Vibration of axially moving beam supported by viscoelastic foundation

doi:10.1007/s10483-017-2170-9

Abstract

Abstract:

In this paper, transverse vibration of an axially moving beam supported by a viscoelastic foundation is analyzed by a complex modal analysis method. The equation of motion is developed based on the generalized Hamilton's principle. Eigenvalues and eigenfunctions are semi-analytically obtained. The governing equation is represented in a canonical state space form, which is defined by two matrix differential operators. The orthogonality of the eigenfunctions and the adjoint eigenfunctions is used to decouple the system in the state space. The responses of the system to arbitrary external excitation and initial conditions are expressed in the modal expansion. Numerical examples are presented to illustrate the proposed approach. The effects of the foundation parameters on free and forced vibration are examined.

Key words: natural frequency, forced vibration, axially moving beam, viscoelastic foundation, complex modal analysis

2010 MSC Number:

O326

Haijuan ZHANG, Jian MA, Hu DING, Liqun CHEN. Vibration of axially moving beam supported by viscoelastic foundation. Applied Mathematics and Mechanics (English Edition), 2017, 38(2): 161-172.

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References

[1] Bhat, R. B., Xistris, G. D., and Sanker, T. S. Dynamic behavior of a moving belt supported on elastic foundation. Journal of Mechanics Design, 104, 143-147(1982)
[2] Perkins, N. C. Linear dynamics of a translating string on an elastic foundation. Journal of Vibration and Acoustics, 112, 2-7(1990)
[3] Wickert, J. A. Response solutions for the vibration of a traveling string on an elastic foundation. Journal of Vibration and Acoustics, 116, 137-139(1994)
[4] Xiong, Y. and Hutton, S. G. Vibration and stability analysis of a multi-guided rotating string. Journal of Sound and Vibration, 169, 669-683(1994)
[5] Tan, C. A. and Zhang, L. Dynamic characteristics of a constrained string translating across an elastic foundation. Journal of Vibration and Acoustics, 116, 318-325(1994)
[6] Riedel, C. H. and Tan, C. A. Dynamic characteristics and mode localization of elastically constrained axially moving strings and beams. Journal of Sound and Vibration, 215, 455-473(1998)
[7] Saeed, H. M. and Festroni, F. Simulation of combined system by periodic structures:the wave transfer matrix approach. Journal of Sound and Vibration, 213, 53-73(1998)
[8] Parker, R. G. Supercritical speed stability of the trivial equilibrium of an axially-moving string on an elastic foundation. Journal of Sound and Vibration, 221, 205-219(1999)
[9] Jha, R. K. and Parker, R. G. Spatial discretization of axially moving media vibration problems. Journal of Vibration and Acoustics, 122, 290-294(2000)
[10] Kartik, V. and Wickert, J. A. Vibration and guiding of moving media with edge weave imperfections. Journal of Sound and Vibration, 291, 419-436(2006)
[11] Yurdda?. A., Özkaya, E., and Boyac?, H. Nonlinear vibrations of axially moving multi-supported strings having non-ideal support condition. Nonlinear Dynamics, 73, 1223-1244(2012)
[12] Yang, X. D., Lim, C. W., and Liew, K. M. Vibration and stability of an axially moving beam on elastic foundation. Advances in Structural Engineering, 13, 241-248(2010)
[13] Ba?atli, S. M., Özkaya, E., and Öz, H. R. Dynamics of axially accelerating beams with an inte rmediate support. Journal of Vibration and Acoustics, 133, 470-476(2011)
[14] Ba?atli, S. M., Özkaya, E., and Öz, H. R. Dynamics of axially accelerating beams with multip le supports. Nonlinear Dynamics, 74, 237-255(2013)
[15] Kural, S. and Özkaya, E. Vibrations of an axially accelerating, multiple supported flexible beam. Structural Engineering and Mechanics, 44, 521-538(2012)
[16] Ghayesh, M. H. Stability and bifurcations of an axially moving beam with an intermediate spring support. Nonlinear Dynamics, 69, 193-210(2012)
[17] Yang, Y., Ding, H., and Chen, L. Q. Dynamic response to a moving load of a Timoshenko beam resting on a nonlinear viscoelastic foundation. Acta Mechanica Sinica, 29, 718-727(2013)
[18] Sun, L. A closed-form solution of a Bernoulli-Euler beam on a viscoelastic foundation under harmonic line loads. Journal of Sound and Vibration, 242, 619-627(2001)
[19] Ghayesh, M. H. Nonlinear transversal vibration and stability of an axially moving viscoelastic string supported by a partial viscoelastic guide. Journal of Sound and Vibration, 314, 757-774(2008)
[20] Ahmadian, M. T., Nasrabadi, V. Y., and Mohammadi, H. Nonlinear transversal vibration of an axially moving viscoelastic string on a viscoelastic guide subjected to mono-frequency excitation. Acta Mechanica, 214, 357-373(2010)
[21] Li, X. J. and Chen, L. Q. Modal analysis of coupled vibration of belt drive systems. Applied Mathematics and Mechanics(English Edition),29(1), 9-13(2008) DOI 10.1007/s10483-008-0102-x
[22] Huang, L. H. and Liu, X. T. Eigenvalue and stability analysis for transverse vibrations of axially moving strings based on Hamiltonian dynamics. Acta Mechanica Sinica, 21, 485-494(2005)
[23] Chung, C. H. and Kao, I. Modeling of axially moving wire with damping:eigenfunctions, orthogonality and applications in slurry wiresaws. Journal of Sound and Vibration, 330, 2947-2963(2011)
[24] Ding, H. and Jean, W. Z. Steady-state responses of pulley-belt systems with a one-way clutch and belt bending stiffness. Journal of Vibration and Acoustics, 136, 041006(2014)
[25] Ding, H., Zhang, G. C., Chen, L. Q., and Yang, S. P. Forced vibrations of supercritically transporting viscoelastic beams. Journal of Vibration and Acoustics, 134, 051007(2012)
[26] Zhang, H. J. and Chen, L. Q. Vibration of an axially moving string supported by a viscoelastic foundation. Acta Mechanica Solida Sinica, 26(3), 221-231(2016)
[27] Yan, Q. Y., Ding, H., and Chen, L. Q. Nonlinear dynamics of an axially moving viscoelastic Timoshenko beam under parametric and external excitations. Applied Mathematics and Mechanics (English Edition), 36(8), 971-984(2015) DOI 10.1007/s10483-015-1966-7
[28] Ding, H. Steady-state responses of a belt-drive dynamical system under dual excitations. Acta Mechanics Sinica, 32, 156-169(2016)