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Simulations of transverse vibrations of an axially moving string: a modified difference approach. (English) Zbl 1329.74301

Summary: A modified finite difference approach to simulate transverse vibrations of an axially moving string is presented. The stress is treated as a new unknown in discretization of the spatial variable. A set of differential-algebraic equations is established based on the discreted governing equation and the constitutive relation. For linear vibrations, a conserved functional is employed to test the algorithm, and the 1, 2, 3, 4-term truncated modal analytical solutions are compared with the numerical solution. For the free nonlinear vibration, a new conserved functional is used to check the algorithm. Effects of the transport speed on the free and forced nonlinear vibrations are numerically investigated.

MSC:

74S20 Finite difference methods applied to problems in solid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
74H45 Vibrations in dynamical problems in solid mechanics
74K05 Strings
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[1] Wickert, J. A.; Mote, C. D., Current research on the vibration and stability of axially moving materials, Shock Vib. Dig., 20, 5, 3-13 (1988)
[2] Abrate, A. S., Vibration of belts and belt drives, Mech. Mach. Theory, 27, 645-659 (1992)
[3] L.Q. Chen, Analysis and control of transverse vibrations of axially moving strings. ASME Appl. Mech. Rev., in press.; L.Q. Chen, Analysis and control of transverse vibrations of axially moving strings. ASME Appl. Mech. Rev., in press.
[4] Fung, R. F.; Huang, J. S.; Chen, Y. C., The transient amplitude of the viscoelastic traveling string: an integral constitutive law, J. Sound Vib., 201, 153-167 (1997)
[5] Zhang, L.; Zu, J. W.; Zhong, Z., Transient response for viscoelastic moving belts using block-by-block method, Int. J. Struct. Stability Dyn., 2, 265-280 (2002)
[6] L.Q. Chen, W.J. Zhao, J.W. Zu, Transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law. J. Sound Vib., to be published.; L.Q. Chen, W.J. Zhao, J.W. Zu, Transient responses of an axially accelerating viscoelastic string constituted by a fractional differentiation law. J. Sound Vib., to be published. · Zbl 1236.74201
[7] L.Q. Chen, W.J. Zhao, A numerical method for simulating transverse vibrations of axially moving strings. Appl. Math. Comp., in press.; L.Q. Chen, W.J. Zhao, A numerical method for simulating transverse vibrations of axially moving strings. Appl. Math. Comp., in press. · Zbl 1299.74079
[8] Brenan, K. E.; Campbell, S. L.; Petzold, L. R., Numerical solution of initial-value problems in differential-algebraic equations (1996), SIAM: SIAM North Holland · Zbl 0844.65058
[9] Wickert, J. A.; Mote, C. D., Classical vibration analysis of axially moving continua, ASME J. Appl. Mech., 57, 738-744 (1990) · Zbl 0724.73125
[10] Renshaw, A. A.; Rahn, C. D.; Wickert, J. A.; Mote, C. D., Energy and conserved functionals for axially moving materials, ASME J. Vib. Acoust., 120, 2, 634-636 (1998)
[11] Chen, L. Q.; Zu, J. W., Energetics and conserved functional of moving materials undergoing transverse nonlinear vibration, ASME J. Vib. Acoust., 126, 3, 452-455 (2004)
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