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Primarily nonlinear effects observed in a driven asymmetrical vibrating wire
1.C. Valette, “The mechanics of vibrating strings,” in Mechanics of Musical Instruments, edited by A. Hirschberg, J. Kergomard, and G. Weinreich (Springer, Wien, 1995), pp. 115–183.
2.C. Valette and C. Cuesta, Mécanique de la Corde Vibrante (Mechanics of the vibrating string) (Hermès, Paris, 1993).
3.A. Watzky, “Non-linear three-dimensional large-amplitude damped free vibration of a stiff elastic stretched string,” J. Sound Vib. 153, 125–142 (1992).
4.A. Watzky, “Sur la vibration non linéaire des fils précontraints (cordes, verges tendes)” [“On the nonlinear vibrations of prestressed wires (strings, stretched rods)”], thèse de l’Université P. et M. Curie, Paris, 1992 (Ph.D. dissertation, University P. and M. Curie, Paris, 1992).
5.T. C. Molteno and N. B. Tufillaro, “An experimental investigation into the dynamics of a string,” Am. J. Phys. 72, 1157–1169 (2004).
6.Reference 1, p. 160, Eqs. 151, 152, and 153 for which bending, twisting, damping, asymmetry, and external driving force are not included.
7.Reference 3, p. 133, Eqs. 4.13a, 4.13b, 4.13c, and 4.14 for which damping, asymmetry, and external driving force are not included.
8.R. J. Hanson, J. M. Anderson, and H. K. Macomber, “Measurements of nonlinear effects in a driven vibrating wire,” J. Acoust. Soc. Am. 96, 1549–1556 (1994).
9.Yellow brass wire 0.55 mm in diameter manufactured by Zuckerman Harpsichord, Stonington, CT.
10.OI refers to the General Electric Photon Coupled Interrupter Module H21B1. See R. J. Hanson, “Optoelectronic detection of string vibration,” Phys. Teach. 25, 165–166 (1987). Experimentation with different values of resistors is needed for optimum performance of different module samples.
11.A photodiode array (PDA) of 128 photodiodes arranged linearly over a range of 2–3 mm is used to sense the horizontal transverse position of the wire. A light source above the wire casts a shadow on the detector mounted below, and a digital algorithm is used to determine the center of the shadow. The PDA is EG&G RETICON Catalog No. RL0128GAG connected to a RC0301LNN Amplifier Board. The algorithm and associated electronics were developed locally.
12.M. Hancock, “The dynamics of musical strings,” Catgut Acoustical Society Journal 1, 8(Series II), 23–35 (1991).
13.W. F. Vinen, “The detection of single quanta circulation in liquid helium II,” Proc. R. Soc. London, Ser. A 260, 218–236 (1961).
14.C. E. Gough, “The theory of string resonances on musical instruments,” Acustica 49, 124–141 (1981). See especially p. 139.
15.The driving frequency is above the low-frequency resonance so the motion due to the low resonance should be out of phase with the driving force. On the other hand, the driving frequency is below the high-frequency resonance so the motion due to the high resonance should be in phase with the driving force. Thus, the two motions differ in phase by 180° along the two characteristic axes.
17.J. A. Elliott, “Intrinsic nonlinear effects in vibrating strings,” Am. J. Phys. 48, 478–480 (1980).
18.C. E. Gough, “The nonlinear free vibration of a damped elastic string,” J. Acoust. Soc. Am. 75, 1770–1776 (1984).
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