rail-bridge systemwith loadmoving speed v = 32ms (a) first beam(b) second beam

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

ANSYSMATLAB

(a)

minus020

minus016

minus012

minus008

minus004

000

004D

eflec

tion

(mm

)

01 02 03 04 0500t (s)

ANSYSMATLAB

(b)

Figure 4 Mid-span deflection time-history for double simply supported rail-bridge system with load moving speed v = 64 ms (a) firstbeam (b) second beam

quadruple simply supported beam system subjected to exter-nal load with different moving speeds (32 ms and 64 ms)were calculated The comparisons between the theoreticallycalculated results and the ANSYS finite element numericalcalculation are shown in Figures 5 and 6 The results alsoshow that the theoretical results are consistent with theresults obtained from the ANSYS finite element numericalcalculation thus validating and rationalizing this theoreticalanalysis model in an effective manner

32 Analysis of Maximum Deflection-Loading Speed for Mid-Span of Double Simply Supported Beam Figure 7 shows themaximum dynamic deflection for the mid-span of doublesimply supported rail-bridge system at different load movingspeeds With the increase in load moving speeds the effectof speed on the dynamic response of the double simply

supported rail-bridge beam system shows a nonlinear rela-tionship a sine curve was observed instead of increasingwith the increase of speed linearly The maximum mid-span deflection for the first simply supported beam showsthe variation of sine curve around the amplitude of 102mm Besides with the increase of speed the period andamplitude of sine curve also increased The maximum mid-span deflection for the second simply supported beam alsoshows the variation of sine curve but unlike the first beamthe mid-span deflection for this beam shows an overallincreasing trend with the increase of speeds

Due to the very low moment of inertia of the firstbeam compared to the second beam the first beam showshierarchical oscillations while the secondary beam does notshow any local oscillations and the mid-span deflectiondynamic response of the first beam is insensitive to the

Shock and Vibration 7

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02 04 06 08 1000t (s)

(a)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus028

minus021

minus014

minus007

000

007

Defl

ectio

n (m

m)

(b)

ANSYSMATLAB

minus024

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

02 04 06 08 1000t (s)

(c)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(d)

Figure 5 Mid-span deflection time-history for quadruple simply supported beam system with load moving speed v = 32 ms (a) first beam(b) second beam (c) third beam (d) fourth beam

speed Relationship between maximum deflection and load-ing speed for mid-span of double simply supported beamwith a higher I1 value is shown in Figures 8-9 The resultsshow that the local oscillation of the first beamrsquos mid-spanmaximumdeflection reduced significantly when themomentof inertia I1 is revalued as 100 I1 and there are barely no localoscillations of the first beamrsquos mid-span maximum deflectionwhen the moment of inertia I1 is taken as 105 I1

Moreover the speeds atwhich themid-spandeflection forthe double beams reaches the peak value are the same withthe appearance of multiple points of resonance (wave crest)and points of cancellation (wave trough) as shown in Table 1

Figures 10 and 11 show the contrast ofmid-span deflectionresponses of the double simply supported rail-bridge beamsystem under different resonances and cancellation speeds

Table 1 and Figures 10-11 show that with the increasein resonance speeds the time-history curves of mid-spandeflection dynamic response of the first beam are consistentwhile the local oscillation of mid-span deflection time-history curves of the second beam significantly reduced thevibration period becomes longer and the mid-span dynamic

response reaches the amplitude when the moving loads actnear the mid-span of double rail-bridge system With theincrease in cancellation speeds the time-history curves ofmid-span deflection dynamic response of the first beamare consistent and when the mid-span deflection dynamicresponse of the second beam reaches the amplitude thedistance between the moving-load position and mid-spanincreases When the load moving speed is 881 ms the mid-spandynamic deflection response of the second beam reachesthe amplitude while the moving loads are on the 13 and 23positions of the beam

33 Analysis of Parameter Influence of Double Simply Sup-ported Rail-Bridge System Based on the analysis in theabove section further study was conducted regarding theeffect of interlayer spring stiffness beam masses and flexuralrigidities on the dynamic responses of the double simplysupported rail-bridge system in this section The movingspeed of loads v = 100 ms was taken as the example to studythe effect of variation of three parameters

8 Shock and Vibration

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

01 02 03 04 0500t (s)

(a)

ANSYSMATLAB

01 02 03 04 0500t (s)

minus028

minus021

minus014

minus007

000

007

Defl

ectio

n (m

m)

(b)

ANSYSMATLAB

minus024

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

(c)

ANSYSMATLAB

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

(d)

Figure 6 Mid-span deflection time-history for quadruple simply supported beam system with load moving speed v = 64 ms (a) first beam(b) second beam (c) third beam (d) fourth beam

First Beam

40 80 120 160 2000v (ms)

096

100

104

108

112

116

120

Defl

ectio

n (m

m)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 7 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam (a) first beam (b)second beam

Shock and Vibration 9

First Beam038

040

042

044

046

048

050

052D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 8 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 100I1 (a) first beam (b) second beam

First Beam

40 80 120 160 2000v (ms)

0060

0063

0066

0069

0072

0075

0078

Defl

ectio

n (m

m)

(a)

Second Beam0040

0044

0048

0052

0056

0060D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(b)

Figure 9 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 105I1 (a) first beam (b) second beam

Table 1 Relationship between resonance (cancellation) speed and amplitude

Speed at the wavecrest (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)

Speed at the wavetrough (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)189 1024 0161 204 1009 0159232 1025 0162 253 1015 0160299 1027 0165 334 1011 0162411 1039 0169 487 1004 0163687 1058 0180 881 0996 0164

Figure 12 show that interlayer stiffness variation has agreater effect on the dynamic response of the first sim-ply supported beam and the mid-span dynamic deflectionamplitude of the first simply supported beam decreasedwith the increase in interlayer stiffness while the interlayer

stiffness has a negligible effect on the mid-span dynamictime-history of the second simply supported beam Thereason is that in the double simply supported rail-bridgesystem the differences in the stiffness of the first and secondbeams are large the flexural rigidity of the first beam is

10 Shock and Vibration

v=189msv=232msv=299ms

v=411msv=687ms

4 8 12 16 20 24 28 320Load position (m)

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

(a)

v=189msv=232msv=299ms

v=411msv=687ms

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(b)

Figure 10 Mid-span deflection time-history for double simply supported rail-bridge system under resonance condition (a) first beam (b)second beam

v=204msv=253msv=334ms

v=487msv=881ms

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(a)

0 4 8 12 16 20 24 28 32

v=204msv=253msv=334ms

v=487msv=881ms

Load position (m)

minus020

minus016

minus012

minus008

minus004

000

Defl

ectio

n (m

m)

(b)

Figure 11 Mid-span deflection time-history for the double simply supported rail-bridge system under cancellation condition (a) first beam(b) second beam

much smaller than that of the second beam and the bendingresistance of the second beam plays a leading role in thebending resistance of the entire system

Figures 13 and 14 show the effects of mass variationper unit length of the first and second beams on mid-span deflection time-history for double simply supportedrail-bridge system respectively Figure 13 shows that underdifferent masses of the first beam the time-history curvesof mid-span deflection dynamic response for the two simplysupported beams are consistent Therefore the effect of massvariation of the first simply supported beam on the mid-spandeflection dynamic response of the double-beam system canbe neglected Figure 14 shows that the mass variation of thesecond simply supported beam has a significant effect on the

mid-span deflection dynamic response of both beams whenthemass per unit length of the second simply supported beamincreases to four times of the original mass the amplitudes ofmid-span deflection dynamic response of the first and secondsimply supported beams both increase significantly with thegrowth rate reaching up to 10 and 45 respectively

Figures 15 and 16 show the effect of flexural rigidityvariations of the first and second simply supported beamson the mid-span deflection time-history for double simplysupported rail-bridge system Figure 15 shows that the mid-span dynamic deflection amplitude of the first simply sup-ported beam significantly decreased with the increase in thestiffness of this beam while the stiffness of the first beamhas a smaller effect on the mid-span dynamic deflection

Shock and Vibration 11

025kr050kr100kr

200kr400kr

minus28

minus24

minus20

minus16

minus12

minus08

minus04

00

04

Defl

ectio

n (m

m)

008 016 024 032000t (s)

(a)

025kr050kr100kr

200kr400kr

008 016 024 032000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 12 Comparison diagram of mid-span deflection time-history for the double simply supported rail-bridge system under differentinterlayer stiffness (a) first beam (b) second beam

100

200400

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025Ｇ1050Ｇ1

Ｇ1

Ｇ1Ｇ1

(a)

025050100

200400

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

Ｇ1Ｇ1

Ｇ1Ｇ1

Ｇ1

(b)

Figure 13 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof first beam (a) first beam (b) second beam

amplitude of the second simply supported beam Underdifferent stiffness of the first beam the mid-span deflectiondynamic time-history curves for the second simply supportedbeam are consistent Figure 16 shows that the flexural rigidityvariations of the second simply supported beam have a sig-nificant effect on the mid-span deflection dynamic responseof double simply supported rail-bridge system In additionthe mid-span deflection dynamic response amplitudes ofboth beams significantly increased with the decrease in theflexural stiffness of second beam In practical applicationsthe dynamic response of the rail-bridge system in high-speedrailways can be decreased by moderately decreasing the massof the second beam and increasing the flexural stiffness ofboth simply supported beams

4 Conclusions

In this study a dynamic analysis model was established for amultiple simply supported beam system subjected to movingloads in combination of finite sin-Fourier transform andnumerical Laplace transform based on Durbin transforman analytical calculation method was developed for thespatial displacement time-domain response of a multiplesimply supported beam systemThe following conclusions aredrawn by comparing the examples and ANSYS finite elementnumerical computation method(1) A simply supported rail-bridge system of high-speedrailways with a span of 32 m in China was taken as theexample and the theoretically calculated results were com-pared with the results obtained from ANSYS finite element

12 Shock and Vibration

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

100

200400

025Ｇ2050Ｇ2

Ｇ2

Ｇ2Ｇ2

(a)

minus030

minus024

minus018

minus012

minus006

000

006

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025050100

200400Ｇ2

Ｇ2

Ｇ2Ｇ2

Ｇ2

(b)

Figure 14 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof second beam (a) first beam (b) second beam

008 016 024 032000t (s)

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02510501

10012001

(a)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

02510501

10012001

(b)

Figure 15 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof first beam (a) first beam (b) second beam

numerical calculation validating the analytical calculationmethod reported in this paper(2) The effect of load moving speed on the dynamicresponse of a double simply supported rail-bridge system ofhigh-speed railways subjected tomoving loads was evaluatedThe results indicate that the double simply supported rail-bridge system has multiple points of resonance and cancel-lation which should be paid attention in engineering(3) For a typical double simply supported rail-bridgesystem of high-speed railway in China because of a largedifference in the flexural rigidity of the first and secondbeams the variation in interlayer stiffness mainly affectsthe dynamic response of the first simply supported beamexhibiting a positive correlation When the mass and flexuralstiffness parameters of first simply supported beam vary they

mainly affect the dynamic response of this layer of simplysupported beam but slightly affect the dynamic responseof second simply supported beam The variation in theparameters of second simply supported beam has a greatereffect on the double simply supported beam system and adecrease in the mass of this layer of simply supported beamand an increase in the flexural rigidity of this beam canboth decrease the dynamic response of the double simplysupported rail-bridge system

Data Availability

The data used to support the findings of this study areincluded within the article

Shock and Vibration 13

minus14

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

05021002

20024002

(a)

008 016 024 032000t (s)

minus048

minus040

minus032

minus024

minus016

minus008

000

008

Defl

ectio

n (m

m)

05021002

20024002

(b)

Figure 16 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof second beam (a) first beam (b) second beam

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51708630)

References

[1] H Ouyang ldquoMoving-load dynamic problems a tutorial (with abrief overview)rdquoMechanical Systems and Signal Processing vol25 no 6 pp 2039ndash2060 2011

[2] D G Duffy ldquoThe response of an infinite railroad track to amoving vibrating massrdquo Journal of Applied Mechanics vol 57no 1 pp 66ndash73 1990

[3] L Fryba Vibration of solids and structures under moving loadsSpringer Science amp Business Media 2013

[4] G Michaltsos D Sophianopoulos and A N Kounadis ldquoTheeffect of a moving mass and other parameters on the dynamicresponse of a simply supported beamrdquo Journal of Sound andVibration vol 191 no 3 pp 357ndash362 1996

[5] P Museros E Moliner and M D Martınez-Rodrigo ldquoFreevibrations of simply-supported beam bridges under movingloads Maximum resonance cancellation and resonant verticalaccelerationrdquo Journal of Sound and Vibration vol 332 no 2 pp326ndash345 2013

[6] P Sniady ldquoDynamic response of a Timoshenko beam to amoving forcerdquo Journal of Applied Mechanics vol 75 no 2Article ID 024503 2008

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

[8] K Y Lee and A A Renshaw ldquoSolution of the moving massproblem using complex eigenfunction expansionsrdquo Journal ofApplied Mechanics vol 67 no 4 pp 823ndash827 2000

[9] U Lee ldquoSeparation between the flexible structure and themoving mass sliding on itrdquo Journal of Sound and Vibration vol209 no 5 pp 867ndash877 1998

[10] S A Eftekhari and A A Jafari ldquoCoupling ritz method andtriangular quadrature rule for moving mass problemrdquo Journalof Applied Mechanics vol 79 no 2 2012

[11] M Dublin andH R Friedrich ldquoForced responses of two elasticbeams interconnected by spring-damper systemsrdquo Journal of theAeronautical Sciences vol 23 pp 824ndash829 1956

[12] J M Seelig and W H Hoppmann ldquoImpact on an elasticallyconnected double-beam systemrdquo Journal of Applied Mechanicsvol 31 pp 621ndash626 1964

[13] P G Kessel ldquoResonances Excited in an Elastically ConnectedDouble-Beam System by a Cyclic Moving LoadrdquoThe Journal ofthe Acoustical Society of America vol 40 no 3 pp 684ndash6871966

[14] S Chonan ldquoDynamical behaviours of elastically connecteddouble-beam systems subjected to an impulsive loadrdquo Bulletinof JSME vol 19 no 132 pp 595ndash603 1976

[15] H V Vu A M Ordonez and B H Karnopp ldquoVibration of adouble-beam systemrdquo Journal of Sound and Vibration vol 229no 4 pp 807ndash822 2000

[16] H J Xu Z H Ding Z R Lu and J K Liu ldquoStructural DamageDetection Using a Modified Artificial Bee Colony AlgorithmrdquoCMES-Computer Modeling in Engineering amp Sciences vol 111pp 335ndash355 2016

[17] Z Oniszczuk ldquoFree transverse vibrations of elastically con-nected simply supported double-beam complex systemrdquo Jour-nal of Sound and Vibration vol 232 no 2 pp 387ndash403 2000

[18] Z Oniszczuk ldquoForced transverse vibrations of an elasticallyconnected complex simply supported double-beam systemrdquoJournal of Sound and Vibration vol 264 no 2 pp 273ndash2862003

[19] Y Q Zhang Y Lu S L Wang and X Liu ldquoVibration andbuckling of a double-beam system under compressive axialloadingrdquo Journal of Sound and Vibration vol 318 no 1-2 pp341ndash352 2008

[20] V Stojanovic and P Kozic ldquoForced transverse vibration ofRayleigh and Timoshenko double-beam system with effect of

14 Shock and Vibration

compressive axial loadrdquo International Journal of MechanicalSciences vol 60 no 1 pp 59ndash71 2012

[21] A Palmeri and S Adhikari ldquoA Galerkin-type state-spaceapproach for transverse vibrations of slender double-beamsystems with viscoelastic inner layerrdquo Journal of Sound andVibration vol 330 no 26 pp 6372ndash6386 2011

[22] Z Zhang X Huang Z Zhang and H Hua ldquoOn the transversevibration of Timoshenko double-beam systems coupled withvarious discontinuitiesrdquo International Journal of MechanicalSciences vol 89 pp 222ndash241 2014

[23] Y X Li Z J Hu and L Z Sun ldquoDynamical behavior of adouble-beam system interconnected by a viscoelastic layerrdquoInternational Journal of Mechanical Sciences vol 105 pp 291ndash303 2016

[24] M Abu-Hilal ldquoDynamic response of a double EulerndashBernoullibeam due to a moving constant loadrdquo Journal of Sound andVibration vol 297 no 3-5 pp 477ndash491 2006

[25] Y Wu and Y Gao ldquoAnalytical Solutions for Simply SupportedViscously Damped Double-Beam System under Moving Har-monic Loadsrdquo Journal of Engineering Mechanics vol 141 no 7p 04015004 2015

[26] Y-H Chen and Z-M Shiu ldquoResonant curves of an elevatedrailway to harmonic moving loadsrdquo International Journal ofStructural Stability and Dynamics vol 4 no 2 pp 237ndash2572004

[27] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of aninfinite tomoshenko beam on a viscoelastic foundation to aharmonic moving loadrdquo Journal of Sound and Vibration vol241 no 5 pp 809ndash824 2001

[28] M FMHussein andH EMHunt ldquoModelling of floating-slabtracks with continuous slabs under oscillating moving loadsrdquoJournal of Sound andVibration vol 297 no 1-2 pp 37ndash54 2006

[29] H P Wang J Li and K Zhang ldquoVibration analysis of themaglev guideway with the moving loadrdquo Journal of Sound andVibration vol 305 no 4-5 pp 621ndash640 2007

[30] V Stojanovic and M Petkovic ldquoMoment Lyapunov exponentsand stochastic stability of a three-dimensional system on elasticfoundation using a perturbation approachrdquo Journal of AppliedMechanics vol 80 no 5 2013

[31] L Jun and H Hongxing ldquoDynamic stiffness vibration analysisof an elastically connected three-beam systemrdquo Applied Acous-tics vol 69 no 7 pp 591ndash600 2008

[32] J Li Y Chen and H Hua ldquoExact dynamic stiffness matrixof a Timoshenko three-beam systemrdquo International Journal ofMechanical Sciences vol 50 no 6 pp 1023ndash1034 2008

[33] S S Rao ldquoNatural vibrations of systems of elastically connectedTimoshenko beams rdquo The Journal of the Acoustical Society ofAmerica vol 55 no 6 pp 1232ndash1237 1974

[34] Q Mao ldquoFree vibration analysis of elastically connectedmultiple-beams by using the Adomianmodified decompositionmethodrdquo Journal of Sound and Vibration vol 331 no 11 pp2532ndash2542 2012

[35] V Stojanovic P Kozic and G Janevski ldquoExact closed-formsolutions for the natural frequencies and stability of elasticallyconnected multiple beam system using Timoshenko and high-order shear deformation theoryrdquo Journal of Sound and Vibra-tion vol 332 no 3 pp 563ndash576 2013

[36] M H Kargarnovin M T Ahmadian and R-A Jafari-Talookolaei ldquoAnalytical solution for the dynamic analysis of adelaminated composite beam traversed by a moving constantforcerdquo Journal of Vibration and Control vol 19 no 10 pp 1524ndash1537 2013

[37] A Ariaei S Ziaei-Rad and M Ghayour ldquoTransverse vibrationof a multiple-Timoshenko beam system with intermediateelastic connections due to a moving loadrdquo Archive of AppliedMechanics vol 81 no 3 pp 263ndash281 2011

[38] Y Povstenko and J Klekot ldquoThe Dirichlet problem for thetime-fractional advection-diffusion equation in a line segmentrdquoBoundary Value Problems Paper No 89 8 pages 2016

[39] F Durbin ldquoNumerical inversion of Laplace transforms anefficient improvement to Dubner and Abatersquos methodrdquo TheComputer Journal vol 17 pp 371ndash376 1974

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Shock and Vibration 5

(a) (b)

Figure 2 Deflections of double simply supported rail-bridge system with different positions of the moving load (a) first beam (b) secondbeam

I2 = 1095m4L1 = L2 = 32mk1 = 60MNm2p1 = 85KN

(27)

Based on the solving programs compiled inMATLAB thevertical deflection of rail and bridge with different positionsof themoving load in the double simply supported rail-bridgesystem when the moving speed of external load v = 100 mswere plotted as shown in Figure 2

The results in the diagram show that the dynamic verticaldeflection values for the rail and bridge in the double simplysupported system subjected to moving loads reached themaximum near the mid-span (middle point of the system)The displacement time-history responses of the mid-span offirst and second simply supported beams effectively reflectthe maximum displacement responses of the double simplysupported beam system of rail-bridge in a high-speed railwaysystem Hence for the following computational analysisthe mid-span displacement responses for the double simplysupported beam system were selected for analysis

To validate the theoretical solution in this paper ANSYSfinite element software was used to conduct numericalsimulation on the dynamic response of the double simplysupported rail-bridge system subjected to moving loads andthe time-history responses for the mid-span displacements ofthe double simply supported rail-bridge system subjected toexternal load with different moving speeds (32 ms and 64ms) were calculated Regarding the FE model the springswere modeled by Combin14 element in ANSYS and thespacing between the Combin14 elements was set as 01mwhich is a relatively small spacing to model the continuoussupport between both the beams Then the results werecompared with the theoretically calculated results and thecomparison results are shown in Figures 3 and 4 The resultsshow that the theoretically calculated values are consistentwith the results obtained from the ANSYS finite elementnumerical calculation

Next as another validation of the theoretical solution inthis paper the dynamic responses of a quadruple beam sys-tem subjected to moving loads were calculated by theoreticalanalysis model and compared with the calculation results ofANSYS finite element software The material and geometricparameters of the quadruple simply supported beam systemare as follows

m1 = 60KgmE1 = 21 times 105MpaI1 = 3217 times 10minus5m4m2 = 1275KgmE2 = 355 times 104MpaI2 = 17 times 10minus3m4m3 = 140125KgmE3 = 30 times 104MpaI3 = 1686 times 10minus3m4m4 = 18300KgmE4 = 345 times 104MpaI4 = 1095m4L1 = L2 = L3 = L4 = 32mk1 = 60MNm2k2 = 900MNm2k3 = 1375MNm2p1 = 85KN

(28)

Similar to the double-beam systemrsquos analysis the time-history responses for the mid-span displacements of the

6 Shock and Vibration

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

02 04 06 08 1000t (s)

(a)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 3Mid-span deflection time-history for double simply supported rail-bridge systemwith loadmoving speed v = 32ms (a) first beam(b) second beam

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

ANSYSMATLAB

(a)

minus020

minus016

minus012

minus008

minus004

000

004D

eflec

tion

(mm

)

01 02 03 04 0500t (s)

ANSYSMATLAB

(b)

Figure 4 Mid-span deflection time-history for double simply supported rail-bridge system with load moving speed v = 64 ms (a) firstbeam (b) second beam

quadruple simply supported beam system subjected to exter-nal load with different moving speeds (32 ms and 64 ms)were calculated The comparisons between the theoreticallycalculated results and the ANSYS finite element numericalcalculation are shown in Figures 5 and 6 The results alsoshow that the theoretical results are consistent with theresults obtained from the ANSYS finite element numericalcalculation thus validating and rationalizing this theoreticalanalysis model in an effective manner

32 Analysis of Maximum Deflection-Loading Speed for Mid-Span of Double Simply Supported Beam Figure 7 shows themaximum dynamic deflection for the mid-span of doublesimply supported rail-bridge system at different load movingspeeds With the increase in load moving speeds the effectof speed on the dynamic response of the double simply

supported rail-bridge beam system shows a nonlinear rela-tionship a sine curve was observed instead of increasingwith the increase of speed linearly The maximum mid-span deflection for the first simply supported beam showsthe variation of sine curve around the amplitude of 102mm Besides with the increase of speed the period andamplitude of sine curve also increased The maximum mid-span deflection for the second simply supported beam alsoshows the variation of sine curve but unlike the first beamthe mid-span deflection for this beam shows an overallincreasing trend with the increase of speeds

Due to the very low moment of inertia of the firstbeam compared to the second beam the first beam showshierarchical oscillations while the secondary beam does notshow any local oscillations and the mid-span deflectiondynamic response of the first beam is insensitive to the

Shock and Vibration 7

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02 04 06 08 1000t (s)

(a)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus028

minus021

minus014

minus007

000

007

Defl

ectio

n (m

m)

(b)

ANSYSMATLAB

minus024

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

02 04 06 08 1000t (s)

(c)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(d)

Figure 5 Mid-span deflection time-history for quadruple simply supported beam system with load moving speed v = 32 ms (a) first beam(b) second beam (c) third beam (d) fourth beam

speed Relationship between maximum deflection and load-ing speed for mid-span of double simply supported beamwith a higher I1 value is shown in Figures 8-9 The resultsshow that the local oscillation of the first beamrsquos mid-spanmaximumdeflection reduced significantly when themomentof inertia I1 is revalued as 100 I1 and there are barely no localoscillations of the first beamrsquos mid-span maximum deflectionwhen the moment of inertia I1 is taken as 105 I1

Moreover the speeds atwhich themid-spandeflection forthe double beams reaches the peak value are the same withthe appearance of multiple points of resonance (wave crest)and points of cancellation (wave trough) as shown in Table 1

Figures 10 and 11 show the contrast ofmid-span deflectionresponses of the double simply supported rail-bridge beamsystem under different resonances and cancellation speeds

Table 1 and Figures 10-11 show that with the increasein resonance speeds the time-history curves of mid-spandeflection dynamic response of the first beam are consistentwhile the local oscillation of mid-span deflection time-history curves of the second beam significantly reduced thevibration period becomes longer and the mid-span dynamic

response reaches the amplitude when the moving loads actnear the mid-span of double rail-bridge system With theincrease in cancellation speeds the time-history curves ofmid-span deflection dynamic response of the first beamare consistent and when the mid-span deflection dynamicresponse of the second beam reaches the amplitude thedistance between the moving-load position and mid-spanincreases When the load moving speed is 881 ms the mid-spandynamic deflection response of the second beam reachesthe amplitude while the moving loads are on the 13 and 23positions of the beam

33 Analysis of Parameter Influence of Double Simply Sup-ported Rail-Bridge System Based on the analysis in theabove section further study was conducted regarding theeffect of interlayer spring stiffness beam masses and flexuralrigidities on the dynamic responses of the double simplysupported rail-bridge system in this section The movingspeed of loads v = 100 ms was taken as the example to studythe effect of variation of three parameters

8 Shock and Vibration

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

01 02 03 04 0500t (s)

(a)

ANSYSMATLAB

01 02 03 04 0500t (s)

minus028

minus021

minus014

minus007

000

007

Defl

ectio

n (m

m)

(b)

ANSYSMATLAB

minus024

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

(c)

ANSYSMATLAB

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

(d)

Figure 6 Mid-span deflection time-history for quadruple simply supported beam system with load moving speed v = 64 ms (a) first beam(b) second beam (c) third beam (d) fourth beam

First Beam

40 80 120 160 2000v (ms)

096

100

104

108

112

116

120

Defl

ectio

n (m

m)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 7 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam (a) first beam (b)second beam

Shock and Vibration 9

First Beam038

040

042

044

046

048

050

052D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 8 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 100I1 (a) first beam (b) second beam

First Beam

40 80 120 160 2000v (ms)

0060

0063

0066

0069

0072

0075

0078

Defl

ectio

n (m

m)

(a)

Second Beam0040

0044

0048

0052

0056

0060D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(b)

Figure 9 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 105I1 (a) first beam (b) second beam

Table 1 Relationship between resonance (cancellation) speed and amplitude

Speed at the wavecrest (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)

Speed at the wavetrough (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)189 1024 0161 204 1009 0159232 1025 0162 253 1015 0160299 1027 0165 334 1011 0162411 1039 0169 487 1004 0163687 1058 0180 881 0996 0164

Figure 12 show that interlayer stiffness variation has agreater effect on the dynamic response of the first sim-ply supported beam and the mid-span dynamic deflectionamplitude of the first simply supported beam decreasedwith the increase in interlayer stiffness while the interlayer

stiffness has a negligible effect on the mid-span dynamictime-history of the second simply supported beam Thereason is that in the double simply supported rail-bridgesystem the differences in the stiffness of the first and secondbeams are large the flexural rigidity of the first beam is

10 Shock and Vibration

v=189msv=232msv=299ms

v=411msv=687ms

4 8 12 16 20 24 28 320Load position (m)

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

(a)

v=189msv=232msv=299ms

v=411msv=687ms

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(b)

Figure 10 Mid-span deflection time-history for double simply supported rail-bridge system under resonance condition (a) first beam (b)second beam

v=204msv=253msv=334ms

v=487msv=881ms

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(a)

0 4 8 12 16 20 24 28 32

v=204msv=253msv=334ms

v=487msv=881ms

Load position (m)

minus020

minus016

minus012

minus008

minus004

000

Defl

ectio

n (m

m)

(b)

Figure 11 Mid-span deflection time-history for the double simply supported rail-bridge system under cancellation condition (a) first beam(b) second beam

much smaller than that of the second beam and the bendingresistance of the second beam plays a leading role in thebending resistance of the entire system

Figures 13 and 14 show the effects of mass variationper unit length of the first and second beams on mid-span deflection time-history for double simply supportedrail-bridge system respectively Figure 13 shows that underdifferent masses of the first beam the time-history curvesof mid-span deflection dynamic response for the two simplysupported beams are consistent Therefore the effect of massvariation of the first simply supported beam on the mid-spandeflection dynamic response of the double-beam system canbe neglected Figure 14 shows that the mass variation of thesecond simply supported beam has a significant effect on the

mid-span deflection dynamic response of both beams whenthemass per unit length of the second simply supported beamincreases to four times of the original mass the amplitudes ofmid-span deflection dynamic response of the first and secondsimply supported beams both increase significantly with thegrowth rate reaching up to 10 and 45 respectively

Figures 15 and 16 show the effect of flexural rigidityvariations of the first and second simply supported beamson the mid-span deflection time-history for double simplysupported rail-bridge system Figure 15 shows that the mid-span dynamic deflection amplitude of the first simply sup-ported beam significantly decreased with the increase in thestiffness of this beam while the stiffness of the first beamhas a smaller effect on the mid-span dynamic deflection

Shock and Vibration 11

025kr050kr100kr

200kr400kr

minus28

minus24

minus20

minus16

minus12

minus08

minus04

00

04

Defl

ectio

n (m

m)

008 016 024 032000t (s)

(a)

025kr050kr100kr

200kr400kr

008 016 024 032000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 12 Comparison diagram of mid-span deflection time-history for the double simply supported rail-bridge system under differentinterlayer stiffness (a) first beam (b) second beam

100

200400

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025Ｇ1050Ｇ1

Ｇ1

Ｇ1Ｇ1

(a)

025050100

200400

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

Ｇ1Ｇ1

Ｇ1Ｇ1

Ｇ1

(b)

Figure 13 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof first beam (a) first beam (b) second beam

amplitude of the second simply supported beam Underdifferent stiffness of the first beam the mid-span deflectiondynamic time-history curves for the second simply supportedbeam are consistent Figure 16 shows that the flexural rigidityvariations of the second simply supported beam have a sig-nificant effect on the mid-span deflection dynamic responseof double simply supported rail-bridge system In additionthe mid-span deflection dynamic response amplitudes ofboth beams significantly increased with the decrease in theflexural stiffness of second beam In practical applicationsthe dynamic response of the rail-bridge system in high-speedrailways can be decreased by moderately decreasing the massof the second beam and increasing the flexural stiffness ofboth simply supported beams

4 Conclusions

In this study a dynamic analysis model was established for amultiple simply supported beam system subjected to movingloads in combination of finite sin-Fourier transform andnumerical Laplace transform based on Durbin transforman analytical calculation method was developed for thespatial displacement time-domain response of a multiplesimply supported beam systemThe following conclusions aredrawn by comparing the examples and ANSYS finite elementnumerical computation method(1) A simply supported rail-bridge system of high-speedrailways with a span of 32 m in China was taken as theexample and the theoretically calculated results were com-pared with the results obtained from ANSYS finite element

12 Shock and Vibration

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

100

200400

025Ｇ2050Ｇ2

Ｇ2

Ｇ2Ｇ2

(a)

minus030

minus024

minus018

minus012

minus006

000

006

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025050100

200400Ｇ2

Ｇ2

Ｇ2Ｇ2

Ｇ2

(b)

Figure 14 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof second beam (a) first beam (b) second beam

008 016 024 032000t (s)

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02510501

10012001

(a)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

02510501

10012001

(b)

Figure 15 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof first beam (a) first beam (b) second beam

numerical calculation validating the analytical calculationmethod reported in this paper(2) The effect of load moving speed on the dynamicresponse of a double simply supported rail-bridge system ofhigh-speed railways subjected tomoving loads was evaluatedThe results indicate that the double simply supported rail-bridge system has multiple points of resonance and cancel-lation which should be paid attention in engineering(3) For a typical double simply supported rail-bridgesystem of high-speed railway in China because of a largedifference in the flexural rigidity of the first and secondbeams the variation in interlayer stiffness mainly affectsthe dynamic response of the first simply supported beamexhibiting a positive correlation When the mass and flexuralstiffness parameters of first simply supported beam vary they

mainly affect the dynamic response of this layer of simplysupported beam but slightly affect the dynamic responseof second simply supported beam The variation in theparameters of second simply supported beam has a greatereffect on the double simply supported beam system and adecrease in the mass of this layer of simply supported beamand an increase in the flexural rigidity of this beam canboth decrease the dynamic response of the double simplysupported rail-bridge system

Data Availability

The data used to support the findings of this study areincluded within the article

Shock and Vibration 13

minus14

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

05021002

20024002

(a)

008 016 024 032000t (s)

minus048

minus040

minus032

minus024

minus016

minus008

000

008

Defl

ectio

n (m

m)

05021002

20024002

(b)

Figure 16 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof second beam (a) first beam (b) second beam

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51708630)

References

[1] H Ouyang ldquoMoving-load dynamic problems a tutorial (with abrief overview)rdquoMechanical Systems and Signal Processing vol25 no 6 pp 2039ndash2060 2011

[2] D G Duffy ldquoThe response of an infinite railroad track to amoving vibrating massrdquo Journal of Applied Mechanics vol 57no 1 pp 66ndash73 1990

[3] L Fryba Vibration of solids and structures under moving loadsSpringer Science amp Business Media 2013

[4] G Michaltsos D Sophianopoulos and A N Kounadis ldquoTheeffect of a moving mass and other parameters on the dynamicresponse of a simply supported beamrdquo Journal of Sound andVibration vol 191 no 3 pp 357ndash362 1996

[5] P Museros E Moliner and M D Martınez-Rodrigo ldquoFreevibrations of simply-supported beam bridges under movingloads Maximum resonance cancellation and resonant verticalaccelerationrdquo Journal of Sound and Vibration vol 332 no 2 pp326ndash345 2013

[6] P Sniady ldquoDynamic response of a Timoshenko beam to amoving forcerdquo Journal of Applied Mechanics vol 75 no 2Article ID 024503 2008

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

[8] K Y Lee and A A Renshaw ldquoSolution of the moving massproblem using complex eigenfunction expansionsrdquo Journal ofApplied Mechanics vol 67 no 4 pp 823ndash827 2000

[9] U Lee ldquoSeparation between the flexible structure and themoving mass sliding on itrdquo Journal of Sound and Vibration vol209 no 5 pp 867ndash877 1998

[10] S A Eftekhari and A A Jafari ldquoCoupling ritz method andtriangular quadrature rule for moving mass problemrdquo Journalof Applied Mechanics vol 79 no 2 2012

[11] M Dublin andH R Friedrich ldquoForced responses of two elasticbeams interconnected by spring-damper systemsrdquo Journal of theAeronautical Sciences vol 23 pp 824ndash829 1956

[12] J M Seelig and W H Hoppmann ldquoImpact on an elasticallyconnected double-beam systemrdquo Journal of Applied Mechanicsvol 31 pp 621ndash626 1964

[13] P G Kessel ldquoResonances Excited in an Elastically ConnectedDouble-Beam System by a Cyclic Moving LoadrdquoThe Journal ofthe Acoustical Society of America vol 40 no 3 pp 684ndash6871966

[14] S Chonan ldquoDynamical behaviours of elastically connecteddouble-beam systems subjected to an impulsive loadrdquo Bulletinof JSME vol 19 no 132 pp 595ndash603 1976

[15] H V Vu A M Ordonez and B H Karnopp ldquoVibration of adouble-beam systemrdquo Journal of Sound and Vibration vol 229no 4 pp 807ndash822 2000

[16] H J Xu Z H Ding Z R Lu and J K Liu ldquoStructural DamageDetection Using a Modified Artificial Bee Colony AlgorithmrdquoCMES-Computer Modeling in Engineering amp Sciences vol 111pp 335ndash355 2016

[17] Z Oniszczuk ldquoFree transverse vibrations of elastically con-nected simply supported double-beam complex systemrdquo Jour-nal of Sound and Vibration vol 232 no 2 pp 387ndash403 2000

[18] Z Oniszczuk ldquoForced transverse vibrations of an elasticallyconnected complex simply supported double-beam systemrdquoJournal of Sound and Vibration vol 264 no 2 pp 273ndash2862003

[19] Y Q Zhang Y Lu S L Wang and X Liu ldquoVibration andbuckling of a double-beam system under compressive axialloadingrdquo Journal of Sound and Vibration vol 318 no 1-2 pp341ndash352 2008

[20] V Stojanovic and P Kozic ldquoForced transverse vibration ofRayleigh and Timoshenko double-beam system with effect of

14 Shock and Vibration

compressive axial loadrdquo International Journal of MechanicalSciences vol 60 no 1 pp 59ndash71 2012

[21] A Palmeri and S Adhikari ldquoA Galerkin-type state-spaceapproach for transverse vibrations of slender double-beamsystems with viscoelastic inner layerrdquo Journal of Sound andVibration vol 330 no 26 pp 6372ndash6386 2011

[22] Z Zhang X Huang Z Zhang and H Hua ldquoOn the transversevibration of Timoshenko double-beam systems coupled withvarious discontinuitiesrdquo International Journal of MechanicalSciences vol 89 pp 222ndash241 2014

[23] Y X Li Z J Hu and L Z Sun ldquoDynamical behavior of adouble-beam system interconnected by a viscoelastic layerrdquoInternational Journal of Mechanical Sciences vol 105 pp 291ndash303 2016

[24] M Abu-Hilal ldquoDynamic response of a double EulerndashBernoullibeam due to a moving constant loadrdquo Journal of Sound andVibration vol 297 no 3-5 pp 477ndash491 2006

[25] Y Wu and Y Gao ldquoAnalytical Solutions for Simply SupportedViscously Damped Double-Beam System under Moving Har-monic Loadsrdquo Journal of Engineering Mechanics vol 141 no 7p 04015004 2015

[26] Y-H Chen and Z-M Shiu ldquoResonant curves of an elevatedrailway to harmonic moving loadsrdquo International Journal ofStructural Stability and Dynamics vol 4 no 2 pp 237ndash2572004

[27] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of aninfinite tomoshenko beam on a viscoelastic foundation to aharmonic moving loadrdquo Journal of Sound and Vibration vol241 no 5 pp 809ndash824 2001

[28] M FMHussein andH EMHunt ldquoModelling of floating-slabtracks with continuous slabs under oscillating moving loadsrdquoJournal of Sound andVibration vol 297 no 1-2 pp 37ndash54 2006

[29] H P Wang J Li and K Zhang ldquoVibration analysis of themaglev guideway with the moving loadrdquo Journal of Sound andVibration vol 305 no 4-5 pp 621ndash640 2007

[30] V Stojanovic and M Petkovic ldquoMoment Lyapunov exponentsand stochastic stability of a three-dimensional system on elasticfoundation using a perturbation approachrdquo Journal of AppliedMechanics vol 80 no 5 2013

[31] L Jun and H Hongxing ldquoDynamic stiffness vibration analysisof an elastically connected three-beam systemrdquo Applied Acous-tics vol 69 no 7 pp 591ndash600 2008

[32] J Li Y Chen and H Hua ldquoExact dynamic stiffness matrixof a Timoshenko three-beam systemrdquo International Journal ofMechanical Sciences vol 50 no 6 pp 1023ndash1034 2008

[33] S S Rao ldquoNatural vibrations of systems of elastically connectedTimoshenko beams rdquo The Journal of the Acoustical Society ofAmerica vol 55 no 6 pp 1232ndash1237 1974

[34] Q Mao ldquoFree vibration analysis of elastically connectedmultiple-beams by using the Adomianmodified decompositionmethodrdquo Journal of Sound and Vibration vol 331 no 11 pp2532ndash2542 2012

[35] V Stojanovic P Kozic and G Janevski ldquoExact closed-formsolutions for the natural frequencies and stability of elasticallyconnected multiple beam system using Timoshenko and high-order shear deformation theoryrdquo Journal of Sound and Vibra-tion vol 332 no 3 pp 563ndash576 2013

[36] M H Kargarnovin M T Ahmadian and R-A Jafari-Talookolaei ldquoAnalytical solution for the dynamic analysis of adelaminated composite beam traversed by a moving constantforcerdquo Journal of Vibration and Control vol 19 no 10 pp 1524ndash1537 2013

[37] A Ariaei S Ziaei-Rad and M Ghayour ldquoTransverse vibrationof a multiple-Timoshenko beam system with intermediateelastic connections due to a moving loadrdquo Archive of AppliedMechanics vol 81 no 3 pp 263ndash281 2011

[38] Y Povstenko and J Klekot ldquoThe Dirichlet problem for thetime-fractional advection-diffusion equation in a line segmentrdquoBoundary Value Problems Paper No 89 8 pages 2016

[39] F Durbin ldquoNumerical inversion of Laplace transforms anefficient improvement to Dubner and Abatersquos methodrdquo TheComputer Journal vol 17 pp 371ndash376 1974

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

6 Shock and Vibration

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

02 04 06 08 1000t (s)

(a)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 3Mid-span deflection time-history for double simply supported rail-bridge systemwith loadmoving speed v = 32ms (a) first beam(b) second beam

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

ANSYSMATLAB

(a)

minus020

minus016

minus012

minus008

minus004

000

004D

eflec

tion

(mm

)

01 02 03 04 0500t (s)

ANSYSMATLAB

(b)

Figure 4 Mid-span deflection time-history for double simply supported rail-bridge system with load moving speed v = 64 ms (a) firstbeam (b) second beam

quadruple simply supported beam system subjected to exter-nal load with different moving speeds (32 ms and 64 ms)were calculated The comparisons between the theoreticallycalculated results and the ANSYS finite element numericalcalculation are shown in Figures 5 and 6 The results alsoshow that the theoretical results are consistent with theresults obtained from the ANSYS finite element numericalcalculation thus validating and rationalizing this theoreticalanalysis model in an effective manner

32 Analysis of Maximum Deflection-Loading Speed for Mid-Span of Double Simply Supported Beam Figure 7 shows themaximum dynamic deflection for the mid-span of doublesimply supported rail-bridge system at different load movingspeeds With the increase in load moving speeds the effectof speed on the dynamic response of the double simply

supported rail-bridge beam system shows a nonlinear rela-tionship a sine curve was observed instead of increasingwith the increase of speed linearly The maximum mid-span deflection for the first simply supported beam showsthe variation of sine curve around the amplitude of 102mm Besides with the increase of speed the period andamplitude of sine curve also increased The maximum mid-span deflection for the second simply supported beam alsoshows the variation of sine curve but unlike the first beamthe mid-span deflection for this beam shows an overallincreasing trend with the increase of speeds

Due to the very low moment of inertia of the firstbeam compared to the second beam the first beam showshierarchical oscillations while the secondary beam does notshow any local oscillations and the mid-span deflectiondynamic response of the first beam is insensitive to the

Shock and Vibration 7

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02 04 06 08 1000t (s)

(a)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus028

minus021

minus014

minus007

000

007

Defl

ectio

n (m

m)

(b)

ANSYSMATLAB

minus024

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

02 04 06 08 1000t (s)

(c)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(d)

Figure 5 Mid-span deflection time-history for quadruple simply supported beam system with load moving speed v = 32 ms (a) first beam(b) second beam (c) third beam (d) fourth beam

speed Relationship between maximum deflection and load-ing speed for mid-span of double simply supported beamwith a higher I1 value is shown in Figures 8-9 The resultsshow that the local oscillation of the first beamrsquos mid-spanmaximumdeflection reduced significantly when themomentof inertia I1 is revalued as 100 I1 and there are barely no localoscillations of the first beamrsquos mid-span maximum deflectionwhen the moment of inertia I1 is taken as 105 I1

Moreover the speeds atwhich themid-spandeflection forthe double beams reaches the peak value are the same withthe appearance of multiple points of resonance (wave crest)and points of cancellation (wave trough) as shown in Table 1

Figures 10 and 11 show the contrast ofmid-span deflectionresponses of the double simply supported rail-bridge beamsystem under different resonances and cancellation speeds

Table 1 and Figures 10-11 show that with the increasein resonance speeds the time-history curves of mid-spandeflection dynamic response of the first beam are consistentwhile the local oscillation of mid-span deflection time-history curves of the second beam significantly reduced thevibration period becomes longer and the mid-span dynamic

response reaches the amplitude when the moving loads actnear the mid-span of double rail-bridge system With theincrease in cancellation speeds the time-history curves ofmid-span deflection dynamic response of the first beamare consistent and when the mid-span deflection dynamicresponse of the second beam reaches the amplitude thedistance between the moving-load position and mid-spanincreases When the load moving speed is 881 ms the mid-spandynamic deflection response of the second beam reachesthe amplitude while the moving loads are on the 13 and 23positions of the beam

33 Analysis of Parameter Influence of Double Simply Sup-ported Rail-Bridge System Based on the analysis in theabove section further study was conducted regarding theeffect of interlayer spring stiffness beam masses and flexuralrigidities on the dynamic responses of the double simplysupported rail-bridge system in this section The movingspeed of loads v = 100 ms was taken as the example to studythe effect of variation of three parameters

8 Shock and Vibration

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

01 02 03 04 0500t (s)

(a)

ANSYSMATLAB

01 02 03 04 0500t (s)

minus028

minus021

minus014

minus007

000

007

Defl

ectio

n (m

m)

(b)

ANSYSMATLAB

minus024

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

(c)

ANSYSMATLAB

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

(d)

Figure 6 Mid-span deflection time-history for quadruple simply supported beam system with load moving speed v = 64 ms (a) first beam(b) second beam (c) third beam (d) fourth beam

First Beam

40 80 120 160 2000v (ms)

096

100

104

108

112

116

120

Defl

ectio

n (m

m)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 7 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam (a) first beam (b)second beam

Shock and Vibration 9

First Beam038

040

042

044

046

048

050

052D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 8 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 100I1 (a) first beam (b) second beam

First Beam

40 80 120 160 2000v (ms)

0060

0063

0066

0069

0072

0075

0078

Defl

ectio

n (m

m)

(a)

Second Beam0040

0044

0048

0052

0056

0060D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(b)

Figure 9 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 105I1 (a) first beam (b) second beam

Table 1 Relationship between resonance (cancellation) speed and amplitude

Speed at the wavecrest (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)

Speed at the wavetrough (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)189 1024 0161 204 1009 0159232 1025 0162 253 1015 0160299 1027 0165 334 1011 0162411 1039 0169 487 1004 0163687 1058 0180 881 0996 0164

Figure 12 show that interlayer stiffness variation has agreater effect on the dynamic response of the first sim-ply supported beam and the mid-span dynamic deflectionamplitude of the first simply supported beam decreasedwith the increase in interlayer stiffness while the interlayer

stiffness has a negligible effect on the mid-span dynamictime-history of the second simply supported beam Thereason is that in the double simply supported rail-bridgesystem the differences in the stiffness of the first and secondbeams are large the flexural rigidity of the first beam is

10 Shock and Vibration

v=189msv=232msv=299ms

v=411msv=687ms

4 8 12 16 20 24 28 320Load position (m)

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

(a)

v=189msv=232msv=299ms

v=411msv=687ms

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(b)

Figure 10 Mid-span deflection time-history for double simply supported rail-bridge system under resonance condition (a) first beam (b)second beam

v=204msv=253msv=334ms

v=487msv=881ms

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(a)

0 4 8 12 16 20 24 28 32

v=204msv=253msv=334ms

v=487msv=881ms

Load position (m)

minus020

minus016

minus012

minus008

minus004

000

Defl

ectio

n (m

m)

(b)

Figure 11 Mid-span deflection time-history for the double simply supported rail-bridge system under cancellation condition (a) first beam(b) second beam

much smaller than that of the second beam and the bendingresistance of the second beam plays a leading role in thebending resistance of the entire system

Figures 13 and 14 show the effects of mass variationper unit length of the first and second beams on mid-span deflection time-history for double simply supportedrail-bridge system respectively Figure 13 shows that underdifferent masses of the first beam the time-history curvesof mid-span deflection dynamic response for the two simplysupported beams are consistent Therefore the effect of massvariation of the first simply supported beam on the mid-spandeflection dynamic response of the double-beam system canbe neglected Figure 14 shows that the mass variation of thesecond simply supported beam has a significant effect on the

mid-span deflection dynamic response of both beams whenthemass per unit length of the second simply supported beamincreases to four times of the original mass the amplitudes ofmid-span deflection dynamic response of the first and secondsimply supported beams both increase significantly with thegrowth rate reaching up to 10 and 45 respectively

Figures 15 and 16 show the effect of flexural rigidityvariations of the first and second simply supported beamson the mid-span deflection time-history for double simplysupported rail-bridge system Figure 15 shows that the mid-span dynamic deflection amplitude of the first simply sup-ported beam significantly decreased with the increase in thestiffness of this beam while the stiffness of the first beamhas a smaller effect on the mid-span dynamic deflection

Shock and Vibration 11

025kr050kr100kr

200kr400kr

minus28

minus24

minus20

minus16

minus12

minus08

minus04

00

04

Defl

ectio

n (m

m)

008 016 024 032000t (s)

(a)

025kr050kr100kr

200kr400kr

008 016 024 032000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 12 Comparison diagram of mid-span deflection time-history for the double simply supported rail-bridge system under differentinterlayer stiffness (a) first beam (b) second beam

100

200400

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025Ｇ1050Ｇ1

Ｇ1

Ｇ1Ｇ1

(a)

025050100

200400

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

Ｇ1Ｇ1

Ｇ1Ｇ1

Ｇ1

(b)

Figure 13 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof first beam (a) first beam (b) second beam

amplitude of the second simply supported beam Underdifferent stiffness of the first beam the mid-span deflectiondynamic time-history curves for the second simply supportedbeam are consistent Figure 16 shows that the flexural rigidityvariations of the second simply supported beam have a sig-nificant effect on the mid-span deflection dynamic responseof double simply supported rail-bridge system In additionthe mid-span deflection dynamic response amplitudes ofboth beams significantly increased with the decrease in theflexural stiffness of second beam In practical applicationsthe dynamic response of the rail-bridge system in high-speedrailways can be decreased by moderately decreasing the massof the second beam and increasing the flexural stiffness ofboth simply supported beams

4 Conclusions

In this study a dynamic analysis model was established for amultiple simply supported beam system subjected to movingloads in combination of finite sin-Fourier transform andnumerical Laplace transform based on Durbin transforman analytical calculation method was developed for thespatial displacement time-domain response of a multiplesimply supported beam systemThe following conclusions aredrawn by comparing the examples and ANSYS finite elementnumerical computation method(1) A simply supported rail-bridge system of high-speedrailways with a span of 32 m in China was taken as theexample and the theoretically calculated results were com-pared with the results obtained from ANSYS finite element

12 Shock and Vibration

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

100

200400

025Ｇ2050Ｇ2

Ｇ2

Ｇ2Ｇ2

(a)

minus030

minus024

minus018

minus012

minus006

000

006

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025050100

200400Ｇ2

Ｇ2

Ｇ2Ｇ2

Ｇ2

(b)

Figure 14 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof second beam (a) first beam (b) second beam

008 016 024 032000t (s)

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02510501

10012001

(a)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

02510501

10012001

(b)

Figure 15 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof first beam (a) first beam (b) second beam

numerical calculation validating the analytical calculationmethod reported in this paper(2) The effect of load moving speed on the dynamicresponse of a double simply supported rail-bridge system ofhigh-speed railways subjected tomoving loads was evaluatedThe results indicate that the double simply supported rail-bridge system has multiple points of resonance and cancel-lation which should be paid attention in engineering(3) For a typical double simply supported rail-bridgesystem of high-speed railway in China because of a largedifference in the flexural rigidity of the first and secondbeams the variation in interlayer stiffness mainly affectsthe dynamic response of the first simply supported beamexhibiting a positive correlation When the mass and flexuralstiffness parameters of first simply supported beam vary they

mainly affect the dynamic response of this layer of simplysupported beam but slightly affect the dynamic responseof second simply supported beam The variation in theparameters of second simply supported beam has a greatereffect on the double simply supported beam system and adecrease in the mass of this layer of simply supported beamand an increase in the flexural rigidity of this beam canboth decrease the dynamic response of the double simplysupported rail-bridge system

Data Availability

The data used to support the findings of this study areincluded within the article

Shock and Vibration 13

minus14

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

05021002

20024002

(a)

008 016 024 032000t (s)

minus048

minus040

minus032

minus024

minus016

minus008

000

008

Defl

ectio

n (m

m)

05021002

20024002

(b)

Figure 16 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof second beam (a) first beam (b) second beam

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51708630)

References

[1] H Ouyang ldquoMoving-load dynamic problems a tutorial (with abrief overview)rdquoMechanical Systems and Signal Processing vol25 no 6 pp 2039ndash2060 2011

[2] D G Duffy ldquoThe response of an infinite railroad track to amoving vibrating massrdquo Journal of Applied Mechanics vol 57no 1 pp 66ndash73 1990

[3] L Fryba Vibration of solids and structures under moving loadsSpringer Science amp Business Media 2013

[4] G Michaltsos D Sophianopoulos and A N Kounadis ldquoTheeffect of a moving mass and other parameters on the dynamicresponse of a simply supported beamrdquo Journal of Sound andVibration vol 191 no 3 pp 357ndash362 1996

[5] P Museros E Moliner and M D Martınez-Rodrigo ldquoFreevibrations of simply-supported beam bridges under movingloads Maximum resonance cancellation and resonant verticalaccelerationrdquo Journal of Sound and Vibration vol 332 no 2 pp326ndash345 2013

[6] P Sniady ldquoDynamic response of a Timoshenko beam to amoving forcerdquo Journal of Applied Mechanics vol 75 no 2Article ID 024503 2008

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

[8] K Y Lee and A A Renshaw ldquoSolution of the moving massproblem using complex eigenfunction expansionsrdquo Journal ofApplied Mechanics vol 67 no 4 pp 823ndash827 2000

[9] U Lee ldquoSeparation between the flexible structure and themoving mass sliding on itrdquo Journal of Sound and Vibration vol209 no 5 pp 867ndash877 1998

[10] S A Eftekhari and A A Jafari ldquoCoupling ritz method andtriangular quadrature rule for moving mass problemrdquo Journalof Applied Mechanics vol 79 no 2 2012

[11] M Dublin andH R Friedrich ldquoForced responses of two elasticbeams interconnected by spring-damper systemsrdquo Journal of theAeronautical Sciences vol 23 pp 824ndash829 1956

[12] J M Seelig and W H Hoppmann ldquoImpact on an elasticallyconnected double-beam systemrdquo Journal of Applied Mechanicsvol 31 pp 621ndash626 1964

[13] P G Kessel ldquoResonances Excited in an Elastically ConnectedDouble-Beam System by a Cyclic Moving LoadrdquoThe Journal ofthe Acoustical Society of America vol 40 no 3 pp 684ndash6871966

[14] S Chonan ldquoDynamical behaviours of elastically connecteddouble-beam systems subjected to an impulsive loadrdquo Bulletinof JSME vol 19 no 132 pp 595ndash603 1976

[15] H V Vu A M Ordonez and B H Karnopp ldquoVibration of adouble-beam systemrdquo Journal of Sound and Vibration vol 229no 4 pp 807ndash822 2000

[16] H J Xu Z H Ding Z R Lu and J K Liu ldquoStructural DamageDetection Using a Modified Artificial Bee Colony AlgorithmrdquoCMES-Computer Modeling in Engineering amp Sciences vol 111pp 335ndash355 2016

[17] Z Oniszczuk ldquoFree transverse vibrations of elastically con-nected simply supported double-beam complex systemrdquo Jour-nal of Sound and Vibration vol 232 no 2 pp 387ndash403 2000

[18] Z Oniszczuk ldquoForced transverse vibrations of an elasticallyconnected complex simply supported double-beam systemrdquoJournal of Sound and Vibration vol 264 no 2 pp 273ndash2862003

[19] Y Q Zhang Y Lu S L Wang and X Liu ldquoVibration andbuckling of a double-beam system under compressive axialloadingrdquo Journal of Sound and Vibration vol 318 no 1-2 pp341ndash352 2008

[20] V Stojanovic and P Kozic ldquoForced transverse vibration ofRayleigh and Timoshenko double-beam system with effect of

14 Shock and Vibration

compressive axial loadrdquo International Journal of MechanicalSciences vol 60 no 1 pp 59ndash71 2012

[21] A Palmeri and S Adhikari ldquoA Galerkin-type state-spaceapproach for transverse vibrations of slender double-beamsystems with viscoelastic inner layerrdquo Journal of Sound andVibration vol 330 no 26 pp 6372ndash6386 2011

[22] Z Zhang X Huang Z Zhang and H Hua ldquoOn the transversevibration of Timoshenko double-beam systems coupled withvarious discontinuitiesrdquo International Journal of MechanicalSciences vol 89 pp 222ndash241 2014

[23] Y X Li Z J Hu and L Z Sun ldquoDynamical behavior of adouble-beam system interconnected by a viscoelastic layerrdquoInternational Journal of Mechanical Sciences vol 105 pp 291ndash303 2016

[24] M Abu-Hilal ldquoDynamic response of a double EulerndashBernoullibeam due to a moving constant loadrdquo Journal of Sound andVibration vol 297 no 3-5 pp 477ndash491 2006

[25] Y Wu and Y Gao ldquoAnalytical Solutions for Simply SupportedViscously Damped Double-Beam System under Moving Har-monic Loadsrdquo Journal of Engineering Mechanics vol 141 no 7p 04015004 2015

[26] Y-H Chen and Z-M Shiu ldquoResonant curves of an elevatedrailway to harmonic moving loadsrdquo International Journal ofStructural Stability and Dynamics vol 4 no 2 pp 237ndash2572004

[27] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of aninfinite tomoshenko beam on a viscoelastic foundation to aharmonic moving loadrdquo Journal of Sound and Vibration vol241 no 5 pp 809ndash824 2001

[28] M FMHussein andH EMHunt ldquoModelling of floating-slabtracks with continuous slabs under oscillating moving loadsrdquoJournal of Sound andVibration vol 297 no 1-2 pp 37ndash54 2006

[29] H P Wang J Li and K Zhang ldquoVibration analysis of themaglev guideway with the moving loadrdquo Journal of Sound andVibration vol 305 no 4-5 pp 621ndash640 2007

[30] V Stojanovic and M Petkovic ldquoMoment Lyapunov exponentsand stochastic stability of a three-dimensional system on elasticfoundation using a perturbation approachrdquo Journal of AppliedMechanics vol 80 no 5 2013

[31] L Jun and H Hongxing ldquoDynamic stiffness vibration analysisof an elastically connected three-beam systemrdquo Applied Acous-tics vol 69 no 7 pp 591ndash600 2008

[32] J Li Y Chen and H Hua ldquoExact dynamic stiffness matrixof a Timoshenko three-beam systemrdquo International Journal ofMechanical Sciences vol 50 no 6 pp 1023ndash1034 2008

[33] S S Rao ldquoNatural vibrations of systems of elastically connectedTimoshenko beams rdquo The Journal of the Acoustical Society ofAmerica vol 55 no 6 pp 1232ndash1237 1974

[34] Q Mao ldquoFree vibration analysis of elastically connectedmultiple-beams by using the Adomianmodified decompositionmethodrdquo Journal of Sound and Vibration vol 331 no 11 pp2532ndash2542 2012

[35] V Stojanovic P Kozic and G Janevski ldquoExact closed-formsolutions for the natural frequencies and stability of elasticallyconnected multiple beam system using Timoshenko and high-order shear deformation theoryrdquo Journal of Sound and Vibra-tion vol 332 no 3 pp 563ndash576 2013

[36] M H Kargarnovin M T Ahmadian and R-A Jafari-Talookolaei ldquoAnalytical solution for the dynamic analysis of adelaminated composite beam traversed by a moving constantforcerdquo Journal of Vibration and Control vol 19 no 10 pp 1524ndash1537 2013

[37] A Ariaei S Ziaei-Rad and M Ghayour ldquoTransverse vibrationof a multiple-Timoshenko beam system with intermediateelastic connections due to a moving loadrdquo Archive of AppliedMechanics vol 81 no 3 pp 263ndash281 2011

[38] Y Povstenko and J Klekot ldquoThe Dirichlet problem for thetime-fractional advection-diffusion equation in a line segmentrdquoBoundary Value Problems Paper No 89 8 pages 2016

[39] F Durbin ldquoNumerical inversion of Laplace transforms anefficient improvement to Dubner and Abatersquos methodrdquo TheComputer Journal vol 17 pp 371ndash376 1974

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Shock and Vibration 7

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02 04 06 08 1000t (s)

(a)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus028

minus021

minus014

minus007

000

007

Defl

ectio

n (m

m)

(b)

ANSYSMATLAB

minus024

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

02 04 06 08 1000t (s)

(c)

ANSYSMATLAB

02 04 06 08 1000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(d)

Figure 5 Mid-span deflection time-history for quadruple simply supported beam system with load moving speed v = 32 ms (a) first beam(b) second beam (c) third beam (d) fourth beam

speed Relationship between maximum deflection and load-ing speed for mid-span of double simply supported beamwith a higher I1 value is shown in Figures 8-9 The resultsshow that the local oscillation of the first beamrsquos mid-spanmaximumdeflection reduced significantly when themomentof inertia I1 is revalued as 100 I1 and there are barely no localoscillations of the first beamrsquos mid-span maximum deflectionwhen the moment of inertia I1 is taken as 105 I1

Moreover the speeds atwhich themid-spandeflection forthe double beams reaches the peak value are the same withthe appearance of multiple points of resonance (wave crest)and points of cancellation (wave trough) as shown in Table 1

Figures 10 and 11 show the contrast ofmid-span deflectionresponses of the double simply supported rail-bridge beamsystem under different resonances and cancellation speeds

Table 1 and Figures 10-11 show that with the increasein resonance speeds the time-history curves of mid-spandeflection dynamic response of the first beam are consistentwhile the local oscillation of mid-span deflection time-history curves of the second beam significantly reduced thevibration period becomes longer and the mid-span dynamic

response reaches the amplitude when the moving loads actnear the mid-span of double rail-bridge system With theincrease in cancellation speeds the time-history curves ofmid-span deflection dynamic response of the first beamare consistent and when the mid-span deflection dynamicresponse of the second beam reaches the amplitude thedistance between the moving-load position and mid-spanincreases When the load moving speed is 881 ms the mid-spandynamic deflection response of the second beam reachesthe amplitude while the moving loads are on the 13 and 23positions of the beam

33 Analysis of Parameter Influence of Double Simply Sup-ported Rail-Bridge System Based on the analysis in theabove section further study was conducted regarding theeffect of interlayer spring stiffness beam masses and flexuralrigidities on the dynamic responses of the double simplysupported rail-bridge system in this section The movingspeed of loads v = 100 ms was taken as the example to studythe effect of variation of three parameters

8 Shock and Vibration

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

01 02 03 04 0500t (s)

(a)

ANSYSMATLAB

01 02 03 04 0500t (s)

minus028

minus021

minus014

minus007

000

007

Defl

ectio

n (m

m)

(b)

ANSYSMATLAB

minus024

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

(c)

ANSYSMATLAB

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

(d)

Figure 6 Mid-span deflection time-history for quadruple simply supported beam system with load moving speed v = 64 ms (a) first beam(b) second beam (c) third beam (d) fourth beam

First Beam

40 80 120 160 2000v (ms)

096

100

104

108

112

116

120

Defl

ectio

n (m

m)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 7 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam (a) first beam (b)second beam

Shock and Vibration 9

First Beam038

040

042

044

046

048

050

052D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 8 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 100I1 (a) first beam (b) second beam

First Beam

40 80 120 160 2000v (ms)

0060

0063

0066

0069

0072

0075

0078

Defl

ectio

n (m

m)

(a)

Second Beam0040

0044

0048

0052

0056

0060D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(b)

Figure 9 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 105I1 (a) first beam (b) second beam

Table 1 Relationship between resonance (cancellation) speed and amplitude

Speed at the wavecrest (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)

Speed at the wavetrough (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)189 1024 0161 204 1009 0159232 1025 0162 253 1015 0160299 1027 0165 334 1011 0162411 1039 0169 487 1004 0163687 1058 0180 881 0996 0164

Figure 12 show that interlayer stiffness variation has agreater effect on the dynamic response of the first sim-ply supported beam and the mid-span dynamic deflectionamplitude of the first simply supported beam decreasedwith the increase in interlayer stiffness while the interlayer

stiffness has a negligible effect on the mid-span dynamictime-history of the second simply supported beam Thereason is that in the double simply supported rail-bridgesystem the differences in the stiffness of the first and secondbeams are large the flexural rigidity of the first beam is

10 Shock and Vibration

v=189msv=232msv=299ms

v=411msv=687ms

4 8 12 16 20 24 28 320Load position (m)

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

(a)

v=189msv=232msv=299ms

v=411msv=687ms

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(b)

Figure 10 Mid-span deflection time-history for double simply supported rail-bridge system under resonance condition (a) first beam (b)second beam

v=204msv=253msv=334ms

v=487msv=881ms

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(a)

0 4 8 12 16 20 24 28 32

v=204msv=253msv=334ms

v=487msv=881ms

Load position (m)

minus020

minus016

minus012

minus008

minus004

000

Defl

ectio

n (m

m)

(b)

Figure 11 Mid-span deflection time-history for the double simply supported rail-bridge system under cancellation condition (a) first beam(b) second beam

much smaller than that of the second beam and the bendingresistance of the second beam plays a leading role in thebending resistance of the entire system

Figures 13 and 14 show the effects of mass variationper unit length of the first and second beams on mid-span deflection time-history for double simply supportedrail-bridge system respectively Figure 13 shows that underdifferent masses of the first beam the time-history curvesof mid-span deflection dynamic response for the two simplysupported beams are consistent Therefore the effect of massvariation of the first simply supported beam on the mid-spandeflection dynamic response of the double-beam system canbe neglected Figure 14 shows that the mass variation of thesecond simply supported beam has a significant effect on the

mid-span deflection dynamic response of both beams whenthemass per unit length of the second simply supported beamincreases to four times of the original mass the amplitudes ofmid-span deflection dynamic response of the first and secondsimply supported beams both increase significantly with thegrowth rate reaching up to 10 and 45 respectively

Figures 15 and 16 show the effect of flexural rigidityvariations of the first and second simply supported beamson the mid-span deflection time-history for double simplysupported rail-bridge system Figure 15 shows that the mid-span dynamic deflection amplitude of the first simply sup-ported beam significantly decreased with the increase in thestiffness of this beam while the stiffness of the first beamhas a smaller effect on the mid-span dynamic deflection

Shock and Vibration 11

025kr050kr100kr

200kr400kr

minus28

minus24

minus20

minus16

minus12

minus08

minus04

00

04

Defl

ectio

n (m

m)

008 016 024 032000t (s)

(a)

025kr050kr100kr

200kr400kr

008 016 024 032000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 12 Comparison diagram of mid-span deflection time-history for the double simply supported rail-bridge system under differentinterlayer stiffness (a) first beam (b) second beam

100

200400

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025Ｇ1050Ｇ1

Ｇ1

Ｇ1Ｇ1

(a)

025050100

200400

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

Ｇ1Ｇ1

Ｇ1Ｇ1

Ｇ1

(b)

Figure 13 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof first beam (a) first beam (b) second beam

amplitude of the second simply supported beam Underdifferent stiffness of the first beam the mid-span deflectiondynamic time-history curves for the second simply supportedbeam are consistent Figure 16 shows that the flexural rigidityvariations of the second simply supported beam have a sig-nificant effect on the mid-span deflection dynamic responseof double simply supported rail-bridge system In additionthe mid-span deflection dynamic response amplitudes ofboth beams significantly increased with the decrease in theflexural stiffness of second beam In practical applicationsthe dynamic response of the rail-bridge system in high-speedrailways can be decreased by moderately decreasing the massof the second beam and increasing the flexural stiffness ofboth simply supported beams

4 Conclusions

In this study a dynamic analysis model was established for amultiple simply supported beam system subjected to movingloads in combination of finite sin-Fourier transform andnumerical Laplace transform based on Durbin transforman analytical calculation method was developed for thespatial displacement time-domain response of a multiplesimply supported beam systemThe following conclusions aredrawn by comparing the examples and ANSYS finite elementnumerical computation method(1) A simply supported rail-bridge system of high-speedrailways with a span of 32 m in China was taken as theexample and the theoretically calculated results were com-pared with the results obtained from ANSYS finite element

12 Shock and Vibration

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

100

200400

025Ｇ2050Ｇ2

Ｇ2

Ｇ2Ｇ2

(a)

minus030

minus024

minus018

minus012

minus006

000

006

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025050100

200400Ｇ2

Ｇ2

Ｇ2Ｇ2

Ｇ2

(b)

Figure 14 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof second beam (a) first beam (b) second beam

008 016 024 032000t (s)

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02510501

10012001

(a)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

02510501

10012001

(b)

Figure 15 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof first beam (a) first beam (b) second beam

numerical calculation validating the analytical calculationmethod reported in this paper(2) The effect of load moving speed on the dynamicresponse of a double simply supported rail-bridge system ofhigh-speed railways subjected tomoving loads was evaluatedThe results indicate that the double simply supported rail-bridge system has multiple points of resonance and cancel-lation which should be paid attention in engineering(3) For a typical double simply supported rail-bridgesystem of high-speed railway in China because of a largedifference in the flexural rigidity of the first and secondbeams the variation in interlayer stiffness mainly affectsthe dynamic response of the first simply supported beamexhibiting a positive correlation When the mass and flexuralstiffness parameters of first simply supported beam vary they

mainly affect the dynamic response of this layer of simplysupported beam but slightly affect the dynamic responseof second simply supported beam The variation in theparameters of second simply supported beam has a greatereffect on the double simply supported beam system and adecrease in the mass of this layer of simply supported beamand an increase in the flexural rigidity of this beam canboth decrease the dynamic response of the double simplysupported rail-bridge system

Data Availability

The data used to support the findings of this study areincluded within the article

Shock and Vibration 13

minus14

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

05021002

20024002

(a)

008 016 024 032000t (s)

minus048

minus040

minus032

minus024

minus016

minus008

000

008

Defl

ectio

n (m

m)

05021002

20024002

(b)

Figure 16 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof second beam (a) first beam (b) second beam

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51708630)

References

[1] H Ouyang ldquoMoving-load dynamic problems a tutorial (with abrief overview)rdquoMechanical Systems and Signal Processing vol25 no 6 pp 2039ndash2060 2011

[2] D G Duffy ldquoThe response of an infinite railroad track to amoving vibrating massrdquo Journal of Applied Mechanics vol 57no 1 pp 66ndash73 1990

[3] L Fryba Vibration of solids and structures under moving loadsSpringer Science amp Business Media 2013

[4] G Michaltsos D Sophianopoulos and A N Kounadis ldquoTheeffect of a moving mass and other parameters on the dynamicresponse of a simply supported beamrdquo Journal of Sound andVibration vol 191 no 3 pp 357ndash362 1996

[5] P Museros E Moliner and M D Martınez-Rodrigo ldquoFreevibrations of simply-supported beam bridges under movingloads Maximum resonance cancellation and resonant verticalaccelerationrdquo Journal of Sound and Vibration vol 332 no 2 pp326ndash345 2013

[6] P Sniady ldquoDynamic response of a Timoshenko beam to amoving forcerdquo Journal of Applied Mechanics vol 75 no 2Article ID 024503 2008

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

[8] K Y Lee and A A Renshaw ldquoSolution of the moving massproblem using complex eigenfunction expansionsrdquo Journal ofApplied Mechanics vol 67 no 4 pp 823ndash827 2000

[9] U Lee ldquoSeparation between the flexible structure and themoving mass sliding on itrdquo Journal of Sound and Vibration vol209 no 5 pp 867ndash877 1998

[10] S A Eftekhari and A A Jafari ldquoCoupling ritz method andtriangular quadrature rule for moving mass problemrdquo Journalof Applied Mechanics vol 79 no 2 2012

[11] M Dublin andH R Friedrich ldquoForced responses of two elasticbeams interconnected by spring-damper systemsrdquo Journal of theAeronautical Sciences vol 23 pp 824ndash829 1956

[12] J M Seelig and W H Hoppmann ldquoImpact on an elasticallyconnected double-beam systemrdquo Journal of Applied Mechanicsvol 31 pp 621ndash626 1964

[13] P G Kessel ldquoResonances Excited in an Elastically ConnectedDouble-Beam System by a Cyclic Moving LoadrdquoThe Journal ofthe Acoustical Society of America vol 40 no 3 pp 684ndash6871966

[14] S Chonan ldquoDynamical behaviours of elastically connecteddouble-beam systems subjected to an impulsive loadrdquo Bulletinof JSME vol 19 no 132 pp 595ndash603 1976

[15] H V Vu A M Ordonez and B H Karnopp ldquoVibration of adouble-beam systemrdquo Journal of Sound and Vibration vol 229no 4 pp 807ndash822 2000

[16] H J Xu Z H Ding Z R Lu and J K Liu ldquoStructural DamageDetection Using a Modified Artificial Bee Colony AlgorithmrdquoCMES-Computer Modeling in Engineering amp Sciences vol 111pp 335ndash355 2016

[17] Z Oniszczuk ldquoFree transverse vibrations of elastically con-nected simply supported double-beam complex systemrdquo Jour-nal of Sound and Vibration vol 232 no 2 pp 387ndash403 2000

[18] Z Oniszczuk ldquoForced transverse vibrations of an elasticallyconnected complex simply supported double-beam systemrdquoJournal of Sound and Vibration vol 264 no 2 pp 273ndash2862003

[19] Y Q Zhang Y Lu S L Wang and X Liu ldquoVibration andbuckling of a double-beam system under compressive axialloadingrdquo Journal of Sound and Vibration vol 318 no 1-2 pp341ndash352 2008

[20] V Stojanovic and P Kozic ldquoForced transverse vibration ofRayleigh and Timoshenko double-beam system with effect of

14 Shock and Vibration

compressive axial loadrdquo International Journal of MechanicalSciences vol 60 no 1 pp 59ndash71 2012

[21] A Palmeri and S Adhikari ldquoA Galerkin-type state-spaceapproach for transverse vibrations of slender double-beamsystems with viscoelastic inner layerrdquo Journal of Sound andVibration vol 330 no 26 pp 6372ndash6386 2011

[22] Z Zhang X Huang Z Zhang and H Hua ldquoOn the transversevibration of Timoshenko double-beam systems coupled withvarious discontinuitiesrdquo International Journal of MechanicalSciences vol 89 pp 222ndash241 2014

[23] Y X Li Z J Hu and L Z Sun ldquoDynamical behavior of adouble-beam system interconnected by a viscoelastic layerrdquoInternational Journal of Mechanical Sciences vol 105 pp 291ndash303 2016

[24] M Abu-Hilal ldquoDynamic response of a double EulerndashBernoullibeam due to a moving constant loadrdquo Journal of Sound andVibration vol 297 no 3-5 pp 477ndash491 2006

[25] Y Wu and Y Gao ldquoAnalytical Solutions for Simply SupportedViscously Damped Double-Beam System under Moving Har-monic Loadsrdquo Journal of Engineering Mechanics vol 141 no 7p 04015004 2015

[26] Y-H Chen and Z-M Shiu ldquoResonant curves of an elevatedrailway to harmonic moving loadsrdquo International Journal ofStructural Stability and Dynamics vol 4 no 2 pp 237ndash2572004

[27] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of aninfinite tomoshenko beam on a viscoelastic foundation to aharmonic moving loadrdquo Journal of Sound and Vibration vol241 no 5 pp 809ndash824 2001

[28] M FMHussein andH EMHunt ldquoModelling of floating-slabtracks with continuous slabs under oscillating moving loadsrdquoJournal of Sound andVibration vol 297 no 1-2 pp 37ndash54 2006

[29] H P Wang J Li and K Zhang ldquoVibration analysis of themaglev guideway with the moving loadrdquo Journal of Sound andVibration vol 305 no 4-5 pp 621ndash640 2007

[30] V Stojanovic and M Petkovic ldquoMoment Lyapunov exponentsand stochastic stability of a three-dimensional system on elasticfoundation using a perturbation approachrdquo Journal of AppliedMechanics vol 80 no 5 2013

[31] L Jun and H Hongxing ldquoDynamic stiffness vibration analysisof an elastically connected three-beam systemrdquo Applied Acous-tics vol 69 no 7 pp 591ndash600 2008

[32] J Li Y Chen and H Hua ldquoExact dynamic stiffness matrixof a Timoshenko three-beam systemrdquo International Journal ofMechanical Sciences vol 50 no 6 pp 1023ndash1034 2008

[33] S S Rao ldquoNatural vibrations of systems of elastically connectedTimoshenko beams rdquo The Journal of the Acoustical Society ofAmerica vol 55 no 6 pp 1232ndash1237 1974

[34] Q Mao ldquoFree vibration analysis of elastically connectedmultiple-beams by using the Adomianmodified decompositionmethodrdquo Journal of Sound and Vibration vol 331 no 11 pp2532ndash2542 2012

[35] V Stojanovic P Kozic and G Janevski ldquoExact closed-formsolutions for the natural frequencies and stability of elasticallyconnected multiple beam system using Timoshenko and high-order shear deformation theoryrdquo Journal of Sound and Vibra-tion vol 332 no 3 pp 563ndash576 2013

[36] M H Kargarnovin M T Ahmadian and R-A Jafari-Talookolaei ldquoAnalytical solution for the dynamic analysis of adelaminated composite beam traversed by a moving constantforcerdquo Journal of Vibration and Control vol 19 no 10 pp 1524ndash1537 2013

[37] A Ariaei S Ziaei-Rad and M Ghayour ldquoTransverse vibrationof a multiple-Timoshenko beam system with intermediateelastic connections due to a moving loadrdquo Archive of AppliedMechanics vol 81 no 3 pp 263ndash281 2011

[38] Y Povstenko and J Klekot ldquoThe Dirichlet problem for thetime-fractional advection-diffusion equation in a line segmentrdquoBoundary Value Problems Paper No 89 8 pages 2016

[39] F Durbin ldquoNumerical inversion of Laplace transforms anefficient improvement to Dubner and Abatersquos methodrdquo TheComputer Journal vol 17 pp 371ndash376 1974

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Submit your manuscripts atwwwhindawicom

8 Shock and Vibration

ANSYSMATLAB

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

01 02 03 04 0500t (s)

(a)

ANSYSMATLAB

01 02 03 04 0500t (s)

minus028

minus021

minus014

minus007

000

007

Defl

ectio

n (m

m)

(b)

ANSYSMATLAB

minus024

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

(c)

ANSYSMATLAB

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

01 02 03 04 0500t (s)

(d)

Figure 6 Mid-span deflection time-history for quadruple simply supported beam system with load moving speed v = 64 ms (a) first beam(b) second beam (c) third beam (d) fourth beam

First Beam

40 80 120 160 2000v (ms)

096

100

104

108

112

116

120

Defl

ectio

n (m

m)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 7 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam (a) first beam (b)second beam

Shock and Vibration 9

First Beam038

040

042

044

046

048

050

052D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 8 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 100I1 (a) first beam (b) second beam

First Beam

40 80 120 160 2000v (ms)

0060

0063

0066

0069

0072

0075

0078

Defl

ectio

n (m

m)

(a)

Second Beam0040

0044

0048

0052

0056

0060D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(b)

Figure 9 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 105I1 (a) first beam (b) second beam

Table 1 Relationship between resonance (cancellation) speed and amplitude

Speed at the wavecrest (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)

Speed at the wavetrough (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)189 1024 0161 204 1009 0159232 1025 0162 253 1015 0160299 1027 0165 334 1011 0162411 1039 0169 487 1004 0163687 1058 0180 881 0996 0164

Figure 12 show that interlayer stiffness variation has agreater effect on the dynamic response of the first sim-ply supported beam and the mid-span dynamic deflectionamplitude of the first simply supported beam decreasedwith the increase in interlayer stiffness while the interlayer

stiffness has a negligible effect on the mid-span dynamictime-history of the second simply supported beam Thereason is that in the double simply supported rail-bridgesystem the differences in the stiffness of the first and secondbeams are large the flexural rigidity of the first beam is

10 Shock and Vibration

v=189msv=232msv=299ms

v=411msv=687ms

4 8 12 16 20 24 28 320Load position (m)

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

(a)

v=189msv=232msv=299ms

v=411msv=687ms

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(b)

Figure 10 Mid-span deflection time-history for double simply supported rail-bridge system under resonance condition (a) first beam (b)second beam

v=204msv=253msv=334ms

v=487msv=881ms

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(a)

0 4 8 12 16 20 24 28 32

v=204msv=253msv=334ms

v=487msv=881ms

Load position (m)

minus020

minus016

minus012

minus008

minus004

000

Defl

ectio

n (m

m)

(b)

Figure 11 Mid-span deflection time-history for the double simply supported rail-bridge system under cancellation condition (a) first beam(b) second beam

much smaller than that of the second beam and the bendingresistance of the second beam plays a leading role in thebending resistance of the entire system

Figures 13 and 14 show the effects of mass variationper unit length of the first and second beams on mid-span deflection time-history for double simply supportedrail-bridge system respectively Figure 13 shows that underdifferent masses of the first beam the time-history curvesof mid-span deflection dynamic response for the two simplysupported beams are consistent Therefore the effect of massvariation of the first simply supported beam on the mid-spandeflection dynamic response of the double-beam system canbe neglected Figure 14 shows that the mass variation of thesecond simply supported beam has a significant effect on the

mid-span deflection dynamic response of both beams whenthemass per unit length of the second simply supported beamincreases to four times of the original mass the amplitudes ofmid-span deflection dynamic response of the first and secondsimply supported beams both increase significantly with thegrowth rate reaching up to 10 and 45 respectively

Figures 15 and 16 show the effect of flexural rigidityvariations of the first and second simply supported beamson the mid-span deflection time-history for double simplysupported rail-bridge system Figure 15 shows that the mid-span dynamic deflection amplitude of the first simply sup-ported beam significantly decreased with the increase in thestiffness of this beam while the stiffness of the first beamhas a smaller effect on the mid-span dynamic deflection

Shock and Vibration 11

025kr050kr100kr

200kr400kr

minus28

minus24

minus20

minus16

minus12

minus08

minus04

00

04

Defl

ectio

n (m

m)

008 016 024 032000t (s)

(a)

025kr050kr100kr

200kr400kr

008 016 024 032000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 12 Comparison diagram of mid-span deflection time-history for the double simply supported rail-bridge system under differentinterlayer stiffness (a) first beam (b) second beam

100

200400

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025Ｇ1050Ｇ1

Ｇ1

Ｇ1Ｇ1

(a)

025050100

200400

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

Ｇ1Ｇ1

Ｇ1Ｇ1

Ｇ1

(b)

Figure 13 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof first beam (a) first beam (b) second beam

amplitude of the second simply supported beam Underdifferent stiffness of the first beam the mid-span deflectiondynamic time-history curves for the second simply supportedbeam are consistent Figure 16 shows that the flexural rigidityvariations of the second simply supported beam have a sig-nificant effect on the mid-span deflection dynamic responseof double simply supported rail-bridge system In additionthe mid-span deflection dynamic response amplitudes ofboth beams significantly increased with the decrease in theflexural stiffness of second beam In practical applicationsthe dynamic response of the rail-bridge system in high-speedrailways can be decreased by moderately decreasing the massof the second beam and increasing the flexural stiffness ofboth simply supported beams

4 Conclusions

In this study a dynamic analysis model was established for amultiple simply supported beam system subjected to movingloads in combination of finite sin-Fourier transform andnumerical Laplace transform based on Durbin transforman analytical calculation method was developed for thespatial displacement time-domain response of a multiplesimply supported beam systemThe following conclusions aredrawn by comparing the examples and ANSYS finite elementnumerical computation method(1) A simply supported rail-bridge system of high-speedrailways with a span of 32 m in China was taken as theexample and the theoretically calculated results were com-pared with the results obtained from ANSYS finite element

12 Shock and Vibration

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

100

200400

025Ｇ2050Ｇ2

Ｇ2

Ｇ2Ｇ2

(a)

minus030

minus024

minus018

minus012

minus006

000

006

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025050100

200400Ｇ2

Ｇ2

Ｇ2Ｇ2

Ｇ2

(b)

Figure 14 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof second beam (a) first beam (b) second beam

008 016 024 032000t (s)

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02510501

10012001

(a)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

02510501

10012001

(b)

Figure 15 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof first beam (a) first beam (b) second beam

numerical calculation validating the analytical calculationmethod reported in this paper(2) The effect of load moving speed on the dynamicresponse of a double simply supported rail-bridge system ofhigh-speed railways subjected tomoving loads was evaluatedThe results indicate that the double simply supported rail-bridge system has multiple points of resonance and cancel-lation which should be paid attention in engineering(3) For a typical double simply supported rail-bridgesystem of high-speed railway in China because of a largedifference in the flexural rigidity of the first and secondbeams the variation in interlayer stiffness mainly affectsthe dynamic response of the first simply supported beamexhibiting a positive correlation When the mass and flexuralstiffness parameters of first simply supported beam vary they

mainly affect the dynamic response of this layer of simplysupported beam but slightly affect the dynamic responseof second simply supported beam The variation in theparameters of second simply supported beam has a greatereffect on the double simply supported beam system and adecrease in the mass of this layer of simply supported beamand an increase in the flexural rigidity of this beam canboth decrease the dynamic response of the double simplysupported rail-bridge system

Data Availability

The data used to support the findings of this study areincluded within the article

Shock and Vibration 13

minus14

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

05021002

20024002

(a)

008 016 024 032000t (s)

minus048

minus040

minus032

minus024

minus016

minus008

000

008

Defl

ectio

n (m

m)

05021002

20024002

(b)

Figure 16 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof second beam (a) first beam (b) second beam

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51708630)

References

[1] H Ouyang ldquoMoving-load dynamic problems a tutorial (with abrief overview)rdquoMechanical Systems and Signal Processing vol25 no 6 pp 2039ndash2060 2011

[2] D G Duffy ldquoThe response of an infinite railroad track to amoving vibrating massrdquo Journal of Applied Mechanics vol 57no 1 pp 66ndash73 1990

[3] L Fryba Vibration of solids and structures under moving loadsSpringer Science amp Business Media 2013

[4] G Michaltsos D Sophianopoulos and A N Kounadis ldquoTheeffect of a moving mass and other parameters on the dynamicresponse of a simply supported beamrdquo Journal of Sound andVibration vol 191 no 3 pp 357ndash362 1996

[5] P Museros E Moliner and M D Martınez-Rodrigo ldquoFreevibrations of simply-supported beam bridges under movingloads Maximum resonance cancellation and resonant verticalaccelerationrdquo Journal of Sound and Vibration vol 332 no 2 pp326ndash345 2013

[6] P Sniady ldquoDynamic response of a Timoshenko beam to amoving forcerdquo Journal of Applied Mechanics vol 75 no 2Article ID 024503 2008

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

[8] K Y Lee and A A Renshaw ldquoSolution of the moving massproblem using complex eigenfunction expansionsrdquo Journal ofApplied Mechanics vol 67 no 4 pp 823ndash827 2000

[9] U Lee ldquoSeparation between the flexible structure and themoving mass sliding on itrdquo Journal of Sound and Vibration vol209 no 5 pp 867ndash877 1998

[10] S A Eftekhari and A A Jafari ldquoCoupling ritz method andtriangular quadrature rule for moving mass problemrdquo Journalof Applied Mechanics vol 79 no 2 2012

[11] M Dublin andH R Friedrich ldquoForced responses of two elasticbeams interconnected by spring-damper systemsrdquo Journal of theAeronautical Sciences vol 23 pp 824ndash829 1956

[12] J M Seelig and W H Hoppmann ldquoImpact on an elasticallyconnected double-beam systemrdquo Journal of Applied Mechanicsvol 31 pp 621ndash626 1964

[13] P G Kessel ldquoResonances Excited in an Elastically ConnectedDouble-Beam System by a Cyclic Moving LoadrdquoThe Journal ofthe Acoustical Society of America vol 40 no 3 pp 684ndash6871966

[14] S Chonan ldquoDynamical behaviours of elastically connecteddouble-beam systems subjected to an impulsive loadrdquo Bulletinof JSME vol 19 no 132 pp 595ndash603 1976

[15] H V Vu A M Ordonez and B H Karnopp ldquoVibration of adouble-beam systemrdquo Journal of Sound and Vibration vol 229no 4 pp 807ndash822 2000

[16] H J Xu Z H Ding Z R Lu and J K Liu ldquoStructural DamageDetection Using a Modified Artificial Bee Colony AlgorithmrdquoCMES-Computer Modeling in Engineering amp Sciences vol 111pp 335ndash355 2016

[17] Z Oniszczuk ldquoFree transverse vibrations of elastically con-nected simply supported double-beam complex systemrdquo Jour-nal of Sound and Vibration vol 232 no 2 pp 387ndash403 2000

[18] Z Oniszczuk ldquoForced transverse vibrations of an elasticallyconnected complex simply supported double-beam systemrdquoJournal of Sound and Vibration vol 264 no 2 pp 273ndash2862003

[19] Y Q Zhang Y Lu S L Wang and X Liu ldquoVibration andbuckling of a double-beam system under compressive axialloadingrdquo Journal of Sound and Vibration vol 318 no 1-2 pp341ndash352 2008

[20] V Stojanovic and P Kozic ldquoForced transverse vibration ofRayleigh and Timoshenko double-beam system with effect of

14 Shock and Vibration

compressive axial loadrdquo International Journal of MechanicalSciences vol 60 no 1 pp 59ndash71 2012

[21] A Palmeri and S Adhikari ldquoA Galerkin-type state-spaceapproach for transverse vibrations of slender double-beamsystems with viscoelastic inner layerrdquo Journal of Sound andVibration vol 330 no 26 pp 6372ndash6386 2011

[22] Z Zhang X Huang Z Zhang and H Hua ldquoOn the transversevibration of Timoshenko double-beam systems coupled withvarious discontinuitiesrdquo International Journal of MechanicalSciences vol 89 pp 222ndash241 2014

[23] Y X Li Z J Hu and L Z Sun ldquoDynamical behavior of adouble-beam system interconnected by a viscoelastic layerrdquoInternational Journal of Mechanical Sciences vol 105 pp 291ndash303 2016

[24] M Abu-Hilal ldquoDynamic response of a double EulerndashBernoullibeam due to a moving constant loadrdquo Journal of Sound andVibration vol 297 no 3-5 pp 477ndash491 2006

[25] Y Wu and Y Gao ldquoAnalytical Solutions for Simply SupportedViscously Damped Double-Beam System under Moving Har-monic Loadsrdquo Journal of Engineering Mechanics vol 141 no 7p 04015004 2015

[26] Y-H Chen and Z-M Shiu ldquoResonant curves of an elevatedrailway to harmonic moving loadsrdquo International Journal ofStructural Stability and Dynamics vol 4 no 2 pp 237ndash2572004

[27] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of aninfinite tomoshenko beam on a viscoelastic foundation to aharmonic moving loadrdquo Journal of Sound and Vibration vol241 no 5 pp 809ndash824 2001

[28] M FMHussein andH EMHunt ldquoModelling of floating-slabtracks with continuous slabs under oscillating moving loadsrdquoJournal of Sound andVibration vol 297 no 1-2 pp 37ndash54 2006

[29] H P Wang J Li and K Zhang ldquoVibration analysis of themaglev guideway with the moving loadrdquo Journal of Sound andVibration vol 305 no 4-5 pp 621ndash640 2007

[30] V Stojanovic and M Petkovic ldquoMoment Lyapunov exponentsand stochastic stability of a three-dimensional system on elasticfoundation using a perturbation approachrdquo Journal of AppliedMechanics vol 80 no 5 2013

[31] L Jun and H Hongxing ldquoDynamic stiffness vibration analysisof an elastically connected three-beam systemrdquo Applied Acous-tics vol 69 no 7 pp 591ndash600 2008

[32] J Li Y Chen and H Hua ldquoExact dynamic stiffness matrixof a Timoshenko three-beam systemrdquo International Journal ofMechanical Sciences vol 50 no 6 pp 1023ndash1034 2008

[33] S S Rao ldquoNatural vibrations of systems of elastically connectedTimoshenko beams rdquo The Journal of the Acoustical Society ofAmerica vol 55 no 6 pp 1232ndash1237 1974

[34] Q Mao ldquoFree vibration analysis of elastically connectedmultiple-beams by using the Adomianmodified decompositionmethodrdquo Journal of Sound and Vibration vol 331 no 11 pp2532ndash2542 2012

[35] V Stojanovic P Kozic and G Janevski ldquoExact closed-formsolutions for the natural frequencies and stability of elasticallyconnected multiple beam system using Timoshenko and high-order shear deformation theoryrdquo Journal of Sound and Vibra-tion vol 332 no 3 pp 563ndash576 2013

[36] M H Kargarnovin M T Ahmadian and R-A Jafari-Talookolaei ldquoAnalytical solution for the dynamic analysis of adelaminated composite beam traversed by a moving constantforcerdquo Journal of Vibration and Control vol 19 no 10 pp 1524ndash1537 2013

[37] A Ariaei S Ziaei-Rad and M Ghayour ldquoTransverse vibrationof a multiple-Timoshenko beam system with intermediateelastic connections due to a moving loadrdquo Archive of AppliedMechanics vol 81 no 3 pp 263ndash281 2011

[38] Y Povstenko and J Klekot ldquoThe Dirichlet problem for thetime-fractional advection-diffusion equation in a line segmentrdquoBoundary Value Problems Paper No 89 8 pages 2016

[39] F Durbin ldquoNumerical inversion of Laplace transforms anefficient improvement to Dubner and Abatersquos methodrdquo TheComputer Journal vol 17 pp 371ndash376 1974

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Submit your manuscripts atwwwhindawicom

Shock and Vibration 9

First Beam038

040

042

044

046

048

050

052D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(a)

Second Beam

40 80 120 160 2000v (ms)

014

016

018

020

022

024

026

028

Defl

ectio

n (m

m)

(b)

Figure 8 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 100I1 (a) first beam (b) second beam

First Beam

40 80 120 160 2000v (ms)

0060

0063

0066

0069

0072

0075

0078

Defl

ectio

n (m

m)

(a)

Second Beam0040

0044

0048

0052

0056

0060D

eflec

tion

(mm

)

40 80 120 160 2000v (ms)

(b)

Figure 9 Relationship between maximum deflection and loading speed for mid-span of double simply supported beam with first beamrsquosmoment of inertia taking as 105I1 (a) first beam (b) second beam

Table 1 Relationship between resonance (cancellation) speed and amplitude

Speed at the wavecrest (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)

Speed at the wavetrough (ms)

Amplitude at themid-span of firstbeam (mm)

Amplitude at themid-span of

second beam (mm)189 1024 0161 204 1009 0159232 1025 0162 253 1015 0160299 1027 0165 334 1011 0162411 1039 0169 487 1004 0163687 1058 0180 881 0996 0164

Figure 12 show that interlayer stiffness variation has agreater effect on the dynamic response of the first sim-ply supported beam and the mid-span dynamic deflectionamplitude of the first simply supported beam decreasedwith the increase in interlayer stiffness while the interlayer

stiffness has a negligible effect on the mid-span dynamictime-history of the second simply supported beam Thereason is that in the double simply supported rail-bridgesystem the differences in the stiffness of the first and secondbeams are large the flexural rigidity of the first beam is

10 Shock and Vibration

v=189msv=232msv=299ms

v=411msv=687ms

4 8 12 16 20 24 28 320Load position (m)

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

(a)

v=189msv=232msv=299ms

v=411msv=687ms

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(b)

Figure 10 Mid-span deflection time-history for double simply supported rail-bridge system under resonance condition (a) first beam (b)second beam

v=204msv=253msv=334ms

v=487msv=881ms

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(a)

0 4 8 12 16 20 24 28 32

v=204msv=253msv=334ms

v=487msv=881ms

Load position (m)

minus020

minus016

minus012

minus008

minus004

000

Defl

ectio

n (m

m)

(b)

Figure 11 Mid-span deflection time-history for the double simply supported rail-bridge system under cancellation condition (a) first beam(b) second beam

much smaller than that of the second beam and the bendingresistance of the second beam plays a leading role in thebending resistance of the entire system

Figures 13 and 14 show the effects of mass variationper unit length of the first and second beams on mid-span deflection time-history for double simply supportedrail-bridge system respectively Figure 13 shows that underdifferent masses of the first beam the time-history curvesof mid-span deflection dynamic response for the two simplysupported beams are consistent Therefore the effect of massvariation of the first simply supported beam on the mid-spandeflection dynamic response of the double-beam system canbe neglected Figure 14 shows that the mass variation of thesecond simply supported beam has a significant effect on the

mid-span deflection dynamic response of both beams whenthemass per unit length of the second simply supported beamincreases to four times of the original mass the amplitudes ofmid-span deflection dynamic response of the first and secondsimply supported beams both increase significantly with thegrowth rate reaching up to 10 and 45 respectively

Figures 15 and 16 show the effect of flexural rigidityvariations of the first and second simply supported beamson the mid-span deflection time-history for double simplysupported rail-bridge system Figure 15 shows that the mid-span dynamic deflection amplitude of the first simply sup-ported beam significantly decreased with the increase in thestiffness of this beam while the stiffness of the first beamhas a smaller effect on the mid-span dynamic deflection

Shock and Vibration 11

025kr050kr100kr

200kr400kr

minus28

minus24

minus20

minus16

minus12

minus08

minus04

00

04

Defl

ectio

n (m

m)

008 016 024 032000t (s)

(a)

025kr050kr100kr

200kr400kr

008 016 024 032000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 12 Comparison diagram of mid-span deflection time-history for the double simply supported rail-bridge system under differentinterlayer stiffness (a) first beam (b) second beam

100

200400

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025Ｇ1050Ｇ1

Ｇ1

Ｇ1Ｇ1

(a)

025050100

200400

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

Ｇ1Ｇ1

Ｇ1Ｇ1

Ｇ1

(b)

Figure 13 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof first beam (a) first beam (b) second beam

amplitude of the second simply supported beam Underdifferent stiffness of the first beam the mid-span deflectiondynamic time-history curves for the second simply supportedbeam are consistent Figure 16 shows that the flexural rigidityvariations of the second simply supported beam have a sig-nificant effect on the mid-span deflection dynamic responseof double simply supported rail-bridge system In additionthe mid-span deflection dynamic response amplitudes ofboth beams significantly increased with the decrease in theflexural stiffness of second beam In practical applicationsthe dynamic response of the rail-bridge system in high-speedrailways can be decreased by moderately decreasing the massof the second beam and increasing the flexural stiffness ofboth simply supported beams

4 Conclusions

In this study a dynamic analysis model was established for amultiple simply supported beam system subjected to movingloads in combination of finite sin-Fourier transform andnumerical Laplace transform based on Durbin transforman analytical calculation method was developed for thespatial displacement time-domain response of a multiplesimply supported beam systemThe following conclusions aredrawn by comparing the examples and ANSYS finite elementnumerical computation method(1) A simply supported rail-bridge system of high-speedrailways with a span of 32 m in China was taken as theexample and the theoretically calculated results were com-pared with the results obtained from ANSYS finite element

12 Shock and Vibration

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

100

200400

025Ｇ2050Ｇ2

Ｇ2

Ｇ2Ｇ2

(a)

minus030

minus024

minus018

minus012

minus006

000

006

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025050100

200400Ｇ2

Ｇ2

Ｇ2Ｇ2

Ｇ2

(b)

Figure 14 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof second beam (a) first beam (b) second beam

008 016 024 032000t (s)

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02510501

10012001

(a)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

02510501

10012001

(b)

Figure 15 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof first beam (a) first beam (b) second beam

numerical calculation validating the analytical calculationmethod reported in this paper(2) The effect of load moving speed on the dynamicresponse of a double simply supported rail-bridge system ofhigh-speed railways subjected tomoving loads was evaluatedThe results indicate that the double simply supported rail-bridge system has multiple points of resonance and cancel-lation which should be paid attention in engineering(3) For a typical double simply supported rail-bridgesystem of high-speed railway in China because of a largedifference in the flexural rigidity of the first and secondbeams the variation in interlayer stiffness mainly affectsthe dynamic response of the first simply supported beamexhibiting a positive correlation When the mass and flexuralstiffness parameters of first simply supported beam vary they

mainly affect the dynamic response of this layer of simplysupported beam but slightly affect the dynamic responseof second simply supported beam The variation in theparameters of second simply supported beam has a greatereffect on the double simply supported beam system and adecrease in the mass of this layer of simply supported beamand an increase in the flexural rigidity of this beam canboth decrease the dynamic response of the double simplysupported rail-bridge system

Data Availability

The data used to support the findings of this study areincluded within the article

Shock and Vibration 13

minus14

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

05021002

20024002

(a)

008 016 024 032000t (s)

minus048

minus040

minus032

minus024

minus016

minus008

000

008

Defl

ectio

n (m

m)

05021002

20024002

(b)

Figure 16 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof second beam (a) first beam (b) second beam

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51708630)

References

[1] H Ouyang ldquoMoving-load dynamic problems a tutorial (with abrief overview)rdquoMechanical Systems and Signal Processing vol25 no 6 pp 2039ndash2060 2011

[2] D G Duffy ldquoThe response of an infinite railroad track to amoving vibrating massrdquo Journal of Applied Mechanics vol 57no 1 pp 66ndash73 1990

[3] L Fryba Vibration of solids and structures under moving loadsSpringer Science amp Business Media 2013

[4] G Michaltsos D Sophianopoulos and A N Kounadis ldquoTheeffect of a moving mass and other parameters on the dynamicresponse of a simply supported beamrdquo Journal of Sound andVibration vol 191 no 3 pp 357ndash362 1996

[5] P Museros E Moliner and M D Martınez-Rodrigo ldquoFreevibrations of simply-supported beam bridges under movingloads Maximum resonance cancellation and resonant verticalaccelerationrdquo Journal of Sound and Vibration vol 332 no 2 pp326ndash345 2013

[6] P Sniady ldquoDynamic response of a Timoshenko beam to amoving forcerdquo Journal of Applied Mechanics vol 75 no 2Article ID 024503 2008

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

[8] K Y Lee and A A Renshaw ldquoSolution of the moving massproblem using complex eigenfunction expansionsrdquo Journal ofApplied Mechanics vol 67 no 4 pp 823ndash827 2000

[9] U Lee ldquoSeparation between the flexible structure and themoving mass sliding on itrdquo Journal of Sound and Vibration vol209 no 5 pp 867ndash877 1998

[10] S A Eftekhari and A A Jafari ldquoCoupling ritz method andtriangular quadrature rule for moving mass problemrdquo Journalof Applied Mechanics vol 79 no 2 2012

[11] M Dublin andH R Friedrich ldquoForced responses of two elasticbeams interconnected by spring-damper systemsrdquo Journal of theAeronautical Sciences vol 23 pp 824ndash829 1956

[12] J M Seelig and W H Hoppmann ldquoImpact on an elasticallyconnected double-beam systemrdquo Journal of Applied Mechanicsvol 31 pp 621ndash626 1964

[13] P G Kessel ldquoResonances Excited in an Elastically ConnectedDouble-Beam System by a Cyclic Moving LoadrdquoThe Journal ofthe Acoustical Society of America vol 40 no 3 pp 684ndash6871966

[14] S Chonan ldquoDynamical behaviours of elastically connecteddouble-beam systems subjected to an impulsive loadrdquo Bulletinof JSME vol 19 no 132 pp 595ndash603 1976

[15] H V Vu A M Ordonez and B H Karnopp ldquoVibration of adouble-beam systemrdquo Journal of Sound and Vibration vol 229no 4 pp 807ndash822 2000

[16] H J Xu Z H Ding Z R Lu and J K Liu ldquoStructural DamageDetection Using a Modified Artificial Bee Colony AlgorithmrdquoCMES-Computer Modeling in Engineering amp Sciences vol 111pp 335ndash355 2016

[17] Z Oniszczuk ldquoFree transverse vibrations of elastically con-nected simply supported double-beam complex systemrdquo Jour-nal of Sound and Vibration vol 232 no 2 pp 387ndash403 2000

[18] Z Oniszczuk ldquoForced transverse vibrations of an elasticallyconnected complex simply supported double-beam systemrdquoJournal of Sound and Vibration vol 264 no 2 pp 273ndash2862003

[19] Y Q Zhang Y Lu S L Wang and X Liu ldquoVibration andbuckling of a double-beam system under compressive axialloadingrdquo Journal of Sound and Vibration vol 318 no 1-2 pp341ndash352 2008

[20] V Stojanovic and P Kozic ldquoForced transverse vibration ofRayleigh and Timoshenko double-beam system with effect of

14 Shock and Vibration

compressive axial loadrdquo International Journal of MechanicalSciences vol 60 no 1 pp 59ndash71 2012

[21] A Palmeri and S Adhikari ldquoA Galerkin-type state-spaceapproach for transverse vibrations of slender double-beamsystems with viscoelastic inner layerrdquo Journal of Sound andVibration vol 330 no 26 pp 6372ndash6386 2011

[22] Z Zhang X Huang Z Zhang and H Hua ldquoOn the transversevibration of Timoshenko double-beam systems coupled withvarious discontinuitiesrdquo International Journal of MechanicalSciences vol 89 pp 222ndash241 2014

[23] Y X Li Z J Hu and L Z Sun ldquoDynamical behavior of adouble-beam system interconnected by a viscoelastic layerrdquoInternational Journal of Mechanical Sciences vol 105 pp 291ndash303 2016

[24] M Abu-Hilal ldquoDynamic response of a double EulerndashBernoullibeam due to a moving constant loadrdquo Journal of Sound andVibration vol 297 no 3-5 pp 477ndash491 2006

[25] Y Wu and Y Gao ldquoAnalytical Solutions for Simply SupportedViscously Damped Double-Beam System under Moving Har-monic Loadsrdquo Journal of Engineering Mechanics vol 141 no 7p 04015004 2015

[26] Y-H Chen and Z-M Shiu ldquoResonant curves of an elevatedrailway to harmonic moving loadsrdquo International Journal ofStructural Stability and Dynamics vol 4 no 2 pp 237ndash2572004

[27] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of aninfinite tomoshenko beam on a viscoelastic foundation to aharmonic moving loadrdquo Journal of Sound and Vibration vol241 no 5 pp 809ndash824 2001

[28] M FMHussein andH EMHunt ldquoModelling of floating-slabtracks with continuous slabs under oscillating moving loadsrdquoJournal of Sound andVibration vol 297 no 1-2 pp 37ndash54 2006

[29] H P Wang J Li and K Zhang ldquoVibration analysis of themaglev guideway with the moving loadrdquo Journal of Sound andVibration vol 305 no 4-5 pp 621ndash640 2007

[30] V Stojanovic and M Petkovic ldquoMoment Lyapunov exponentsand stochastic stability of a three-dimensional system on elasticfoundation using a perturbation approachrdquo Journal of AppliedMechanics vol 80 no 5 2013

[31] L Jun and H Hongxing ldquoDynamic stiffness vibration analysisof an elastically connected three-beam systemrdquo Applied Acous-tics vol 69 no 7 pp 591ndash600 2008

[32] J Li Y Chen and H Hua ldquoExact dynamic stiffness matrixof a Timoshenko three-beam systemrdquo International Journal ofMechanical Sciences vol 50 no 6 pp 1023ndash1034 2008

[33] S S Rao ldquoNatural vibrations of systems of elastically connectedTimoshenko beams rdquo The Journal of the Acoustical Society ofAmerica vol 55 no 6 pp 1232ndash1237 1974

[34] Q Mao ldquoFree vibration analysis of elastically connectedmultiple-beams by using the Adomianmodified decompositionmethodrdquo Journal of Sound and Vibration vol 331 no 11 pp2532ndash2542 2012

[35] V Stojanovic P Kozic and G Janevski ldquoExact closed-formsolutions for the natural frequencies and stability of elasticallyconnected multiple beam system using Timoshenko and high-order shear deformation theoryrdquo Journal of Sound and Vibra-tion vol 332 no 3 pp 563ndash576 2013

[36] M H Kargarnovin M T Ahmadian and R-A Jafari-Talookolaei ldquoAnalytical solution for the dynamic analysis of adelaminated composite beam traversed by a moving constantforcerdquo Journal of Vibration and Control vol 19 no 10 pp 1524ndash1537 2013

[37] A Ariaei S Ziaei-Rad and M Ghayour ldquoTransverse vibrationof a multiple-Timoshenko beam system with intermediateelastic connections due to a moving loadrdquo Archive of AppliedMechanics vol 81 no 3 pp 263ndash281 2011

[38] Y Povstenko and J Klekot ldquoThe Dirichlet problem for thetime-fractional advection-diffusion equation in a line segmentrdquoBoundary Value Problems Paper No 89 8 pages 2016

[39] F Durbin ldquoNumerical inversion of Laplace transforms anefficient improvement to Dubner and Abatersquos methodrdquo TheComputer Journal vol 17 pp 371ndash376 1974

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

10 Shock and Vibration

v=189msv=232msv=299ms

v=411msv=687ms

4 8 12 16 20 24 28 320Load position (m)

minus12

minus10

minus08

minus06

minus04

minus02

00

02D

eflec

tion

(mm

)

(a)

v=189msv=232msv=299ms

v=411msv=687ms

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(b)

Figure 10 Mid-span deflection time-history for double simply supported rail-bridge system under resonance condition (a) first beam (b)second beam

v=204msv=253msv=334ms

v=487msv=881ms

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

4 8 12 16 20 24 28 320Load position (m)

(a)

0 4 8 12 16 20 24 28 32

v=204msv=253msv=334ms

v=487msv=881ms

Load position (m)

minus020

minus016

minus012

minus008

minus004

000

Defl

ectio

n (m

m)

(b)

Figure 11 Mid-span deflection time-history for the double simply supported rail-bridge system under cancellation condition (a) first beam(b) second beam

much smaller than that of the second beam and the bendingresistance of the second beam plays a leading role in thebending resistance of the entire system

Figures 13 and 14 show the effects of mass variationper unit length of the first and second beams on mid-span deflection time-history for double simply supportedrail-bridge system respectively Figure 13 shows that underdifferent masses of the first beam the time-history curvesof mid-span deflection dynamic response for the two simplysupported beams are consistent Therefore the effect of massvariation of the first simply supported beam on the mid-spandeflection dynamic response of the double-beam system canbe neglected Figure 14 shows that the mass variation of thesecond simply supported beam has a significant effect on the

mid-span deflection dynamic response of both beams whenthemass per unit length of the second simply supported beamincreases to four times of the original mass the amplitudes ofmid-span deflection dynamic response of the first and secondsimply supported beams both increase significantly with thegrowth rate reaching up to 10 and 45 respectively

Figures 15 and 16 show the effect of flexural rigidityvariations of the first and second simply supported beamson the mid-span deflection time-history for double simplysupported rail-bridge system Figure 15 shows that the mid-span dynamic deflection amplitude of the first simply sup-ported beam significantly decreased with the increase in thestiffness of this beam while the stiffness of the first beamhas a smaller effect on the mid-span dynamic deflection

Shock and Vibration 11

025kr050kr100kr

200kr400kr

minus28

minus24

minus20

minus16

minus12

minus08

minus04

00

04

Defl

ectio

n (m

m)

008 016 024 032000t (s)

(a)

025kr050kr100kr

200kr400kr

008 016 024 032000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 12 Comparison diagram of mid-span deflection time-history for the double simply supported rail-bridge system under differentinterlayer stiffness (a) first beam (b) second beam

100

200400

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025Ｇ1050Ｇ1

Ｇ1

Ｇ1Ｇ1

(a)

025050100

200400

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

Ｇ1Ｇ1

Ｇ1Ｇ1

Ｇ1

(b)

Figure 13 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof first beam (a) first beam (b) second beam

amplitude of the second simply supported beam Underdifferent stiffness of the first beam the mid-span deflectiondynamic time-history curves for the second simply supportedbeam are consistent Figure 16 shows that the flexural rigidityvariations of the second simply supported beam have a sig-nificant effect on the mid-span deflection dynamic responseof double simply supported rail-bridge system In additionthe mid-span deflection dynamic response amplitudes ofboth beams significantly increased with the decrease in theflexural stiffness of second beam In practical applicationsthe dynamic response of the rail-bridge system in high-speedrailways can be decreased by moderately decreasing the massof the second beam and increasing the flexural stiffness ofboth simply supported beams

4 Conclusions

In this study a dynamic analysis model was established for amultiple simply supported beam system subjected to movingloads in combination of finite sin-Fourier transform andnumerical Laplace transform based on Durbin transforman analytical calculation method was developed for thespatial displacement time-domain response of a multiplesimply supported beam systemThe following conclusions aredrawn by comparing the examples and ANSYS finite elementnumerical computation method(1) A simply supported rail-bridge system of high-speedrailways with a span of 32 m in China was taken as theexample and the theoretically calculated results were com-pared with the results obtained from ANSYS finite element

12 Shock and Vibration

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

100

200400

025Ｇ2050Ｇ2

Ｇ2

Ｇ2Ｇ2

(a)

minus030

minus024

minus018

minus012

minus006

000

006

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025050100

200400Ｇ2

Ｇ2

Ｇ2Ｇ2

Ｇ2

(b)

Figure 14 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof second beam (a) first beam (b) second beam

008 016 024 032000t (s)

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02510501

10012001

(a)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

02510501

10012001

(b)

Figure 15 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof first beam (a) first beam (b) second beam

numerical calculation validating the analytical calculationmethod reported in this paper(2) The effect of load moving speed on the dynamicresponse of a double simply supported rail-bridge system ofhigh-speed railways subjected tomoving loads was evaluatedThe results indicate that the double simply supported rail-bridge system has multiple points of resonance and cancel-lation which should be paid attention in engineering(3) For a typical double simply supported rail-bridgesystem of high-speed railway in China because of a largedifference in the flexural rigidity of the first and secondbeams the variation in interlayer stiffness mainly affectsthe dynamic response of the first simply supported beamexhibiting a positive correlation When the mass and flexuralstiffness parameters of first simply supported beam vary they

mainly affect the dynamic response of this layer of simplysupported beam but slightly affect the dynamic responseof second simply supported beam The variation in theparameters of second simply supported beam has a greatereffect on the double simply supported beam system and adecrease in the mass of this layer of simply supported beamand an increase in the flexural rigidity of this beam canboth decrease the dynamic response of the double simplysupported rail-bridge system

Data Availability

The data used to support the findings of this study areincluded within the article

Shock and Vibration 13

minus14

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

05021002

20024002

(a)

008 016 024 032000t (s)

minus048

minus040

minus032

minus024

minus016

minus008

000

008

Defl

ectio

n (m

m)

05021002

20024002

(b)

Figure 16 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof second beam (a) first beam (b) second beam

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51708630)

References

[1] H Ouyang ldquoMoving-load dynamic problems a tutorial (with abrief overview)rdquoMechanical Systems and Signal Processing vol25 no 6 pp 2039ndash2060 2011

[2] D G Duffy ldquoThe response of an infinite railroad track to amoving vibrating massrdquo Journal of Applied Mechanics vol 57no 1 pp 66ndash73 1990

[3] L Fryba Vibration of solids and structures under moving loadsSpringer Science amp Business Media 2013

[4] G Michaltsos D Sophianopoulos and A N Kounadis ldquoTheeffect of a moving mass and other parameters on the dynamicresponse of a simply supported beamrdquo Journal of Sound andVibration vol 191 no 3 pp 357ndash362 1996

[5] P Museros E Moliner and M D Martınez-Rodrigo ldquoFreevibrations of simply-supported beam bridges under movingloads Maximum resonance cancellation and resonant verticalaccelerationrdquo Journal of Sound and Vibration vol 332 no 2 pp326ndash345 2013

[6] P Sniady ldquoDynamic response of a Timoshenko beam to amoving forcerdquo Journal of Applied Mechanics vol 75 no 2Article ID 024503 2008

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

[8] K Y Lee and A A Renshaw ldquoSolution of the moving massproblem using complex eigenfunction expansionsrdquo Journal ofApplied Mechanics vol 67 no 4 pp 823ndash827 2000

[9] U Lee ldquoSeparation between the flexible structure and themoving mass sliding on itrdquo Journal of Sound and Vibration vol209 no 5 pp 867ndash877 1998

[10] S A Eftekhari and A A Jafari ldquoCoupling ritz method andtriangular quadrature rule for moving mass problemrdquo Journalof Applied Mechanics vol 79 no 2 2012

[11] M Dublin andH R Friedrich ldquoForced responses of two elasticbeams interconnected by spring-damper systemsrdquo Journal of theAeronautical Sciences vol 23 pp 824ndash829 1956

[12] J M Seelig and W H Hoppmann ldquoImpact on an elasticallyconnected double-beam systemrdquo Journal of Applied Mechanicsvol 31 pp 621ndash626 1964

[13] P G Kessel ldquoResonances Excited in an Elastically ConnectedDouble-Beam System by a Cyclic Moving LoadrdquoThe Journal ofthe Acoustical Society of America vol 40 no 3 pp 684ndash6871966

[14] S Chonan ldquoDynamical behaviours of elastically connecteddouble-beam systems subjected to an impulsive loadrdquo Bulletinof JSME vol 19 no 132 pp 595ndash603 1976

[15] H V Vu A M Ordonez and B H Karnopp ldquoVibration of adouble-beam systemrdquo Journal of Sound and Vibration vol 229no 4 pp 807ndash822 2000

[16] H J Xu Z H Ding Z R Lu and J K Liu ldquoStructural DamageDetection Using a Modified Artificial Bee Colony AlgorithmrdquoCMES-Computer Modeling in Engineering amp Sciences vol 111pp 335ndash355 2016

[17] Z Oniszczuk ldquoFree transverse vibrations of elastically con-nected simply supported double-beam complex systemrdquo Jour-nal of Sound and Vibration vol 232 no 2 pp 387ndash403 2000

[18] Z Oniszczuk ldquoForced transverse vibrations of an elasticallyconnected complex simply supported double-beam systemrdquoJournal of Sound and Vibration vol 264 no 2 pp 273ndash2862003

[19] Y Q Zhang Y Lu S L Wang and X Liu ldquoVibration andbuckling of a double-beam system under compressive axialloadingrdquo Journal of Sound and Vibration vol 318 no 1-2 pp341ndash352 2008

[20] V Stojanovic and P Kozic ldquoForced transverse vibration ofRayleigh and Timoshenko double-beam system with effect of

14 Shock and Vibration

compressive axial loadrdquo International Journal of MechanicalSciences vol 60 no 1 pp 59ndash71 2012

[21] A Palmeri and S Adhikari ldquoA Galerkin-type state-spaceapproach for transverse vibrations of slender double-beamsystems with viscoelastic inner layerrdquo Journal of Sound andVibration vol 330 no 26 pp 6372ndash6386 2011

[22] Z Zhang X Huang Z Zhang and H Hua ldquoOn the transversevibration of Timoshenko double-beam systems coupled withvarious discontinuitiesrdquo International Journal of MechanicalSciences vol 89 pp 222ndash241 2014

[23] Y X Li Z J Hu and L Z Sun ldquoDynamical behavior of adouble-beam system interconnected by a viscoelastic layerrdquoInternational Journal of Mechanical Sciences vol 105 pp 291ndash303 2016

[24] M Abu-Hilal ldquoDynamic response of a double EulerndashBernoullibeam due to a moving constant loadrdquo Journal of Sound andVibration vol 297 no 3-5 pp 477ndash491 2006

[25] Y Wu and Y Gao ldquoAnalytical Solutions for Simply SupportedViscously Damped Double-Beam System under Moving Har-monic Loadsrdquo Journal of Engineering Mechanics vol 141 no 7p 04015004 2015

[26] Y-H Chen and Z-M Shiu ldquoResonant curves of an elevatedrailway to harmonic moving loadsrdquo International Journal ofStructural Stability and Dynamics vol 4 no 2 pp 237ndash2572004

[27] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of aninfinite tomoshenko beam on a viscoelastic foundation to aharmonic moving loadrdquo Journal of Sound and Vibration vol241 no 5 pp 809ndash824 2001

[28] M FMHussein andH EMHunt ldquoModelling of floating-slabtracks with continuous slabs under oscillating moving loadsrdquoJournal of Sound andVibration vol 297 no 1-2 pp 37ndash54 2006

[29] H P Wang J Li and K Zhang ldquoVibration analysis of themaglev guideway with the moving loadrdquo Journal of Sound andVibration vol 305 no 4-5 pp 621ndash640 2007

[30] V Stojanovic and M Petkovic ldquoMoment Lyapunov exponentsand stochastic stability of a three-dimensional system on elasticfoundation using a perturbation approachrdquo Journal of AppliedMechanics vol 80 no 5 2013

[31] L Jun and H Hongxing ldquoDynamic stiffness vibration analysisof an elastically connected three-beam systemrdquo Applied Acous-tics vol 69 no 7 pp 591ndash600 2008

[32] J Li Y Chen and H Hua ldquoExact dynamic stiffness matrixof a Timoshenko three-beam systemrdquo International Journal ofMechanical Sciences vol 50 no 6 pp 1023ndash1034 2008

[33] S S Rao ldquoNatural vibrations of systems of elastically connectedTimoshenko beams rdquo The Journal of the Acoustical Society ofAmerica vol 55 no 6 pp 1232ndash1237 1974

[34] Q Mao ldquoFree vibration analysis of elastically connectedmultiple-beams by using the Adomianmodified decompositionmethodrdquo Journal of Sound and Vibration vol 331 no 11 pp2532ndash2542 2012

[35] V Stojanovic P Kozic and G Janevski ldquoExact closed-formsolutions for the natural frequencies and stability of elasticallyconnected multiple beam system using Timoshenko and high-order shear deformation theoryrdquo Journal of Sound and Vibra-tion vol 332 no 3 pp 563ndash576 2013

[36] M H Kargarnovin M T Ahmadian and R-A Jafari-Talookolaei ldquoAnalytical solution for the dynamic analysis of adelaminated composite beam traversed by a moving constantforcerdquo Journal of Vibration and Control vol 19 no 10 pp 1524ndash1537 2013

[37] A Ariaei S Ziaei-Rad and M Ghayour ldquoTransverse vibrationof a multiple-Timoshenko beam system with intermediateelastic connections due to a moving loadrdquo Archive of AppliedMechanics vol 81 no 3 pp 263ndash281 2011

[38] Y Povstenko and J Klekot ldquoThe Dirichlet problem for thetime-fractional advection-diffusion equation in a line segmentrdquoBoundary Value Problems Paper No 89 8 pages 2016

[39] F Durbin ldquoNumerical inversion of Laplace transforms anefficient improvement to Dubner and Abatersquos methodrdquo TheComputer Journal vol 17 pp 371ndash376 1974

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Shock and Vibration 11

025kr050kr100kr

200kr400kr

minus28

minus24

minus20

minus16

minus12

minus08

minus04

00

04

Defl

ectio

n (m

m)

008 016 024 032000t (s)

(a)

025kr050kr100kr

200kr400kr

008 016 024 032000t (s)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

(b)

Figure 12 Comparison diagram of mid-span deflection time-history for the double simply supported rail-bridge system under differentinterlayer stiffness (a) first beam (b) second beam

100

200400

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025Ｇ1050Ｇ1

Ｇ1

Ｇ1Ｇ1

(a)

025050100

200400

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

Ｇ1Ｇ1

Ｇ1Ｇ1

Ｇ1

(b)

Figure 13 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof first beam (a) first beam (b) second beam

amplitude of the second simply supported beam Underdifferent stiffness of the first beam the mid-span deflectiondynamic time-history curves for the second simply supportedbeam are consistent Figure 16 shows that the flexural rigidityvariations of the second simply supported beam have a sig-nificant effect on the mid-span deflection dynamic responseof double simply supported rail-bridge system In additionthe mid-span deflection dynamic response amplitudes ofboth beams significantly increased with the decrease in theflexural stiffness of second beam In practical applicationsthe dynamic response of the rail-bridge system in high-speedrailways can be decreased by moderately decreasing the massof the second beam and increasing the flexural stiffness ofboth simply supported beams

4 Conclusions

In this study a dynamic analysis model was established for amultiple simply supported beam system subjected to movingloads in combination of finite sin-Fourier transform andnumerical Laplace transform based on Durbin transforman analytical calculation method was developed for thespatial displacement time-domain response of a multiplesimply supported beam systemThe following conclusions aredrawn by comparing the examples and ANSYS finite elementnumerical computation method(1) A simply supported rail-bridge system of high-speedrailways with a span of 32 m in China was taken as theexample and the theoretically calculated results were com-pared with the results obtained from ANSYS finite element

12 Shock and Vibration

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

100

200400

025Ｇ2050Ｇ2

Ｇ2

Ｇ2Ｇ2

(a)

minus030

minus024

minus018

minus012

minus006

000

006

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025050100

200400Ｇ2

Ｇ2

Ｇ2Ｇ2

Ｇ2

(b)

Figure 14 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof second beam (a) first beam (b) second beam

008 016 024 032000t (s)

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02510501

10012001

(a)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

02510501

10012001

(b)

Figure 15 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof first beam (a) first beam (b) second beam

numerical calculation validating the analytical calculationmethod reported in this paper(2) The effect of load moving speed on the dynamicresponse of a double simply supported rail-bridge system ofhigh-speed railways subjected tomoving loads was evaluatedThe results indicate that the double simply supported rail-bridge system has multiple points of resonance and cancel-lation which should be paid attention in engineering(3) For a typical double simply supported rail-bridgesystem of high-speed railway in China because of a largedifference in the flexural rigidity of the first and secondbeams the variation in interlayer stiffness mainly affectsthe dynamic response of the first simply supported beamexhibiting a positive correlation When the mass and flexuralstiffness parameters of first simply supported beam vary they

mainly affect the dynamic response of this layer of simplysupported beam but slightly affect the dynamic responseof second simply supported beam The variation in theparameters of second simply supported beam has a greatereffect on the double simply supported beam system and adecrease in the mass of this layer of simply supported beamand an increase in the flexural rigidity of this beam canboth decrease the dynamic response of the double simplysupported rail-bridge system

Data Availability

The data used to support the findings of this study areincluded within the article

Shock and Vibration 13

minus14

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

05021002

20024002

(a)

008 016 024 032000t (s)

minus048

minus040

minus032

minus024

minus016

minus008

000

008

Defl

ectio

n (m

m)

05021002

20024002

(b)

Figure 16 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof second beam (a) first beam (b) second beam

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51708630)

References

[1] H Ouyang ldquoMoving-load dynamic problems a tutorial (with abrief overview)rdquoMechanical Systems and Signal Processing vol25 no 6 pp 2039ndash2060 2011

[2] D G Duffy ldquoThe response of an infinite railroad track to amoving vibrating massrdquo Journal of Applied Mechanics vol 57no 1 pp 66ndash73 1990

[3] L Fryba Vibration of solids and structures under moving loadsSpringer Science amp Business Media 2013

[4] G Michaltsos D Sophianopoulos and A N Kounadis ldquoTheeffect of a moving mass and other parameters on the dynamicresponse of a simply supported beamrdquo Journal of Sound andVibration vol 191 no 3 pp 357ndash362 1996

[5] P Museros E Moliner and M D Martınez-Rodrigo ldquoFreevibrations of simply-supported beam bridges under movingloads Maximum resonance cancellation and resonant verticalaccelerationrdquo Journal of Sound and Vibration vol 332 no 2 pp326ndash345 2013

[6] P Sniady ldquoDynamic response of a Timoshenko beam to amoving forcerdquo Journal of Applied Mechanics vol 75 no 2Article ID 024503 2008

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

[8] K Y Lee and A A Renshaw ldquoSolution of the moving massproblem using complex eigenfunction expansionsrdquo Journal ofApplied Mechanics vol 67 no 4 pp 823ndash827 2000

[9] U Lee ldquoSeparation between the flexible structure and themoving mass sliding on itrdquo Journal of Sound and Vibration vol209 no 5 pp 867ndash877 1998

[10] S A Eftekhari and A A Jafari ldquoCoupling ritz method andtriangular quadrature rule for moving mass problemrdquo Journalof Applied Mechanics vol 79 no 2 2012

[11] M Dublin andH R Friedrich ldquoForced responses of two elasticbeams interconnected by spring-damper systemsrdquo Journal of theAeronautical Sciences vol 23 pp 824ndash829 1956

[12] J M Seelig and W H Hoppmann ldquoImpact on an elasticallyconnected double-beam systemrdquo Journal of Applied Mechanicsvol 31 pp 621ndash626 1964

[13] P G Kessel ldquoResonances Excited in an Elastically ConnectedDouble-Beam System by a Cyclic Moving LoadrdquoThe Journal ofthe Acoustical Society of America vol 40 no 3 pp 684ndash6871966

[14] S Chonan ldquoDynamical behaviours of elastically connecteddouble-beam systems subjected to an impulsive loadrdquo Bulletinof JSME vol 19 no 132 pp 595ndash603 1976

[15] H V Vu A M Ordonez and B H Karnopp ldquoVibration of adouble-beam systemrdquo Journal of Sound and Vibration vol 229no 4 pp 807ndash822 2000

[16] H J Xu Z H Ding Z R Lu and J K Liu ldquoStructural DamageDetection Using a Modified Artificial Bee Colony AlgorithmrdquoCMES-Computer Modeling in Engineering amp Sciences vol 111pp 335ndash355 2016

[17] Z Oniszczuk ldquoFree transverse vibrations of elastically con-nected simply supported double-beam complex systemrdquo Jour-nal of Sound and Vibration vol 232 no 2 pp 387ndash403 2000

[18] Z Oniszczuk ldquoForced transverse vibrations of an elasticallyconnected complex simply supported double-beam systemrdquoJournal of Sound and Vibration vol 264 no 2 pp 273ndash2862003

[19] Y Q Zhang Y Lu S L Wang and X Liu ldquoVibration andbuckling of a double-beam system under compressive axialloadingrdquo Journal of Sound and Vibration vol 318 no 1-2 pp341ndash352 2008

[20] V Stojanovic and P Kozic ldquoForced transverse vibration ofRayleigh and Timoshenko double-beam system with effect of

14 Shock and Vibration

compressive axial loadrdquo International Journal of MechanicalSciences vol 60 no 1 pp 59ndash71 2012

[21] A Palmeri and S Adhikari ldquoA Galerkin-type state-spaceapproach for transverse vibrations of slender double-beamsystems with viscoelastic inner layerrdquo Journal of Sound andVibration vol 330 no 26 pp 6372ndash6386 2011

[22] Z Zhang X Huang Z Zhang and H Hua ldquoOn the transversevibration of Timoshenko double-beam systems coupled withvarious discontinuitiesrdquo International Journal of MechanicalSciences vol 89 pp 222ndash241 2014

[23] Y X Li Z J Hu and L Z Sun ldquoDynamical behavior of adouble-beam system interconnected by a viscoelastic layerrdquoInternational Journal of Mechanical Sciences vol 105 pp 291ndash303 2016

[24] M Abu-Hilal ldquoDynamic response of a double EulerndashBernoullibeam due to a moving constant loadrdquo Journal of Sound andVibration vol 297 no 3-5 pp 477ndash491 2006

[25] Y Wu and Y Gao ldquoAnalytical Solutions for Simply SupportedViscously Damped Double-Beam System under Moving Har-monic Loadsrdquo Journal of Engineering Mechanics vol 141 no 7p 04015004 2015

[26] Y-H Chen and Z-M Shiu ldquoResonant curves of an elevatedrailway to harmonic moving loadsrdquo International Journal ofStructural Stability and Dynamics vol 4 no 2 pp 237ndash2572004

[27] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of aninfinite tomoshenko beam on a viscoelastic foundation to aharmonic moving loadrdquo Journal of Sound and Vibration vol241 no 5 pp 809ndash824 2001

[28] M FMHussein andH EMHunt ldquoModelling of floating-slabtracks with continuous slabs under oscillating moving loadsrdquoJournal of Sound andVibration vol 297 no 1-2 pp 37ndash54 2006

[29] H P Wang J Li and K Zhang ldquoVibration analysis of themaglev guideway with the moving loadrdquo Journal of Sound andVibration vol 305 no 4-5 pp 621ndash640 2007

[30] V Stojanovic and M Petkovic ldquoMoment Lyapunov exponentsand stochastic stability of a three-dimensional system on elasticfoundation using a perturbation approachrdquo Journal of AppliedMechanics vol 80 no 5 2013

[31] L Jun and H Hongxing ldquoDynamic stiffness vibration analysisof an elastically connected three-beam systemrdquo Applied Acous-tics vol 69 no 7 pp 591ndash600 2008

[32] J Li Y Chen and H Hua ldquoExact dynamic stiffness matrixof a Timoshenko three-beam systemrdquo International Journal ofMechanical Sciences vol 50 no 6 pp 1023ndash1034 2008

[33] S S Rao ldquoNatural vibrations of systems of elastically connectedTimoshenko beams rdquo The Journal of the Acoustical Society ofAmerica vol 55 no 6 pp 1232ndash1237 1974

[34] Q Mao ldquoFree vibration analysis of elastically connectedmultiple-beams by using the Adomianmodified decompositionmethodrdquo Journal of Sound and Vibration vol 331 no 11 pp2532ndash2542 2012

[35] V Stojanovic P Kozic and G Janevski ldquoExact closed-formsolutions for the natural frequencies and stability of elasticallyconnected multiple beam system using Timoshenko and high-order shear deformation theoryrdquo Journal of Sound and Vibra-tion vol 332 no 3 pp 563ndash576 2013

[36] M H Kargarnovin M T Ahmadian and R-A Jafari-Talookolaei ldquoAnalytical solution for the dynamic analysis of adelaminated composite beam traversed by a moving constantforcerdquo Journal of Vibration and Control vol 19 no 10 pp 1524ndash1537 2013

[37] A Ariaei S Ziaei-Rad and M Ghayour ldquoTransverse vibrationof a multiple-Timoshenko beam system with intermediateelastic connections due to a moving loadrdquo Archive of AppliedMechanics vol 81 no 3 pp 263ndash281 2011

[38] Y Povstenko and J Klekot ldquoThe Dirichlet problem for thetime-fractional advection-diffusion equation in a line segmentrdquoBoundary Value Problems Paper No 89 8 pages 2016

[39] F Durbin ldquoNumerical inversion of Laplace transforms anefficient improvement to Dubner and Abatersquos methodrdquo TheComputer Journal vol 17 pp 371ndash376 1974

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

12 Shock and Vibration

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

100

200400

025Ｇ2050Ｇ2

Ｇ2

Ｇ2Ｇ2

(a)

minus030

minus024

minus018

minus012

minus006

000

006

Defl

ectio

n (m

m)

008 016 024 032000t (s)

025050100

200400Ｇ2

Ｇ2

Ｇ2Ｇ2

Ｇ2

(b)

Figure 14 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different massesof second beam (a) first beam (b) second beam

008 016 024 032000t (s)

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

02510501

10012001

(a)

minus020

minus016

minus012

minus008

minus004

000

004

Defl

ectio

n (m

m)

008 016 024 032000t (s)

02510501

10012001

(b)

Figure 15 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof first beam (a) first beam (b) second beam

numerical calculation validating the analytical calculationmethod reported in this paper(2) The effect of load moving speed on the dynamicresponse of a double simply supported rail-bridge system ofhigh-speed railways subjected tomoving loads was evaluatedThe results indicate that the double simply supported rail-bridge system has multiple points of resonance and cancel-lation which should be paid attention in engineering(3) For a typical double simply supported rail-bridgesystem of high-speed railway in China because of a largedifference in the flexural rigidity of the first and secondbeams the variation in interlayer stiffness mainly affectsthe dynamic response of the first simply supported beamexhibiting a positive correlation When the mass and flexuralstiffness parameters of first simply supported beam vary they

mainly affect the dynamic response of this layer of simplysupported beam but slightly affect the dynamic responseof second simply supported beam The variation in theparameters of second simply supported beam has a greatereffect on the double simply supported beam system and adecrease in the mass of this layer of simply supported beamand an increase in the flexural rigidity of this beam canboth decrease the dynamic response of the double simplysupported rail-bridge system

Data Availability

The data used to support the findings of this study areincluded within the article

Shock and Vibration 13

minus14

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

05021002

20024002

(a)

008 016 024 032000t (s)

minus048

minus040

minus032

minus024

minus016

minus008

000

008

Defl

ectio

n (m

m)

05021002

20024002

(b)

Figure 16 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof second beam (a) first beam (b) second beam

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51708630)

References

[1] H Ouyang ldquoMoving-load dynamic problems a tutorial (with abrief overview)rdquoMechanical Systems and Signal Processing vol25 no 6 pp 2039ndash2060 2011

[2] D G Duffy ldquoThe response of an infinite railroad track to amoving vibrating massrdquo Journal of Applied Mechanics vol 57no 1 pp 66ndash73 1990

[3] L Fryba Vibration of solids and structures under moving loadsSpringer Science amp Business Media 2013

[4] G Michaltsos D Sophianopoulos and A N Kounadis ldquoTheeffect of a moving mass and other parameters on the dynamicresponse of a simply supported beamrdquo Journal of Sound andVibration vol 191 no 3 pp 357ndash362 1996

[5] P Museros E Moliner and M D Martınez-Rodrigo ldquoFreevibrations of simply-supported beam bridges under movingloads Maximum resonance cancellation and resonant verticalaccelerationrdquo Journal of Sound and Vibration vol 332 no 2 pp326ndash345 2013

[6] P Sniady ldquoDynamic response of a Timoshenko beam to amoving forcerdquo Journal of Applied Mechanics vol 75 no 2Article ID 024503 2008

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

[8] K Y Lee and A A Renshaw ldquoSolution of the moving massproblem using complex eigenfunction expansionsrdquo Journal ofApplied Mechanics vol 67 no 4 pp 823ndash827 2000

[9] U Lee ldquoSeparation between the flexible structure and themoving mass sliding on itrdquo Journal of Sound and Vibration vol209 no 5 pp 867ndash877 1998

[10] S A Eftekhari and A A Jafari ldquoCoupling ritz method andtriangular quadrature rule for moving mass problemrdquo Journalof Applied Mechanics vol 79 no 2 2012

[11] M Dublin andH R Friedrich ldquoForced responses of two elasticbeams interconnected by spring-damper systemsrdquo Journal of theAeronautical Sciences vol 23 pp 824ndash829 1956

[12] J M Seelig and W H Hoppmann ldquoImpact on an elasticallyconnected double-beam systemrdquo Journal of Applied Mechanicsvol 31 pp 621ndash626 1964

[13] P G Kessel ldquoResonances Excited in an Elastically ConnectedDouble-Beam System by a Cyclic Moving LoadrdquoThe Journal ofthe Acoustical Society of America vol 40 no 3 pp 684ndash6871966

[14] S Chonan ldquoDynamical behaviours of elastically connecteddouble-beam systems subjected to an impulsive loadrdquo Bulletinof JSME vol 19 no 132 pp 595ndash603 1976

[15] H V Vu A M Ordonez and B H Karnopp ldquoVibration of adouble-beam systemrdquo Journal of Sound and Vibration vol 229no 4 pp 807ndash822 2000

[16] H J Xu Z H Ding Z R Lu and J K Liu ldquoStructural DamageDetection Using a Modified Artificial Bee Colony AlgorithmrdquoCMES-Computer Modeling in Engineering amp Sciences vol 111pp 335ndash355 2016

[17] Z Oniszczuk ldquoFree transverse vibrations of elastically con-nected simply supported double-beam complex systemrdquo Jour-nal of Sound and Vibration vol 232 no 2 pp 387ndash403 2000

[18] Z Oniszczuk ldquoForced transverse vibrations of an elasticallyconnected complex simply supported double-beam systemrdquoJournal of Sound and Vibration vol 264 no 2 pp 273ndash2862003

[19] Y Q Zhang Y Lu S L Wang and X Liu ldquoVibration andbuckling of a double-beam system under compressive axialloadingrdquo Journal of Sound and Vibration vol 318 no 1-2 pp341ndash352 2008

[20] V Stojanovic and P Kozic ldquoForced transverse vibration ofRayleigh and Timoshenko double-beam system with effect of

14 Shock and Vibration

compressive axial loadrdquo International Journal of MechanicalSciences vol 60 no 1 pp 59ndash71 2012

[21] A Palmeri and S Adhikari ldquoA Galerkin-type state-spaceapproach for transverse vibrations of slender double-beamsystems with viscoelastic inner layerrdquo Journal of Sound andVibration vol 330 no 26 pp 6372ndash6386 2011

[22] Z Zhang X Huang Z Zhang and H Hua ldquoOn the transversevibration of Timoshenko double-beam systems coupled withvarious discontinuitiesrdquo International Journal of MechanicalSciences vol 89 pp 222ndash241 2014

[23] Y X Li Z J Hu and L Z Sun ldquoDynamical behavior of adouble-beam system interconnected by a viscoelastic layerrdquoInternational Journal of Mechanical Sciences vol 105 pp 291ndash303 2016

[24] M Abu-Hilal ldquoDynamic response of a double EulerndashBernoullibeam due to a moving constant loadrdquo Journal of Sound andVibration vol 297 no 3-5 pp 477ndash491 2006

[25] Y Wu and Y Gao ldquoAnalytical Solutions for Simply SupportedViscously Damped Double-Beam System under Moving Har-monic Loadsrdquo Journal of Engineering Mechanics vol 141 no 7p 04015004 2015

[26] Y-H Chen and Z-M Shiu ldquoResonant curves of an elevatedrailway to harmonic moving loadsrdquo International Journal ofStructural Stability and Dynamics vol 4 no 2 pp 237ndash2572004

[27] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of aninfinite tomoshenko beam on a viscoelastic foundation to aharmonic moving loadrdquo Journal of Sound and Vibration vol241 no 5 pp 809ndash824 2001

[28] M FMHussein andH EMHunt ldquoModelling of floating-slabtracks with continuous slabs under oscillating moving loadsrdquoJournal of Sound andVibration vol 297 no 1-2 pp 37ndash54 2006

[29] H P Wang J Li and K Zhang ldquoVibration analysis of themaglev guideway with the moving loadrdquo Journal of Sound andVibration vol 305 no 4-5 pp 621ndash640 2007

[30] V Stojanovic and M Petkovic ldquoMoment Lyapunov exponentsand stochastic stability of a three-dimensional system on elasticfoundation using a perturbation approachrdquo Journal of AppliedMechanics vol 80 no 5 2013

[31] L Jun and H Hongxing ldquoDynamic stiffness vibration analysisof an elastically connected three-beam systemrdquo Applied Acous-tics vol 69 no 7 pp 591ndash600 2008

[32] J Li Y Chen and H Hua ldquoExact dynamic stiffness matrixof a Timoshenko three-beam systemrdquo International Journal ofMechanical Sciences vol 50 no 6 pp 1023ndash1034 2008

[33] S S Rao ldquoNatural vibrations of systems of elastically connectedTimoshenko beams rdquo The Journal of the Acoustical Society ofAmerica vol 55 no 6 pp 1232ndash1237 1974

[34] Q Mao ldquoFree vibration analysis of elastically connectedmultiple-beams by using the Adomianmodified decompositionmethodrdquo Journal of Sound and Vibration vol 331 no 11 pp2532ndash2542 2012

[35] V Stojanovic P Kozic and G Janevski ldquoExact closed-formsolutions for the natural frequencies and stability of elasticallyconnected multiple beam system using Timoshenko and high-order shear deformation theoryrdquo Journal of Sound and Vibra-tion vol 332 no 3 pp 563ndash576 2013

[36] M H Kargarnovin M T Ahmadian and R-A Jafari-Talookolaei ldquoAnalytical solution for the dynamic analysis of adelaminated composite beam traversed by a moving constantforcerdquo Journal of Vibration and Control vol 19 no 10 pp 1524ndash1537 2013

[37] A Ariaei S Ziaei-Rad and M Ghayour ldquoTransverse vibrationof a multiple-Timoshenko beam system with intermediateelastic connections due to a moving loadrdquo Archive of AppliedMechanics vol 81 no 3 pp 263ndash281 2011

[38] Y Povstenko and J Klekot ldquoThe Dirichlet problem for thetime-fractional advection-diffusion equation in a line segmentrdquoBoundary Value Problems Paper No 89 8 pages 2016

[39] F Durbin ldquoNumerical inversion of Laplace transforms anefficient improvement to Dubner and Abatersquos methodrdquo TheComputer Journal vol 17 pp 371ndash376 1974

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Shock and Vibration 13

minus14

minus12

minus10

minus08

minus06

minus04

minus02

00

02

Defl

ectio

n (m

m)

008 016 024 032000t (s)

05021002

20024002

(a)

008 016 024 032000t (s)

minus048

minus040

minus032

minus024

minus016

minus008

000

008

Defl

ectio

n (m

m)

05021002

20024002

(b)

Figure 16 Comparison diagram of mid-span deflection time-history for double simply supported rail-bridge system under different stiffnessof second beam (a) first beam (b) second beam

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51708630)

References

[1] H Ouyang ldquoMoving-load dynamic problems a tutorial (with abrief overview)rdquoMechanical Systems and Signal Processing vol25 no 6 pp 2039ndash2060 2011

[2] D G Duffy ldquoThe response of an infinite railroad track to amoving vibrating massrdquo Journal of Applied Mechanics vol 57no 1 pp 66ndash73 1990

[3] L Fryba Vibration of solids and structures under moving loadsSpringer Science amp Business Media 2013

[4] G Michaltsos D Sophianopoulos and A N Kounadis ldquoTheeffect of a moving mass and other parameters on the dynamicresponse of a simply supported beamrdquo Journal of Sound andVibration vol 191 no 3 pp 357ndash362 1996

[5] P Museros E Moliner and M D Martınez-Rodrigo ldquoFreevibrations of simply-supported beam bridges under movingloads Maximum resonance cancellation and resonant verticalaccelerationrdquo Journal of Sound and Vibration vol 332 no 2 pp326ndash345 2013

[6] P Sniady ldquoDynamic response of a Timoshenko beam to amoving forcerdquo Journal of Applied Mechanics vol 75 no 2Article ID 024503 2008

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

[8] K Y Lee and A A Renshaw ldquoSolution of the moving massproblem using complex eigenfunction expansionsrdquo Journal ofApplied Mechanics vol 67 no 4 pp 823ndash827 2000

[9] U Lee ldquoSeparation between the flexible structure and themoving mass sliding on itrdquo Journal of Sound and Vibration vol209 no 5 pp 867ndash877 1998

[10] S A Eftekhari and A A Jafari ldquoCoupling ritz method andtriangular quadrature rule for moving mass problemrdquo Journalof Applied Mechanics vol 79 no 2 2012

[11] M Dublin andH R Friedrich ldquoForced responses of two elasticbeams interconnected by spring-damper systemsrdquo Journal of theAeronautical Sciences vol 23 pp 824ndash829 1956

[12] J M Seelig and W H Hoppmann ldquoImpact on an elasticallyconnected double-beam systemrdquo Journal of Applied Mechanicsvol 31 pp 621ndash626 1964

[13] P G Kessel ldquoResonances Excited in an Elastically ConnectedDouble-Beam System by a Cyclic Moving LoadrdquoThe Journal ofthe Acoustical Society of America vol 40 no 3 pp 684ndash6871966

[14] S Chonan ldquoDynamical behaviours of elastically connecteddouble-beam systems subjected to an impulsive loadrdquo Bulletinof JSME vol 19 no 132 pp 595ndash603 1976

[15] H V Vu A M Ordonez and B H Karnopp ldquoVibration of adouble-beam systemrdquo Journal of Sound and Vibration vol 229no 4 pp 807ndash822 2000

[16] H J Xu Z H Ding Z R Lu and J K Liu ldquoStructural DamageDetection Using a Modified Artificial Bee Colony AlgorithmrdquoCMES-Computer Modeling in Engineering amp Sciences vol 111pp 335ndash355 2016

[17] Z Oniszczuk ldquoFree transverse vibrations of elastically con-nected simply supported double-beam complex systemrdquo Jour-nal of Sound and Vibration vol 232 no 2 pp 387ndash403 2000

[18] Z Oniszczuk ldquoForced transverse vibrations of an elasticallyconnected complex simply supported double-beam systemrdquoJournal of Sound and Vibration vol 264 no 2 pp 273ndash2862003

[19] Y Q Zhang Y Lu S L Wang and X Liu ldquoVibration andbuckling of a double-beam system under compressive axialloadingrdquo Journal of Sound and Vibration vol 318 no 1-2 pp341ndash352 2008

[20] V Stojanovic and P Kozic ldquoForced transverse vibration ofRayleigh and Timoshenko double-beam system with effect of

14 Shock and Vibration

compressive axial loadrdquo International Journal of MechanicalSciences vol 60 no 1 pp 59ndash71 2012

[21] A Palmeri and S Adhikari ldquoA Galerkin-type state-spaceapproach for transverse vibrations of slender double-beamsystems with viscoelastic inner layerrdquo Journal of Sound andVibration vol 330 no 26 pp 6372ndash6386 2011

[22] Z Zhang X Huang Z Zhang and H Hua ldquoOn the transversevibration of Timoshenko double-beam systems coupled withvarious discontinuitiesrdquo International Journal of MechanicalSciences vol 89 pp 222ndash241 2014

[23] Y X Li Z J Hu and L Z Sun ldquoDynamical behavior of adouble-beam system interconnected by a viscoelastic layerrdquoInternational Journal of Mechanical Sciences vol 105 pp 291ndash303 2016

[24] M Abu-Hilal ldquoDynamic response of a double EulerndashBernoullibeam due to a moving constant loadrdquo Journal of Sound andVibration vol 297 no 3-5 pp 477ndash491 2006

[25] Y Wu and Y Gao ldquoAnalytical Solutions for Simply SupportedViscously Damped Double-Beam System under Moving Har-monic Loadsrdquo Journal of Engineering Mechanics vol 141 no 7p 04015004 2015

[26] Y-H Chen and Z-M Shiu ldquoResonant curves of an elevatedrailway to harmonic moving loadsrdquo International Journal ofStructural Stability and Dynamics vol 4 no 2 pp 237ndash2572004

[27] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of aninfinite tomoshenko beam on a viscoelastic foundation to aharmonic moving loadrdquo Journal of Sound and Vibration vol241 no 5 pp 809ndash824 2001

[28] M FMHussein andH EMHunt ldquoModelling of floating-slabtracks with continuous slabs under oscillating moving loadsrdquoJournal of Sound andVibration vol 297 no 1-2 pp 37ndash54 2006

[29] H P Wang J Li and K Zhang ldquoVibration analysis of themaglev guideway with the moving loadrdquo Journal of Sound andVibration vol 305 no 4-5 pp 621ndash640 2007

[30] V Stojanovic and M Petkovic ldquoMoment Lyapunov exponentsand stochastic stability of a three-dimensional system on elasticfoundation using a perturbation approachrdquo Journal of AppliedMechanics vol 80 no 5 2013

[31] L Jun and H Hongxing ldquoDynamic stiffness vibration analysisof an elastically connected three-beam systemrdquo Applied Acous-tics vol 69 no 7 pp 591ndash600 2008

[32] J Li Y Chen and H Hua ldquoExact dynamic stiffness matrixof a Timoshenko three-beam systemrdquo International Journal ofMechanical Sciences vol 50 no 6 pp 1023ndash1034 2008

[33] S S Rao ldquoNatural vibrations of systems of elastically connectedTimoshenko beams rdquo The Journal of the Acoustical Society ofAmerica vol 55 no 6 pp 1232ndash1237 1974

[34] Q Mao ldquoFree vibration analysis of elastically connectedmultiple-beams by using the Adomianmodified decompositionmethodrdquo Journal of Sound and Vibration vol 331 no 11 pp2532ndash2542 2012

[35] V Stojanovic P Kozic and G Janevski ldquoExact closed-formsolutions for the natural frequencies and stability of elasticallyconnected multiple beam system using Timoshenko and high-order shear deformation theoryrdquo Journal of Sound and Vibra-tion vol 332 no 3 pp 563ndash576 2013

[36] M H Kargarnovin M T Ahmadian and R-A Jafari-Talookolaei ldquoAnalytical solution for the dynamic analysis of adelaminated composite beam traversed by a moving constantforcerdquo Journal of Vibration and Control vol 19 no 10 pp 1524ndash1537 2013

[37] A Ariaei S Ziaei-Rad and M Ghayour ldquoTransverse vibrationof a multiple-Timoshenko beam system with intermediateelastic connections due to a moving loadrdquo Archive of AppliedMechanics vol 81 no 3 pp 263ndash281 2011

[38] Y Povstenko and J Klekot ldquoThe Dirichlet problem for thetime-fractional advection-diffusion equation in a line segmentrdquoBoundary Value Problems Paper No 89 8 pages 2016

[39] F Durbin ldquoNumerical inversion of Laplace transforms anefficient improvement to Dubner and Abatersquos methodrdquo TheComputer Journal vol 17 pp 371ndash376 1974

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

14 Shock and Vibration

compressive axial loadrdquo International Journal of MechanicalSciences vol 60 no 1 pp 59ndash71 2012

[21] A Palmeri and S Adhikari ldquoA Galerkin-type state-spaceapproach for transverse vibrations of slender double-beamsystems with viscoelastic inner layerrdquo Journal of Sound andVibration vol 330 no 26 pp 6372ndash6386 2011

[22] Z Zhang X Huang Z Zhang and H Hua ldquoOn the transversevibration of Timoshenko double-beam systems coupled withvarious discontinuitiesrdquo International Journal of MechanicalSciences vol 89 pp 222ndash241 2014

[23] Y X Li Z J Hu and L Z Sun ldquoDynamical behavior of adouble-beam system interconnected by a viscoelastic layerrdquoInternational Journal of Mechanical Sciences vol 105 pp 291ndash303 2016

[24] M Abu-Hilal ldquoDynamic response of a double EulerndashBernoullibeam due to a moving constant loadrdquo Journal of Sound andVibration vol 297 no 3-5 pp 477ndash491 2006

[25] Y Wu and Y Gao ldquoAnalytical Solutions for Simply SupportedViscously Damped Double-Beam System under Moving Har-monic Loadsrdquo Journal of Engineering Mechanics vol 141 no 7p 04015004 2015

[26] Y-H Chen and Z-M Shiu ldquoResonant curves of an elevatedrailway to harmonic moving loadsrdquo International Journal ofStructural Stability and Dynamics vol 4 no 2 pp 237ndash2572004

[27] Y-H Chen Y-H Huang and C-T Shih ldquoResponse of aninfinite tomoshenko beam on a viscoelastic foundation to aharmonic moving loadrdquo Journal of Sound and Vibration vol241 no 5 pp 809ndash824 2001

[28] M FMHussein andH EMHunt ldquoModelling of floating-slabtracks with continuous slabs under oscillating moving loadsrdquoJournal of Sound andVibration vol 297 no 1-2 pp 37ndash54 2006

[29] H P Wang J Li and K Zhang ldquoVibration analysis of themaglev guideway with the moving loadrdquo Journal of Sound andVibration vol 305 no 4-5 pp 621ndash640 2007

[30] V Stojanovic and M Petkovic ldquoMoment Lyapunov exponentsand stochastic stability of a three-dimensional system on elasticfoundation using a perturbation approachrdquo Journal of AppliedMechanics vol 80 no 5 2013

[31] L Jun and H Hongxing ldquoDynamic stiffness vibration analysisof an elastically connected three-beam systemrdquo Applied Acous-tics vol 69 no 7 pp 591ndash600 2008

[32] J Li Y Chen and H Hua ldquoExact dynamic stiffness matrixof a Timoshenko three-beam systemrdquo International Journal ofMechanical Sciences vol 50 no 6 pp 1023ndash1034 2008

[33] S S Rao ldquoNatural vibrations of systems of elastically connectedTimoshenko beams rdquo The Journal of the Acoustical Society ofAmerica vol 55 no 6 pp 1232ndash1237 1974

[34] Q Mao ldquoFree vibration analysis of elastically connectedmultiple-beams by using the Adomianmodified decompositionmethodrdquo Journal of Sound and Vibration vol 331 no 11 pp2532ndash2542 2012

[35] V Stojanovic P Kozic and G Janevski ldquoExact closed-formsolutions for the natural frequencies and stability of elasticallyconnected multiple beam system using Timoshenko and high-order shear deformation theoryrdquo Journal of Sound and Vibra-tion vol 332 no 3 pp 563ndash576 2013

[36] M H Kargarnovin M T Ahmadian and R-A Jafari-Talookolaei ldquoAnalytical solution for the dynamic analysis of adelaminated composite beam traversed by a moving constantforcerdquo Journal of Vibration and Control vol 19 no 10 pp 1524ndash1537 2013

[37] A Ariaei S Ziaei-Rad and M Ghayour ldquoTransverse vibrationof a multiple-Timoshenko beam system with intermediateelastic connections due to a moving loadrdquo Archive of AppliedMechanics vol 81 no 3 pp 263ndash281 2011

[38] Y Povstenko and J Klekot ldquoThe Dirichlet problem for thetime-fractional advection-diffusion equation in a line segmentrdquoBoundary Value Problems Paper No 89 8 pages 2016

[39] F Durbin ldquoNumerical inversion of Laplace transforms anefficient improvement to Dubner and Abatersquos methodrdquo TheComputer Journal vol 17 pp 371ndash376 1974

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom