Double elastic pendulum problem 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 double elastic pendulum.

The equations that will be integrated here are derived in this article.

The ordinary differential equation system being solved is

\begin{aligned} \begin{bmatrix} \ddot{r}_1 \\ \ddot{r}_2 \\ \ddot{\theta}_1 \\ \ddot{\theta}_2 \end{bmatrix} &= \begin{bmatrix} 1 & \dfrac{m_2\cos{\Delta}}{m_1+m_2} & 0 & -\dfrac{m_2r_2\sin{\Delta}}{m_1+m_2} \\ \cos{\Delta} & 1 & r_1\sin{\Delta} & 0 \\ 0 & \dfrac{m_2\sin{\Delta}}{(m_1+m_2)r_1} & 1 & \dfrac{m_2r_2\cos{\Delta}}{(m_1+m_2)r_1} \\ -\dfrac{\sin{\Delta}}{r_2} & 0 & \dfrac{r_1\cos{\Delta}}{r_2} & 1 \end{bmatrix}^{-1} \begin{bmatrix} r_1\dot{\theta}_1^2-g\sin{\theta_1} + \dfrac{m_2}{m_1+m_2}\left(r_2\dot{\theta}_2^2\cos{\Delta} + 2\dot{r}_2\dot{\theta}_2\sin{\Delta}\right) + \dfrac{Q_{r_1}-k_1(r_1-l_1)}{m_1+m_2}\\ r_2\dot{\theta}_2^2-g\sin{\theta_2} + r_1\dot{\theta}_1^2\cos{\Delta} - 2\dot{r}_1\dot{\theta}_1\sin{\Delta} + \dfrac{Q_{r_2}-k_2(r_2-l_2)}{m_2} \\ -\dfrac{2\dot{r}_1\dot{\theta}_1}{r_1} - \dfrac{g\cos{\theta_1}}{r_1} - \dfrac{m_2}{(m_1+m_2)r_1}\left[2\dot{r}_2\dot{\theta}_2\cos{\Delta} -r_2\dot{\theta}_2^2\sin{\Delta}\right] + \dfrac{Q_{\theta_1}}{(m_1+m_2)r_1^2} \\ -\dfrac{2\dot{r}_2\dot{\theta}_2}{r_2}- \dfrac{g\cos{\theta_2}}{r_2} - \dfrac{2\dot{r}_1\dot{\theta}_1\cos{\Delta}}{r_2} - \dfrac{r_1\dot{\theta}_1^2\sin{\Delta}}{r_2} + \dfrac{Q_{\theta_2}}{m_2r_2^2} \end{bmatrix}. \end{aligned}

Simulation parameter form.
Parameter	Value	Explanation
$g$ :		Gravitational acceleration in metres (m) per second (s) squared ( $\mathrm{m}\cdot \mathrm{s}^{-2}$ ).
$l_1$ :		Rest length (m) of pendulum 1.
$l_2$ :		Rest length (m) of pendulum 2.
$m_1$ :		Mass (kilograms or kg) of pendulum bob 1.
$m_2$ :		Mass (kg) of pendulum bob 2.
$k_1$ :		Spring coefficient for the first pendulum.
$k_2$ :		Spring coefficient for the second pendulum.
$b_1$ :		Linear dissipation coefficient for pendulum 1.
$b_2$ :		Linear dissipation coefficient for pendulum 2.
$c_1$ :		Quadratic dissipation coefficient for pendulum 1.
$c_2$ :		Quadratic dissipation coefficient for pendulum 2.
$t_f$ :		End time (s) for the simulation.
$r_{1,0}$ :		Initial value of $r_1$ (m).
$\dot{r}_{1,0}$ :		Initial value of $\dot{r}_1$ in $\mathrm{m}\cdot \mathrm{s}^{-1}$ .
$r_{2,0}$ :		Initial value of $r_2$ (m).
$\dot{r}_{2,0}$ :		Initial value of $\dot{r}_2$ in $\mathrm{m}\cdot \mathrm{s}^{-1}$ .
$\theta_{1,0}$ :		Initial value of $\theta_1$ in radians (r).
$\dot{\theta}_{1,0}$ :		Initial value of $\dot{\theta}_1$ in $\mathrm{r}\cdot \mathrm{s}^{-1}$ .
$\theta_{2,0}$ :		Initial value of $\theta_2$ in r.
$\dot{\theta}_{2,0}$ :		Initial value of $\dot{\theta}_2$ in $\mathrm{r}\cdot \mathrm{s}^{-1}$ .
$\epsilon$ :		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.
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Double elastic pendulum problem solver

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