# 7.5.5 Deducing bridge parameters by adding masses

One way to estimate some key parameters of the bridge model is to clamp the feet in a suitable vice, then track how the frequencies of rocking and bouncing change when a small concentrated mass is added. We will explain the idea based on the rocking resonance of the bridge — a simpler version of the same argument can be applied to the bouncing mode. The procedure to be described is an extension of one suggested by violin maker Andreas Hudelmayer.

Our model of the bridge rocking mode is sketched in Fig. 1. The top portion of the bridge is regarded as a rigid body, able to rotate about an effective pivot point a distance $L$ below the top point of the bridge. This rotation is restrained by a torsional spring of stiffness $K$ at the pivot, and the moment of inertia of the top portion of the bridge about the pivot is $I$.

We now add a small additional mass $\Delta m$, on the centre line a distance $b$ below the top. The original rocking resonance frequency $\omega_0$ is given by

$$f_0 = 2 \pi \sqrt{\dfrac{K}{I}} . \tag{1}$$

The reduced frequency allowing for the effect of the added mass is

$$f_1 = 2 \pi \sqrt{\dfrac{K}{I+ \Delta m (L-b)^2}} \tag{2}$$

on the assumption that the mode shape has not changed as a result of the added mass. In particular, we assume that the effective pivot point has not moved, so that the same length $L$ still applies and the moment of inertia $I$ is unchanged.

Eliminating $K$ between these two equations gives

$$\left(\dfrac{f_0}{f_1}\right)^2=1+ \dfrac{\Delta m (L-b)^2}{I} . \tag{3}$$

Now recall from the discussion around Fig. 9 in section 7.5 that the clearest fit to the measured admittance was obtained when the moment of inertia $I$ was expressed in terms of an effective point mass at the top of the bridge so that

$$I=m_e L^2 . \tag{4}$$

Now if we take the particular case of equation (3) in which $b=0$, we obtain a simple result that is independent of $L$:

$$\left(\dfrac{f_0}{f_1}\right)^2=1+ \dfrac{\Delta m}{m_e} \tag{5}$$

so that

$$m_e = \dfrac{f_1^2}{f_0^2 – f_1^2} \Delta m . \tag{6}$$

In other words, for a known value of $\Delta m$ we can use the measured frequencies with and without the added mass to give a direct estimate of the effective mass defined by equation (4).

If we want to estimate the value of $L$, we need to work a little harder. One approach is to move the same mass $\Delta m$ to a different, non-zero, value of $b$, and measured the resonance frequency again. If $f_1$ is the frequency for $b=0$ and $f_2$ is the frequency with the new value of $b$, then

$$\left(\dfrac{f_0}{f_2}\right)^2=1+ \dfrac{\Delta m (L-b)^2}{m_e L^2}, \tag{7}$$

so eliminating $\Delta m/m_e$ using equation (5) gives

$$\left(\dfrac{f_0}{f_2}\right)^2-1 = \left[ \left(\dfrac{f_0}{f_1}\right)^2 – 1 \right] \left(1-\dfrac{b}{L} \right)^2 \tag{8}$$

so that $L$ is determined by

$$1-\dfrac{b}{L}= \sqrt{\dfrac{f_1^2 (f_0^2 – f_2^2)}{f_2^2 (f_0^2 – f_1^2)}} . \tag{9}$$