Pendulum Model

Pendulum(; driven = true, name, defaults)

Create an ODESystem representing a damped, driven or undriven pendulum, depending on the value of driven (defaults to true, i.e., driven pendulum).

The equation used in this model is $\ddot{θ} + 2 ζ ω_0 \dot{θ} + ω_0^2 \sin(θ) = τ,$ where $θ$ is the angle counter-clockwise from the downward equilibrium, $ζ$ is the damping parameter, $ω_0$ is the resonant angular frequency, and $τ$ is the input torque divided by the moment of inertia of the pendulum around the pivot (driven = false sets $τ ≡ 0$).

The name of the ODESystem is name.

Users may optionally provide default values of the parameters through defaults: a vector of the default values for [ζ, ω_0].

Example

@named pendulum = Pendulum(driven = false)
pendulum = structural_simplify(pendulum)

x0 = π * rand(2)
p = rand(2)
τ = 1 / prod(p)

prob = ODEProblem(pendulum, x0, 15τ, p)
sol = solve(prob, Tsit5())

# Check that the undriven pendulum fell to the downward equilibrium
θ_end, ω_end = sol.u[end]
x_end, y_end = sin(θ_end), -cos(θ_end)

sqrt(sum(abs2, [x_end, y_end, ω_end] .- [0, -1, 0])) < 1e-4

plot_pendulum(θ, t; title)
plot_pendulum(sol; title, N, angle_symbol)

Plot the pendulum's trajectory.

Arguments

• θ: The angle of the pendulum at each time step.
• t: The time steps.
• sol: The solution to the ODE problem.

Keyword arguments

• title: The title of the plot; defaults to no title (i.e., title="").
• N: The number of points to plot; when using θ and t, uses length(t); defaults to 500 when using sol.
• angle_symbol: The symbol of the angle in sol; defaults to :θ.