In Chapter 3 we discussed the linear vibrations of plates and shells, and it is now appropriate to examine nonlinear effects. In all cases these arise from mechanisms very similar to those causing nonlinearity in the behavior of ideal strings, as discussed above. In the case of the extensional modes of curved shells, tension forces enter directly and are linearly proportional to displacement. In addition, however, and for both the nonextensional modes of shells and the necessarily nonextensional transverse modes of plates, there are always tension forces proportional to the squares and higher powers of the displacements. It is from these that nonlinear effects arise.
The range of possible plate and shell geometries is so large that we cannot survey it here. It is, however, only in the case of the rather thin shells of gongs that significant nonlinear effects can be heard, because only in this case do we encounter adequately large vibration amplitudes. We can understand something of the general behavior by considering just the case of a spherical-cap shell (Grossman et al., 1969, Fletcher, 1985). In the case of a fiat circular plate, it can be shown (Fletcher, 1985) that the frequency of the lowest axisymmetric mode for a plate of thickness h varies with amplitude a approximately as
(5.47) where wo is the mode frequency at infinitesimal amplitude. The origin of this frequency shift is just the same as that for the string discussed above.
The reason for the dependence on plate thickness is the fact that this quantity determines the relative importance of stiffness and tension forces in plate behavior. The behavior of a gong with a fiat central vibrating sec- tion is thus rather like that of a string, and the pitch glides downwards as the vibration decays after an initial vigorous excitation. The actual vi- bration, and thus the radiated sound, contains not only the shifted normal mode frequencies of the plate but also all odd harmonics of those mode frequencies, generated by distortions of each modal waveform-there are no distinct modes corresponding to these added frequencies.
A more interesting case is that of a spherical cap shell of height H and thickness h. The behavior is now more complicated because the vibration is asymmetrical about its rest point. Once again we consider only the lowest axisymmetric mode. The results calculated by Fletcher (1985) are shown in Fig. 5.3 and depend not only on the ratio af H between the vibration am-
5.7. Nonlinear Effects in Plates and Shells 149
1.4
1.2
h/H 1.0
w
Wo ...
0.8 '
' ' \
0.2\
0.6 \
0.4
0 2
a H
FIGURE 5.3. Calculated frequency w for the lowest axisymmetric mode of a spherical-cap shell of dome height H and thickness h when vibrating with ampli- tude a. The mode frequency for infinitesimal vibrations is w0 . The broken curve shows the limited range of stable vibrations for a moderately thin everted shell (Fletcher, 1985).
plitude and the shell height, but also on the geometrical quantity h/ H. If the shell is very thin, so that h/ H ô 1, then nonlinear tension effects dom- inate the behavior. As the vibration amplitude is increased to approach H, the mode frequency falls to about half its small-amplitude value, while fur- ther increase in amplitude causes it to rise again. For progressively thicker shells, the nonlinearity is less important because of the dominance of bend- ing stiffness, and very thick shells with h >.> H behave essentially as flat plates.
This behavior in a musical instrument is not generally welcome in West- ern music, but has an important place in Chinese opera, where contrasting upward and downward gliding gongs are used for dramatic effect. As with other nonlinear systems, each mode is accompanied by harmonics caused by modal distortion, the amplitude of the mth harmonic of a given mode varying initially as the mth power of the amplitude of its fundamental. For spherical-cap gongs, however, this simple relation ceases to hold once the vibration amplitude becomes comparable with the dome height H (Rossing and Fletcher, 1983).
In addition to these pitch-glide effects, plates and shells can exhibit a variety of more complex behaviors when excited to large amplitude. Among
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150 5. Nonlinear Systems
these we may note a transfer of energy to higher modes at places where the shell shape has a sharp curvature (Legge and Fletcher, 1987), a phenomenon that is used to great musical effect in the large Chinese tamtam, familiar in Western orchestras, and to which we return in Chapter 20. Under certain circumstances, and particularly when driven sinusoidally, shells can also exhibit period doubling, or higher multiplication, and a transition to chaotic behavior (Legge and Fletcher, 1989). This may be important to the sound of orchestral cymbals, for which the vibration may actually be chaotic. We return to this in Chapter 20.
References
Beyer, R.T. (1974). "Nonlinear Acoustics," pp. 6Q-90. U.S. Naval Sea Systems Command.
Bogoliubov, N.N., and Mitropolsky, Y.A. (1961). "Asymptotic Methods in the Theory of Non-linear Oscillations." Hindustan, New Delhi, and Gordon &
Breach, New York.
Carrier, G.F. (1945). On the nonlinear vibration problem of the elastic string.
Q. Appl. Math. 3, 157-165.
Elliott, J.A. (1980). Intrinsic nonlinear effects in vibrating strings. Am. J. Phys.
48, 478-480.
Fletcher, N.H. (1978). Mode locking in nonlinearly excited inharmonic musical oscillators. J. Acoust. Soc. Am. 64, 1566-1569.
Fletcher, N.H. (1985). Nonlinear frequency shifts in quasispherical-cap shells:
Pitch glide in Chinese gongs. J. Acoust. Soc. Am. 78, 2069-2073.
Gough, C. (1981). The theory of string resonances on musical instruments.
Acustica 49, 124-141.
Gough, C. (1984). The nonlinear free vibration of a damped elastic string. J.
Acoust. Soc. Am. 75, 177Q-1776.
Grossman, P.L., Koplik, B., and Yu, Y-Y. (1969). Nonlinear vibration of shallow spherical shells. J. Appl. Mech. 36, 451-458.
Hagedorn, P. (1995). Mechanical oscillations. In "Mechanics of Musical Instru- ments," ed. A. Hirschberg, J. Kergomard and G. Weinreich. Springer-Verlag, Vienna and New York, pp. 7-78.
Hanson, R.J., Anderson, J.M., and Macomber, H.K. (1994). Measurement of nonlinear effects in a driven vibrating wire. J. Acoust. Soc. Am. 96, 1549- 1556.
Keefe, D.H., and Laden, B. (1991). Correlation dimension of woodwind multiphonic tones. J. Acoust. Soc. Am. 90, 1754-1765.
Lauterborn, W., and Parlitz, U. (1988). Methods of chaos physics and their application to acoustics. J. Acoust. Soc. Am. 84, 1975-1993.
Legge, K.A., and Fletcher, N.H. (1984). Nonlinear generation of missing modes on a vibrating string. J. Acoust. Soc. Am. 76, 5-12.
Legge, K.A., and Fletcher, N.H. (1987). Non-linear mode coupling in symmet- rically kinked bars. J. Sound Vibr. 118, 23-34.
Legge, K.A., and Fletcher, N.H. (1989). Nonlinearity, chaos, and the sound of shallow gongs. J. Acoust. Soc. Am. 86, 2439-2443.
References 151 Mettin, R., Parlitz, U., and Lauterborn, W. (1993). Bifurcation structure of the driven Van der Pol oscillator. Int. J. Bifurc. Chaos 3, 1529- 1555.
Moon, F.C. (1992). "Chaotic and Fractal Dynamics. An Introduction for Applied Scientists and Engineers." John Wiley, New York.
Morse, P.M., and Ingard, K.U. (1968). "Theoretical Acoustics," pp. 828-882.
McGraw-Hill, New York. Reprinted 1986, Princeton Univ. Press, Princeton, New Jersey.
Muller, G., and Lauterborn, W. (1996). The bowed string as a nonlinear dynamical system. Acustica 82, 657-664.
Popp, K., and Stelter, P. (1990). Stick-slip vibrations and chaos. Phil. Trans.
Roy. Soc. Land. A332, 89-105.
Prosperetti, A. (1976). Subharmonics and ultraharmonics in the forced oscilla- tions of weakly nonlinear systems. Am. J. Phys. 44, 548-554.
Rossing, T.D., and Fletcher, N.H. (1983). Nonlinear vibrations in plates and gongs. J. Acoust. Soc. Am. 73, 345-351.
Tufillaro, N.B. (1989). Nonlinear and chaotic string vibrations. Am. J. Phys.
57, 408-414.
Ueda, Y. (1979). Randomly transitional phenomena in the system governed by Duffing's equation. J. Statistical Phys. 20, 181-196.
Valette, C. (1995) The mechanics of vibrating strings. In "Mechanics of Musical Instruments," ed. A. Hirschberg, J Kergomard and G. Weinreich, Springer- Verlag, Vienna and New York, pp. 115-183.
Vander Pol, B. (1927). Forced oscillations in a circuit with non-linear resistance.
Phil. Mag. 7(3), 65-80.
Vander Pol, B. (1934). The nonlinear theory of electric oscillations. Proc. I.R.E.
22, 1051-1086.
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Part II
Sound Waves
6
Sound Waves in Air
The sensation we call sound is produced primarily by variations in air pressure that are detected by their mechanical effect on the tympana (ear drums) of our auditory system. Motion of each tympanum is communicated through a linked triplet of small bones to the fluid inside a spiral cavity, the cochlea, where it induces nerve impulses from sensory hair cells in contact with a thin membrane (the basilar membrane). Any discussion of details of the physiology and psychophysics of the hearing process would take us too far afield here. The important point is the dominance of air pressure variation in the mechanism of the hearing process. Direct communication of vibration through the bones of the head to the cochlea is possible, if the vibrating object is in direct contact with the head, and intense vibrations at low frequencies can be felt by nerve transducers in other parts of the body, for example in the case of low organ notes, but this is not part of the primary sense of hearing.
The human sense of hearing extends from about 20 Hz to about 20 kHz, though the sensitivity drops substantially for frequencies below about 100 Hz or above 10 kHz. This frequency response is understandably well matched to human speech, most of the energy of which lies between 100 Hz and 10 kHz, with the information content of vowel sounds con- centrated in the range of 300 Hz-3 kHz and the information content of consonants mostly lying above about 1 kHz. Musical sounds have been evolved to stimulate the sense of hearing over its entire range, but again most of the interesting information lies in the range of 100 Hz-3 kHz.
Since the ears respond to pressure only in their immediate vicinity, we devote this and the following chapter to a discussion of the way in which pressure variations-sound waves-propagate through the air and to the way in which vibrating objects couple to the air and excite sound waves.
155
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156 6. Sound Waves in Air