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Technical Brief

Analytical Solutions for Stable and Unstable Period-1 Motions in a Periodically Forced Oscillator With Quadratic Nonlinearity

[+] Author and Article Information
Albert C. J. Luo

e-mail: aluo@siue.edu

Bo Yu

Department of Mechanical and Industrial Engineering,
Southern Illinois University Edwardsville,
Edwardsville, IL 62026-1805

Contributed by the Design Engineering Division of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received June 13, 2012; final manuscript received January 14, 2013; published online April 24, 2013. Assoc. Editor: Philip Bayly.

J. Vib. Acoust 135(3), 034503 (Apr 24, 2013) (5 pages) Paper No: VIB-12-1178; doi: 10.1115/1.4023834 History: Received June 13, 2012; Revised January 14, 2013

In this note, a closed-form solution of periodic motions in a periodically forced oscillator with quadratic nonlinearity is presented without any small parameters. The perturbation method is based on one harmonic term plus perturbation modification, and the traditional harmonic balance is to arbitrarily select harmonic terms with constant coefficients. If harmonic terms are not enough included in the approximate solution, such a solution is not an appropriate, analytical solution for periodic motions, and some analytical solutions cannot be caught.

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Figures

Grahic Jump Location
Fig. 1

Analytical prediction of period-1 motions from two harmonic terms (HB2): (a) a0, (b) A1, (c) A2. Parameters: (δ=0.05, α=10.0, β=5.0, Q0=4.5).

Grahic Jump Location
Fig. 2

Excitation amplitudes effect on periodic motions from two harmonic terms (HB2) for Q0=1.5,2.5,3.5,4.0,5.0: (a) a0, (b) A1, and (c) A2. Parameters: (δ=0.05, α=10.0, β=5.0, Q0=4.5).

Grahic Jump Location
Fig. 3

Analytical prediction of period-1 motions from the 30 harmonic terms (HB30): (a) a0, (b) A1, (c) A2, (d) A30. Parameters: (δ=0.05, α=10.0, β=5.0, Q0=4.5).

Grahic Jump Location
Fig. 4

Analytical and numerical solutions of period-1 motions (HB30): (a) stable (Ω=6.5), (b) stable (Ω=1.65), (c) unstable (Ω=6.45), and (d) unstable (Ω=1.40)

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