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Research Papers

# Parametric Resonance of a Two Degrees-of-Freedom System Induced by Bounded Noise

[+] Author and Article Information
Jinyu Zhu

Department of Civil and Environmental Engineering, Faculty of Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canadaj7zhu@engmail.uwaterloo.ca

W.-C. Xie

Department of Civil and Environmental Engineering, Faculty of Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

Ronald M. C. So, X. Q. Wang

Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

J. Appl. Mech 76(4), 041007 (Apr 22, 2009) (13 pages) doi:10.1115/1.2999427 History: Received January 25, 2008; Revised September 16, 2008; Published April 22, 2009

## Abstract

The dynamic stability of a two degrees-of-freedom system under bounded noise excitation with a narrowband characteristic is studied through the determination of moment Lyapunov exponents. The partial differential eigenvalue problem governing the moment Lyapunov exponent is established. For weak noise excitations, a singular perturbation method is employed to obtain second-order expansions of the moment Lyapunov exponents and Lyapunov exponents, which are shown to be in good agreement with those obtained using Monte Carlo simulation. The different cases when the system is in subharmonic resonance, combination additive resonance, and combined resonance in the absence of noise, respectively, are considered. The effects of noise and frequency detuning on the parametric resonance are investigated.

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## Figures

Figure 1

Lyapunov exponent for β1=β2=0 and ν0=2ω1

Figure 2

Moment Lyapunov exponent for β1=β2=0, and ν0=2ω1(K=12)

Figure 3

Lyapunov exponent for β1=β2=0, and ν0=ω1+ω2

Figure 4

Moment Lyapunov exponent for β1=β2=0 and ν0=ω1+ω2(K=20)

Figure 5

Moment Lyapunov exponent for β1=0.2, β2=0.1, and ν0=ω1+ω2

Figure 6

Moment Lyapunov exponent for β1=0.2, β2=0.1, and ν0=ω1+ω2

Figure 7

Lyapunov exponent for β1=0.2, β2=0.1, and ν0=ω1+ω2

Figure 8

Moment Lyapunov exponent for β1=β2=0.1, and ν0=ω1+ω2

Figure 9

Moment Lyapunov exponent for β1=0.2, β2=0.1, and ν0=ω1+ω2

Figure 10

Lyapunov exponent for β1=β2=0, ω1=ω2, and ν0=2ω1

Figure 11

Lyapunov exponent for β1=β2=0, ω1=ω2, and ν0=2ω1

Figure 12

Lyapunov exponent for β1=β2=0.2, ω1=ω2, and ν0=2ω1

Figure 13

Moment Lyapunov exponent for β1=β2=0, ω1=ω2, and ν0=2ω1(K=20)

Figure 14

Moment Lyapunov exponent for β1=β2=0.2, ω1=ω2, and ν0=2ω1

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