All Problems

Mechanical Oscillations

Problem 4.71

A point performs damped oscillations with frequency ω\omega and damping coefficient β.\beta . Find the velocity amplitude of the point as a function of time tt if at the moment t=0t=0 (a) its displacement amplitude is equal to a0a_{0}; (b) the displacement of the point x(0)=0x(0)=0 and its velocity projection vx(0)=x˙0v_{x}(0)=\dot{x}_{0}.

Reveal Answer
(a) v(t)=a0ω2+β2e−βtv(t)=a_{0} \sqrt{\omega^{2}+\beta^{2}} \mathrm{e}^{-\beta t} (b) v(t)=∣x˙0∣1+(β/ω)2e−βtv(t)=\left|\dot{x}_{0}\right| \sqrt{1+(\beta / \omega)^{2}} \mathrm{e}^{-\beta t}
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