Simple model of a piano soundboard

Journal of the Acoustical Society of America

Volumn 102, Issue 2, 1997, Pages 1159-1168

  Giordano, N ^a  

^a PURDUE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ACOUSTIC IMPEDANCE; ACOUSTICS; ANISOTROPY; ARTICLE; ELASTICITY; MATHEMATICAL MODEL; PRIORITY JOURNAL; VIBRATION;

EID: 0030744846     PISSN: 00014966     EISSN: None     Source Type: Journal    
DOI: 10.1121/1.419868     Document Type: Article

References (45)

1. 1A. Chaigne and A. Askenfelt, "Numerical simulations of piano strings. I. Physical model for a struck string using finite difference methods," J. Acoust. Soc. Am. 95, 1112 (1994).
2. 2A. Chaigne and A. Askenfelt, "Numerical simulations of piano strings. I. Comparisons with measurements and systematic exploration of some hammer-string parameters," J. Acoust. Soc. Am. 95, 1631 (1994).
3. 3I. Nakamura, "The vibrational character of the piano soundboard," Proc. 11th ICA, Paris (1983), Vol. 4, p. 385.
4. 4J. Kindel and I-C. Wang, "Modal analysis and finite element analysis of a piano soundboard," in Proceedings of the 5th International Modal Analysis Conference (Union College, Schenectady, NY, 1987), p. 1545.
5. 5J. Kindel, "Modal analysis and finite element analysis of a piano soundboard," Master's thesis, University of Cincinnati, 1989.
6. 6While not strictly intended to apply to piano soundboards, the work of Ref. 7 on the vibrations of orthotropic plates is also quite relevant to the problems considered in the present paper.
7. 7G. W. Caldersmith, "Vibrations of orthotropic rectangular plates," Acustica 56, 144 (1984); G. Caldersmith and T. D. Rossing, "Determination of modal coupling in vibrating rectangular plates," Appl. Acoust. 17, 33 (1984).
8. 7G. W. Caldersmith, "Vibrations of orthotropic rectangular plates," Acustica 56, 144 (1984); G. Caldersmith and T. D. Rossing, "Determination of modal coupling in vibrating rectangular plates," Appl. Acoust. 17, 33 (1984).
9. 8H. L. Schwab, "Finite element analysis of a guitar soundboard," Catgut Acoust. Soc. 24, 13 (1975);
10. H. L. Schwab and K. C. Chen, "Finite element analysis of a guitar soundboard - Part II," Catgut Acoust. Soc. 25, 13 (1976).
11. 9O. Christensen and B. B. Vistisen, "Simple model for low-frequency guitar function," J. Acoust. Soc. Am. 68, 758 (1980).
12. 10O. Christensen, "An oscillator model for analysis of guitar sound pressure response," Acustica 54, 289 (1984).
13. 11A. Chaigne, "On the use of finite differences for musical synthesis. Application to plucked string instruments," J. Acoust. 5, 181 (1992).
14. 12See, for example, B. E. Richardson, G. W. Roberts, and G. P. Walker, "Numerical modeling of two violin plates," J. Catgut Acoust. Soc. 47, 12 (1987).
15. 13K. Wogram, "Acoustical research on pianos. Part I: Vibrational characteristics of the soundboard," Das Musikinstrument 24, 694 (1980); 24, 776 (1980); 24, 872 (1980).
16. 13K. Wogram, "Acoustical research on pianos. Part I: Vibrational characteristics of the soundboard," Das Musikinstrument 24, 694 (1980); 24, 776 (1980); 24, 872 (1980).
17. 13K. Wogram, "Acoustical research on pianos. Part I: Vibrational characteristics of the soundboard," Das Musikinstrument 24, 694 (1980); 24, 776 (1980); 24, 872 (1980).
18. 14Note that while Ref. 13 plotted the sound-pressure level, we have chosen here to plot the sound power level. This affects the vertical scale, but not the overall shape of this log-log plot.
19. 15H. Suzuki, "Vibration and sound radiation of a piano soundboard," J. Acoust. Soc. Am. 80, 1573 (1986).
20. 16N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments (Springer-Verlag, New York, 1991).
21. 17A. H. Benade, Fundamentals of Musical Acoustics (Oxford U. P., New York, 1976).
22. 18G. Weinreich, "Coupled piano strings," J. Acoust. Soc. Am. 62, 1474 (1977).
23. 19H. A. Conklin, Jr., "Design and tone in the mechanoacoustic piano. Part II. Piano structure," J. Acoust. Soc. Am. 100, 695 (1996).
24. 20The frequency at which this falloff begins depends somewhat on the position on the soundboard where the impedance is measured, and was observed (Ref. 13) to vary by a factor of ∼2. However, above this point |Z\ decreased at approximately the same rate with frequency at all locations on the soundboard.
25. 21D. W. Martin, "Decay rates of piano tones," J. Acoust. Soc. Am. 19, 535 (1947).
26. 22T. C. Hundley, H. Benioff, and D. W. Martin, "Factors contributing to the multiple rate of piano tone decay," J. Acoust. Soc. Am. 64, 1303 (1978).
27. 23We should also note that Ref. 19 has shown the results of impedance measurements which do not show any significant decrease in the impedance at high frequencies. However, those results extend to only about 5000 Hz, so they may be consistent with those of Wogram (Ref. 13). In any event, it would be very desirable to re-examine the experimental situation, as will be discussed further below.
28. 24Conklin (Ref. 19) gives a particularly instructive discussion of soundboard design and behavior.
29. 25K. Wogram, in The Acoustics of the Piano, edited by A. Askenfelt (Royal Swedish Academy of Music Publication No. 64, Stockholm, 1990), p. 83.
30. 26L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergammon, London, 1959).
31. 27There is a substantial literature (see, for example, Ref. 28 and references contained therein) on the vibrations of plates whose thickness varies with position, since structures of this type serve many practical functions. In most cases the analysis of such stepped plates, which are similar in some respects to the model with ribs which we will discuss below, is limited to plates with a single step. Additional steps can be accommodated within the theory, but only with significant computational complications which end up making the problem essentially a numerical one. Given our ultimate goal of modeling a semirealistic soundboard, it therefore seems reasonable to take a numerical approach from the outset.
32. 28I. E. Harik, X. Liu, and N. Balakrishnan, "Analytic solution to free vibration of rectangular plates," J. Sound Vib. 153, 51 (1992).
33. 29N. Giordano, Computational Physics (Prentice-Hall, Upper Saddle River, NJ, 1997).
34. 30Other details of the calculation, such as the choice of time step, etc., follow standard practice, as discussed in Refs. 29, 11, and 1.
35. 31V. Bucur, Acoustics of Wood (CRC, Boca Raton, 1995).
36. 32L. Cremer and M. Heckl, Structure-Borne Sound (Springer-Verlag, New York, 1973).
37. 33For most upright pianos, including the one studied in Ref. 13 for which schematic results are given in Fig. 1, the grain of the soundboard runs diagonally, rather than parallel to one of the edges as we assume here. This is one of the details that we will assume can be neglected in a "first" approximation.
38. 34S. G. Lekhnitskii, Anisotropic Plates (Gordon and Breach, New York, 1968).
39. 35For simplicity we will not include the bridges in our model. Since the bridges generally run approximately parallel to the stiff direction (in our case x), we expect that they will affect the behavior much less than the ribs.
40. 36Ideally one would like to have the spatial step size in a calculation be much smaller than any of the important length scales in the problem. However, using a significantly smaller value of Δx would add greatly to the computational demands, but not, we think, much to the physical insight.
41. xy by this amount did not have a significant effect on the results.
42. 38P. M. Morse and K. U. Ingard, Theoretical Acoustics (Princeton U. P., Princeton, NJ, 1986).
43. 39It is known from Ref. 40 that air loading has a quite significant effect on the modes of a timpani membrane. However, given its different thickness and mass density, we would not expect air loading to be nearly as important for a piano soundboard.
44. 40R. S. Christian, R. E. Davis, A. Tubis, C. A. Anderson, R. I. Mills, and T. D. Rossing, "Effects of air loading on timpani membrane vibrations," J. Acoust. Soc. Am. 76, 1336 (1984).
45. 41It has been pointed out by G. Weinreich (private communication) that other factors connected with the decreasing length of the strings may also play a major role in producing faster decays at high frequencies. This would appear to be a (currently unsettled) question which requires more experimental work.