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Technical Brief

Evaluation of Groebner Basis Methodology as an Aid to Harmonic Balance

[+] Author and Article Information
Jane Liu

Department of Civil and Environmental Engineering,
Tennessee Technological University,
Cookeville, TN 38505
e-mail: jliu@tntech.edu

John Peddieson

Department of Mechanical Engineering,
Tennessee Technological University,
Cookeville, TN 38505

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 5, 2013; final manuscript received October 10, 2013; published online December 24, 2013. Assoc. Editor: Mohammed Daqaq.

J. Vib. Acoust 136(2), 024502 (Dec 24, 2013) (4 pages) Paper No: VIB-13-1310; doi: 10.1115/1.4026213 History: Received September 05, 2013; Revised October 10, 2013

A preliminary exploration is carried out of the utility of the Groebner basis method for implementing the harmonic balance analysis of nonlinear steady state vibrations. The method is found to be worthy of further investigation.

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References

Cochelin, B., and Vergez, C., 2009, “A High Order Purely Frequency-Based Harmonic Balance Formulation for Continuation of Periodic Solutions,” J. Sound Vib., 324, pp. 243–262. [CrossRef]
Guskov, M., and Thouverez, F., 2012, “Harmonic Balance-Based Approach for Quasi-Periodic Motions and Stability Analysis,” ASME J. Vibr. Acoust., 134(3), p. 031003. [CrossRef]
Grolet, A., and Thouverez, F., 2012, “Free and Forced Vibration Analysis of a Nonlinear System With Cyclic Symmetry: Application to a Simplified Model,” J. Sound Vib., 331, pp. 2911–2928. [CrossRef]
Stanton, S., Owens, B., and Mann, B., 2012, “Harmonic Balance Analysis of the Bistable Piezoelectric Inertial Generator,” J. Sound Vib., 331, pp. 3617–3627. [CrossRef]
Peng, Z., Meng, G., Lang, Z., Zhang, W., and Chu, F., 2012, “Study of the Effects of Cubic Nonlinear Damping on Vibration Isolations Using Harmonic Balance Method,” Int. J. Non-Linear Mech., 47, pp. 1073–1080. [CrossRef]
Alvarez, J., Meraz, M., Valdes-Parada, F., and Alvarez-Ramirez, J., 2012, “First-Harmonic Balance Analysis for Fast Evaluation of Periodic Operation of Chemical Processes,” Chem. Eng. Sci., 74, pp. 256–265. [CrossRef]
Peng, Z., Lang, Z., and Billings, S., 2007, “Resonances and Resonant Frequencies for a Class of Nonlinear Systems,” J. Sound Vib., 300, pp. 993–1014. [CrossRef]
Hironaka, H., 1964, “Resolution of Singularities on an Algebraic Variety Over a Field of Characteristic Zero: I,” Ann. Math., 79(1), pp. 109–203. [CrossRef]
Hironaka, H., 1964, “Resolution of Singularities on an Algebraic Variety Over a Field of Characteristic Zero: II,” Ann. Math., 79(2), pp. 205–326. [CrossRef]
Buchberger, B., 1965, “An Algorithm for Finding a Basis for the Residue Class Ring of a Zero-Dimensional Polynomial Ideal,” Ph.D. thesis, University of Innsbruck, Innsbruck, Austria (in German).
Cox, D., Little, J., and O'Shea, D., 2007, Ideals, Varieties, and Algorithms, 3rd ed., Springer, New York.
Jacobsen, L., and Ayre, R., 1958, Engineering Vibrations, McGraw-Hill, New York.

Figures

Grahic Jump Location
Fig. 1

Amplitudes versus frequency ratio for e = 0.01 (solid line: one-harmonic A1; diamonds: two-harmonic A1; circles: two-harmonic A3)

Grahic Jump Location
Fig. 2

Amplitudes versus frequency ratio for e = 0.1 (solid line: one-harmonic A1; diamonds: two-harmonic A1; circles: two-harmonic A3)

Grahic Jump Location
Fig. 3

Amplitudes versus frequency ratio for e = 1.0 (solid line: one-harmonic A1; diamonds: two-harmonic A1; circles: two-harmonic A3)

Grahic Jump Location
Fig. 4

Amplitudes versus frequency ratio for e = 1.5 (solid line: one-harmonic A1; diamonds: two-harmonic A1; circles: two-harmonic A3)

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