Lessons · shm

Damping: when the wobble dies

Regime 2 — calculus does what algebra cannot. The acceleration isn't constant, so the algebra formulas don't apply; calculus is the only road in, and SymPy proves the closed form solves the equation of motion.

Regime 2 · calculus does more Math machine-derived & checked by SymPy 2 modeling assumptions (author-asserted)

Add friction to the mass on a spring. As the damping grows the motion changes character — oscillating, then just barely returning, then crawling back. Where is the boundary, and what is the fastest a system can return to rest?

Natural angular frequency

\omega_0 = \sqrt{k/m}

4 1/s

Critical damping

b_c = 2\sqrt{km}

8 kg/s

Newton's law with a spring and a drag force

m\,x'' = -k\,x - b\,x' \;\;\Longrightarrow\;\; x'' + 2\gamma x' + \omega_0^2 x = 0

Restoring force $-kx$ plus a velocity-dependent damping $-bx'$ . Here $\gamma = b/2m$ and $\omega_0^2 = k/m$ .

back-substitute each form into $m x'' + b x' + k x = 0$ ; check the initial conditions; check $dE/dt = -b v^2$

✓ Underdamped form $e^{-\gamma t}(\cos,\sin)$ solves $x'' + 2\gamma x' + \omega_0^2 x = 0$ . [structural]
✓ Critical form $(x_0 + \dots\,t)\,e^{-\gamma t}$ solves $x'' + 2\gamma x' + \gamma^2 x = 0$ (where $\omega_0 = \gamma$ ). [structural]
✓ Overdamped form $e^{-\gamma t}(\cosh,\sinh)$ solves $x'' + 2\gamma x' + (\gamma^2 - s^2)x = 0$ . [structural]
✓ It matches the initial position $x(0) = x_0$ . [structural]
✓ It matches the initial velocity $x'(0) = v_0$ . [structural]
✓ Energy dissipates at exactly $\tfrac{dE}{dt} = -b v^2$ (here per unit mass, $b/m = 2\gamma$ ). [structural]

Dimensional homogeneity: checked by SymPy (holds).

✗ Common misconception: “More damping always makes a system return to rest faster.”

Past critical damping, adding more damping makes the return SLOWER — the overdamped system crawls back to equilibrium. The quickest return without overshoot is exactly at critical damping ( $\zeta = 1$ ); that is why door closers and instrument needles are tuned to it.

Modeling assumptions — author-asserted, disclosed not discharged

Linear (viscous) damping: the resistive force is -b·x' (proportional to velocity).
Ideal Hooke's-law spring; the mass is a point.

The stacked graph, fully annotated

A static rendering (Matplotlib) at the default parameters — the interactive version is in the Graph tab above.

Formulas used

Hover a formula to preview its reference entry; click to open it in the reference (or the concept graph):