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Research Papers

Floquet-Based Analysis of General Responses of the Mathieu Equation

[+] Author and Article Information
Gizem Acar

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: dilbergi@egr.msu.edu

Brian F. Feeny

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: feeny@egr.msu.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 20, 2015; final manuscript received March 29, 2016; published online May 25, 2016. Assoc. Editor: Marco Amabili.

J. Vib. Acoust 138(4), 041017 (May 25, 2016) (9 pages) Paper No: VIB-15-1332; doi: 10.1115/1.4033341 History: Received August 20, 2015; Revised March 29, 2016

Solutions to the linear unforced Mathieu equation, and their stabilities, are investigated. Floquet theory shows that the solution can be written as a product between an exponential part and a periodic part at the same frequency or half the frequency of excitation. In the current work, an approach combining Floquet theory with the harmonic balance method is investigated. A Floquet solution having an exponential part with an unknown exponential argument and a periodic part consisting of a series of harmonics is assumed. Then, performing harmonic balance, frequencies of the response are found and stability of the solution is examined over a parameter set. The truncated solution is consistent with an existing infinite series solution for the undamped case. The truncated solution is then applied to the damped Mathieu equation and parametric excitation with two harmonics.

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Figures

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Fig. 1

Transition curves for the n = 2 Floquet-based approximation, infinite series solution, and the n = 2 Hill's determinant solution

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Fig. 2

Analytically predicted response frequencies for δ = 0.8, n = 2

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Fig. 3

Numerical and theoretical (n = 2) solutions of the undamped Mathieu equation for n = 2 (a) ω = 0.6, δ = 0.4, (b) ω = 0.8, δ = 0.6, (c) ω = 1.3, δ = 0.8, and (d) ω = 2.5, δ = 0.7

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Fig. 4

FFTs of numerical and theoretical (n = 2) solutions of the undamped Mathieu equation for n = 2 (a) ω = 0.6, δ = 0.4, (b) ω = 0.8, δ = 0.6, (c) ω = 1.3, δ = 0.8, and (d) ω = 2.5, δ = 0.7

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Fig. 5

Free response and FFT plots for ω = 0.6, δ = 0.4, with n = 3 harmonics (a) time response for ω = 0.6, δ = 0.4 and (b) FFT for ω = 0.6, δ = 0.4

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Fig. 6

Transition curves for the damped Mathieu equation for ζ = 0.005, ζ = 0.025, and ζ = 0.05, approximated with n = 2

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Fig. 7

Decay and growth factors for ζ = 0.05 and δ = 0.8

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Fig. 8

Numerical and theoretical (n = 2) solutions of the damped Mathieu equation (a) ω = 0.8, δ = 0.6, ζ = 0.025, (b) ω = 1, δ = 0.3, ζ = 0.025, (c) ω = 1.3, δ = 0.8, ζ = 0.025, and (d) ω = 2, δ = 0.15, ζ = 0.05

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Fig. 9

FFT of numerical and theoretical (n = 2) stable solutions of the damped Mathieu equation (a) ω = 0.8, δ = 0.6, ζ = 0.025, (b) ω = 1, δ = 0.3, ζ = 0.025, (c) ω = 1.3, δ = 0.8, ζ = 0.025, and (d) ω = 2, δ = 0.15, ζ = 0.05

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Fig. 10

Numerical and theoretical solutions of the two-frequency Mathieu equation (a) response for ω = 0.8, δ = 0.6, γ = 0.5, n = 3, (b) response for ω = 1.5, δ = 0.5, γ = 1, n = 2, (c) FFT for ω = 0.8, δ = 0.6, γ = 0.5, n = 3, and (d) FFT for ω = 1.5, δ = 0.5, γ = 1, n = 2

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