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Chin. Phys. B, 2014, Vol. 23(6): 064301    DOI: 10.1088/1674-1056/23/6/064301

Cumulative solutions of nonlinear longitudinal vibration in isotropic solid bars

Qian Zu-Wen
Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China
Abstract  Based on the strain invariant relationship and taking the high-order elastic energy into account, a nonlinear wave equation is derived, in which the excitation, linear damping, and the other nonlinear terms are regarded as the first-order correction to the linear wave equation. To solve the equation, the biggest challenge is that the secular terms exist not only in the fundamental wave equation but also in the harmonic wave equation (unlike the Duffing oscillator, where they exist only in the fundamental wave equation). In order to overcome this difficulty and to obtain a steady periodic solution by the perturbation technique, the following procedures are taken: (i) for the fundamental wave equation, the secular term is eliminated and therefore a frequency response equation is obtained; (ii) for the harmonics, the cumulative solutions are sought by the Lagrange variation parameter method. It is shown by the results obtained that the second- and higher-order harmonic waves exist in a vibrating bar, of which the amplitude increases linearly with the distance from the source when its length is much more than the wavelength; the shift of the resonant peak and the amplitudes of the harmonic waves depend closely on nonlinear coefficients; there are similarities to a certain extent among the amplitudes of the odd- (or even-) order harmonics, based on which the nonlinear coefficients can be determined by varying the strain and measuring the amplitudes of the harmonic waves in different locations.
Keywords:  nonlinear acoustics      compliance nonlinearity      mesoscopic nonlinearity      Duffing'      s oscillator  
Received:  09 September 2013      Revised:  27 October 2013      Published:  15 June 2014
PACS:  43.25.+y (Nonlinear acoustics)  
  43.35.+d (Ultrasonics, quantum acoustics, and physical effects of sound)  
  91.60.Lj (Acoustic properties)  
Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11274337).
Corresponding Authors:  Qian Zu-Wen     E-mail:

Cite this article: 

Qian Zu-Wen Cumulative solutions of nonlinear longitudinal vibration in isotropic solid bars 2014 Chin. Phys. B 23 064301

[1] Gist G A 1994 J. Acoust. Soc. Am. 96 1158
[2] Meegan G D, Johnson P A, Guyer R A and McCall K R 1993 J. Acoust. Soc. Am. 94 3387
[3] Johnson P L, Bernard Z and Rasolofosaon P N J 1996 J. Geophys. Res. 101(B5) 11533
[4] Van Den Abeele, Koen E-A and Johnson P A 1996 J. Acoust. Soc. Am. 99 3334
[5] Pasqualini D, Heitmann K, TenCate J, Tencate J A, Habib S, Higdon D and Johnson P A 2007 J. Geophy. Res. 112(B1) B01204
[6] TenCate J A, Pasqualini D, Habib S, Heitmann K, Higdon D and Johnson P A 2004 Phys. Rev. Lett. 93 065501
[7] Murnaghan F D 1951 Finite Deformation of an Elastic Solid (New York: Wiley Dover)
[8] Breazeal M A and Jacob P 1984 Physical Acoustics, Vol. 17, ed. Mason W P and Thurston R N (New York: Academic) pp. 2-60
[9] Smith R T, Stern R and Stephens W B 1966 J. Acoust. Soc. Am. 40 1002
[10] Wang R Q, Li Y C and Zhang J D 1988 Proceedings of the Third Western/Pacific Regional Acoustics Conference (WESTPAC III88) Shanghai, China, p. 203
[11] Qian Z W and Jiang W H 1998 Proc. 16th International Congress on Acoustics and 135th Meeting Acoustical Society of America, Vol. III, ed. Kuhl P K and Crum L A, Seattle, Washington, USA, p. 1727
[12] Liu C, Feng X, He M, Yang B J and Zhao J G 2000 Prog. Geophys. 15 68 (in Chinese)
[13] Demonico S N 1977 Geophysics 42 1339
[14] Winkler K W and Mulphy W F 1984 J. Acoust. Soc. Am. 76 820
[15] Winkler K W 1983 J. Geophys. Research 88 9493
[16] Green Jr R E 1973 Ultrasonic Investigation of Mechanical Properties in Treatise on Materials Science and Technology, Vol. 3 (New York: Academic Press)
[17] Qian Z W 2009 Nonlinear Acoustics, 2nd edn. (Beijing: Science Press) Chap. 17 (in Chinese)
[18] Nayfeh A H and Mook D T 1979 Nonlinear Oscillations (Toronto: John Wiley) Chap. 4
[19] Chen Y S 2002 Nonlinear Vibration (Beijing: Higher Education Press) Chap. 7 (in Chinese)
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