Engineering Acoustics/Moving Resonators

Moving Resonators
Consider the situation shown in the figure below. We have a typical Helmholtz resonator driven by a massless piston which generates a sinusoidal pressure $$P_G$$, however the cavity is not fixed in this case. Rather, it is supported above the ground by a spring with compliance $$C_M$$. Assume the cavity has a mass $$M_M$$.

Recall the Helmholtz resonator (see Module #9). The difference in this case is that the pressure in the cavity exerts a force on the bottom of the cavity, which is now not fixed as in the original Helmholtz resonator. This pressure causes a force that acts upon the cavity bottom. If the surface area of the cavity bottom is $$S_C$$, then Newton's Laws applied to the cavity bottom give

$$\sum{F} = p_CS_C - \frac{x}{C_M} = M_M\ddot{x} \Rightarrow p_CS_C = \left[\frac{1}{j\omega C_M} + j\omega M_M\right]u$$

In order to develop the equivalent circuit, we observe that we simply need to use the pressure (potential across $$C_A$$) in the cavity to generate a force in the mechanical circuit. The above equation shows that the mass of the cavity and the spring compliance should be placed in series in the mechanical circuit. In order to convert the pressure to a force, the transformer is used with a ratio of $$1:S_C$$.

Example
A practical example of a moving resonator is a marimba. A marimba is a similar to a xylophone but has larger resonators that produce deeper and richer tones. The resonators (seen in the picture as long, hollow pipes) are mounted under an array of wooden bars which are struck to create tones. Since these resonators are not fixed, but are connected to the ground through a stiffness (the stand), it can be modeled as a moving resonator. Marimbas are not tunable instruments like flutes or even pianos. It would be interesting to see how the tone of the marimba changes as a result of changing the stiffness of the mount.

For more information about the acoustics of marimbas see http://www.mostlymarimba.com/techno1.html

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