Journal of Engineering and Natural Sciences Mühendislik ve Fen Bilimleri Dergisi VIBRATION OF VISCOELASTIC BEAMS SUBJECTED TO MOVING HARMONIC LOADS

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1 Journal of Engineering and Natural Sciences Mühendisli ve Fen Bilileri Dergisi Siga 004/3 VIBRATION OF VISCOEASTIC BEAMS SUBJECTED TO MOVING HARMONIC OADS Turgut KOCATÜRK *, Mesut ŞİMŞEK Departent of Civil Engineering, Yildiz Technical University, Yıldız-İSTANBU Geliş/Received: Kabul/Accepted: ABSTRACT The transverse vibration of a bea with interediate point constraints subjected to a oving haronic load is analyzed within the fraewor of the Bernoulli-Euler bea theory. The agrange equations are used for exaining the dynaic response of beas subjected to the oving haronic load. The constraint conditions of supports are taen into account by using agrange ultipliers. In the study, for applying the agrange equations, trial function denoting the deflection of the bea is expressed in the polynoial for. By using the agrange equations, the proble is reduced to the solution of a syste of algebraic equations. The syste of algebraic equations is solved by using the direct tie integration ethod of Newar [8]. Results of nuerical siulations are presented for various cobinations of constant axial velocity, excitation frequency, nuber of point supports and various values of daping coefficient. Keywords: Forced vibrations of bea, free vibrations of bea, oving haronic load HAREKETİ HARMONİK YÜKER ETKİSİNDEKİ VİSKOEASTİK KİRİŞERİN TİTREŞİMİ ÖZET Bu çalışada hareetli haroni yüler etisindei irişlerin enine titreşileri Bernoulli-Euler iriş teorisi çerçevesinde inceleniştir. Problein çözüü için agrange denleleri ullanılıştır. Problede esnet şartları agrange çarpanları ullanılara sağlanıştır. Çalışada, agrange denlelerinin uygulanası için irişin yerdeğiştirelerini ifade eden çözü fonsiyonunun oluşturulasında polinolar ullanılıştır. agrange denleleri ullanılara proble cebri denle sisteinin çözüüne indirgeniştir. Bu denle sistei Newar [8] yöntei ullanılara çözülüştür. Problede irişin yerdeğiştireleri, hareetli haroni yüün freansı ve hızı, çeşitli sönü oranları ve açılı sayısı için sayısal olara inceleniştir. Anahtar Sözcüler: Zorlanış iriş titreşileri, serbest iriş titreşileri, hareetli haroni yü 1. INTRODUCTION Transverse vibration of beas subjected to oving loads has been an interesting research topic for long years. Vibrations of this ind occur in any branches of engineering, for exaple in bridges and railways. Many ethods have been presented for response prediction, but only the notable ones cited here. The earliest wor on the the behaviour of a single-span bea subjected to a constant oving haronic load was reported by Tiosheno and Young [1]. Fryba [] presented various analytical solutions for vibration probles of siple and continuous beas under oving loads in his boo. H.P. ee [3] utilized Hailton s principle to solve the dynaic response of a bea with interediate point constraints subjected to a oving load by using the * Sorulu Yazar/Corresponding Autor:e-ail: ocatur@yildiz.edu.tr, Tel: (01) /

2 Vibration of Viscoelastic Beas Subjected to... vibration odes of a siply supported bea as the assued odes. Abu-Hilal and Mohsen [4] studied the dynaic response of elastic hoogenous isotropic beas with different boundary conditions subjected to a constant force travelling with accelerating, decelerating and constant velocity types of otion. Zheng et al. [5] considered the dynaic response of the continuous beas subjected to oving loads by using the odified bea vibration functions. Dugush and Eisenberg [6] exained vibrations of non-unifor continuous beas under oving loads by using both the odal analysis ethod and the direct integration ethod. Zhu and aw [7] analyzed the dynaic response of a continuous bea under oving loads using Hailton s principle and eigenpairs obtained by the Ritz ethod. In the present study, the agrange equations are used for exaining the dynaic response of viscoelastic beas subjected to a oving haronic load with constant axial speed. The constraint conditions of the supports are taen into account by using agrange ultipliers. In the study, for applying the agrange equations, the trial function denoting the deflection of the bea is expressed in the polynoial for. By using the agrange equations, the proble is reduced to a syste of algebraic equations. This syste of algebraic equations is solved by using the direct tie integration ethod of Newar [8]. The convergence of the study is based on the nuerical values obtained for various nubers of polynoial ters. Results in this paper are readily applicable for further investigation in this field.. THEORY AND FORMUATIONS A continuous viscoelastic Bernoulli-Euler bea with N point supports subjected to a oving haronic load is depicted in Fig 1. The considered bea has a unifor cross sectional area, and its length is. The bea is constrained against vertical displaceents at various points. The constraint conditions are satisfied by using agrange ultipliers. A oving haronic load Qt ( ) is applied in the y direction fro left to right with prescribed constant speed in the axial direction. The assuptions ade in the following forulation are that transverse deflections are sall so that the dynaic behaviour of the bea is governed by the Bernoulli-Euler bea theory. Moreover, all the transverse deflections occur in the sae plane, defined by the x and y axes. The y axis is chosen at the idpoint of the total length of the bea as shown in Fig. 1. Q (t) x Q 1 3 N y x / / Figure 1. A continuous bea with N point supports subjected to a oving haronic load According to the Bernoulli-Euler bea theory, the elastic strain energy of the bea at any tie in Cartesian coordinates due to bending is 1 wxt (, ) x U = EI( x) dx (1) where E, I( x ) and wxt (, ) are the Young s odulus and the oent of inertia of the cross section of the bea and the displaceent function of the bea. 117

3 T. Kocatür, M. Şişe Siga 004/3 Neglecting the rotatory inertia effects, the inetic energy of the bea at any tie is 1 wxt (, ) T = ρ A( x) dx t () where ρ and A(x) are the ass density and the cross-section area of the bea. The Kelvin odel for the aterial is used. In this case, the dissipation function of the bea at any tie is 1 wxt (, ) i x R = r d x ri = γ EI( x) (4) where r i and γ are the coefficient of internal daping of the viscoelastic bea and proportionality constant of internal daping, respectively. The potential energy of the external force Q () t at any tie is () ( Q ) V = Q t w x,t (5a) ( ) sin ( Ω ) Qt = P t (5b) where P is the aplitude of the oving haronic force, Ω is the excitation frequency, coordinate of the oving haronic load at any tie t and expressed as x = v.t / ; Q (3) x Q is the xq ; 0 t (6) v where v is the axial velocity of the oving haronic load. The functional of the proble is I = T ( U + V ) (7) It is nown that soe expressions satisfying geoetrical boundary conditions are chosen for w( x, t ) and by using the agrange equations, the natural boundary conditions are also satisfied. Therefore, by using the agrange equations and by assuing the displaceent w( x, t ) to be representable by a linear series of adissible functions and adjusting the coefficients in the series to satisfy the agrange equations, an approxiate solution is found for the displaceent function. For applying the agrange equations, the trial function w( x, t ) is approxiated by 0 1 M space-dependent polynoial ters x, x, x,..., x and tie-dependent generalized displaceent coordinates A () t. Thus = 0 M w( x,t) = A ( t) x (8) where w( x,t ) is the dynaic response of the bea subjected to the oving haronic load. The constraint conditions of the supports are satisfied by using the agrange ultipliers. The constraint conditions are ( Si t) λ w x, = 0, i = 13,,,..., N (9) i 118

4 Vibration of Viscoelastic Beas Subjected to... where xsi denotes the location of the i th support, N denotes the nuber of the point supports. In Eq. (9), λ i quantities are the agrange ultipliers which are the support reactions in the considered proble. The agrange ultipliers forulation of the considered proble necessities the construction of the agrangian functional; (, ) = I + λ i w x Si t, i = 13,,,..., N (10) which attains its stationary value at the solution ( ( Si, ), i ) wx t λ. The generalized daping force Q D r can be obtained fro the dissipation function by differentiating R with respect to A R D =, = 13,,,..., M+ N (11) A Q r Then, using the agrange equations d + QD = 0 r A dt A, = 13,,,..., M+ N (1) where the dot above is the derivative with respect to tie, and introducing A λ, i = 13,,,..., N (13) = M + i i yield the following equation; [ A]{ A } + [ B]{ A } + [ C]{ A } { D} Where =, = 13,,..., M+ N (14) A = EI( x)( x ) ( x ) d x,, = 13,,..., M = S A x, = 1,...,M ; = M,..., M + N = S A x, = M,..., M + N; = 1,..., M A = 0,, = M,..., M + N B = ( ) ( ) r x x dx,, = 13,,..., M i B = 0,,= M,...,M + N C = ρ Axxx ( ) dx,, = 13,,..., M C = 0,,= M,...,M + N D = Qx Q, = 13,,..., M D = 0, = M,..., M + N (15) 119

5 T. Kocatür, M. Şişe Siga 004/3 where ( x ) is the second derivative of the on tie, but { D} depends on tie; naely x. [ A ], [ B ], [ ] x Q depens on tie. C are the atrices that do not depend For free vibration analysis, the tie-dependent generalized displaceent coordinates can be expressed as follows: i t A () t = A e ω (16) By substituting Eq. (16) into Eq. (14) and taing the daping atrix of the bea [ B ] and the external forces atrix { D } as zero in Eq. (14), this situation results in a set of linear hoogeneous equations that can be expressed in the following atrix for; [ A]{ A } ω [ C]{ A } { 0} = (17) where ω is the natural frequency of the considered bea. By using the direct tie integration ethod of Newar [8], Eq. (14) is solved and A, A, A and λ i coefficients are obtained for any tie t. Then, the displaceents, velocities and accelerations at the considered point and tie can be deterined by using Eq. (8). 3. NUMERICA RESUTS A nuber of nuerical exaples are presented to deonstrate the versatility, accuracy and efficiency of the present ethod. The obtained results are in good agreeent with the previously published results where applicable. In the following figures, x Q is the distance between the oving haronic load and the idpoint of the bea. At this stage, a convergence study is carried out. For this purpose, the natural frequencies of the considered bea are deterined by calculating the eigenvalues ω i of the frequency Eq. (17). In Table 1, the calculated natural frequencies are copared with those of Tiosheno and Young [1] and Fyrba []. The convergence is tested by taing the nuber of the polynoial ters 4, 5, 6, 8, 10, 1, 14. It is seen that the present converged values show excellent agreeent with those of Tiosheno and Young [1] and Fyrba []. Table 1. Convergence study of the natural frequencies ω i (rad/s) of the bea and coparison of the obtained results with the existing exact results Nuber of Polynoial ω 1 ω ω 3 ω 4 Ters Present Study References [1] and [] It is observed fro Table 1 that, the natural frequencies decrease as the nuber of the 10

6 Vibration of Viscoelastic Beas Subjected to... polynoial ters increases: It eans that the convergence to the exact value is fro above. By increasing the nuber of the polynoial ters, the exact value can be approached fro above. It should be reebered that energy ethods always overestiate the fundaental frequency, so with ore refined analyses, the exact value can be approached fro above. Convergence study indicates that the calculated values are converged to within three significant figures. In all of the calculations, the daping ratio ξ is taen as 0 0, 0 05, 0 10 and diensionless daping coefficient which is given in reference [4] as follows is considered. γ1+ γω ξ = (18) ω In Eq. (18), γ 1 is the proportionality constant of the external daping, γ is the proportionality constant of the internal daping and ω is the natural circular frequency of the th ode, respectively. It is nown that external daping is very sall with respect to internal daping. Therefore, the external daping is ignored in this study. In the case of daping by using Eqs. (4) and (18), the daping coefficient r i is obtained. Fro here on, the nuber of the polynoial ters is taen as 1 in all of the nuerical investigations A Single-Span Bea A single-span bea with siply-supported ends is considered. The cross sectional area A and the 3 ass density ρ are and 7700 g / 3, respectively. The total length is 1 and Young s odulus E is MPa ( EI = N ). The deflection at the centre of the 3 span due to oving haronic load w is noralized by the static deflection D ( D = P / 48 EI ). The nuerical integration is perfored using Gaussian quadrature. The frequency ratio β is taen as defined in reference [4] as follows Ω β = (19) ω 1 where ω1 is the natural frequency at the first ode of vibration of a siply supported bea calculated fro the Eq. (17), has a value of rad/s. The effect of the daping is represented by the daping ratio ξ = 0 0, 0 05, 0 10 in the calculations. In Figs. -5, the noralized ( w/d ) deflections at the center of a single-span bea are shown. These deflections occur due to oving haronic load travelling at constant axial velocity, v = 15 5, v = 39, v = 78 and v = 155 /s, for various values of ξ. These results are copared with those given in references [1,3,4] for oving load and oving haronic load. Good agreeent is observed. Figs. -5 show the effects of the velocity, the excitation frequency and daping for the single-span bea. In all these figures, the effect of daping is clear for all cases where an increase in daping yields, in general, a decrease in the response. It is seen fro the Figs. -5 that for the sall values of velocity, the excitation frequency has ore iportant effect on the behaviour of the bea, i.e; for especially v = 15 5 /s. The axiu diensionless absolute displaceent of the bea is increased by increasing the values of β until β = 1. The above entioned displaceent reaches a axiu value at resonance ( β = 1 ), and then with the increase in β, it decreases. The case of resonance is uch ore visible for this velocity. Because, as the values of velocity increases, the acting tie of the load on the bea becoes 11

7 T. Kocatür, M. Şişe Siga 004/3 shorter. Therefore, at high values of the velocity, the load leaves the bea without copleting its one-period. It is seen fro the obtained results that in the case of β = 0, the axiu dynaic deflection at the centre of the span is associated with speed v = 78 /s. Moreover, it should also be pointed out that negative displaceent eans tension stresses at the top of the bea. The diensionless deflection for a single-span bea is independent of the agnitude of the force. In the case of oving load, naely β = 0, the obtained results are excellent agreeent with those of ee [3]. Figure. Noralized deflections at the centre of a single-span bea varying β for v = 15 5 /s, ( ) ξ = 0 0, ( ) ξ = 0 05, ( ) ξ =

8 Vibration of Viscoelastic Beas Subjected to... Figure 3. Noralized deflections at the centre of a single-span bea varying β for v = 39 /s, ( ) ξ = 0 0, ( ) ξ = 0 05, ( ) ξ =

9 T. Kocatür, M. Şişe Siga 004/3 Figure 4. Noralized deflections at the centre of a single-span bea varying β for v = 78 /s, ( ) ξ = 0 0, ( ) ξ = 0 05, ( ) ξ =

10 Vibration of Viscoelastic Beas Subjected to... Figure 5. Noralized deflections at the centre of a single-span bea varying β for v = 155 /s, ( ) ξ = 0 0, ( ) ξ = 0 05, ( ) ξ = Two-Span Bea A two-span viscoelastic bea with siply-supported ends and a siple support at the iddle of the span of the bea is considered. The geoetrical and the physical properties of the bea are 15

11 T. Kocatür, M. Şişe Siga 004/3 the sae of those of the first proble s. The fundeental natural frequency of the bea at the first ode is rad/s which is obtained fro the equation [17]. The daping ratio is 3 assued to be ξ = The deflections are noralized by the deflection D ( D = P / 48 EI ) of the single-span bea. Figs. 6-7 show the noralized deflections under the oving haronic load for a two-span viscoelastic bea. In the case of oving load, naely β = 0, the present results are in perfect agreeent with those given in references [3] and [5]. Fig. 6 shows the dynaic response of a two-span bea for different values of the axial velocity of the load for β = 1 and ξ = For the considered paraeters, it is noticed that in the low velocities of oving haronic load, i.e., v = 15 5 /s, the bea has uch ore higher axiu dynaic displaceent than in the fast velocities of the oving haronic load and there is a drastic decrease in the dynaic displaceent by increasing the value of the velocity. Fig. 7 shows the effect of the excitation frequency Ω represented in the calculations by the frequency ratio β for a two-span bea, where the velocity is held constant ( v =15 5 /s). It is clear fro Fig. 7 that the circular frequency Ω has visible effect on the shape of the dynaic displaceent. Higher excitation frequency leads to sharper shapes and larger axiu deflections until a value of β. By increasing β until a certain value, the displaceents increase. However, after that certain value of β, the displaceents decrease with the increase in β. This situation is observed fro Fig. 7 Additionally, the dynaic displaceents for two-span bea are very sall with respect to the single-span bea. Figure 6. Noralized deflections under oving haronic load for two-span bea for v = 15 5, v = 39, v = 78, v = 155 /s, ξ = 0 05, β = 1 16

12 Vibration of Viscoelastic Beas Subjected to... Figure 7. Noralized deflections under the oving haronic load for two-span bea for β = 0 0, 0 5, 0 50, 1 0, 0, ξ = 0 05, v = 15 5 /s 4. CONCUSIONS The dynaic deflections of beas subjected to a oving haronic load with a constant velocity have been investigated. To use the agrange equations with the trial function in the polynoial for and to satisfy the constraint conditions by the use of agrange ultipliers is a very good 17

13 T. Kocatür, M. Şişe Siga 004/3 way for studying the dynaic behavior of continuous beas subjected to a oving haronic load. Nuerical calculations have been conducted to clarify the effects of the three iportant paraeters, the axial velocity of the oving haronic load, the excitation frequency of the oving load and the daping of the viscoelastic bea. It is observed fro the investigations that the axial velocity of the load, the frequency of the load, the daping of the viscoelastic bea have a very iportant effect on the deflections. All of the obtained results are very accurate and ay be useful for designing structural and echanical systes under oving haronic loads. REFERENCES [1] Tiosheno S., Young D. H. Vibration Probles in Engineering, Third edition, Van Nostrand Copany, New Yor, , [] Fryba., Vibration of solids and Structures Under Moving oads, Noordhoff International Publishing, Groningen, 13-43, 197. [3] ee H. P., Dynaic response of a bea with interediate point constraints subject to a oving load Journal of Sound and Vibration, 171, , [4] Abu-Hilal M., Mohsen M., Vibration of beas with general boundary conditions due to a oving haronic load, Journal of Sound and Vibartion, 3, , 000. [5] Zheng D. Y., Cheung Y. K., Au F. T. K., Cheng Y. S., Vibration of ulti-span nonunifor beas under oving loads by using odified bea vibration functions, Journal of Sound and Vibration, 1, , [6] Dugush Y. A., Eisenberg M., Vibrations of non-unifor continuous beas under oving loads, Journal of Sound and Vibration, 54, , 00. [7] Zhu X. Q, aw S. S., Precise tie-step integration for the dynaic response of a continuous bea under oving loads, Journal of Sound and Vibration, 40, , 000. [8] Newar N. M., A ethod of coputation for structural dynaics, ASCE Eng. Mech. Div., 85, 67-94,