Acta Mechanica Solida Sinica

, Volume 22, Issue 3, pp 267–275

# Nonlinear dynamics of axially accelerating viscoelastic beams based on differential quadrature

Article

## Abstract

This paper investigates nonlinear dynamical behaviors in transverse motion of an axially accelerating viscoelastic beam via the differential quadrature method. The governing equation, a nonlinear partial-differential equation, is derived from the viscoelastic constitution relation using the material derivative. The differential quadrature scheme is developed to solve numerically the governing equation. Based on the numerical solutions, the nonlinear dynamical behaviors are identified by use of the Poincaré map and the phase portrait. The bifurcation diagrams are presented in the case that the mean axial speed and the amplitude of the speed fluctuation are respectively varied while other parameters are fixed. The Lyapunov exponent and the initial value sensitivity of the different points of the beam, calculated from the time series based on the numerical solutions, are used to indicate periodic motions or chaotic motions occurring in the transverse motion of the axially accelerating viscoelastic beam.

## Key words

nonlinear partial-differential equation numerical solution chaos bifurcation differential quadrature

## References

1. [1]
Chen, S.H., Huang, J.L. and Sze, K.Y., Multidimensional Lindstedt-Poincaré method for nonlinear vibration of axially moving beams. Journal of Sound and Vibration, 2007, 306: 1–11.
2. [2]
Chen, S.H. and Huang, J.L., On internal resonance of nonlinear vibration of axially moving beams. Acta Mechanica Sinica, 2005, 37: 57–63 (in Chinese).Google Scholar
3. [3]
Ravindra, B. and Zhu, W.D., Low dimensional chaotic response of axially accelerating continuum in the supercritical regime. Archive of Applied Mechanics, 1998, 68: 195–205.
4. [4]
Marynowski, K. and Kapitaniak, T., Kelvin-Voigt versus Burgers internal damping in modeling of axially moving viscoelastic web. International Journal of Non-Linear Mechanics, 2002, 37: 1147–1161.
5. [5]
Marynowski, K., Non-linear vibrations of an axially moving viscoelastic web with time-dependent tension. Chaos, Solitons and Fractals, 2004, 21: 481–490.
6. [6]
Yang, X.D. and Chen, L.Q., Bifurcation and chaos of an axially accelerating viscoelastic beam. Chaos, Solitons and Fractals, 2005, 23(1): 249–258.
7. [7]
Chen, L.Q. and Yang, X.D., Transverse nonlinear dynamics of axially accelerating viscoelastic beams based on 4-term Galerkin truncation. Chaos, Solitons and Fractals, 2006, 27(3): 748–757.
8. [8]
Hu, Q.Q., Lim, C.W. and Chen, L.Q., Nonlinear vibration of a cantilever with a Derjaguin-Muller-Toporov Contact End, International Journal of Structure Stability and Dynamics., 2008, 8(1): 25–40.
9. [9]
Bert, C.W. and Malik, M., The differential quadrature method in computational mechanics: A review. Applied Mechanics Review, 1996, 49: 1–28.
10. [10]
Shu, C., Differential Quadrature and Its Application in Engineering. Berlin: Spring, 2001.Google Scholar
11. [11]
Zhang, W., Wen, H.B. and Yao, M.H., Periodic and chaotic oscillation of a parametrically excited viscoelastic moving belt with 1:3 iinternal resonance. Acta Mechanica Sinica, 2004, 36: 443–454 (in Chinese).Google Scholar
12. [12]
Yang, X.D. and Chen, L.Q., Non-linear forced vibration of axially moving viscoelastic beams. Acta Mechanica Solida Sinica, 2006, 19(4): 365–373.
13. [13]
Mockensturm, E.M. and Guo, J., Nonlinear vibration of parametrically excited, viscoelastic, axially moving strings. ASME Journal of Applied Mechanics, 2005, 347: 347–380.
14. [14]
Chen, L.Q., Chen, H. and Lim, C.W., Asymptotic analysis of axially accelerating viscoelastic strings, International Journal of Engineering Science, 2008, 46(10): 976–985.
15. [15]
Serletis, A. Shahmoradi, A. and Serletis, D., Effect of noise on estimation of Lyapunov exponents from a time series. Chaos, Solitons and Fractals, 2007, 32: 883–887.
16. [16]
Wolf, A., Swift, J.B., Swinney, H.L. and Vastano, J.A., Determining Lyapunov exponents from a time series. Physica D, 1985, 16: 285–317.
17. [17]
Wang, X. and Bert, C.W., A new approach in applying differential quadrature to static and free vibration analyses of beams and plates. Journal of Sound and Vibration, 1993, 162: 566–572.