A loudspeaker cone and its flexible suspension can be modelled as a rigid mass on a…

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Electricity, Magnetism & Electromagnetism · Simple, damped and driven oscillations

Mathematical Methods for Physics, Ch Second-order linear equations: damping, driving and resonance

Problem

A loudspeaker cone and its flexible suspension can be modelled as a rigid mass on a linear spring with a linear damper: the moving mass (cone plus coil) is

m = 12\,\mathrm{g}

, the suspension behaves as a spring of stiffness

k = 4200\,\mathrm{N\,m^{-1}}

, and mechanical losses in the suspension act as a damper with coefficient

b = 0.90\,\mathrm{kg\,s^{-1}}

, so that

m\ddot{x} + b\dot{x} + kx = F(t)

for the cone's displacement

x

from equilibrium,

F(t)

being the force from the voice coil. ((a)) Write this equation in the standard form

\ddot{x} + \gamma\dot{x} + \omega_0^2x = F(t)/m

and state

\omega_0

and

\gamma

. ((b)) An electrical LCR circuit obeys

L_{\mathrm{c}}\ddot{q} + R\dot{q} + q/C = V(t)

. A designer wants a circuit with the same

\omega_0

and

\gamma

as the cone and chooses the inductor to be

L_{\mathrm{c}} = 2.0\,\mathrm{mH}

. Find the values of

R

and

C

this requires.

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