Damped Oscillation
Calculate and plot a graph of the displacement for damped oscillations subject to a resistive force proportional to velocity.
Display a graph spanning 4 natural periods.
・Damped oscillation occurs when resistance is low (e.g., ω0 = 5, κ = 1)
・At the critical point (κ = ω0), there is no oscillation and the system decays most rapidly (e.g., ω0 = 5, κ = 5)
・With high resistance, there is no oscillation and the system decays slowly (e.g., ω0 = 5, κ = 20)
・Damped oscillation occurs when resistance is low (e.g., ω0 = 5, κ = 1)
・At the critical point (κ = ω0), there is no oscillation and the system decays most rapidly (e.g., ω0 = 5, κ = 5)
・With high resistance, there is no oscillation and the system decays slowly (e.g., ω0 = 5, κ = 20)
| \(\\\hspace{80px}{\large \frac{d^2 x}{dt^2}}+2\kappa{\large \frac{dx}{dt}}+{\omega_{\small 0}}^2 x=0\\\) |
| \(\normalsize Damped\ Oscillation\\ (1)\ equation\hspace{20px} {\large \frac{d^2 x}{dt^2}}+2\kappa{\large \frac{dx}{dt}}+{\omega_{\small 0}}^2 x=0\\ \hspace{30px}\omega_{\small 0}:\ undamped\ angular\ frequency\\ \hspace{30px}\kappa:\ resistance\ coefficient\\ (2)\ if\ \kappa<\omega_{\small 0},\hspace{20px}\omega_{\small d}=\sqrt{{\omega_{\small 0}}^2-\kappa^2}\\ \hspace{20px}x=x_{\small 0}e^{-\kappa t}\left\{ \cos(\omega_{\small d}t)+{\large\frac{\kappa}{\omega_{\small d}}} \sin(\omega_{\small d}t)\right\}\\ (3)\ if\ \kappa=\omega_{\small 0},\hspace{20px}x=x_{\small 0}(1+\omega_{\small 0}t)e^{-\omega_{\small 0}t}\\ (4)\ if\ \kappa>\omega_{\small 0},\hspace{20px}\omega_{\small d}=\sqrt{\kappa^2-{\omega_{\small 0}}^2}\\ \hspace{25px} x= x_{0} \frac{{\small(\omega_d+\kappa)}e^{(\omega_d-\kappa)t}+{\small(\omega_d-\kappa)}e^{-(\omega_d+\kappa)t}}{2\omega_d}\\ \) |
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