Double 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 double pendulum. See this article for the derivation of the equations of motion.

Figure 1: Diagram of the double pendulum.

The equations being solved in this webpage are

\begin{aligned} &\ddot{\theta}_1 = \dfrac{-1}{\left(\dfrac{m_{1r}}{12} + m_{1b}+m_{2b}\right)l_1} \left[m_{2b}l_2 ( \ddot{\theta}_2\cos{(\theta_1-\theta_2)} +\dot{\theta}_2^2\sin{(\theta_1-\theta_2)}) + g \cos{\theta_1}\left(\dfrac{m_{1r}}{2} +m_{2r} +m_{1b} + m_{2b}\right) -(b_{1b} + c_{1b} l_1 \dot{\theta}_1)l_1 \dot{\theta}_1 \right.\\ &\left.+\left(b_{2b}+c_{2b}\sqrt{l_1^2 \dot{\theta}_1^2 + l_2^2 \dot{\theta}_2^2 +2l_1 l_2\dot{\theta}_1 \dot{\theta}_2 \cos{(\theta_1-\theta_2)}}\right)(l_1 \dot{\theta}_1 + l_2 \dot{\theta}_2 \cos{(\theta_1-\theta_2)}) +\left(b_{1r} + \dfrac{c_{1r}l_1 \dot{\theta}_1}{2}\right) \dfrac{l_1 \dot{\theta}_1}{4} \right.\\ &\left.+\left(b_{2r} + c_{2r}\sqrt{l_1^2 \dot{\theta}_1^2 + \dfrac{l_2^2 \dot{\theta}_2^2}{4} + l_2 \dot{\theta}_1 \dot{\theta}_2 \cos{(\theta_1 -\theta_2)}}\right)\left(l_1 \dot{\theta}_1 + \dfrac{l_2\dot{\theta}_2 \cos{\left(\theta_1 - \theta_2\right)}}{2}\right)\right]\\ &\ddot{\theta}_2 = \dfrac{1}{\left(\dfrac{m_{2r}}{12} + m_{2b}\right)l_2 - \dfrac{m_{2b}^2l_2\cos^2{(\theta_1-\theta_2)}}{\left(\dfrac{m_{1r}}{12} + m_{1b}+m_{2b}\right)}}\left[\dfrac{m_{2b}\cos{(\theta_1-\theta_2)}}{\left(\dfrac{m_{1r}}{12} + m_{1b}+m_{2b}\right)}\left[m_{2b}l_2\dot{\theta}_2^2\sin{(\theta_1-\theta_2)} + g \cos{\theta_1}\left(\dfrac{m_{1r}}{2} +m_{2r} +m_{1b} + m_{2b}\right) -(b_{1b} + c_{1b} l_1 \dot{\theta}_1)l_1 \dot{\theta}_1 \right.\right.\\ &\quad\left.\left.+\left(b_{2b}+c_{2b}\sqrt{l_1^2 \dot{\theta}_1^2 + l_2^2 \dot{\theta}_2^2 +2l_1 l_2\dot{\theta}_1 \dot{\theta}_2 \cos{(\theta_1-\theta_2)}}\right)(l_1 \dot{\theta}_1 + l_2 \dot{\theta}_2 \cos{(\theta_1-\theta_2)})+\left(b_{1r} + \dfrac{c_{1r}l_1 \dot{\theta}_1}{2}\right) \dfrac{l_1 \dot{\theta}_1}{4} +\left(b_{2r} + c_{2r}\sqrt{l_1^2 \dot{\theta}_1^2 + \dfrac{l_2^2 \dot{\theta}_2^2}{4} + l_2 \dot{\theta}_1 \dot{\theta}_2 \cos{(\theta_1 -\theta_2)}}\right)\left(l_1 \dot{\theta}_1 \right.\right.\right. \\ &\quad\left.\left.\left.+ \dfrac{l_2\dot{\theta}_2 \cos{\left(\theta_1 - \theta_2\right)}}{2}\right)\right] + m_{2b}(l_1\dot{\theta}_1^2\sin{(\theta_1-\theta_2)}-g\cos{\theta_2}) -\dfrac{1}{4}\left(b_{2r} + c_{2r}\sqrt{l_1^2 \dot{\theta}_1^2 + \dfrac{l_2^2 \dot{\theta}_2^2}{4} + l_1 l_2 \dot{\theta}_1 \dot{\theta}_2 \cos{(\theta_1 -\theta_2)}}\right)(2l_1 \dot{\theta}_1 \cos{(\theta_1-\theta_2)}+ l_2 \dot{\theta}_2)\right.\\ &\quad\left.-\left(b_{2b}+c_{2b}\sqrt{l_1^2 \dot{\theta}_1^2 + l_2^2 \dot{\theta}_2^2 +2l_1l_2\dot{\theta}_1 \dot{\theta}_2 \cos{(\theta_1-\theta_2)}}\right)(l_1 \dot{\theta}_1 \cos{(\theta_1-\theta_2)} + l_2 \dot{\theta}_2)\right]. \end{aligned}

Simulation parameter form.
Parameter	Value	Explanation
$g$ :		Gravitational acceleration (in $\mathrm{m}\cdot \mathrm{s}^{-2}$ ).
$l_1$ :		Length of pendulum 1 in metres.
$l_2$ :		Length of pendulum 2 in metres.
$m_{1b}$ :		Mass of pendulum bob 1 in kilograms.
$m_{1r}$ :		Mass of pendulum rod 1 in kilograms.
$m_{2b}$ :		Mass of pendulum bob 2 in kilograms.
$m_{2r}$ :		Mass of pendulum rod 2 in kilograms.
$b_{1b}$ :		Linear dissipation coefficient of pendulum bob 1.
$b_{1r}$ :		Linear dissipation coefficient of pendulum rod 1.
$c_{1b}$ :		Quadratic dissipation coefficient of pendulum bob 1.
$c_{1r}$ :		Quadratic dissipation coefficient of pendulum rod 1.
$b_{2b}$ :		Linear dissipation coefficient for pendulum bob 2.
$b_{2r}$ :		Linear dissipation coefficient for pendulum rod 2.
$c_{2b}$ :		Quadratic dissipation coefficient for pendulum bob 2.
$c_{2r}$ :		Quadratic dissipation coefficient for pendulum rod 2.
$t_f$ :		End time for the simulation in seconds.
$\theta_{1,0}$ :		Initial value of $\theta_1$ in radians.
$\dot{\theta}_{1,0}$ :		Initial value of $\dot{\theta}_1$ in radians per second.
$\theta_{2,0}$ :		Initial value of $\theta_2$ in radians.
$\dot{\theta}_{2,0}$ :		Initial value of $\dot{\theta}_2$ in radians per second.
$\epsilon$ :		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.

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