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Mechanical Oscillations

Problem 4.69

A point performs damped oscillations with frequency ω\omega and damping coefficient β\beta according to the law (4.1 b).(4.1 \mathrm{~b}) . Find the initial amplitude a0a_{0} and the initial phase α\alpha if at the mòment t=0t=0 the displacement of the point and its velocity projection are equal to (a) x(0)=0x(0)=0 and vx(0)=x˙0v_ {x}(0)=\dot{x}_ {0} (b) x(0)=x0x(0)=x_ {0} and vx(0)=0v_ {x}(0)=0

Reveal Answer
4.69. (a) a0=∣x˙0∣ω,α={−π/2, when x˙0>0,+π/2, when x˙0<0;a_{0}=\frac{\left|\dot{x}_{0}\right|}{\omega}, \quad \alpha=\left\{\begin{array}{ll}-\pi / 2, & \text { when } & \dot{x}_{0}>0, \\ +\pi / 2, & \text { when } & \dot{x}_{0}<0 ;\end{array}\right. (b) a0=a_{0}= =∣x0∣1+(β/ω)2,α=arctan⁡(−β/ω),=\left|x_{0}\right| \sqrt{1+(\beta / \omega)^{2}}, \quad \alpha=\arctan (-\beta / \omega), with −π/2<α<0,-\pi / 2<\alpha<0, if x0>0x_{0}>0 and π/2<α<π,\pi / 2<\alpha<\pi, if x0<0x_{0}<0
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