# Damping harmonics motion common harmonics * $x(t)$ : $x_mcos(\omega t+\phi)$ * $v(t)$ : $-\omega x_msin(\omega t+\phi)$ * $a(t)$ : $-\omega^2x_mcos(\omega t+\phi)$ ## Damping force * Damping force : $F_d = D_1(\text{backward force})$ * Damping coefficient : $b$ $$F_d = -bv$$ $$F_{net} = F_d+F_s$$ $$ma = -kx-bv$$ $$m\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx = 0$$ $$m\ddot x+b\dot x+kx = 0$$ * let $x(t) = e^{πœ†t}$ , $πœ†$ is solution * then $V = \dot x = \frac{de^{πœ†t}}{dt} =πœ†e^{πœ†t}$ , $a = \ddot x = \frac{d^2e^{πœ†t}}{dt^2} = πœ†^2e^{πœ†t}$ * and $e^{πœ†t} \not= 0$ $$mπœ†^2e^{πœ†t}+bπœ†e^{πœ†t}+ke^{πœ†t} = 0$$ :::success $$mπœ†^2+bπœ†+k = 0$$ ::: ## Overdamping, critical damping, underdamping $$mπœ†^2+bπœ†+k = 0$$ $$πœ† = \frac{-b\pm\sqrt{b^2-4mk}}{2m}$$ when $πœ†$ is a real number : $b^2-4mk\ge 0$ $$πœ†_1 = \frac{-b+\sqrt{b^2-4mk}}{2m}$$ $$πœ†_2 = \frac{-b-\sqrt{b^2-4mk}}{2m}$$ $πœ†_1$ & $πœ†_2 < 0$ , then $\mid πœ†_2\mid>\mid πœ†_1\mid$ $x_1(t) = C_1e^{-πœ†_1t}$ (major) $x_2(t) = C_2e^{-πœ†_2t}$ (minor) $$x(t) = x_1(t)+x_2(t) = C_2e^{-πœ†_2t}+C_1e^{-πœ†_1t}$$ $$x(0) = C_2e^{-πœ†_20}+C_1e^{-πœ†_10}$$ $$v(0) = C_2πœ†e^{-πœ†_20}+C_1ue^{-πœ†_10}$$ $$C_1+C_2 = x_i$$ $$C_1πœ†_1+C_2πœ†_2 = v_i$$ $$C_1 = x_i-C_2$$ $$(x_i-C_2)πœ†_1+C_2πœ†_2 = v_i$$ $$x_iπœ†_1-C_2πœ†_1+C_2πœ†_2 = v_i$$ $$C_2(-πœ†_1+πœ†_2) = v_i-x_iπœ†_1$$ :::success $$\frac{v_i-x_iπœ†_1}{πœ†_1-πœ†_2}\longrightarrow(C_2)$$ ::: $$C_1 = x_i-\frac{v_i-x_iπœ†_1}{πœ†_1-πœ†_2}$$ :::success $$\frac{x_iπœ†_2-v_i}{πœ†_1-πœ†_2}\longrightarrow(C_1)$$ ::: * $b^2-4mk = 0$ : critical damping $$πœ† = \frac{-b}{2m}$$ :::success $$x(t) = Ce^{πœ†t} = Ce^{\frac{-bt}{2m}}$$ ::: when $πœ†$ is not a real number : $b^2-4mk < 0$ ( Underdamped ) ## Damped oscillation $$mπœ†^2+bπœ†+k = 0$$ let $2𝛾 = \frac bm,\omega_i^2 = \frac km$ $𝛾$ is the decay rate, in the reciprocal of the time units of the independent variable $t$ $$πœ†^2+2π›Ύπœ†+\omega_i^2 = 0$$ $$πœ† = -𝛾\pm i\omega'$$ $$πœ† = \frac{-2𝛾\pm\sqrt{4𝛾^2-4\omega_i^2}}{2}$$ $$πœ† = -𝛾\pm i\sqrt{𝛾^2-\omega_i^2}$$ $$(πœ†+𝛾)i = \sqrt{\omega_i^2-𝛾^2} = \omega'$$ $$\omega'= \sqrt{\frac{k}{m}-\frac{b^2}{4m^2}}$$ $$x(t) = e^{πœ†t} = e^{-t𝛾\pm it\sqrt{\omega_i^2-𝛾^2}}$$ by Euler's formula : $e^{i\theta} = \cos(\theta)+i\sin(\theta)$ $$ x(t) = e^{(-\gamma)t} \left( C_1 e^{i\omega' t} + C_2 e^{-i\omega' t} \right) $$ $$ x(t) = e^{-\gamma t} \left( C_1 (\cos(\omega' t) + i \sin(\omega' t)) + C_2 (\cos(\omega' t) - i \sin(\omega' t)) \right) $$ $$ x(t) = e^{-\gamma t} \left( (C_1 + C_2) \cos(\omega' t) + i (C_1 - C_2) \sin(\omega' t) \right) $$ let $C'_1 = C_1 + C_2$ , $C'_2 = i (C_1 - C_2)$ $$x(t) = e^{-𝛾t}(C_1'cos(\omega't)+C_2'sin(\omega't))$$ by Trigonometric Identities :::success $$= e^{-\frac {bt}{2m}}x_mcos(\omega't+\phi)$$ ::: ## Damping coefficient 1. $b = 2m\sqrt{(\frac{k}{m}-\frac{4\pi^2}{T^2})}$ $$\omega' = \frac{2\pi}{T} = \sqrt{\frac{k}{m}-\frac{b^2}{4m^2}}$$ $$\frac{b^2}{4m^2} = \frac{k}{m}-\frac{4\pi^2}{T^2}$$ $$\frac{b}{2m} = \sqrt{(\frac{k}{m}-\frac{4\pi^2}{T^2})}$$ :::success $$b = 2m\sqrt{(\frac{k}{m}-\frac{4\pi^2}{T^2})}$$ ::: 2. $b = \frac{-2m\cdot ln(x)}{t}$ $$e^{-\frac {bt}{2m}} = x$$ $$-\frac {bt}{2m} = ln(x)$$ :::success $$b = \frac{-2m\cdot ln(x)}{t}$$ ::: 3.$b = 2m\sqrt{\omega_i^2-\omega'^2}$ $$2𝛾 = \frac bm$$ $$b = 2m𝛾$$ $$\sqrt{\omega_i^2-𝛾^2} = \omega'$$ :::success $$b = 2m\sqrt{\omega_i^2-\omega'^2}$$ ::: * critical damping ($\omega' = 0$) $$b = 2m\sqrt{\omega_i^2-0}$$ $$b = 2m\omega_i$$ $$b = 2m\sqrt{\frac km}$$ :::success $$b = 2\sqrt{km}$$ ::: ## Graph ![damping](https://hackmd.io/_uploads/SkpZZ00S1l.png)