Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions

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• 作者：You-Qi Tang ; Deng-Bo Zhang ; Jia-Ming Gao
• 关键词：Nonlinearity ; Parametric and 3 ; 1 internal resonance ; Axially accelerating viscoelastic beam ; Longitudinally varying tension ; Nonhomogeneous boundary condition ; Method of multiple scales ; Differential quadrature scheme
• 刊名：Nonlinear Dynamics
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：83
• 期：1-2
• 页码：401-418
• 全文大小：1,658 KB
• 参考文献：1.脰z, H.R., Pakdemirli, M., Boyaci, H.: Non-linear vibrations and stability of an axially moving beam with time-dependent velocity. Int. J. Non-Linear Mech. 36, 107鈥?15 (2001)CrossRef
  2.Chen, L.Q., Ding, H., Lim, C.W.: Principal parametric resonance of axially accelerating viscoelastic beams: multi-scale analysis and differential quadrature verification. Shock Vib. 19, 527鈥?43 (2012)CrossRef
  3.Chen, L.Q., Yang, X.D.: Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models. Int. J. Solids Struct. 42, 37鈥?0 (2005)CrossRef
  4.Sahoo, B., Panda, L.N., Pohit, G.: Parametric and internal resonances of an axially moving beam with time-dependent velocity. Model. Simul. Eng. 2013, 919517 (2013)
  5.Ghayesh, M.H., Balar, S.: Non-linear parametric vibration and stability of axially moving visco-elastic Rayleigh beams. Int. J. Solids Struct. 45, 6451鈥?467 (2008)CrossRef
  6.Tang, Y.Q., Chen, L.Q., Yang, X.D.: Parametric resonance of axially moving Timoshenko beams with time-dependent speed. Nonlinear Dyn. 58, 715鈥?24 (2009)CrossRef
  7.Ghayesh, M.H., Khadem, S.E.: Rotary inertia and temperature effects on non-nonlinear vibration, steady-state response and stability of an axially moving beam with time-dependent velocity. Int. J. Mech. Sci. 50, 389鈥?04 (2008)CrossRef
  8.Ghayesh, M.H., Balar, S.: Non-linear parametric vibration and stability analysis for two dynamic modes of axially moving Timoshenko beams. Appl. Math. Model. 34, 2850鈥?859 (2010)CrossRef
  9.Ravindra, B., Zhu, W.D.: Low dimensional chaotic response of axially accelerating continuum in the supercritical regime. Arch. Appl. Mech. 68, 195鈥?05 (1998)CrossRef
  10.Ghakraborty, G., Mallik, A.K.: Stability of an accelerating beam. J. Sound Vib. 227, 309鈥?20 (1999)CrossRef
  11.Ding, H., Chen, L.Q.: Nonlinear dynamics of axially accelerating viscoelastic beams based on differential quadrature. Acta Mech. Solida Sin. 22, 267鈥?75 (2009)CrossRef
  12.Chen, L.H., Zhang, W., Yang, F.H.: Nonlinear dynamics of higher dimensional system for an axially accelerating viscoelastic beam. J. Sound Vib. 329, 5321鈥?345 (2010)CrossRef
  13.Chen, L.Q., Tang, Y.Q.: Parametric stability of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions. ASME J. Vib. Acoust. 134, 011008 (2012)CrossRef
  14.Chen, L.Q., Tang, Y.Q.: Combination and principal parametric resonances of axially accelerating viscoelastic beams: recognition of longitudinally varying tensions. J. Sound Vib. 330, 5598鈥?614 (2011)CrossRef
  15.Tang, Y.Q., Chen, L.Q., Zhang, H.J., Yang, S.P.: Stability of axially accelerating viscoelastic Timoshenko beams: recognition of longitudinally varying tensions. Mech. Mach. Theory 62, 31鈥?0 (2013)CrossRef
  16.Mote Jr, C.D.: A study of band saw vibrations. J. Frankl. Inst. 276, 430鈥?44 (1965)CrossRef
  17.Riedel, C.H., Tan, C.A.: Coupled, forced response of an axially moving strip with internal resonance. Int. J. Non-Linear Mech. 37, 101鈥?16 (2002)CrossRef
  18.Chen, L.Q., Tang, Y.Q., Zu, J.W.: Nonlinear transverse vibration of axially accelerating strings with exact internal resonances and longitudinally varying tensions. Nonlinear Dyn. 76, 1443鈥?468 (2014)CrossRef
  19.Panda, L.N., Kar, R.C.: Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances. Nonlinear Dyn. 49, 9鈥?0 (2007)CrossRef
  20.Panda, L.N., Kar, R.C.: Nonlinear dynamics of a pipe conveying pulsating fluid with combination, principal parametric and internal resonances. J. Sound Vib. 309, 375鈥?06 (2008)
  21.Sze, K.Y., Chen, S.H., Huang, J.L.: The incremental harmonic balance method for nonlinear vibration of axially moving beams. J. Sound Vib. 281, 611鈥?26 (2005)CrossRef
  22.Chen, S.H., Huang, J.L., Sze, K.Y.: Multidimensional lindstedt-poincar茅 method for nonlinear vibration of axially moving beams. J. Sound Vib. 306, 1鈥?1 (2007)CrossRef
  23.Huang, J.L., Su, R.K.L., Li, W.H., Chen, S.H.: Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances. J. Sound Vib. 330, 471鈥?85 (2011)CrossRef
  24.Ghayesh, M.H.: Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance. Int. J. Mech. Sci. 53, 1022鈥?037 (2011)CrossRef
  25.Ghayesh, M.H., Kafiabad, H.A., Reid, T.: Sub- and super-critical nonlinear dynamics of a harmonically excited axially moving beam. Int. J. Solids Struct. 49, 227鈥?43 (2012)CrossRef
  26.Ghayesh, M.H., Kazemirad, S., Amabili, M.: Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance. Mech. Mach. Theory 52, 18鈥?4 (2012)CrossRef
  27.Tang, Y.Q., Chen, L.Q.: Nonlinear free transverse vibrations of in-plane moving plates: without and with internal resonances. J. Sound Vib. 330, 110鈥?26 (2011)CrossRef
  28.Tang, Y.Q., Chen, L.Q.: Primary resonance in forced vibrations of in-plane translating viscoelastic plates with 3:1 internal resonance. Nonlinear Dyn. 69, 159鈥?72 (2012)CrossRef
  29.Tang, Y.Q., Chen, L.Q.: Parametric and internal resonances of in-plane accelerating viscoelastic plates. Acta Mech. 223, 415鈥?31 (2012)CrossRef
  30.Wang, Y.Q., Liang, L., Guo, X.H.: Internal resonance of axially moving laminated circular cylindrical shells. J. Sound Vib. 332, 6434鈥?450 (2013)CrossRef
  31.Mockensturm, E.M., Guo, J.: Nonlinear vibration of parametrically excited, viscoelastic, axially moving strings. ASME J. Appl. Mech. 72, 374鈥?80 (2005)CrossRef
  32.Wickert, J.A.: Non-linear vibration of a traveling tensioned beam. Int. J. Non-Linear Mech. 27, 503鈥?17 (1992)CrossRef
  33.Chen, L.Q., Zu, J.W.: Solvability condition in multi-scale analysis of gyroscopic continua. J. Sound Vib. 309, 338鈥?42 (2008)CrossRef
  34.Bert, C.W., Malik, M.: The differential quadrature method in computational mechanics: a review. Appl. Mech. Rev. 49, 1鈥?8 (1996)CrossRef
  35.Malik, M., Bert, C.W.: Implementing multiple boundary conditions in the DQ solution of higher-order PDE鈥檚: application to free vibration of plates. Int. J. Numer. Meth. Eng. 39, 1237鈥?258 (1996)CrossRef
  36.Bracewell, R.N.: The Fourier Transform and Its Applications. McGraw-Hill, New York (2000)
  
• 作者单位：You-Qi Tang (1)
  Deng-Bo Zhang (1)
  Jia-Ming Gao (1)
  
  1. School of Mechanical Engineering, Shanghai Institute of Technology, Shanghai, 201418, China
  
• 刊物类别：Engineering
• 刊物主题：Vibration, Dynamical Systems and Control
  Mechanics
  Mechanical Engineering
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-269X

文摘

In this paper, parametric and 3:1 internal resonance of axially moving viscoelastic beams on elastic foundation are analytically and numerically investigated. The beam is restricted by viscous damping force. The beam鈥檚 material obeys the Kelvin model in which the material time derivative is used. The governing equations of coupled planar vibration and the associated boundary conditions are derived from the generalized Hamilton principle. The effects of the nonhomogeneous boundary conditions due to the viscoelasticity are highlighted, while the boundary conditions are assumed to be homogeneous in previous studies. In small but finite stretching problems, the equation is simplified into a governing equation of transverse nonlinear vibration. It is a nonlinear integro-partial differential equation with time-dependent and space-dependent coefficients. The dependence of the tension on the finite axial support rigidity is also modeled. The method of multiple scales is directly applied to establish the solvability conditions. The nonlinear steady-state oscillating response along with the stability and bifurcation of the beam is investigated. A detailed study is carried out to determine the influence of the viscoelastic coefficient and the viscous damping coefficient on dynamic behavior of the system. The numerical calculations by the differential quadrature scheme confirm the approximate analytical results. Keywords Nonlinearity Parametric and 3:1 internal resonance Axially accelerating viscoelastic beam Longitudinally varying tension Nonhomogeneous boundary condition Method of multiple scales Differential quadrature scheme