Scientia Iranica A (2016) 23(1), 133{141

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

A new simpli�ed formula in prediction of the resonancevelocity for multiple masses traversing a thin beam

R. Afghani Khoraskania, M. Mo�db, S. Eftekhar Azamc andM. Ebrahimzadeh Hassanabadid;�

a. Faculty of Architecture and Urban Planning, Shahid Beheshti University, Tehran, Iran.b. Department of Civil Engineering, Sharif Univeristy of Technology, Tehran, Iran.c. Politecnico di Milano, Dipartimento di Ingegneria Civile e Ambientale, Piazza L. da Vinci 32, 20133 Milano, Italy.d. Department of Structural Engineering, Road, Housing, and Urban Development Research Center, Tehran, Iran.

Received 29 September 2014; received in revised form 11 June 2015; accepted 25 July 2015

KEYWORDSEuler-Bernoulli beam;Dynamic response;Series of travelingmasses;Resonance;Simpli�ed formula.

Abstract. In this article, transverse vibration of an Euler-Bernoulli beam carrying aseries of traveling masses is analyzed. A semi-analytical approach based on eigenfunctionexpansion method is employed to achieve the dynamic response of the beam. The inertiaof the traveling masses changes the fundamental period of the base beam. Therefore,a comprehensive parametric survey is required to reveal the resonance velocity of thetraversing inertial loads. In order to facilitate resonance detection for engineeringpractitioners, a new simpli�ed formula is proposed to approximate the resonance velocity.

© 2016 Sharif University of Technology. All rights reserved.

1. Introduction

The investigations on moving load dynamic problemsare said to be initiated early in the 19th century,ever since the �rst railway bridges were built [1].Since then, numerous studies have discovered di�erentaspects of this subject in structural dynamics [2-5].The term moving load emphasizes the time-varyingposition of an applied load on a speci�c structuralmember/mechanical device [6-13].

The railways [14-16], bridges [17-25], ground [26],and pavements [27] could be mentioned as the well-known cases of civil structures in uenced by travelingloads. There exists a variety of sources producingmoving loads on the structures such as the pedestri-

*. Corresponding author. Tel.: +98 21 88255942;Fax: +98 21 88254842E-mail addresses: r.afghani@sbu.ac.ir (R. AfghaniKhoraskani); mo�d@sharif.edu (M. Mo�d);s.eftekharazam@stru.polimi.it (S. Eftekhar Azam);m.ebrahimzadeh@bhrc.ac.ir, and mohsen.eb.h@gmail.com(M. Ebrahimzadeh Hassanabadi)

ans [19,22,25], trains [14,15], aircrafts [3,4,11], or themoving vehicles [27].

The numerical explorations of bridges have beenwidely accomplished by the researchers assuming abeam element to establish the governing equations ofmotion [1,7,8,12,28-35]. Prediction of the dynamicperformance of a bridge under a moving load wouldalso be necessary in the health monitoring of thestructure [36,37] or the assessment of design parametersspectra [8,11,20, 24].

A moving load dynamic problem could be sim-ulated either by the moving force [3,38-40] or themoving mass [6,8,11,12,18,20,24,29,41,42] modeling as-sumptions. The moving force is a highly simpli�edmodel in which the vehicle/bridge interaction force islimited to the weight of structure. In contrast to themoving force approach, the moving mass frameworkyields valid results for a wider range of parametersleading to a more realistic assessment of the basestructure dynamics; this is due to the considerationof inertial interaction of the traveling load and its sub-structure in the moving mass modeling approach.

134 R. Afghani Khoraskani et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 133{141

An issue of considerable interest in the bridgeengineering is the detection of bridge resonant statescaused by the multiple traversing loads [20,39-41]. Theresonance leads to the ballast degradation, derailment,deterioration of passenger ride comfort, and increasein the maintenance costs [39]. Accordingly, it wouldbe useful to predict the resonance velocity of a seriesof moving loads traversing the bridge; in this regard,the moving loads traversing the bridge at resonancevelocity lead to resonant vibration of the structure.

Regarding the available literature on the vibrationof bridges under moving vehicles, the prediction ofmoving masses resonance velocity is yet an issue opento discussion in the multiple moving mass problem.Dealing with the multiple moving forces, the systemmatrices are stationary and known. Hence, the exactresonance velocity could be analytically computed.However, in the case of the moving mass problem,matrices of the system continuously vary in time;therefore, it is not simple to analytically derive anexact formula for the resonance velocity as that of themultiple moving forces.

In this study, the problem of a series of identicaland equally spaced moving masses travelling with thesame speed along a beam-type structure is focused.This is frequently presumed as a model of a trainpassing over a bridge. Eigenfunction expansion methodis exploited to capture the base-beam de ections. Anew simpli�ed formula is proposed to predict resonancevelocity of the traversing masses. The formula is es-sentially based on the assumption that introducing theinertia of travelling masses into the mass matrix of theevolution equations will change the fundamental periodof the structure; therefore, it results in a di�erentresonance velocity than that obtained by neglectingthe inertia of the masses (moving force approximation).The accuracy of the introduced relation is assessedthrough extensive numerical analyses, where the pro-posed formula shows a very close agreement with thetrue resonance.

2. Problem formulation

A uniform, continuous Euler-Bernoulli beam is con-sidered as shown in Figure 1. It is assumed thatboth bending sti�ness of the base beam (EI) and themass per unit length (�A) are constant throughoutthe beam. Permanent contact condition between the

Figure 1. A beam excited by multiple moving masses.

moving masses and the structure is assumed during theperiod of loads movement on the beam. Regarding theEuler-Bernoulli Beam theory, the constitutive equationof motion could be written as [43,44]:8>>><>>>:

EI @4

@x4W (x; t) + �A @2

@t2W (x; t)= �PN

k=1 �kPk� (x�Xk(t)) ;

Pk = Mk

�g + � d

2

dt2W (Xk(t); t)�;

(1)

where W is the beam de ection and its positivedirection is assumed to obey the positive direction ofthe y axis in Figure 1 (i.e. + "); the horizontal axis xis assumed to correspond to the beam neutral axis; � ismass per unit of volume; A is the cross sectional areaof the beam; and t signi�es time. Moreover, Mk, g, and�() are mass of the kth traveling object, gravitationalacceleration, and Dirac delta, respectively. Xk(t) andPk are the location and the contact force of the kthtraveling mass, and N is the number of the movingloads. The function �k is de�ned as:

�k =

8<:1; 0 � Xk(t) � l0; Xk(t) < 0 or l < Xk(t)

(2)

The parameter � allows for convenient switching be-tween moving mass and moving force frameworks. � =0 corresponds to the simpli�ed moving force simulationand by assuming � = 1, the full contribution of movingloads inertial interaction with the base beam could beconsidered in the mathematical model (moving mass).A semi analytical procedure based on eigenfunctionexpansion method could be adopted to treat Eq. (1).To this end, the beam equation of free vibration shouldbe tackled to extract the eigenfunctions:

EI@4wj(x)@x4 = �A!2

jwj(x); (3)

where wj(x) denotes the jth shape function, and !j isthe jth natural frequency. W (x; t) in Eq. (1) could bereplaced by a series expansion as:

W (x; t) =1Xj=1

aj(t)wj(x) �nXj=1

aj(t)wj(x): (4)

Substituting Eq. (4) into Eq. (1) yields:nXj=1

��aj(t)!2

j +d2

dt2aj(t)

��Awj(x)

�= �

NXk=1

�k�Mkg + �Mk

nXj=1

d2=dt2faj(t)wj

(Xk(t))g�� (x�Xk(t)) : (5)

By regarding the orthonormal shape functions charac-

R. Afghani Khoraskani et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 133{141 135

teristic, i.e.:Z l

0�Awi(x)wj(x)dx = �ij =

(0; i 6= j1; i = j;

(6)

in which the shape functions are normalized by massper unit length of the beam �A. Multiplying both sidesof Eq. (5) by wi(x), and then integrating over the beamlength yields:�

!2i ai(t) +

d2

dt2ai(t)

�= �

NXk=1

�k�Mkg

+ �Mk

nXj=1

d2=dt2 [aj(t)wj(Xk(t))]�

wi (Xk(t)) : (7)

The expanded form of d2

dt2 [aj(t)wj(Xk(t))] is:

d2

dt2(aj(t)wj(Xk)) = wj(Xk)

d2

dt2aj(t)

+ 2��

@wj(x)@x

�dXk

dt

�x=Xk

ddtaj(t)

+��

@2wj(x)@x2

��dXk

dt

�2

+�@wj(x)@x

��d2Xk

dt2

��x=Xk

aj(t): (8)

By assuming uniform moving masses Mk = M withconstant velocity v as depicted in Figure 1, Eq. (8)could be simpli�ed to:

wj(Xk)d2

dt2aj(t) + 2v

�@@xwj(x)

�x=Xk

ddtaj(t)

+ v2�@2

@x2wj(x)�x=Xk

aj(t);

and the matrix from explanation of Eq. (7) could bewritten as:

M(t)d2

dt2a(t) + C(t)

ddt

a(t) + K(t)a(t) = F(t); (9)

where:

M(t) = [�ij ]n�n + �NXk=1

Mk(t); (10)

Mk(t) =�mkij(t)

�n�n = �kM [wi(x)wj(x)]x=Xk ; (11)

C(t) = �NXk=1

Ck(t); (12)

Ck(t) =�ckij(t)

�n�n

= �kM�2vwi(x)

�@@xwj(x)

��x=Xk

; (13)

K(t) =�!2i �ij

�n�n + �

NXk=1

Kk(t); (14)

Kk(t) =�Kkij(t)

�n�n

= �M�v2wi(x)

@2

@x2wj(x)�x=Xk

; (15)

F(t) =NXk=1

Fk(t); (16)

Fk(t) =�fki (t)

�n�1 = ��kMg [wi (Xk(t))] ; (17)

a(t) = [ai(t)]n�1 : (18)

The second order coupled ODEs, set in Eq. (9), can bereformulated to a reduced �rst order set of equationsas follows:

ddt

Q(t) = A(t)Q(t) + G(t);

Q(t0) = Q0; (19)

where:

A(t) =�

0n�n In�n�M�1K �M�1C

�2n�2n

; (20)

Q(t) =�

a(t)ddta(t)

�2n�1

; (21)

G(t) =�

0n�1M�1F

�2n�1

: (22)

Eq. (19) can be solved by using matrix exponential [45].In this study, the �rst three vibration modes are usedin the analysis.

3. The resonance velocity

When a set of sequential moving masses travels alongthe beam length, at a certain velocity, the beam isexpected to start to resonate. The most frequentresonance case is related to the coincidence of frequencycontent of excitation with the vibration frequency ofthe �rst de ection mode of the whole system. Inthe case that the inertial e�ects of the masses arenegligible, the �rst natural frequency of the systemcan be reasonably assumed as that of the beam alone.

136 R. Afghani Khoraskani et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 133{141

However, the fundamental frequency of the base beamwill change due to the inertial e�ects of the movingmasses added to the system. To calculate the resonancevelocity, due to the complications introduced by theinertia of the masses, these variations have usually beenignored; hence, the equations employed to predict theresonance velocity, previously, are frequently based onthe moving force presumption.

In this study, the resonance velocity derived viamoving force is modi�ed so as to account for the inertiale�ects of the moving objects traveling over a beam-typestructure. To accomplish this objective, the possiblechanges in the natural frequency of the system dueto the inertial e�ects of the moving masses have beenincluded in the analyses as well.

The resonance velocity of multiple moving forcestraveling along the beam could be expressed as:

vp =lm!1

2j�; (23)

where lm is the spacing between the moving masses(Figure 1), and !1 denotes the fundamental frequencyof the base beam. Given the load spacing, lm, theresonance speed for the series of moving forces becomesproportionate to 1

j , where j = 1; 2; :::; which impliesthat resonance will take place at speeds starting fromthe primary resonance velocity lm!1

2� in descendingvalues. To obtain the modi�ed fundamental frequencyof the beam carrying a series of moving inertial loads,it is assumed that the mass of the system is increasedby an amount equal to k:�:�:A:l, while the sti�nessof the system remains constant; where � is the massratio of each moving object with respect to the beammass, and k is the maximum number of objects that canbe simultaneously placed on the beam and will causethe maximum static de ection in the beam. In otherwords, a beam with the same exural sti�ness, EI, butwith an increased uniformly distributed mass should beconsidered to approximate the fundamental frequencyof the system. Consequently, assuming a beam withmass per unit volume of:

�0 = (k:� + 1):�; (24)

and regarding the fact that the �rst natural frequencyof a thin beam is proportional to

qEI�A [46], the funda-

mental frequency of the assumed beam with increasedmass density, �0, could be attained by a modi�cationfactor:8><>:!

01 = �!1;

� =q

11+k� :

(25)

By applying this modi�cation factor to Eq. (23), theresonance velocity for multiple moving masses could be

approximately simpli�ed as follows:

vres =r

11 + k�

vp: (26)

Considering a simply supported single span beam !i =� i�l

�2qEI�A , for the largest resonance velocity (primary

resonance velocity), Eq. (26) reads:

vres =r

11 + k�

sEI�A

�lm2l2

!: (27)

4. Numerical examples

A single span simply-supported Euler-Bernoulli beamis considered for the numerical examples. The bound-ary conditions of the beam are W (0; t) = @2

@x2W (0; t) =0 and W (l; t) = @2

@x2W (l; t) = 0. Therefore, normalizedeigenfunctions of the beam, considering Eq. (6), couldbe given by wi(x) =

q2�Al sin

� i�xl

�. The coupled dy-

namic system of beam and moving masses are supposedto be at rest at the beginning moment of analysis t = 0;therefore, zero initial condition is considered for Eq. (1)i.e. W (x; 0) = 0 and @

@tW (x; t)jt=0 = 0. A customarystandard normalization scheme [4,5,8] is adopted inthe representation of results by which the outputs aredescribed with non-dimensional relative parameters.The de ection of the beam at its midspan is selected asrepresentative for the quantities of the beam response;moreover, throughout the article, the maximum staticde ection caused by a single load in the set of therelevant travelling masses, Wstat = �Mgl3

48EI , is used topresent normalized de ection, WN = W (0:5l; t)=Wstat.Such normalization provides a better understanding ofthe dynamic ampli�cation factor, DAF= max(WN ),which is the maximum value of beam normalizedde ection, WN , during the forced vibration of thebeam.

To numerically determine the critical velocity, arange of values of vp has been considered, and thedynamic ampli�cation factors are obtained for eachvelocity at which the series of masses pass over thebeam. The spectral analyses are presented in Figure 2,wherein the abscissa denotes normalized velocity v=vpand the vertical axis represents the associated DynamicAmpli�cation Factor (DAF). In Figure 2, the resultsof the aforementioned analyses are confronted for thetwo cases of moving forces and moving masses. Theresonance velocity in this manner is the velocity atwhich the peaks of DAF take place. For the seriesof moving forces, the maximum of DAF peak appearsat the primary resonance velocity, i.e. v = vp whichelucidates Eq. (23). As mentioned before, resonancewill take place at speeds starting from a primary

R. Afghani Khoraskani et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 133{141 137

Figure 2. Dynamic ampli�cation factor at the midspanof the beam at various travelling velocities. Moving forceversus moving mass, N = 15, lm = 0:3l.

resonance velocity, then, repeating in descending valuesproportional to 1

j of the primary resonance, wherej = 1; 2; :::. The spike which appears at v = 0:5vpis associated with the secondary resonance velocity.A shifting trend of the spectral curves towards lowerresonance frequency components could be observedas the mass of the moving objects increases. Thiswould strictly justify the fact that the loads inertia cannotably dominate either the primary or the secondaryresonance velocities.

In case the ratio of the mass is not small ascompared to the beam mass, the resonance velocityobtained for moving forces will be of poor accuracyfor the multiple moving mass problem; for instance, incase the ratio of the mass of travelling objects and thebeam mass � = M

�Al are equal to 0.15, the resonancevelocity turns out to be 0:83 lvp or, equivalently, � =0:831. Therefore, as the mass ratio of the travellingmasses increases, the modi�cation in the moving forcesresonance velocity becomes increasingly signi�cant.

Without any loss of generality, let place our focuson the resonance speed at which the maximum peak of

Figure 3. Dynamic ampli�cation factor at the midspanof the beam at various travelling velocities. Moving forceversus moving mass, N = 15, lm = 0:3l.

DAF, or, in other words, the so-called herein primaryresonance velocity is produced. In Figure 3, thelater DAF spectral analysis is inspected for a series ofmoving masses, but instead, with small mass ratio withconcentration on the primary resonance. It could beobserved that even for relatively light moving masses,namely � = 0:02, the discrepancy of the moving forceand moving mass at resonant states is not negligible.

To allow an assessment of the proposed formulafor the resonance velocity with the numerical analyses,in a series of examples, the modi�cation factors resultedfrom the numerical simulations are compared with theproposed simpli�ed formula in Eq. (26), and the resultsof the comparison are presented in Table 1. It couldbe observed that the resonance velocities obtainedfrom Eq. (26) are in reasonable agreement with theresonance velocity attained by the spectral peaks innumerical analyses.

When there are 15 and 25 loads spaced by lm =0:3l, for the mass ratio � = 0:2, the di�erence betweenthe two resonance velocities is 1.3%. It could beperceived that the change in the number of moving

Table 1. Modi�cation factors obtained by Eq. (26) and the spectra.

Number ofmoving

masses (N)

loadspacing

(lm)k �

� resultedfrom

Eq. (25)

� resultedfrom

spectra

15 lm = 0:3l 30.10 0.877 0.8840.15 0.830 0.8310.20 0.791 0.785

15 lm = 0:6l 20.10 0.913 0.9210.15 0.877 0.8890.20 0.845 0.851

25 lm = 0:3l 30.10 0.877 0.8740.15 0.830 0.8240.20 0.791 0.775

138 R. Afghani Khoraskani et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 133{141

Figure 4. Time histories of the dynamic ampli�cationfactor at the midspan of a beam subjected to N = 25travelling masses at the true resonance velocity, thepredicted resonance velocity, and the resonance velocityobtained from the moving force assumptions; lm = 0:3l,� = 0:2.

masses from N = 15 to N = 25 is in uential in thecomputation of the true resonance velocities throughthe numerical method; the reason for this can beunderstood from Figure 4, wherein the time historyof the structural dynamic response for a moving masstraveling at its true resonance velocity is depicted. Att = 0, the �rst mass is at the left end of the beam,i.e. x = 0. The beam experiences steady state loadingscheme due to the stream of dynamic moving objectsacting on the beam as long as the third mass entersthe beam. This steady state dynamic loading continuesuntil the Nth load reaches the left end of the beam. Inthis regard, by increasing the total number of movingmasses N , the steady state part of the response woulddominate the numerically computed true resonancevelocity for which the predicted true resonance wouldtend towards its asymptotic state for N = 1. Thesimpli�ed formula is not sensitive to the total numberof masses. It predicts the same value independent ofthis number, as does the original formula for resonancevelocity of the set of moving forces.

In Figure 4, it can also be observed that althoughthe behavior of the system at the resonance velocitypredicted by the Eq. (26) does not exactly matchthe behavior of the system at the resonance velocity,it is far closer than the behavior of the system atthe resonance velocity predicted without including theinertia of the masses in analysis (vp).

In Figure 5, the same time histories of Figure 4are re-plotted assuming di�erent weights of the movingmasses, i.e. � = 0:1. It could be seen that the valuesof the time histories and the maximum response forthe case of true resonance velocity and the predictedresonance velocity via Eq. (26) are in close agreement,while the moving force is noticeably error-prone (T1denotes the fundamental period of the beam).

According to Table 1, the precision of the simpli-�ed formula could be a�ected by the maximum numberof masses which can be simultaneously placed on the

Figure 5. Time histories of the beam dynamic responsesubject to N = 25 travelling masses at the true resonancevelocity, the predicted resonance velocity, and theresonance velocity obtained from the moving forceassumptions; lm = 0:3l, � = 0:1.

beam and the mass of the moving objects, i.e. k, whichitself is dependent on spacing of the loads. With largespacing of the loads, k reduces and it is important tomention that the stream of moving masses should travelat higher velocities to reach the primary resonancevelocity. Therefore, the impact of the convectiveterms of the moving masses transverse acceleration (asmentioned in Eq. (8)) signi�cantly increases. On theother hand, larger masses could amplify these e�ects.Accordingly, the simpli�ed formula would feature smalldiscrepancies with true resonance velocity.

In some analytical investigations, it has been em-phasized that the moving mass and moving oscillatorcould be in close agreement for some real cases ofvehicle bridge coupled systems [9]. A comparison of thepresent method provided by moving mass model withthe corresponding moving oscillator problem (Figure 6)is depicted in Figure 7. The method of investigationemployed is that proposed by Yang et al. [47]. Itcan be concluded that the present technique is of goodengineering level of agreement for suspension systems

having =pK=M!1

� 2:0.

5. Conclusion

In this paper, the constitutive equation of an Euler-Bernoulli beam acted upon by multiple moving massesis studied. A critical velocity is proposed in termsof the modi�ed fundamental period of the structure,span length, and spacing of the moving masses. The

Figure 6. A beam traversed by sequential travelingoscillators.

R. Afghani Khoraskani et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 133{141 139

Figure 7. Dynamic ampli�cation factor of the beam atvarious travelling velocities. Moving mass versus movingoscillator, N = 15, c = 0, lm = 0:3l, � = 0:20.

proposed resonance velocity is based on the resonancevelocity of a beam subjected to a series of movingforces, which is recti�ed by a relevant modi�cationfactor. The results for approximate modi�cation factorappear to be in reasonable agreement with the trueones obtained from numerical assessments. Moreover,regarding the vehicle suspension with � 2:0, a closeagreement could be observed with the present method.

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Biographies

Roham Afghani Khoraskani received his PhD de-gree from the Department of Building and Environ-mental Science and Technology (BEST) at Politecnicodi Milano, Italy. He received his BS degree fromIsfahan University of Technology (IUT) and his MSDegree from Sharif University of Technology (SUT),Iran. Currently, he is an Assistant Professor at Faculty

of Architecture and Urban Planning, Shahid BeheshtiUniversity, Tehran, Iran.

Massood Mo�d is Professor of Civil and Structuralengineering. His research interests are in the area ofstructural analysis and structural dynamics.

Saeed Eftekhar Azam received his PhD degreefrom the Department of Structural engineering atPolitecnico di Milano, Italy. He received his BS degreefrom Tehran University and his MS degree from SharifUniversity of Technology (SUT), Iran. His researchinterests include dynamics of structures, structural sys-tem identi�cation, and structural health monitoring.

Mohsen Ebrahimzadeh Hassanabadi received hisMS degree from the Department of Structural Engi-neering at Babol University of Technology, Iran, andhis BS from University of Tehran, Iran. Currently, heis a researcher at BHRC.