Shanghai University
Shanghai Institute of Applied Mathematics and Mechanics
Nonlinear forced vibration of a viscoelastic buckled beam
with 2:1 internal resonance
Liu-Yang Xiong, Guo-Ce Zhang, Hu Ding and Li-Qun Chen
Shanghai institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China
Abstract
Nonlinear dynamics of a viscoelastic buckled beam subjected to primary resonance in the presence of internal resonance is investigated.
Appropriate choice of system parameters provides the condition for 2:1 internal resonance. The ordinary differential equations of the two mode
shapes are established using the Galerkin method. The multiple scales method is applied to derive the modulation-phase equations. Steady state
solutions of the system as well as their stability are examined. The double-jump, the saturation phenomenon and the non-periodic region
phenomena are observed illustrating the influence of internal resonance. The validity range of the analytical approximations is assessed by
comparing the analytical approximate results with a numerical solution by the Runge-Kutta method. The unstable regions in the internal
resonance are explored via numerical simulations.
Fig. 1: A buckled beam with external harmonic excitation
Fig. 3: Effect of external excitation amplitude: (a) amplitudefrequency response curves with different amplitude of external
excitation; (b) saturation phenomenon
Result & Discussion
1. First primary resonance
Saturation phenomenon: the amplitude activated by
external excitation does not change eventually with the
growing of excitation amplitude.
3. Numerical Verification
Fig. 2: Amplitude-frequency
response curves with different
parameters: (a) different
external excitation amplitudes
(b) different viscoelastic
damping coefficients and (c)
different axial loads
Internal Resonance
Jumping Phenomena
The height of the two resonance peaks and the bandwidth
of the resonance are increasing with larger external
excitation amplitude and smaller viscoelastic damping.
The changes of axial load will bring up offset of the
nonlinear resonance response curve. The system is
not perfectly tuned.
There is no stable solution near the perfect first primary
resonance at specific parameter combinations.
2. Second primary resonance
Fig. 4: Comparison of solutions at quarter point obtained by
numerical method and approximate analytical method: (a)
amplitude-frequency response; (b) time history response
The results calculated by Runge-Kutta method and
those obtained by the multiple scale method are in
basic agreement .
There is the possibility
that the chaotic solution
occurs in the unstable
regions near the perfect
first primary resonance .
Fig. 5: Poincaré maps for =0.00002, b=0.004