The Damped Nonlinear Pendulum

ω

0

0.9

γ

0.3

t

15.

plot type

stream

density

θ(0) = 1.25 θ'(0) = -0.40

The plots show the motion of a harmonic oscillator with damping, in phase space on the left and as a function of time on the right, with the position of the pendulum in the top-right corner. The equation of motion is

′′

θ

(t)+γ

′

θ

(t)+

2

ω

0

sinθ(t)0

, where

ω

0

is the natural frequency and

γ

is the damping constant. This equation does not take the form of the usual approximation

sin(θ)≈θ

.

Nonlinear analogs of underdamping and overdamping can be observed.

The second-order equation can be solved by splitting it into two first-order equations.