To understand how the behaviour of a soundpost is affected by the various contact stiffnesses at its ends, it is useful to start with the simplest problem. Instead of looking at the interaction of the post with flexible top and back plates, we will see what happens when the post is held between two flat, parallel, rigid surfaces. For that problem, the axial motion of the post is decoupled from the bending motion, and we can consider them separately. The two cases are illustrated in Fig. 1. The linear springs in the left-hand sketch have stiffness $k_p$, while the rotational springs in the right-hand sketch have stiffness $K_p$.
We first consider axial motion, at frequency $\omega$. The displacement of a point at vertical position $z$ is $w_p(z) e^{i \omega t}$, and it satisfies the differential equation
$$E_p \dfrac{d^2 w_p}{d z^2}= -\rho_p \omega^2 w_p \tag{1}$$
where $E_p$ is the Young’s modulus and $\rho_p$ is the density. The post has length $L_p$ and radius $a_p$ so that the cross-section area is $A_p=\pi a_p^2$. The general solution to this equation takes the form
$$w_p = P \cos \alpha z + Q \sin \alpha z \tag{2}$$
where
$$\alpha = \omega \sqrt{\dfrac{\rho_p}{E_p}} \tag{3}$$
is the wavenumber.
The boundary condition at $z=0$ is
$$k_p w_p= E_p A_p \dfrac{d w_p}{d z} \tag{4}$$
so that
$$k_p P = Q E_p A_p \alpha . \tag{5}$$
The corresponding boundary condition at $z=L_p$ is
$$k_p w_p= – E_p A_p \dfrac{d w_p}{d z} \tag{6}$$
so that
$$P \cos \alpha L_p + Q \sin \alpha L_p $$
$$=-\dfrac{E_p A_p \alpha}{k_p}\left[ -P \sin \alpha L_p + Q \cos \alpha L_p \right] . \tag{7}$$
Equating the values of $P/Q$ given by equations (5) and (7) and simplifying leads to the condition
$$\tan \alpha L_p = \dfrac{2 \lambda \alpha}{\lambda^2 \alpha^2 – 1} \tag{8}$$
where
$$\lambda = \dfrac{E_p A_p}{k_p} . \tag{9}$$
The solutions for $\alpha$ of this equation give the natural frequencies of axial motion of the post, via equation (3).
Solving numerically for a wide range of values of $k_p$, using the parameter values given in section 7.6.1, gives the results shown in Fig. 2. Notice that this plot covers a wider frequency range than usual, in order to see the behaviour clearly. When the contact stiffness is very low, the lowest resonance involves the post bouncing as a rigid body between the two springs. But as the stiffness increases, the frequency of this resonance rises across the entire audible frequency range, until with very high stiffness the post is constrained between fixed ends. The mode shape then involves a half-wavelength of deformation between nodal points at the ends, with a frequency near 27 kHz.
This is exactly the same resonance frequency as that of the second mode when the stiffness is very low. The reason is that as the contact stiffness tends to zero, the post has free ends. The next mode then consists of a half-wavelength of deformation, but in the cosine phase rather than the sine phase of the first mode with high stiffness.
We can now look at bending motion of the soundpost, related to the right-hand sketch in Fig. 1. For bending in the $x-z$ plane, the horizontal displacement of the post is $u_p(z)e^{i \omega t}$, and it obeys the differential equation
$$E_p I_p \dfrac{\partial^4 u_p}{\partial z^4}= \rho_p A_p \omega^2 u_p \tag{10}$$
where $I_p=\pi a_p^4/4$ is the second moment of area. The general solution can be written
$$u_p=P \cos \alpha z + Q \sin \alpha z + R \cosh \alpha z + S \sinh \alpha z \tag{11}$$
where now the wavenumber $\alpha$ is given by
$$\alpha^4 = \omega^2 \dfrac{\rho_p A_p}{E_p I_p} . \tag{12}$$
We have two boundary conditions at each end of the post. At $z=0$ we have
$$u_p=0 \mathrm{~~and~~} E_p I_p \dfrac{d^2 u_p}{dz} = K_p \dfrac{d u_p}{dz} \tag{13}$$
which lead to
$$P+R=0 \tag{14}$$
and
$$\alpha^2 E_p I_p (-P+R)=K_p \alpha (Q+S) \tag{15}$$
so that
$$R=-P \mathrm{~~and~~} S=-2 \mu P – Q \tag{16}$$
where
$$\mu = \dfrac{E_p I_p \alpha}{K_p} . \tag{17}$$
At $z=L_p$ we have
$$u_p=0 \mathrm{~~and~~} E_p I_p \dfrac{d^2 u_p}{dz} = -K_p \dfrac{d u_p}{dz} . \tag{18}$$
Substituting and working through some rather tedious algebra, these four boundary conditions lead to the equation
$$\cos \alpha L_p \cosh \alpha L_p – \mu^2 \sin \alpha L_p \sinh \alpha L_p$$
$$+ 2 \mu \cos \alpha L_p \sinh \alpha L_p – 2 \mu \sin \alpha L_p \cosh \alpha L_p = 1 \tag{19}$$
which governs the natural frequencies.
Solving numerically for the roots $\alpha$ and then deducing frequencies from equation (12) leads to the results plotted in Fig. 3. When the torsional spring stiffness is very low the post behaves like a pinned-pinned beam, but when the stiffness becomes very large it behaves like a clamped-clamped beam. These two limits determine the pattern: each frequency rises from the pinned-pinned value to the clamped-clamped value as the stiffness increases.
There is one more useful calculation to do in connection with the torsional spring stiffness. It is very hard to have an intuitive feel for a torsional spring stiffness, and it is useful to link it to the axial stiffness. If we imagine a perfectly-fitted post, then the contact stiffness should be uniformly distributed over the cross-section of the post (on a length scale bigger than the individual asperity contacts). Suppose this stiffness is $s$ per unit area. The axial stiffness is then simply the aggregate over the section:
$$k_p=A_p s . \tag{20}$$
We can now calculate the torsional stiffness, by a suitable integration. Figure 4 shows a pair of strips of width $dx$, symmetrically distributed at a distance $x$ from the centre of the post. The half-length of each strip is $\sqrt{a_p^2-x^2}$. In response to a rotation by $\theta$ the compression at position $x$ is $x \theta$, so the combined moment from the two strips is
$$2x \times x \theta \times 2s\sqrt{a_p^2-x^2} dx \tag{21}$$
and the total torsional stiffness follows by integration:
$$K_p \theta = \theta \int_0^{a_p}{4sx^2 \sqrt{a_p^2-x^2} dx} . \tag{22}$$
Substituting $x=a_p \sin \phi$,
$$K_p=4sa_p^4 \int_0^{\pi/2}{ \sin^2 \phi \cos^2 \phi~d\phi} . \tag{23}$$
This integral is included in standard tables of integrals (e.g. [1]), so
$$K_p = 4s a_p^4 \left[-\dfrac{1}{8}\left(\dfrac{1}{4} \sin 4 \phi ~-~ \phi\right) \right]_0^{\pi/2}$$
$$=\dfrac{\pi s a_p^4}{4} = \dfrac{a_p^2 k_p}{4} . \tag{24}$$
Replotting the results from Figs. 2 and 3, but with $K_p$ expressed in terms of $k_p$ using this formula, gives Fig. 4. The transition for bending modes occurs at somewhat higher stiffness than the transition for axial modes, but they lie in a similar range.
It would obviously be useful to estimate plausible values for these contact spring stiffnesses. We can do that via a simple experiment, making use of Figs. 3 and 6. Figure 3 shows that the lowest bending resonance of the soundpost, when confined between rigid surfaces, is a sensitive function of the torsional contact stiffness. So we can take a violin soundpost, cut the ends to be accurately parallel, then trap it between the flat steel jaws of a vice. The frequency of the lowest bending resonance can be measured, as the vice is progressively tightened.
To calibrate the properties of this particular soundpost, it is useful to estimate the first frequency with perfectly clamped ends. Perfect clamping is, however, difficult to achieve; but there is a simple alternative. The theory of bending vibration of beams tells us that the frequencies of a beam clamped at both ends are exactly the same as those of the same beam when both ends are free, like a xylophone bar. So the soundpost can be supported on soft rubber bands at the nodal points of the lowest bending mode, approximately 1/4 of the way in from each end, then it can be tapped and the lowest frequency found using an FFT.
Figure 6 shows some results. The blue stars show the data of Fig. 3, scaled to give the frequency ratio to the lowest pinned-pinned mode — the frequency we would expect from a tilted post, or the limiting frequency as the torsional contact stiffness goes to zero. The upper limit, with high contact stiffness, is the first frequency of the clamped-clamped post (indicated by the black line). Theory tells us that this frequency ratio should be 2.27. Equating this to our measured free-free frequency we can deduce the pinned-pinned frequency.
Now the three red lines show frequency ratios measured with three different degrees of tightness of the vice. The left-hand line has just enough pressure to hold the post in place. The middle one is the result of tightening the vice to give the kind of pressure that might be representative of a normal soundpost in a violin. The right-hand line shows the result of tightening the vice way beyond anything that would be encountered in a violin. Where the middle red line crosses the line of blue stars gives us the estimate we want, for a plausible value of torsional contact stiffness to use as the datum in our simulation model. Scaling this using equation (24) gives the corresponding value for axial contact stiffness, on the assumption that the post ends are well-fitted so that the distribution of contact stiffness is uniform.
There is one remaining comment to make about the results of this experiment. If the boundary conditions at the end of the post had remained the same as the vice was tightened, we would expect the frequencies to go down, not up. If the post had been put under tension, like a string, we would obviously expect the frequencies to go up. Here we have a compressive force rather than tension, so frequencies should go down. This effect has been ignored in the theory developed in this section because, as the results of the vice experiment have shown, it is overwhelmed in practice by the effect of contact stiffness rising, thereby changing the boundary conditions. However, the measured frequencies of the post in the vice will presumably have been reduced a bit by the effect, so we should regard the positions as the red lines in Fig. 6 as only approximate. But we were only aiming for an order-of-magnitude estimate of the contact stiffness, and this argument should have provided that.
[1] I. S. Gradshteyn and I. M. Ryzhik, “Table of integrals, series and products”, Academic Press (1994)