Simulations of piano strings and hammers can be carried out by the same approach described in earlier sections. The hammer contacts the string at a single point through a nonlinear contact spring satisfying a power-law force-compression relation
$$F=K C^\alpha \tag{1}$$
where $F$ is the contact force, $C$ is the compression of the hammer felt, $\alpha$ is a non-dimensional exponent and $K$ is a magnitude scaling factor. The hammer can be modelled either as a single concentrated mass, or using the “drumstick” model from section 12.1 which can allow hammer resonances. The string is described in modal terms, including the effect of inharmonicity due to bending stiffness but not including any representation of coupling to the piano soundboard. For the purposes of these simple simulations, all string modes are given the same Q-factor, with the value 2000. All modes of the strings and hammers are represented by IIR digital filters, for computational efficiency.
The simulation model makes use of measurements from the Broadwood piano seen in Figs. 6, 7 and 11 of section 12.2, augmented by some values from the literature where I was not able to do direct measurements. The parameter values relevant to all the C notes are listed in Table 1. The frequency is the nominal fundamental frequency based on equal temperament. We need not go into the subtleties of real piano tuning for the purposes of these simulations, which are aimed at illustrating the physics rather than reproducing in detail the sound of any particular piano.
All strings have a steel monofilament carrying the tension, and the bass strings are over-wrapped with copper. The diameter of the steel core was measured, and for over-wrapped strings the outer diameter of the wrapping was also measured. The mass per unit length was then calculated using standard values for the densities of steel and copper $(7800 \mathrm{kg/m}^3$ and $8960 \mathrm{kg/m}^3$ respectively). No allowance was made for the fact that the wrappings stopped short of the ends of the vibrating length of each string. The masses listed in the table are the total vibrating mass of each string, taking account of the multiple strings used for most notes. The bending stiffness of each string was estimated from the diameter of the steel core, using a standard formula for a circular string and assuming that the Young’s modulus of steel is 210 GPa. For the bass strings, the influence of the copper wrapping was ignored.
The position on each string where the hammer strikes was measured, and expressed as a fraction of the vibrating string length: this is the “Striking ratio” listed in Table 1. It was not possible to measure the hammer masses directly, so plausible values for the effective concentrated masses of the hammers have been estimated from values given by Conklin [1] and Hall and Askenfelt [2]. The final two rows of Table 1 give values of two ratios that will be used later: the first is the ratio of the hammer mass to the vibrating string mass, the second is the ratio of tension to length (expressed in kN/m).
Parameters for the nonlinear stiffness from equation (1) are listed in Table 2: they are taken from measurements by Hall and Askenfelt [2], later used in simulation studies by Chaigne and Askenfelt [3]. I haven’t given a unit for the magnitude constant $K$, because the units are peculiar and different in each case because of the differing exponents: but the values are all appropriate to consistent SI units. For a linear spring with $\alpha = 1$ the corresponding units of $K$ would have been Pa.
In order to resolve fine details during the bouncing process, the simulations were run at a high sampling rate. The chosen rate was 176.4 kHz: this apparently peculiar value was chosen because it is 4 times the standard CD audio rate of 44.1 kHz, so that when down-sampling was needed in order to produce sound examples it could be done without further processing by taking every 4th sample. The quantity used for all sound examples is the “bridge force”, as used in earlier demonstrations. In anticipation of the need for such sound examples, the number of string modes was in all cases restricted to those with frequencies below the corresponding Nyquist frequency, 22.05 kHz, in order to avoid issues of aliasing.
[1] H. A. Conklin, “Design and tone in the mechanoacoustic piano. Part I. Piano hammers and tonal effects”, Journal of the Acoustical Society of America 99, 3286–3296, (1996).
[2] Donald E. Hall and Anders Askenfelt, “Piano string excitation V: Spectra for real hammers and strings”, Journal of the Acoustical Society of America 83, 1627—1638 (1988)
[3] Antoine Chaigne and Anders Askenfelt, “Numerical simulations of piano strings. II Comparisons with measurements and systematic exploration of some hammer-string parameters” Journal of the Acoustical Society of America 95, 1631–1640 (1994).