--- jupyter: jupytext: formats: ipynb,md text_representation: extension: .md format_name: markdown format_version: '1.3' jupytext_version: 1.18.1 kernelspec: display_name: Julia 1.12 language: julia name: julia-1.12 --- # Pendulum ## Equations of motion The dimensional equation for a pendulum with the constraints - no friction - rigid rod - vertically movable anchor point with prescribed acceleration $a_0(t)$ with amplitude $c^{*}$ reads $$ \frac{\text{d}^2\theta}{\text{d} t^{*2}} = - \frac{a_0(t^{*})}{l^{*}} \sin(\theta) - \frac{g^{*}}{l^{*}} \sin(\theta) $$ In the following, we will solely be concerned with a prescribed sinusoidal motion. Its acceleration is given by $$ a_0(t^{*}) = c^{*} \sin(k^{*} t^{*}) $$ with amplitude $c^{*}$ and wave number $k^{*}$. For non-dimensionalisation, we introduce a time scale $T$, which yields $$ \frac{1}{T^2}\frac{\text{d}^2\theta}{\text{d} t^2} = - \frac{c^{*}\sin(k^{*} T t) + g^{*}}{l^{*}} \sin(\theta). $$ Moreover, the following non-dimensional parameters are introduced. $$ \tau := T^{-1} \sqrt{l^{*}/ g^{*}} \quad\text{(gravitational time scale)}$$ $$ c := \frac{c^{*}}{g^{*}} \quad\text{(relative anchor point acceleration)}$$ $$ k := k^{*}T \quad\text{(anchor point oscillation time scale)}$$ The non-dimensional equation then reads $$ \frac{\text{d}^2\theta}{\text{d} t^{2}} = - \frac{1}{\tau^2} (c \sin(k t) + 1) \sin(\theta), $$ which is consistent with the Buckingham theorem given five dimensional variables ($t^*$,$g^*$,$c^*$,$k^*$,$l^*$) and two dimensions (time, length). ## Numerical method We define the angular velocity of the pendulum $\Omega$ to reformulate our problem into a set of two first-order ordinary differential equations, $$ \frac{\text{d}\Omega}{\text{d}t} = - \frac{1}{\tau^2} (c \sin(k t) + 1) \sin(\theta), $$ $$ \text{d}\theta / \text{d}t = \Omega, $$ which can be solved numerically. We employ the classical Runge-Kutta method (RK4) for $\text{d}\mathbf{x}/\text{d}t = f(\mathbf{x},t)$, i.e. $$ \mathbf{x}_{n+1} = \mathbf{x}_n + \frac{\Delta t}{6} ( r_1 + 2 r_2 + 2 r_3 + r_4 ),$$ $$ r_1 = f(t_n, \mathbf{x}_n),$$ $$ r_2 = f(t_n + \Delta t /2, \mathbf{x}_n + r_1 \Delta t /2),$$ $$ r_3 = f(t_n + \Delta t /2, \mathbf{x}_n + r_2 \Delta t /2),$$ $$ r_4 = f(t_n + \Delta t, \mathbf{x}_n + r_3 \Delta t),$$ where $\mathbf{x} := (\theta,\Omega,t)^T$ is the state vector of our problem. This enables us to get a discrete-in-time map of the form $$ \mathbf{x}_{n+1} = f(\mathbf{x}_n,\mathbf{p}) $$ with $\mathbf{x}$ and $\mathbf{p}$ being the state and parameter vectors, respectively. Note that $\mathbf{p}$ contains the parameters of the physical system, as well as the numerical parameter ($\Delta t$) since both affect the evolution of the discrete numerical system. ```julia struct State θ :: Float64 Ω :: Float64 t :: Float64 end struct Parameter τ :: Float64 c :: Float64 k :: Float64 Δt :: Float64 end function dsmap(x::State,p::Parameter) f = (x::State,p::Parameter) -> -(1.0/p.τ^2)*(p.c*sin(p.k*x.t)+1)*sin(x.θ) coeff = [p.Δt/6.0,p.Δt/3.0,p.Δt/3.0,p.Δt/6.0] step = [0.0,p.Δt/2.0,p.Δt/2.0,p.Δt] rΩ = f(x,p) rθ = x.Ω Ωnp1 = x.Ω + coeff[1]*rΩ θnp1 = x.θ + coeff[1]*rθ for ii = 2:length(coeff) rΩ = f(State(x.θ+rθ*step[ii],NaN,x.t+step[ii]),p) rθ = x.Ω + rΩ*step[ii] θnp1 += coeff[ii]*rθ Ωnp1 += coeff[ii]*rΩ end return State(θnp1,Ωnp1,x.t+p.Δt) end function dsrun(x::State,p::Parameter,nstep::Int) X = Vector{State}(undef,nstep) for ii = 1:nstep x = dsmap(x,p) X[ii] = x end return X end ``` ## Simple pendulum First we analyse the dynamics of the pendulum without anchor point movement, e.g. $a_0 = 0$ (in general) or $c = 0$ (for sinusoidal). We visualise the time series in terms of angle vs time. In this simplified case, we can easily visualise the entire state space, since it is two-dimensional, i.e. visualise angular velocity vs angle. ```julia using Plots function plot_evolution_state_simple(X::Vector{State},p::Parameter) t = getproperty.(X,:t) θ = mod.((getproperty.(X,:θ).+π),2π).-π Ω = getproperty.(X,:Ω) p1 = plot( t/τ,θ/π, xlabel="t/τ",ylabel="θ/π", seriestype=:scatter,marker=:circle,markersize=0.01, legend=false ) p2 = plot( θ/π,Ω*τ/π, xlabel="θ/π",ylabel="Ωτ/π", xlims=(-1.1,1.1),ylims=(-1.1,1.1), seriestype=:scatter,marker=:circle,markersize=0.01, aspect_ratio=1, legend=false ) plot(p1,p2;layout=(1,2)) end ``` ### Periodic swing We initialise the pendulum at a 90° angle with no initial velocity. This results in a periodic swinging motion which is very well captured by our numerical method. The state space is a circle whose centre is the stable equilibrium ($\theta=0, \Omega=0$). ```julia # Initial conditions θ0 = π/2 Ω0 = 0.0 t0 = 0.0 x = State(θ0,Ω0,t0) # Parameter τ = 1.0 c = 0.0 k = 0.0 Δt = 0.01 p = Parameter(τ,c,k,Δt) # Simulate nstep = 10000 X = dsrun(x,p,nstep) plot_evolution_state_simple(X,p) ``` ### Unstable equilibrium Next we initialise at the unstable equilibrium point ($\theta=\pi,\Omega=0$) and iterate in order to observe whether computational errors cause disturbances which lead to instability. The unstable state can be sustained for reasonable time using our numerical method and a sufficiently small time step. ```julia # Initial conditions θ0 = π Ω0 = 0.0 t0 = 0.0 x = State(θ0,Ω0,t0) # Parameter τ = 1.0 c = 0.0 k = 0.0 Δt = 0.01 p = Parameter(τ,c,k,Δt) # Simulate nstep = 10000 X = dsrun(x,p,nstep) plot_evolution_state_simple(X,p) ``` ### Heteroclinics Now we add a small disturbance to the unstable equilibrium. We observe the state of the system traveling along the heteroclinics between the stable and unstable equilibrium points. ```julia # Initial conditions θ0 = π*(1-1e-6) Ω0 = 0.0 t0 = 0.0 x = State(θ0,Ω0,t0) # Parameter τ = 1.0 c = 0.0 k = 0.0 Δt = 0.01 p = Parameter(τ,c,k,Δt) # Simulate nstep = 10000 X = dsrun(x,p,nstep) plot_evolution_state_simple(X,p) ``` ## Pendulum with anchor point movement ```julia # Initial conditions θ0 = π/2 Ω0 = 0.0 t0 = 0.0 x = State(θ0,Ω0,t0) # Parameter τ = 1.0 c = 0.75 k = 1.0 Δt = 0.01 p = Parameter(τ,c,k,Δt) # Simulate nstep = 100000 X = dsrun(x,p,nstep) plot_evolution_state_simple(X,p) ```