# Damped oscillation Calculator

## Calculates a table of the displacement of the damped oscillation and draws the chart.

 κ＜ω0 (underdamping): Oscillation. The amplitude decreases exponentially with time.κ=ω0 (critical damping): No oscillation. The amplitude decreases quickly.κ＞ω0 (overdamping): No oscillation. The amplitude decreases slowly.$\vspace{5}\\\hspace{80}{\large \frac{d^2 x}{dt^2}}+2\kappa{\large \frac{dx}{dt}}+{\omega_{\small 0}}^2 x=0\\$
 Undamped angular frequency ω0 Resistance coefficient κ [ Initial displacement x0 ] [ Repetition frequency 50 100 200 ]
 $\normal Damped\ Oscillation(1)\ equation\hspace{20} {\large \frac{d^2 x}{dt^2}}+2\kappa{\large \frac{dx}{dt}}+{\omega_{\small 0}}^2 x=0\\\hspace{30}\omega_{\small 0}:\ undamped\ angular\ frequency\\\hspace{30}\kappa:\ resistance\ coefficient\\[10](2)\ if\ \kappa<\omega_{\small 0},\hspace{20}\omega_{\small d}=\sqrt{{\omega_{\small 0}}^2-\kappa^2}\\\hspace{20}x=x_{\small 0}e^{-\kappa t}\left{cos(\omega_{\small d}t)+{\large\frac{\kappa}{\omega_{\small d}}}sin(\omega_{\small d}t)\right}\\[10](3)\ if\ \kappa=\omega_{\small 0},\hspace{20}x=x_{\small 0}(1+\omega_{\small 0}t)e^{-\omega_{\small 0}t}\\[10](4)\ if\ \kappa>\omega_{\small 0},\hspace{20}\omega_{\small d}=\sqrt{\kappa^2-{\omega_{\small 0}}^2}\\\hspace{25} x= x_{0} \frac{{\small(\omega_d+\kappa)}e^{(\omega_d-\kappa)t}+{\small(\omega_d-\kappa)}e^{-(\omega_d+\kappa)t}}{2\omega_d}\\$

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