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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Harne, R. L., and Wang, K. W., 2014, “On the Fundamental and Superharmonic Effects in Bistable Energy Harvesting,” J. Intell. Mater. Syst. Struct., 25(8), pp. 937–950. [CrossRef]
Harne, R. L., and Wang, K. W., 2013, “A Review of the Recent Research on Vibration Energy Harvesting Via Bistable Systems,” Smart Mater. Struct., 22(2), p. 023001. [CrossRef]
Stanton, S. C., McGehee, C. C., and Mann, B. P., 2010, “Nonlinear Dynamics for Broadband Energy Harvesting: Investigation of a Bistable Piezoelectric Inertial Generator,” Phys. D, 239(10), pp. 640–653. [CrossRef]
Erturk, A., and Inman, D. J., 2011, “Broadband Piezoelectric Power Generation on High-Energy Orbits of the Bistable Duffing Oscillator With Electromechanical Coupling,” J. Sound Vib., 330(10), pp. 2339–2353. [CrossRef]
Masana, R., and Daqaq, M. F., 2012, “Energy Harvesting in the Super-Harmonic Frequency Region of a Twin-Well Oscillator,” J. Appl. Phys., 111(4), p. 044501. [CrossRef]
Panyam, M., Masana, R., and Daqaq, M. F., 2014, “On Approximating the Effective Bandwidth of Bi-Stable Energy Harvesters,” Int. J. Non-Linear Mech., 67, pp. 153–163. [CrossRef]
Manevitch, L. I., Sigalov, G., Romeo, F., Bergman, L. A., and Vakakis, A. F., 2014, “Dynamics of a Linear Oscillator Coupled to a Bistable Light Attachment: Analytical Study,” ASME J. Appl. Mech., 81(4), p. 041011. [CrossRef]
Romeo, F., Sigalov, G., Bergman, L. A., and Vakakis, A. F., 2015, “Dynamics of a Linear Oscillator Coupled to a Bistable Light Attachment: Numerical Study,” ASME J. Comput. Nonlinear Dyn., 10(1), p. 011007. [CrossRef]
AL-Shudeifat, M. A., 2014, “Highly Efficient Nonlinear Energy Sink,” Nonlinear Dynam., 76(4), pp. 1905–1920. [CrossRef]
Carrella, A., Brennan, M. J., Waters, T. P., and Shin, K., 2008, “On the Design of a High-Static–Low-Dynamic Stiffness Isolator Using Linear Mechanical Springs and Magnets,” J. Sound Vib., 315(3), pp. 712–720. [CrossRef]
Liu, X., Huang, X., and Hua, H., 2013, “On the Characteristics of a Quasi-Zero Stiffness Isolator Using Euler Buckled Beam as Negative Stiffness Corrector,” J. Sound Vib., 332(14), pp. 3359–3376. [CrossRef]
Johnson, D. R., Harne, R. L., and Wang, K. W., 2014, “A Disturbance Cancellation Perspective on Vibration Control Using a Bistable Snap-Through Attachment,” ASME J. Vib. Acoust., 136(3), p. 031006. [CrossRef]
He, J. H., 2006, “Some Asymptotic Methods for Strongly Nonlinear Equations,” Int. J. Mod. Phys. B, 20(10), pp. 1141–1199. [CrossRef]
Kovacic, I., and Brennan, M. J., 2011, The Duffing Equation: Nonlinear Oscillators and Their Behaviour, Wiley, Chichester, UK.
Peng, Z. K., Lang, Z. Q., Billings, S. A., and Tomlinson, G. R., 2008, “Comparisons Between Harmonic Balance and Nonlinear Output Frequency Response Function in Nonlinear System Analysis,” J. Sound Vib., 311(1–2), pp. 56–73. [CrossRef]
Leung, A. Y. T., Guo, Z., and Yang, H. X., 2012, “Residue Harmonic Balance Analysis for the Damped Duffing Resonator Driven by a van der Pol Oscillator,” Int. J. Mech. Sci., 63(1), pp. 59–65. [CrossRef]
Dai, H. H., Schnoor, M., and Atluri, S. N., 2012, “A Simple Collocation Scheme for Obtaining the Periodic Solutions of the Duffing Equation, and Its Equivalence to the High Dimensional Harmonic Balance Method: Subharmonic Oscillations,” Comput. Model. Eng. Sci., 84(5), pp. 459–497. [CrossRef]
Yuste, S. B., 1991, “Comments on the Method of Harmonic Balance in Which Jacobi Elliptic Functions Are Used,” J. Sound Vib., 145(3), pp. 381–390. [CrossRef]
Yuste, S. B., 1992, “‘Cubication’ of Non-Linear Oscillators Using the Principle of Harmonic Balance,” Int. J. Non-Linear Mech., 27(3), pp. 347–356. [CrossRef]
Hu, H., and Tang, J. H., 2006, “Solution of a Duffing-Harmonic Oscillator by the Method of Harmonic Balance,” J. Sound Vib., 294(3), pp. 637–639. [CrossRef]
Blair, K. B., Krousgrill, C. M., and Farris, T. N., 1997, “Harmonic Balance and Continuation Techniques in the Dynamic Analysis of Duffing's Equation,” J. Sound Vib., 202(5), pp. 717–731. [CrossRef]
Yuste, S. B., and Bejarano, J. D., 1986, “Construction of Approximate Analytical Solutions to a New Class of Nonlinear Oscillator Equations,” J. Sound Vib., 110(2), pp. 347–350. [CrossRef]
Gilmore, R. J., and Steer, M. B., 1991, “Nonlinear Circuit Analysis Using the Method of Harmonic Balance—A Review of the Art. Part I. Introductory Concepts,” Int. J. Microwave Millimeter-Wave Comput.-Aided Eng., 1(1), pp. 22–37. [CrossRef]
Rizzoli, V., Mastri, F., and Masotti, D., 1994, “General Noise Analysis of Nonlinear Microwave Circuits by the Piecewise Harmonic-Balance Technique,” IEEE Trans. Microwave Theory Tech., 42(5), pp. 807–819. [CrossRef]
Mickens, R. E., 1984, “Comments on the Method of Harmonic Balance,” J. Sound Vib., 94(3), pp. 456–460. [CrossRef]
Mickens, R. E., 1986, “A Generalization of the Method of Harmonic Balance,” J. Sound Vib., 111(3), pp. 515–518. [CrossRef]
Szemplinska-Stupnicka, W., and Jerzy, R., 1993, “Steady States in the Twin-Well Potential Oscillator: Computer Simulations and Approximate Analytical Studies,” Chaos, 3(3), pp. 375–385. [CrossRef] [PubMed]
Genesio, R., and Tesi, A., 1992, “Harmonic Balance Methods for the Analysis of Chaotic Dynamics in Nonlinear Systems,” Automatica, 28(3), pp. 531–548. [CrossRef]
Raghothama, A., and Narayanan, S., 1999, “Bifurcation and Chaos in Geared Rotor Bearing System by Incremental Harmonic Balance Method,” J. Sound Vib., 226(3), pp. 469–492. [CrossRef]
Yasuda, K., Kamakura, S., and Watanabe, K., 1988, “Identification of Nonlinear Multi-Degree-of-Freedom Systems: Presentation of an Identification Technique,” JSME Int. J., Ser. III, 31(1), pp. 8–14. http://ci.nii.ac.jp/els/110002504864.pdf?id=ART0002767980&type=pdf&lang=en&host=cinii&order_no=&ppv_type=0&lang_sw=&no=1433450107&cp=
Feeny, B. F., Yuan, C. M., and Cusumano, J. P., 2001, “Parametric Identification of an Experimental Magneto-Elastic Oscillator,” J. Sound Vib., 247(5), pp. 785–806. [CrossRef]
Hayashi, C., 1953, “Forced Oscillations With Nonlinear Restoring Force,” J. Appl. Phys., 24(2), pp. 198–206. [CrossRef]
Urabe, M., 1969, “Numerical Investigation of Subharmonic Solutions to Duffing's Equation,” Publ. Res. Inst. Math. Sci., 5(1), pp. 79–112. [CrossRef]
Guckenheimer, J., and Holmes, P., 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York.


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. 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. 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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