Parametric resonance and nonlinear string vibrations

American Journal of Physics

Volumn 72, Issue 6, 2004, Pages 758-766

  Rowland, David R ^a  

^a UNIVERSITY OF QUEENSLAND   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 2642531707     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1645281     Document Type: Article

References (32)

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2. R. Berthet, A. Petrosyan, and B. Roman, "An analog experiment of the parametric instability," Am. J. Phys. 70, 744-749 (2002).
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4. N. B. Tufillaro, "Torsional parametric oscillations in wires," Eur. J. Phys. 11, 122-124 (1990).
5. J. W. Strutt (Baron Rayleigh), The Theory of Sound (Dover, New York, 1945), 2nd ed., Vol. 1, Article 68b.
6. C. V. Raman, "Some remarkable cases of resonance," Phys. Rev. 35, 449-458 (1912).
7. Semi-stable means that the steady-state solution is stable to changes in the driving frequency in one direction (that is, increasing or decreasing), but unstable to changes in the other (see Sec. II and Fig. 2 for further details).
8. H. Kidachi and H. Onogi, "Note on the stability of the nonlinear Mathieu equation," Prog. Theor. Phys. 98, 755-773 (1997).
9. T. Matsuda, "On Melde's String," Prog. Theor. Phys. 100, 1287-1292 (1998).
10. -1 times those that are measured in the laboratory, and so is potentially confusing.
11. D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations (Clarendon, Oxford, 1977), Chap. 8.
12. G. Blanch, in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by M. Abramowitz and I. A. Stegun (National Bureau of Standards, Washington, DC, 1965), Chap. 20.
13. L. D. Landau and E. M. Lifschitz, Mechanics (Pergamon, London, 1960), Sec. 27.
14. J. C. Gutíerrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, "Mathieu functions, a visual approach," Am. J. Phys. 71, 233-242 (2003).
15. Equation (16) shows that if u(0) = 0 and u̇(0) = 0, then u(T)≡0. This condition reveals one of the features of parametric excitation, namely that a parametrically excited mode will only grow if it is "seeded" by random fluctuations in the properties of the system. This seeding is done mathematically by assuming that u(0) is a small but nonzero value (see Pigs. 3-5).
16. A. Watzky, "Non-linear three-dimensional large-amplitude damped free vibration of a stiff elastic stretched string," J. Sound Vib. 153, 125-142 (1992).
17. P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968), Sec. 14.3.
18. G. R. Baldock and T. Bridgeman, Mathematical Theory of Wave Motion (Ellis-Horword, Chichester, 1981), Sec. 1.10.
19. J. A. Elliott, "Intrinsic nonlinear effects in vibrating strings," Am. J. Phys. 48, 478-480 (1980); A. Alippi and A. Bettucci, "Nonlinear strings as bistable elements in acoustic wave propagation," Phys. Rev. Lett. 63, 1230-1232 (1989).
20. J. A. Elliott, "Intrinsic nonlinear effects in vibrating strings," Am. J. Phys. 48, 478-480 (1980); A. Alippi and A. Bettucci, "Nonlinear strings as bistable elements in acoustic wave propagation," Phys. Rev. Lett. 63, 1230-1232 (1989).
21. D. R. Rowland and C. Pask, "The missing wave momentum mystery," Am. J. Phys. 67, 378-388 (1999).
22. R. J. Hanson, J. M. Anderson, and H. K. Macomber, "Measurement of nonlinear effects in a driven vibrating wire," J. Acoust. Soc. Am. 96, 1549-1556 (1994).
23. K. A. Legge and N. H. Fletcher, "Nonlinear generation of missing modes on a vibrating string," J. Acoust. Soc. Am. 76, 5-12 (1984).
24. That this simplification is not unreasonable follows from the fact that although transverse motion on a string generally becomes tubular rather than planar, the stability analysis in Ref. 9 reveals that the stability result in Eq. (12a) for planar motion also holds when the motion is allowed to be nonplanar. Furthermore, although the theoretical analysis in Ref. 6 has some limitations, reasonable agreement between experimental results and theory based on the assumption of planar motion was reported.
25. x, still is.
26. A. W. Leissa and A. M. Saad, "Large amplitude vibrations of strings," Trans. ASME 61, 296-301 (1994).
27. These solutions are referred to as pseudo-unstable because like unstable solutions, perturbations from the equilibrium solution U(T)=0 initially grow exponentially. However, unlike unstable solutions and more like stable solutions, they do not do so without bound and, like stable solutions, periodically return to the perturbation value. (As shown in Ref. 8, the pseudo-unstable solutions travel on closed rather than open curves in the phase plane, although these curves do not remain in the vicinity of the equilibrium solution.)
28. H. Kono, Master Thesis, Osaka Kyoiko University, 1994.
29. 0∼0.017-0. 05kg-wt.
30. N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments (Springer-Verlag, New York, 1991), Chap. 6.
31. J. A. Elliott, "Nonlinear resonance in vibrating strings," Am. J. Phys. 50, 1148-1150 (1982).
32. Reference 29, Chap. 12.