1.4 Damped oscillations

Damped oscillations

Consider a simple harmonic oscillator that has friction, then the equations of motion must be changed with the addition of a friction term. So we write $$m\frac{d^{2}x}{d{t}^2}=-kx-b\frac{dx}{dt}$$ where $b\frac{dx}{dt}$ is the friction term. Rearranging we obtain: $$m\frac{d^{2}x}{d{t}^2}+b\frac{dx}{dt}+kx=0$$ or $$\frac{d^{2}x}{d{t}^2}+\gamma \frac{dx}{dt}+{\omega }_0^{2}x=0$$ Where $\gamma =\frac{b}{m}$ and ${\omega }_0^2=\frac{k}{m}$ Assume a solution of form $$x=A{e}^{i(pt+\alpha )}$$ substitute into equation and get $$(-p^2+ip\gamma +{\omega }_0^2)A{e}^{i(pt+\alpha )}=0$$ so $$-p^2+ip\gamma +{\omega }_0^2=0$$ $p$ must have real and imaginary parts, so rewrite: $p=\omega +is$ $$p^2={\omega }^2+2i\omega s-s^2$$ So the equation $$-p^2+ip\gamma +{\omega }_0^2=0$$ becomes upon substitution: $$-{\omega }^2-2i\omega s+s^2+i\omega \gamma -s\gamma +{\omega }_0^2=0$$ This equation implies that the real and imaginary parts are each zero.Separatethe real and imaginary partsImaginary parts give: $${\displaystyle \begin{array}{c}-2\omega s+\omega \gamma =0\\ s=\frac{\gamma }{2}\end{array}}$$ From Real parts get $$-{\omega }^2+s^2-s\gamma +{\omega }_0^2=0$$ or $$-{\omega }^2+\frac{{\gamma }^2}{4}-\frac{\gamma }{2}\gamma +{\omega }_0^2=0$$ $$-{\omega }^2-\frac{{\gamma }^2}{4}+{\omega }_0^2=0$$ Which rearranges to $${\omega }^2={\omega }_0^2-\frac{{\gamma }^2}{4}$$ Thus the solution becomes $$x=A{e}^{-\gamma t/2}{e}^{i(\omega t+\alpha )}$$ where $$\omega =\sqrt{{\omega }_0^2-\frac{{\gamma }^2}{4}}$$ Note that this has assumed a frictional damping force. For a more complicated damping force, the result would be different.