Elastic pendulum solver

This webpage uses the Runge-Kutta-Fehlberg fourth-order method with fifth-order error checking (RKF45) to approximate the solution to the problem of the elastic pendulum

\begin{aligned} \dfrac{d^{2}r}{dt^2} &= r \dot{\theta}^2 - \dfrac{\mu_1}{M_1} g\sin{\theta} + \dfrac{Q_r-k(r-l_0)}{M_1} \\ \dfrac{d^{2} \theta}{dt^2} &= -\dfrac{2\dot{r}\dot{\theta}}{r} - \dfrac{\mu_1 g\cos{\theta}}{M_1 r} + \dfrac{Q_{\theta}}{M_1r^2}. \end{aligned}

Where

$r$ is the length of the pendulum rod (in metres).
$\theta$ is the angle of the pendulum relative to the positive $x$ -axis.
$g$ is the acceleration due to gravity (in metres per second squared).
$l_0$ is the rest length of the pendulum.
$Q_{r} = -(b_b+c_b v_b)\dot{r} - (b_r + c_r v_r)\dfrac{\dot{r}}{4}$ .
$v_b = \sqrt{\dot{r}^2+r^2\dot{\theta}^2}$ and $v_r = \dfrac{1}{2}v_b$ .
$Q_{\theta} = -(b_b+c_b v_b)r^2\dot{\theta} - (b_r + c_r v_r)\dfrac{r^2\dot{\theta}}{4}$ .

In this calculation, we assumed that the dissipative forces on the rod could be calculated using a centre-of-mass approach.

Simulation parameter form.
Parameter	Value	Explanation
$g$ :		Acceleration due to gravity in metres (m) per second (s) squared ( $\mathrm{m}\cdot \mathrm{s}^{-2}$ ).
$l_0$ :		Rest length (m) of pendulum rod.
$k$ :		Spring coefficient of pendulum.
$m_{b}$ :		Mass (kilograms or kg) of pendulum bob.
$m_{r}$ :		Mass (kilograms or kg) of pendulum rod.
$b_{b}$ :		Linear dissipation coefficient for the pendulum bob.
$b_{r}$ :		Linear dissipation coefficient for the pendulum rod.
$c_{b}$ :		Quadratic dissipation coefficient for the pendulum bob.
$c_{r}$ :		Quadratic dissipation coefficient for the pendulum rod.
$t_f$ :		End time (s) for the simulation.
$r_0$ :		Initial value of $r$ (m).
$\dot{r}_{0}$ :		Initial value of $\dot{r}$ ( $\mathrm{m}\cdot \mathrm{s}^{-1}$ ).
$\theta_{0}$ :		Initial value of $\theta$ in radians (r).
$\dot{\theta}_{0}$ :		Initial value of $\dot{\theta}$ ( $\mathrm{r}\cdot \mathrm{s}^{-1}$ ).
$\epsilon$ :		Error tolerance.
$\mathrm{Tol}_{\mathrm{type}}$ :		Tolerance type, can be either absolute (0) or relative (1).
$h_{\mathrm{Initial}}$ :		Initial step size.
$h_{\mathrm{Min}}$ :		Minimum allowed step size.
$\Delta t$ :		Time increment for skipping ahead in animation.
$t_1$ :		Time you want to skip ahead to in animation when you press the skip button.
$\mathrm{Width}$ :		Width (in px) of Plotly windows used for plotting and animation below.
$\mathrm{Height}$ :		Height (in px) of Plotly windows used for plotting and animation below.
$t_\mathrm{Scale}$ :		Proportion of animation time passed per real time. $t_{\mathrm{Scale}}=1.0$ means animation and real time match. $t_{\mathrm{Scale}}<1.0$ means the animation is going more slowly than real time. $t_{\mathrm{Scale}}>1.0$ means it is going more rapidly.

© Brenton Horne. I do not claim to be infallible, so feel free to open issues or pull requests on GitHub if you notice flaws or things that can be improved.
I favour Oxford spelling, so if you notice I generally use different spelling to Australian spelling, that is probably why.
Homepage. Source code: https://github.com/fusion809/fusion809.github.io.
Website built with Franklin.jl.

Elastic pendulum solver

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