Engineering Handbook/Oscillation


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! Oscillation !! Formula $$\frac{d^2x}{dt^2} + \frac{k}{m} = 0$$ $$s^2 + \frac{k}{m} = 0$$ $$s = \pm \sqrt{-\frac{k}{m}} t $$ $$s = \pm j\omega t $$ $$x = e^{(\pm j\omega t)} $$ $$x = e^{(j\omega t)} + e^{(-j\omega t)}$$ $$x = A Sin \omega t$$ $$\frac{d^2y}{dt^2} + \frac{k}{m} = 0$$ $$s^2 + \frac{k}{m} = 0$$ $$s = \pm \sqrt{-\frac{k}{m}} t $$ $$s = \pm j\omega t $$ $$y = e^{(\pm j\omega t)}$$ $$y = e^{(j\omega t)} + e^{(-j\omega t)}$$ $$y = A Sin \omega t$$ $$\theta(t) = \theta_0\cos\left(\sqrt{g\over \ell}t\right) \quad\quad\quad\quad |\theta_0| \ll 1$$
 * --> || $$-Fx = F $$ $$ -kx = m \frac{d^2x}{dt^2}$$
 * --> || $$-Fx = F $$ $$ -kx = m \frac{d^2x}{dt^2}$$
 * Simple_harmonic_oscillator.gif || $$-Fy = F $$ $$ -ky = m \frac{d^2y}{dt^2}$$
 * Simple_harmonic_oscillator.gif || $$-Fy = F $$ $$ -ky = m \frac{d^2y}{dt^2}$$
 * Simple_pendulum_height.svg || $${\mathrm{d}^2\theta\over \mathrm{d}t^2}+{g\over \ell}\theta=0$$
 * Simple_pendulum_height.svg || $${\mathrm{d}^2\theta\over \mathrm{d}t^2}+{g\over \ell}\theta=0$$
 * }