A mass on a damped spring obeys

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations

Mathematical Methods for Physics, Ch Fourier transforms, spectra and convolution

Problem

\ddot x + \gamma\dot x + \omega_0^2 x = f(t)/m

, with

m = 0.20\,\mathrm{kg}

\omega_0 = 50\,\mathrm{rad\,s^{-1}}

and

\gamma = 4.0\,\mathrm{s^{-1}}

. It is driven by

f(t) = F_0\cos(\omega t)

with

F_0 = 3.0\,\mathrm{N}

and

\omega = 48\,\mathrm{rad\,s^{-1}}

. In the steady state the response is a single component,

x(t) = \mathrm{Re}(Xe^{-i\omega t})

, with complex amplitude

X

, driven by

F_0e^{-i\omega t}

. Using the rule

\mathrm{d}/\mathrm{d}t \to -i\omega

that the derivative theorem gives for a component varying as

e^{-i\omega t}

, obtain the peak displacement amplitude

|X|

without integrating the equation directly.

Browse the problem library