We can learn something interesting about the design of a set of piano hammers by looking at the simplest model that has some hope of applying to real hammers. We will assume that the hammer hits the string (or group of strings) at a single point. For the purpose of analysing how the hammer bounces, we will neglect the damping of string modes, surely a small effect on the short time-scales of bouncing. More seriously, we will also neglect the effect of bending stiffness in the string in order to obtain simple mathematical results. However, note that both effects are fully included in the numerical simulations to be used later.

The model is thus an ideal string of length $L$, tension $T$, mass per unit length $m$ and transverse displacement $w(x,t)$, where $t$ is time and $x$ is distance along the string measured from one end. Note that for the case of a multiple string group on a piano, both the tension and the mass per unit length should be taken as the aggregate across the group. The boundary conditions need not be specified for the purposes of this analysis: so for example it could apply to a string coupled to a soundboard. The hammer has mass $M$ and makes contact with the string at position $x=\beta L$ through a nonlinear contact spring satisfying the power-law relation introduced earlier:

$$F=K (w_1-w_2)^\alpha \tag{1}$$

where $F$ is the force, $K$ is a magnitude constant, $\alpha$ is the power-law exponent, $w_1=w(\beta L)$ and $w_2$ is the displacement of the hammer mass. This equation applies only when $w_1 > w_2$ so that the spring is in compression; otherwise, the force is zero.

The governing equation for the string during hammer contact is thus

$$m \dfrac{\partial^2 w}{\partial t^2} – T \dfrac{\partial^2 w}{\partial x^2} = – \delta(x – \beta L) K (w_1-w_2)^\alpha \tag{2}$$

where the right-hand side involves the Dirac delta function to describe the spatial distribution of the force applied to the string by the hammer. We can introduce non-dimensional variables

$$\hat{x}=\dfrac{x}{L}, \mathrm{~~~~} \hat{t}=\dfrac{t}{\tau} \tag{3}$$

where $\tau$ is the nominal period defined by

$$\tau=\dfrac{2L}{c}=2L\sqrt{\dfrac{m}{T}} . \tag{4}$$

Equation (2) then becomes

$$\dfrac{m}{\tau^2} \dfrac{\partial^2 w}{\partial \hat{t}^2} – \dfrac{T}{L^2} \dfrac{\partial^2 w}{\partial \hat{x}^2} = – \dfrac{1}{L}\delta(\hat{x} – \beta) K (w_1-w_2)^\alpha \tag{5}$$

where we have made use of the scaling property of the delta function:

$$\delta(L \xi)=\dfrac{1}{L} \delta(\xi). \tag{6}$$

Rearranging,

$$\dfrac{\partial^2 w}{\partial \hat{t}^2} – \dfrac{T \tau^2}{m L^2} \dfrac{\partial^2 w}{\partial \hat{x}^2} = – \delta(\hat{x} – \beta) \dfrac{\tau^2}{L m}K (w_1-w_2)^\alpha . \tag{7}$$

Making use of equation (4),

$$\dfrac{T \tau^2}{m L^2}=4$$ and $$\dfrac{\tau^2}{L m}=\dfrac{4L}{T} \tag{8}$$

so that

$$\dfrac{\partial^2 w}{\partial \hat{t}^2} – 4 \dfrac{\partial^2 w}{\partial \hat{x}^2} = – \delta(\hat{x} – \beta) \dfrac{4L}{T}K (w_1-w_2)^\alpha . \tag{9}$$

The governing equation for the hammer mass during string contact is

$$M \dfrac{\partial^2 w_2}{\partial t^2} = K (w_1-w_2)^\alpha \tag{10}$$

so that

$$\dfrac{M}{\tau^2} \dfrac{\partial^2 w_2}{\partial \hat{t}^2} = K (w_1-w_2)^\alpha$$

or

$$\dfrac{M}{Lm} \dfrac{\partial^2 w_2}{\partial \hat{t}^2} = \dfrac{\tau^2}{Lm}K (w_1-w_2)^\alpha = \dfrac{4L}{T}K (w_1-w_2)^\alpha . \tag{11}$$

Taking this equation together with equation (9) for the string, we see that in terms of the non-dimensional variables $\hat{x}$ and $\hat{t}$ the problem of hammer bouncing and the consequent string vibration only depends on the two parameters

$$\dfrac{M}{Lm} \mathrm{~~~and~~~} \dfrac{K}{T/L} , \tag{12}$$

together with the dimensionless $\alpha$ and $\beta$. So for a given power-law exponent and striking point, provided we normalise $K$ by $T/L$ and the hammer mass by the total vibrating string mass $Lm$, a suitably scaled hammer striking *any* string should produce the same waveform of contact force (including any multiple bounces), provided the time axis is expressed in units of the string period $\tau$. This scaling law is implicit in the analysis of Hall [1], but it does not seem to have been noticed that the argument applies equally well to the case of nonlinear contact stiffness.

The consequence of this scaling law is that if we compute the behaviour of a set of hammers of different mass and stiffness coefficient for a particular string, we can immediately extend those results to other strings by making use of the non-dimensional variables. The proviso is that the striking position $\beta$ and the nonlinear exponent $\alpha$ stay the same. The data tabulated in Table 1 of section 12.2.1 shows that $\beta$ does not vary all that much across the range of a piano, so perhaps we can get away with using an intermediate value by taking our reference case at the note $C_4$, “middle C”.

But the measured values of $\alpha$ vary significantly, and a similar average value is not good enough. However, simulation results suggest empirically that we can use a simple trick to get at least a first approximation to the behaviour over the whole range. This empirical trick is illustrated by Figs. 1—3. These show three versions of a plot related to Fig. 15 of section 12.2, computed using the three different measured values of $\alpha$. For each plot, the note $C_4$ has been “played” by a $20 \times 20$ grid of hammers with different masses and stiffness coefficients $K$. The range of masses is linearly spaced, from very small values up to twice the actual mass of the $C_4$ hammer. The coefficient $K$ has been varied over a much wider range, and a logarithmic spacing has been used to display this, again with the nominal value for $C_4$ in the middle. For each combination, the total time of contact between hammer and string has been computed, and used to colour-shade the plot.

If you compare the three plots, it is immediately clear that the patterns look very similar, but progressively shifted upwards as $\alpha$ increases from each plot to the next. Choose a landmark you can recognise in all three plots, and count the squares: you will find you need a shift of about 4 squares between Figs. 1 and 2, and a shift of about 8 squares between Figs. 2 and 3. Bearing in mind the logarithmic vertical scale, these shifts correspond to scaling the value of $K$ by a certain factor, with different values for the two cases.

There could still be a snag. Figures 1—3 all used the same hammer striking speed, 2.5 m/s. If we used a different striking speed, surely the nonlinear stiffness will mean that the same shifts will not work? The comparison has been repeated using speeds 1 m/s and 0.5 m/s, these three speeds between them covering the main range of normal piano playing. All three cases can be approximated by shifts as we have just discussed, and remarkably the required shift doesn’t vary very much with striking speed. An average over all three cases suggests a compromise shift of 3.5 squares between Figs. 1 and 2, and 8.5 squares between Figs. 2 and 3. These correspond to scaling factors for $K$ of 5.5 and 62, respectively. Those values have been used to generate the plots shown in Figs. 20–22 of section 12.2, which give an overview of the bouncing behaviour of all the hammers on a typical piano.

[1] Donald E. Hall, “Piano string excitation III: General solution for a soft narrow hammer”, *Journal of the Acoustical Society of America* **81**, 547—555 (1987).