Mass-spring with friction

Example 2 — Mass-spring with friction (damped)

We add a viscous friction force $-\gamma m\dot x$ proportional to the velocity: $\ddot x + 2\gamma\dot x + \omega_0^2 x = 0.$ Characteristic equation $\lambda^2 + 2\gamma\lambda + \omega_0^2 = 0$, with $\Delta = 4\gamma^2 - 4\omega_0^2$. There are three regimes:

• underdamped ($\gamma<\omega_0$, $\Delta<0$): $\lambda = -\gamma\pm i\sqrt{\omega_0^2-\gamma^2}$, an oscillation that dies away exponentially, $x(t) = e^{-\gamma t}\bigl[A\cos\omega' t + B\sin\omega' t\bigr]$ with $\omega' = \sqrt{\omega_0^2-\gamma^2}$;
• critically damped ($\gamma = \omega_0$, $\Delta=0$): $x(t) = (A+Bt)e^{-\gamma t}$. It is the optimal regime — no oscillation, return to equilibrium in the shortest time — ideal for car shock absorbers;
• overdamped ($\gamma>\omega_0$, $\Delta>0$): $\lambda$ real and negative, pure and slow exponential decay.

Links

Topics: Differential equations
Concepts: Second-order linear ODE · Characteristic equation · Harmonic oscillator · Damping
Methods: Harmonic oscillator ODE
Skills: Modelling · Reasoning by cases · Solving equations