All Problems Mechanical Oscillations
Problem 4.71
A point performs damped oscillations with frequency \(\omega\) and damping coefficient \(\beta .\) Find the velocity amplitude of the point as a function of time \(t\) if at the moment \(t=0\) (a) its displacement amplitude is equal to \(a_{0}\); (b) the displacement of the point \(x(0)=0\) and its velocity projection \(v_{x}(0)=\dot{x}_{0}\).
(a) \(v(t)=a_{0} \sqrt{\omega^{2}+\beta^{2}} \mathrm{e}^{-\beta t}\) (b) \(v(t)=\left|\dot{x}_{0}\right| \sqrt{1+(\beta / \omega)^{2}} \mathrm{e}^{-\beta t}\)