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Abstract：This paper investigates transverse vibrations of finite EulerBernoulli beams resting on viscoelastic foundations.It studies natural frequencies and dynamic response of an elastic beam subjected to a harmonic load.The differential quadrature methods（DQ） are applied directly to the governing equations of the free and the forced vibrations that are two partial differential equations.Under the simple supported boundary condition，the natural frequencies of the transverse vibrations are calculated，and compared with the results of the complex modal analysis method.The natural frequencies and dynamic response are numerically studied.The numerical results obtained with the DQ are in good agreement with that of the complex mode analysis methods for the first seven orders，but with the growth of the orders，the small quantitative differences between them increase.Numerical results also illustrate that the beam vibrates with same frequency of external load under harmonic motion，after a short transient response.
Key words：viscoelastic foundation； natural frequency； dynamic response； differential quadrature
methods
CLC number： O 32 Document code： A Article ID： 10005137（2015）02013206
1 Introduction
Vibration analysis of elastic EulerBernoulli beams resting on different types of foundations is of great interest in the area of transportation and civil engineering.The vibration problem of elastic beams resting on viscoelastic foundation displays viscous characters，and the solution becomes difficult.Most of literatures report free vibration analysis of beam on Pasternak foundation with approximate solutions.In most of the published researches on the topic of a finite beam resting on a foundation，the foundation is assumed as linear elastic one，and the problems are mainly studied by approximate methods such as finite element method[1]，transfer matrix method[2]，RayleighRitz method [3]，differential quadrature methods[4]，and Laplace transform technique[5].
It is worth noticing that all studies mentioned above are about elastic foundation，without accounting for any damping factor in foundation.But in fact，the effects of damping factors are an important research topic for vibration of elastic beams on foundation.In recent years，researchers began to study vibration of elastic beams resting on viscoelastic three parameters foundation.Free transverse vibration of beams on viscoelastic Winkler and Pasternak foundation are studied with complex mode analysis methods[68]，but most of the published works on this specific problem are focused on dynamic response of such a system subjected to a moving load[910]. The differential quadrature（DQ） method is a powerful tool for dealing with dynamical problems.It was introduced for structural analysis by Bert [11]，and since then it has been successfully employed for the analysis of vibration of beams[12].The dynamic response problem of elastic beams lying on viscoelastic foundation displays viscous characters，and the solution becomes difficult.To the author′s best knowledge，there were no literatures on the dynamic response of beams on viscoelastic Pasternak foundation excited by harmonic loads.In the present work，using the DQ method，the natural frequencies and dynamic responses are analyzed.The numerical results obtained by the DQ are compared with the complex modal analysis methods.
The present paper is organized as follows.Section 2 establishes the governing equation for the transverse free vibration of a finite elastic EulerBernoulli beam on viscoelastic foundation.Section 3 develops the DQ schemes to discretize the governing equation.Section 4 presents numerical examples for investigations on the effects of the relevant beam and foundation parmeters on the natural frequencies and the dynamic responses.Section 5 ends the paper with the concluding remarks.
5 Conclusions
This paper investigates transverse vibrations of finite EulerBernoulli beams resting on threeparameter viscoelastic foundations.It studies natural frequencies and dynamic response of an elastic beam subjected to a harmonic load.The differential quadrature methods are applied directly to the governing equations of the free and the forced vibrations which are two partial differential equations.Under the simply supported boundary condition，the natural frequencies of the transverse vibrations are compared with the results of the complex modal analysis method.The natural frequencies and dynamic response are numerically studied.The numerical results obtained with the differential quadrature are in good agreement with that of the complex mode methods for the first seven orders，but with the growth of the orders，the small quantitative differences between them increase.Numerical results also illustrate that the beam vibrates with same frequency of external load under harmonic motion，after a short transient response.
References：
[1] YOKOYAMA T.Vibration analysis of Timoshenko beamColumns on twoparameter elastic foundations[J].Computers & Structures，1996，61（6）：95-1007.
[2] IRIE T，YAMADA G，TAKAHASHI I.Vibration and stability of a nonuniform Timoshenko beam subjected to a flower force[J].Journal of Sound and Vibration，1980，70：503-512. [3] GUTIERREZ R H，LAURA P A A，ROSSI R E.Fundamental frequency of vibration of a Timoshenko beam of non uniform thickness[J].Journal of Sound and Vibration，1991，145：241-245.
[4] MALEKZADEH P，KARAMI G.DQEM for free vibration analysis of Timoshenko beams on elastic foundations [J].Computational Mechanics，2003，31：219-228.
[5] MAGRAB E B.Natural Frequencies and Mode Shapes of Timoshenko beams with attachmens[J].Journal of Vibration and Control，2007，13（7）：905-934.
[6] PENG L，CHEN C X.Analysis of vibrations of beams on viscoelastic winkler foundations[J].Journal of Shanghai Normal University （Natural Sciences），2012，41（6）：586-589.
[7] PENG L，DING H，CHEN L Q.Complex Modal Analysis of Vibrations of beams on viscoelastic Pasternak foundations[J].Journal of Vibration and Shock，2013，32（2）：143-146.
[8] PENG L，DING H，CHEN L Q.Transverse Free Vibration of a Timoshenko Beam Rested on Threeparameter Viscoelastic Foundation[J].Noise and Vibration Control，2013，33（5）：107-110.
[9] KARGARNOVIN M H，YOUNESIAN D.Dynamics of Timoshenko beams on Pasternak foundation under moving load[J].Mechanics Research Communication，2004，31：713-723.
[10] ALIM F F.Dynamic analysis of beams on viscoelastic foundation[J].European Journal of Mechanics A/Solids，2009，28：469-476.
[11] BERT C W，MALIK M.Differential quadrature method in computational mechanics：A review[J].Applied Mechanics Reviews，1996，49：1-27.
[12] PENG L，DING H，CHEN L Q.Transverse Free Vibration of a Beam Resting on a Threeparameter Viscoelastic Foundation[J].Journal of Vibration and Shock，2014，33（1）：101-105.
[13] TANG Y Q.Transverse Vibrations of Axially Moving beams and InPlane Moving Plates：Modeling and Analysis （in Chinese）[D].Shanghai：Shanghai University，2011.
[14] ANSARI M，ESMAILZADEH E，YOUNESIAN D.Internalexternal resonance of beams on nonlinear viscoelastic foundation traversed by moving load[J].Nonlinear Dynamics，2010，61：163-182.
摘要：探讨了黏弹性地基上有限长EulerBernoulli梁的横向振动.主要研究梁的固有频率和简谐均布荷载作用下的动力响应.将微分求积方法（DQ）直接应用于自由与受迫振动控制方程中.在简支边界条件下，得到横向自由振动的固有频率，并与复模态分析方法的结果进行比较.数值结果表明DQ与复模态分析方法得到的前七阶频率值高度吻合，但随着阶数的增长，两种方法数值间的微小差异值增大.数值结果还表明， 在均布简谐荷载作用下，经过短暂的瞬态响应后，梁的振动频率与外部荷载振动频率一致.
关键词：黏弹性地基； 固有频率； 动力响应； 微分求积法
（责任编辑：顾浩然）
Key words：viscoelastic foundation； natural frequency； dynamic response； differential quadrature
methods
CLC number： O 32 Document code： A Article ID： 10005137（2015）02013206
1 Introduction
Vibration analysis of elastic EulerBernoulli beams resting on different types of foundations is of great interest in the area of transportation and civil engineering.The vibration problem of elastic beams resting on viscoelastic foundation displays viscous characters，and the solution becomes difficult.Most of literatures report free vibration analysis of beam on Pasternak foundation with approximate solutions.In most of the published researches on the topic of a finite beam resting on a foundation，the foundation is assumed as linear elastic one，and the problems are mainly studied by approximate methods such as finite element method[1]，transfer matrix method[2]，RayleighRitz method [3]，differential quadrature methods[4]，and Laplace transform technique[5].
It is worth noticing that all studies mentioned above are about elastic foundation，without accounting for any damping factor in foundation.But in fact，the effects of damping factors are an important research topic for vibration of elastic beams on foundation.In recent years，researchers began to study vibration of elastic beams resting on viscoelastic three parameters foundation.Free transverse vibration of beams on viscoelastic Winkler and Pasternak foundation are studied with complex mode analysis methods[68]，but most of the published works on this specific problem are focused on dynamic response of such a system subjected to a moving load[910]. The differential quadrature（DQ） method is a powerful tool for dealing with dynamical problems.It was introduced for structural analysis by Bert [11]，and since then it has been successfully employed for the analysis of vibration of beams[12].The dynamic response problem of elastic beams lying on viscoelastic foundation displays viscous characters，and the solution becomes difficult.To the author′s best knowledge，there were no literatures on the dynamic response of beams on viscoelastic Pasternak foundation excited by harmonic loads.In the present work，using the DQ method，the natural frequencies and dynamic responses are analyzed.The numerical results obtained by the DQ are compared with the complex modal analysis methods.
The present paper is organized as follows.Section 2 establishes the governing equation for the transverse free vibration of a finite elastic EulerBernoulli beam on viscoelastic foundation.Section 3 develops the DQ schemes to discretize the governing equation.Section 4 presents numerical examples for investigations on the effects of the relevant beam and foundation parmeters on the natural frequencies and the dynamic responses.Section 5 ends the paper with the concluding remarks.
5 Conclusions
This paper investigates transverse vibrations of finite EulerBernoulli beams resting on threeparameter viscoelastic foundations.It studies natural frequencies and dynamic response of an elastic beam subjected to a harmonic load.The differential quadrature methods are applied directly to the governing equations of the free and the forced vibrations which are two partial differential equations.Under the simply supported boundary condition，the natural frequencies of the transverse vibrations are compared with the results of the complex modal analysis method.The natural frequencies and dynamic response are numerically studied.The numerical results obtained with the differential quadrature are in good agreement with that of the complex mode methods for the first seven orders，but with the growth of the orders，the small quantitative differences between them increase.Numerical results also illustrate that the beam vibrates with same frequency of external load under harmonic motion，after a short transient response.
References：
[1] YOKOYAMA T.Vibration analysis of Timoshenko beamColumns on twoparameter elastic foundations[J].Computers & Structures，1996，61（6）：95-1007.
[2] IRIE T，YAMADA G，TAKAHASHI I.Vibration and stability of a nonuniform Timoshenko beam subjected to a flower force[J].Journal of Sound and Vibration，1980，70：503-512. [3] GUTIERREZ R H，LAURA P A A，ROSSI R E.Fundamental frequency of vibration of a Timoshenko beam of non uniform thickness[J].Journal of Sound and Vibration，1991，145：241-245.
[4] MALEKZADEH P，KARAMI G.DQEM for free vibration analysis of Timoshenko beams on elastic foundations [J].Computational Mechanics，2003，31：219-228.
[5] MAGRAB E B.Natural Frequencies and Mode Shapes of Timoshenko beams with attachmens[J].Journal of Vibration and Control，2007，13（7）：905-934.
[6] PENG L，CHEN C X.Analysis of vibrations of beams on viscoelastic winkler foundations[J].Journal of Shanghai Normal University （Natural Sciences），2012，41（6）：586-589.
[7] PENG L，DING H，CHEN L Q.Complex Modal Analysis of Vibrations of beams on viscoelastic Pasternak foundations[J].Journal of Vibration and Shock，2013，32（2）：143-146.
[8] PENG L，DING H，CHEN L Q.Transverse Free Vibration of a Timoshenko Beam Rested on Threeparameter Viscoelastic Foundation[J].Noise and Vibration Control，2013，33（5）：107-110.
[9] KARGARNOVIN M H，YOUNESIAN D.Dynamics of Timoshenko beams on Pasternak foundation under moving load[J].Mechanics Research Communication，2004，31：713-723.
[10] ALIM F F.Dynamic analysis of beams on viscoelastic foundation[J].European Journal of Mechanics A/Solids，2009，28：469-476.
[11] BERT C W，MALIK M.Differential quadrature method in computational mechanics：A review[J].Applied Mechanics Reviews，1996，49：1-27.
[12] PENG L，DING H，CHEN L Q.Transverse Free Vibration of a Beam Resting on a Threeparameter Viscoelastic Foundation[J].Journal of Vibration and Shock，2014，33（1）：101-105.
[13] TANG Y Q.Transverse Vibrations of Axially Moving beams and InPlane Moving Plates：Modeling and Analysis （in Chinese）[D].Shanghai：Shanghai University，2011.
[14] ANSARI M，ESMAILZADEH E，YOUNESIAN D.Internalexternal resonance of beams on nonlinear viscoelastic foundation traversed by moving load[J].Nonlinear Dynamics，2010，61：163-182.
摘要：探讨了黏弹性地基上有限长EulerBernoulli梁的横向振动.主要研究梁的固有频率和简谐均布荷载作用下的动力响应.将微分求积方法（DQ）直接应用于自由与受迫振动控制方程中.在简支边界条件下，得到横向自由振动的固有频率，并与复模态分析方法的结果进行比较.数值结果表明DQ与复模态分析方法得到的前七阶频率值高度吻合，但随着阶数的增长，两种方法数值间的微小差异值增大.数值结果还表明， 在均布简谐荷载作用下，经过短暂的瞬态响应后，梁的振动频率与外部荷载振动频率一致.
关键词：黏弹性地基； 固有频率； 动力响应； 微分求积法
（责任编辑：顾浩然）