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Simulation of Sound Production Mechanism (Acoustical Research on the Piano, Part 2)

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Simulation of Sound Production Mechanism (Acoustical Research on the Piano, Part 2) J. Acoust. Soc. Jpn.(E) 14, 2 (1993) Simulation of sound production mechanism (Acoustical research on the piano, Part 2) Isao Nakamura Teikyo University of Technology, 2289 Uruido, Ichihara, Chiba, 290-01 Japan (Received 8 April 1992) 【English translation of the same articie in J. Acoust. Soc. Jpn.(J)37,65-75(1981)】 The mechanism of sound generation on the piano can be divided into four stages :•\(a) transmission of energy from a hammer to a set of strings, (b) propagation of energy in the strings, (c) transition of energy from the strings to the soundboard through the bridge, and (d) acoustic radiation of the energy from the soundboard. This process is represented by an electrical circuit model, each string being represented by a transmis sionline, and the quantitative relationship of the model are calculated by a computer. When a hammer strikes a string, a nearly half-sinusoidal pulse is generated, and this propagates along the transmission line, and then is reflected by the impedances of the bridge and the fixed end of the string. The propagation and reflection change the waveform of the initial pulse and reduce its amplitude. The driving velocity of the soundboard is produced by applying the sum of the forces of all the strings to the driving pointimpedance. This driving velocity produces a sound pressure, and its waveform is determined by the transmission characteristics of the soundboard. An artificial piano sound can be produced from a calculated waveform through a D-A converter. When the fundamental frequency of each string in a set is slightly detuned , they produce beats in each partial making an inharmonic sound. The lower partials in the produced sound change relatively slowly, while the higher partials change relatively rapidly. The amplitude of the initial part of the sound ('initial sound') decays rapidly, while that of the sustained part ('aftersound') decays slowly. This explains how a delicate timbre of the piano sound is produced. Keywords:Piano, Sound production mechanism, Equivalent circuit, Simulation, Numerical experiment PACS number:43.75. Mn, 43.40. Cw sistently,it is not easy to find factors that affect 1. INTRODUCTION subtle sound of each instrument. On the contrary, A traditional method to find physical factors re it is useful, in finding the relationship between the presentingthe sound characteristics of a musical physical construction of a musical instrument and instrument has been to analyse a sound itself by its sound, to understand the characteristics of the using an instrument such as a frequency analyzer sound by synthesizing it with a computer. This or a sonogram. In the last decade or so these in method can be used for testing a theory of sound strumentshave been replaced by a computer so production mechanism, so which is also useful for that the sound is digitally represented and a digital the traditional method of studying musical instru signal processing is applied to it. Although these ments.Although there have been many studies methods are effective in understanding various con using computers to study the sound production ventional(non-electronic) musical instruments con mechanism of a musical instrument, no attempt 73 J. Acoust. Soc. Jpn.(E) 14, 2 (1993) has been made to generate sound in the way describ between the hammer and strings couses the string edabove. In 1971, Hiller and Ruiz 1) produced vibration.The hammer initially moves with the musical sounds by solving their differential equa string, keeping contact, but the contact pressure tions,but they did not directly relate their study to a becomes zero again when the hammer starts moving musical instrument. backward and the displacements of the hammer This paper describes the relationship between the and the strings become equal. They are naturally mechanism of piano sound production and the char separate later when the speed of the string decays. acteristicsof the produced sound by representing the The vertical component (to the soundboard) of the mechanism with an equivalent circuit, by calculating tension of the string gives .a driving force to the the circuit equations with constants (obtained from bridge. actual music instruments) using a computer, and by The soundboard with a thickness of several milli analysing the results. If necessary, it is possible metersis made of non-isotropic wooden pieces by to listen to the output as a sound through a D-A gluing them. Sound ribs are also glued on the converter. Measured waveforms are analysed by soundboard with equal intervals across the direction computer after they have been converted to digital of fibres of the wooden pieces. The bridge consists data through an A-D converter. of two parts, one for a low-frequency region, and the other one for a high-and middle-frequency 2. SOUND PRODUCING MECHANISM region, both parts being fixed on the soundboard. OF PIANO AND ITS EQUIVALENT The soundboard system is driven by a force at the CIRCUIT bridge, and a velocity wave occurs on the board, Hammers, strings and the soundboard (including then the sound wave is radiated from it. Therefore, its bridge) of a piano are components determining its vibrations of a string with the soundboard decay sound production. Figure 1 shows a model of a faster than those of the string alone. set of these components. Figure 2 shows a block Let us represent the above-described sound diagram of the sound production mechanism taking productionmechanism by using an equivalent cir a 'triple string system' in which three identical strings cuitbased on the mobility analogy (force-current are struck simultaneously by a single hammer. A and velocity-voltage). When the elasticity of a hammer is made of wood covered by felt, and this string is taken into account, the vibartion of a string strikes a set of strings when a corresponding key is with the elasticity and tension is represented by pressed. A piano string with given a high tension, which is supported by the frame, links with the (1) soundboard through the bridge so that the vibration where T is the tension of the string, r is the frictional of the string is transmitted to the soundboard. When resistance of air, S is the cross sectional area of the a hammer strikes a string, the contact pressure string, ƒÏ is the density of the string, E is the Young's modulus of the string, and r is the radius of gyration of the string. Equation (1) can be represented by a simultaneous 5-element first-order partial differen tialequation,2) as Fig. 1 Model of sound production mecha nisumof a piano keyboard. (2) Fig. 2 Block diagram of sound production mechanism of a keyboard with triple string. 74 I. NAKAMURA:SIMULATION OF SOUND PRODUCTION MECHANISM OF PIANO Table 1 Analogical parameters in mechanical and electrical systems. where f=T•Ýy/•Ýx is a vertical (to the soundboard) ed in parallel with C and G respectively. component of the tension of the string, Xx is the x- Since the boundaries of the system are approxi component of the stress per a unit area which is matelyon the supporting points of a piano string perpendicular to the x-axis, AC is that in the y-com which are represented by ponent,or the x-component of the stress per unit area on a plane which is perpendicular to the y-axis, (4) v is the velocity in the y-direction, u is the velocity therefore in the x-direction, and ƒÂ=ƒÏS is the linear density. (5) When the mobility analogy shown in Table 1 is These are shown in the two end-terminals of the applied to Eq.(2), circuit in Fig. 3(a). When the impedance analogy shown in Table 1 is used, (3) (6) Figure 3(a) shows a graphical representation of Eq. (3), and this is a two-line distributed-constant circuit, where the symbol of a crossed transformer repre Then an equivalent circuit shown in Fig. 3 (b) is sentsan ideal current-transformer having a ratio of obtained. A symbol of a crossed transformer 1:1. If the radiation impedance of a piano string again represents an ideal transformer having a ratio is taken into account, an additional mass and a of 1:1. The circuit shown in Fig. 3 (b) can also be radiation resistance of the string should be connect- obtained by applying a dual transformation to the circuit shown in Fig. 3 (a). If a system becomes complicated by adding other elements, such as a hammer system and the soundboard system , the mobility analogy is easier than the impedance analogy, since the former has the same form as the mechanical system. (a) When an input voltage V=V0eiƒÖt is given to a two-line distributed-constant circuit in the mobility analogy, its output voltage at a distance x can be expressed as V=V0e-ax+j(ƒÖt-ƒÀx). We can determine the transmission constants ƒ¿ and ƒÀ as follows . The phase constant ƒÀ is given by (b) Fig. 3 Equivalent circuit elements for a (7) string with elasticity.(a) Mobility analogy;(b) Impedance analogy. Assuming 75 J. Acoust. Soc. Jpn.(E) 14, 2 (1993) in a higher frequency region due to the sound radia- tion. Now, let a phase constant, which goes and re turnsa distance l and produces a phase difference of 2nƒÎ , be ƒÀn. Then (12) (a) (b) Let a frequency in this equation be fn, then from Fig. 4 Frequency dependence of propaga tionconstants.(a) Phase constant;(b) this equation and Eq.(9), Attenuation constant. (8) Therefore, the frequencies of partials are (9) (13) (10) where When the radiation conductance AƒÖ2 is taken into (14) account, and Eq.(8) is applied, the attenuation Equation (13) coincides with Fletcher's equation, 3,4) constant ƒ¿ is given by (11) (15) Figure 4 illustrates outlines of Eqs.(10) and (11).
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