Technical Brief

Harmonic Balance Analysis of Snap-Through Orbits in an Undamped Duffing Oscillator

[+] Author and Article Information
Smruti R. Panigrahi

Dynamics and Vibrations Research Laboratory,
Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48823
e-mail: smruti@msu.edu

Brian F. Feeny

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48823
e-mail: feeny@egr.msu.edu

Alejandro R. Diaz

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48823
e-mail: diaz@egr.msu.edu

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 7, 2014; final manuscript received May 15, 2015; published online July 9, 2015. Assoc. Editor: Mohammed Daqaq.

J. Vib. Acoust 137(6), 064502 (Dec 01, 2015) (5 pages) Paper No: VIB-14-1430; doi: 10.1115/1.4030718 History: Received November 07, 2014; Revised May 15, 2015; Online July 09, 2015

A simple nonlinear undamped Duffing oscillator has been studied for its snap-through behavior at large-amplitude vibrations. We present an algorithm that uses the harmonic balance (HB) method to find amplitude and frequency relationships in two- and three-term approximations for solutions that lie outside the separatrix in the phase space. Trends of the approximate solution properties are examined with reference to an analysis of the limit as the trajectory approaches the separatrix.

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Grahic Jump Location
Fig. 1

This plot approximates the solution very near and outside the separatrix. The period T is infinite on the separatrix. As the periodic orbit approaches the separatrix, the peaks can be idealized by Dirac delta functions in the derivation of the Fourier series.

Grahic Jump Location
Fig. 2

Phase plots show solutions using HB and numerical approaches. At x0=1.65062 corresponding to k5=0.01, the three-term HB solution almost perfectly matches the numerical solution of Eq. (2). Also shown in the plot are solutions closer to the separatrix at x0=1.50675 and x0=1.45933 corresponding to k5=0.02 and k5=0.03, respectively.

Grahic Jump Location
Fig. 3

This plot shows the relationship between the amplitude of the first-, third-, and fifth-harmonics versus the initial condition x0. The thick line corresponds to a solution with three odd harmonics and the thin line corresponds to a solution with two odd harmonics. The dots are from a numerical solution. We see that for an initial condition x0 greater than 1.6, the amplitude ratio of the third-harmonic (k3) does not change much for the two-term HB solution compared to that of the three-term solution. The amplitude ratios of the fifth to the leading order harmonics (k5) shows the weakening of the amplitude of the fifth-harmonic as we go farther away from the separatrix.

Grahic Jump Location
Fig. 4

The frequency ω is plotted versus the initial condition x0 for approximate solutions with one, two, and three odd harmonics. The plot suggests a linear trend when x0 is large. The numerical results confirm the accuracy of the HB solutions with three odd harmonics. The ω limits show that the three-term harmonic approximation captures solutions closer to the separatrix, and of lower frequency, than the one- and two-term solutions.




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