The bridge models presented in section 7.5.1 rely on the $2 \times 2$ matrix of admittances on the violin body at the bridge-foot positions. In order for the model to work with maximum accuracy, the measurement of this body matrix needs to be very careful. The features of most interest, the two “hills”, lie up in the kilohertz range, so we would like to be assured of accurate results up to 8 kHz or more. This is quite challenging: several factors that can usually be ignored in vibration measurements need to be considered.

The first factor is the mass of the two small accelerometers used to make the measurement. The type used here has mass 0.3 g: small by normal standards, but enough to have a visible effect on the measurements at the highest frequencies of interest. Luckily, there is a standard trick which allows (in principle) the effect of these masses to be compensated.

It relies on a general result governing the behaviour when two subsystems are joined together to form a single system: in our case, one subsystem is the violin body and the other is the pair of masses. Figure 1 shows a sketch of the situation, in general terms: two subsystems A” and “B” are first shown separated, and then coupled together at two points. The forces and velocities at these two points before and after coupling are defined in the figure.

When the subsystems are separated, each pair of forces is related to the corresponding velocities by a $2 \times 2$ matrix of admittances, which we can call $Y_A$ and $Y_B$ for the two subsystems:

$$\begin{bmatrix} v^{(A)}_1 \\ v^{(A)}_2 \end{bmatrix} = Y_A \begin{bmatrix} F^{(A)}_1 \\ F^{(A)}_2 \end{bmatrix} \tag{1}$$

and

$$\begin{bmatrix} v^{(B)}_1 \\ v^{(B)}_2 \end{bmatrix} = Y_B \begin{bmatrix} F^{(B)}_1 \\ F^{(B)}_2 \end{bmatrix} . \tag{2}$$

When the subsystems are joined together, the velocities at the two coupling points must be the same:

$$\begin{bmatrix} v^{(A)}_1 \\ v^{(A)}_2 \end{bmatrix} = \begin{bmatrix} v^{(B)}_1 \\ v^{(B)}_2 \end{bmatrix} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} \tag{3}$$

while the total applied force at each coupling point must be the sum of the force needed to drive subsystem A and the force needed to drive subsystem B:

$$\begin{bmatrix} F_1 \\ F_2 \end{bmatrix} = \begin{bmatrix} F^{(A)}_1 \\ F^{(A)}_2 \end{bmatrix} + \begin{bmatrix} F^{(B)}_1 \\ F^{(B)}_2 \end{bmatrix} . \tag{4}$$

It follows from eqs. (1-3) that

$$\begin{bmatrix} F_1 \\ F_2 \end{bmatrix} = Y_A^{-1}\begin{bmatrix} v^{(A)}_1 \\ v^{(A)}_2 \end{bmatrix} + Y_B^{-1} \begin{bmatrix} v^{(B)}_1 \\ v^{(B)}_2 \end{bmatrix} = \left(Y_A^{-1} + Y_B^{-1} \right) \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}. \tag{5}$$

But the coupled system is characterised by another $2 \times 2$ matrix of admittances $Y_{coup}$, which satisfies

$$\begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = Y_{coup} \begin{bmatrix} F_1 \\ F_2 \end{bmatrix} \tag{6}$$

so we can deduce that

$$Y_{coup}^{-1} = Y_A^{-1} + Y_B^{-1} \tag{7}$$

which is the result we need. Note that although we have described this in terms of two coupling points, the final result could be trivially extended to any number of coupling points — the matrices simply become bigger as the number of coupling points increases.

To apply this to our particular problem, subsystem A is the violin body, and subsystem B is the pair of separate accelerometer masses $m_{acc}$. The measurement is of the coupled system, so we deduce that

$$Y_{measured}^{-1} = Y_{body}^{-1} + Y_{masses}^{-1} \tag{8}$$

where

$$Y_{masses}^{-1} = \begin{bmatrix} i \omega m_{acc}&0\\0&i \omega m_{acc} \end{bmatrix} \tag{9}$$

and so to compensate for the masses we simply need

$$Y_{body} = \left[Y_{measured}^{-1}\ -\ Y_{masses}^{-1} \right]^{-1}. \tag{10}$$

Unfortunately, when applied in practice a snag appears. In order to test the method thoroughly, two measurements of the body matrix of the test violin were made: one with just the accelerometers in place, the second with a further mass 0.3 g added to each accelerometer using modelling clay. The test of the compensation approach is to make the second measurement look exactly like the first. Figure 2 shows $Y_{22}$ from both measurements. It can be seen that deviations of around 5 dB occur at high frequencies. Figure 3 shows the same comparison, after applying equation (10) to the measurement with the additional added mass. The “compensation” has made the disparity worse!

The problem was tracked down to a phase error in the measurement system. Sensors and their associated electronics can introduce a frequency-dependent phase shift, which does not matter if only the magnitude of the response is plotted. But the mass compensation using eq. (10) involves manipulations of the complex frequency responses, not just the magnitudes, and it is sensitive to phase accuracy.

After a little experimentation, it was found that the problem could be sufficiently addressed by modifying the measured phase by a simple time delay of a few microseconds. When the compensation was then applied, the result shown in Fig. 4 was obtained: almost perfect agreement between the red and blue curves is now seen. The same procedure was then followed with the first measurement, with some confidence that this would now compensate for the accelerometer masses to a satisfactory degree.

There is a bonus for going through this exercise. Our goal is to predict admittance, or other transfer functions, using the measured body matrix together with a theoretical model of the bridge. That process, like the mass compensation, involves a mixture of theoretical expressions with measurements, and it too is sensitive to phase errors in the measurements. Discovering the phase issue and finding a way to minimise the consequences improves the accuracy of the predicted transfer functions.