The Pendulum and Energy

A pendulum is the simplest robot dynamics model, and a swinging arm joint, a walking leg, and a hanging load all behave like one. As it swings, height energy turns into speed energy and back. Friction slowly drains the total.

Release angle Length Damping

Motion and energy

For a mass on a light rod of length $L$ with damping $b$:

$$\ddot\theta = -\frac{g}{L}\sin\theta - b\,\dot\theta \qquad E = \underbrace{\tfrac{1}{2} m L^2 \dot\theta^2}_{\text{kinetic}} + \underbrace{m g L (1 - \cos\theta)}_{\text{potential}}$$

For small swings the period is close to $T = 2\pi\sqrt{L/g}$, independent of how far it swings.

What to notice

At the top of each swing, all the energy is potential and the speed is zero. At the bottom, it is all kinetic. With damping, the total shrinks each swing, and the red "lost" bar grows as energy turns into heat. The phase portrait, angle against angular speed, traces a closed loop without damping and a spiral into the center with it. Release it near 180° and the period grows much longer than the small-swing formula, because the pendulum lingers near the top.