= Damped pendulum
{title2=$\ddot\theta+c\dot\theta+\sin\theta=0$}
A <damped pendulum> is a <simple pendulum> whose <mechanical energy> is removed by a resisting <torque>. Here the model is undriven, with a fixed pivot, a <point mass> $m>0$ at distance $\ell>0$, uniform <gravitational acceleration> $g>0$, and resisting <torque> $-b\,d\theta/dt$ with constant $b>0$. The <angle> $\theta$ is measured from the downward vertical in <radians>. An ideal viscous rotary <dashpot> realizes this torque law; tangential <linear drag> $-\alpha v$ on the bob gives the same equation with $b=\alpha\ell^2$. The <moment of inertia> is $m\ell^2$, so
$$
m\ell^2\frac{d^2\theta}{dt^2}+b\frac{d\theta}{dt}+mg\ell\sin\theta=0,\qquad
\omega_0=\sqrt{\frac g\ell},\quad \gamma=\frac{b}{m\ell^2},\quad \tau=\omega_0t,\quad c=\frac\gamma{\omega_0}>0.
$$
Here $t$ is physical <time>, $\omega_0$ is the undamped small-amplitude <angular frequency>, and $\gamma$ has units of inverse <time>. In the normalized equation $\ddot\theta+c\dot\theta+\sin\theta=0$, dots mean differentiation with respect to $\tau$, not $t$. The parameter $c$ is dimensionless. This separates the chosen damping law from the <small-angle approximation>: retaining $\sin\theta$ allows finite <angles> even when the resisting <torque> is linear in <angular velocity>.
\Image[/damped-pendulum-decay-observation.png]
{title=Proposed viscous-damper experiment and illustrative small-angle decay: angle versus time and successive peak ratios, not measured data}
To motivate the model, release a pendulum without a push and record how its swing shrinks. A published example by Gintautas and Hubler uses a rotary sensor built from a computer mouse to measure the frequency and damping coefficient of a <physical pendulum>. https://arxiv.org/abs/0901.4319[Gintautas and Hubler, A simple, low-cost, data-logging pendulum built from a computer mouse, 2009]{ref} The following is a proposed experiment with a chosen viscous rotary <dashpot>; the plotted observations are predictions, not a reproduction of their measurements.
Mount a compact bob on a light rigid rod with a fixed pivot, couple the pivot to an adjustable viscous rotary <dashpot>, and measure $\ell$ from the pivot to the bob's centre. First record decay with the <dashpot> disconnected, then repeat with it connected at several settings; this establishes how much decay is already due to other losses. Keep the bob, rod, camera, and release <angle> unchanged between these trials. The simple model requires rod and damper-rotor <moments of inertia> to be negligible compared with $m\ell^2$. Otherwise measure the total <moment of inertia> $I$ and the distance $d$ to the assembly's centre of mass: the <physical pendulum> equation becomes $I\,d^2\theta/dt^2+b\,d\theta/dt+Mgd\sin\theta=0$, where $M$ is the total moving <mass>. A damper that also exerts an elastic restoring <torque> would require another term.
Place a camera perpendicular to the swing plane, include the pivot and bob centre in every frame, and determine their coordinates relative to the resting vertical. For horizontal and downward separations $x,y$, use $\theta=\operatorname{atan2}(x,y)$; record the frame timestamps rather than assuming a frame rate. Release from rest at about $10^\circ$, record through several complete swings, and repeat the release. For every same-side turning point, tabulate its <time> $t_n$ and angular <amplitude> $A_n=|\theta(t_n)|$. Measure the <oscillation period> from differences $t_{n+1}-t_n$; comparing several periods exposes drift. Interpolating around each observed extremum reduces frame discretization error. Check that late peaks still exceed the angular resolution; taking the <natural logarithm> of tiny, uncertain peaks amplifies their relative errors.
In the <small-angle approximation>, the <damped pendulum> becomes a <damped harmonic oscillator>. For $0<\gamma<2\omega_0$ and release from rest at $\theta_0$, its exact linear-model solution is
$$
\theta_{\mathrm{lin}}(t)=\theta_0e^{-\gamma t/2}\left(\cos\omega_dt+\frac{\gamma}{2\omega_d}\sin\omega_dt\right),\qquad
\omega_d=\sqrt{\omega_0^2-\frac{\gamma^2}{4}},\qquad T_d=\frac{2\pi}{\omega_d}.
$$
The decaying sinusoidal envelope has coefficient $|\theta_0|\omega_0/\omega_d$; because release is from rest, its successive same-side extrema occur at $t_n=nT_d$ and have $A_n=|\theta_0|e^{-\gamma nT_d/2}$. Thus the envelope and the sequence of actual peak heights are different descriptions of the same trace. The linear damping regimes and this solution are derived in https://farside.ph.utexas.edu/teaching/336k/Newton/node19.html[Richard Fitzpatrick, Damped Oscillatory Motion, equations 82–93].
The experiment tests two predictions independently: nearly equal same-side peak spacing and a straight line when plotting $\ln(A_n/A_0)$ against $t_n-t_0$. Its predicted slope is $-\gamma/2$, giving
$$
\gamma=\frac{2}{t_{n+1}-t_n}\ln\frac{A_n}{A_{n+1}},\qquad b=m\ell^2\gamma,\qquad
\omega_d^2+\frac{\gamma^2}{4}=\frac g\ell.
$$
Use several peaks and repeated trials rather than a single ratio, and compare the frequency identity with independently measured $\ell$ and $g$. For the illustrative choices $\ell=1.00\,\mathrm m$, $g=9.81\,\mathrm{m/s^2}$, $c=0.20$, and $\theta_0=10^\circ$, the linear model gives $\gamma\approx0.6264\,\mathrm{s^{-1}}$, $T_d\approx2.0162\,\mathrm s$, and $A_{n+1}/A_n\approx0.5318$. The first two positive peaks after release are therefore approximately $5.318^\circ$ and $2.828^\circ$. The undamped small-amplitude <oscillation period> is about $2.0061\,\mathrm s$: substantial loss per swing need not imply a large change in period.
Repeat at a larger release <angle> to distinguish two limitations. The relative error in replacing $\sin\theta$ by $\theta$ is about $\theta^2/6$ for small <angles>: about $0.51\%$ at $10^\circ$, but about $17.3\%$ at $60^\circ$ when compared with $\theta$. Finite-amplitude swing spacing can therefore change as the swing shrinks even if $b$ is constant. Separately, <friction> at the pivot or <quadratic drag> can invalidate the assumed linear resisting <torque>; exponential peak decay should be checked, not imposed on all recorded motion. Peters discusses why real pendulum damping can require nonlinear terms. https://arxiv.org/abs/physics/0306081[Randall D. Peters, Nonlinear Damping of the 'Linear' Pendulum, 2003]{ref} Repeatability, residual pivot motion, changes in damper <temperature>, and finite camera resolution are useful checks on this proposed measurement.
The normalized <equilibrium points> are $(\theta,\dot\theta)=(k\pi,0)$. Even multiples of $\pi$ are attracting: their <linearization> has <eigenvalues> $(-c\pm\sqrt{c^2-4})/2$, giving a <stable focus> for $0<c<2$, repeated negative <eigenvalues> at $c=2$, and a <stable node> for $c>2$. These are the underdamped, critically damped, and overdamped small-displacement regimes. Odd multiples of $\pi$ have <eigenvalues> $(-c\pm\sqrt{c^2+4})/2$ of opposite signs and are <saddle equilibria>. The <energy dissipation of a damped pendulum> explains why trajectories in the <phase portrait> spiral inward near a <stable focus> instead of closing into the recurring orbit of an undamped <simple pendulum>.
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