Hambric Vibroacoustic Demonstrators

If we assume the mass in a simple harmonic oscillator is a circular piston within a rigid-walled baffle we can couple its motion to the impedance of a surrounding acoustic medium. Simply adding the acoustic resistance to the oscillator damping and the acoustic reactance to the oscillator mass gives us a coupled vibro-acoustic system.

This simple coupled system is the basis of loudspeaker design. The goal is a flat frequency response across a specific frequency range. Large pistons are used for low frequency sound and small ones for high frequency sound. The approach is to first tune the fundamental oscillator resonance below the lower limit of a desired frequency range. For example, a typical mid-range speaker produces optimal sound between about 250 and 2,000 Hz. The fundamental resonance would be targeted at roughly half the lower limit, or 125 Hz. Next, the diameter of the piston is tuned to set its coincidence frequency slightly above the upper limit. As I'll explain below (under Radiated Sound Power) tuning the resonance and coincidence frequencies this way will produce a flat radiated sound power transfer function across the desired frequency range.

The square of the oscillator mobility (normal velocity/force) is shown in the bottom plot. The mobility includes the effects of fluid loading. The radiation resistance and reactance of the air around the oscillator are shown in the middle plot. The reactance is divided by frequency to convert it to mass, which is added to the oscillator mass. The resistance is added to the oscillator damping. Both quantities change with frequency, with resistance increasing until it reaches a peak, followed by a constant high frequency value of ρcA (the product of the piston area and the acoustic characteristic impedance ρc). The reactance (in terms of mass loading) has a high low frequency limit, and then decreases with increasing frequency.

The radiation resistance (power/square of velocity), along with the oscillator vibration, determines the radiated sound power. Examine the frequency dependence of the square of the mobility above the fundamental resonance frequency - it decreases as the square of frequency. Now examine the frequency dependence of the resistance (the blue curve) below the coincidence frequency - it increases as the square of frequency. The product of the two cancels out the frequency dependencies, producing a flat radiated power response. This only works up to coincidence, beyond which the resistance flattens to a constant value, and the sound power rolls off with frequency.

Adjust the mass and spring constant to shift the fundamental resonance frequency. However, be careful not to use too much mass! Note how the vibration response, and therefore sound power, is directly proportional to mass. Add too much mass, and you reduce the output sound power. Adjust the piston radius to change the resistance and reactance. The larger the radius the larger the surface area and the higher the impedance. Watch the title above the plots to see how the added mass and damping, as well as the fluid-loaded resonance frequency change as you adjust radius. Finally, adjust the oscillator damping to change the response at resonance. Most loudspeakers have very high damping (or low 'Q' factors) to attenuate sound amplification at and around resonance. Finally, I've included some presets for typical loudspeaker parameters. You can also go back to the initial defaults.

This is a simple oscillator model, and does not include any flexural resonances/modes of vibration within the piston. These modes of vibration also affect the vibration and radiated power response in the form of small peaks and valleys. You'll see them in speaker performance curves. Other demonstrators will show how modes of vibration interact with adjacent acoustic spaces. Also, most loudspeakers have backing cavities within the enclosures they are mounted in. These cavities usually add stiffness (not mass), particularly at low frequencies. The added stiffness shifts the fundamental resonance frequency higher. Finally, there are electrical impedances within a loudspeaker that can affect the overall vibration and sound power response.