Passive Acoustic Radiators Justin Yates, Wittenberg University Spring 2014

Outline Review of Passive Acoustic Radiator (PAR or PR) Model of PAR system as a damped, driven oscillator Solutions to the system and their physical meanings Designing a PAR Questions

Review We can model the passive cone of a PAR as a mass on a spring The oscillator then, is both driven and damped β x k F(t)

Review PR’s need to be tuned Tuning can be most easily achieved by changing the mass Masses (m) Restoring spring (k)

Damped Harmonic Oscillator Modeling the PR as a damped, driven oscillator gives us the 2 nd order linear differential equation for the position of the cone; This equation comes from the equation of motion for the driving force of the oscillator Where I have simplified by dividing by m and substituting equivalent coefficients. The driving force in a PAR is actually a sinusoidal function of time so Eq. (1)

Discussion The general solution is It is important to note that the solution consists of two distinct parts. Of the two terms, one is transient and the other is harmonic, or steady. TransientHarmonic

Physical Meaning The full solution and its parts

Designing a PAR We can take what we have learned about the physical meanings of the parts of the general solution to examine the ideal conditions for a PAR system. – General solution – Damping coeff. and the quality factor – The “notch”

Designing a PAR A PAR is designed to exploit the harmonic term of the general solution. The typical frequency range for bass notes is around Hz. These notes, in most music, are long and persistent. This is unlike the higher frequencies in speech or percussion. The transient term corresponds to abrupt changes in the system’s frequency. Speech and percussion are examples of sounds that are mostly transient. They change the system frequency but die off quickly. The low bass notes on the other hand, are long repetitive beats that make them easy to exploit for the resonant term.

Designing a PAR PAR’s work best with a low Q-factor. Increasing β is going to decrease the Q-factor. Another way to think about desired Q and β, is with the transient term. Oscillations die out in the absence of a driving force according to the exponential in the term When designing a PAR you do not want the system to continue to “ring” long after a note is over. We want this term to die out quickly. This means that a properly designed PAR should not have β << ω 0 to keep Q low.

Frequency vs. Output Red—ported cabinet Green—Sealed cabinet Blue—PAR (ω 0 = about 15Hz) The PAR is being driven above resonance here and the notch occurs below the range of human hearing 20

Frequency vs. Output The notch occurs at the resonant frequency of the PAR The plot of the notch is a superposition of the PAR’s frequency response curve with the active driver’s response curve Changing β does not change the resonant frequency of the PAR but it will affect the amplitude of its response Making β larger will decrease the amplitude of the PAR at resonance and decrease the depth of the notch

Frequency vs. Output Frequency (Hz) Amplitude (dB) β = 4 β = 8 ω 0 = 20

Further Exploration Effects of: – box size on output – Surface area of cone

Questions?

References Passive Radiator Systems. (2010, August 16). Retrieved Febuary 1, 2014, from DIY Subwoofers: Transient (acoustics). (2014, January). Retrieved January 30, 2014, from Wikipedia: cs%29 Taylor, J. R. (2005). Classical Mechanics. University Science Books. William E. Boyce, R. C. (2009). Elementary Differential Equations and Boundary Value Problems. John Wiley and Sons Inc.