This is an extension from the Driven Pendulum, which is an extension from the Simple Pendulum. In the simple pendulum, I demonstrated how to solve the equations of motion using the Parker-Sochacki method and demonstrated that a higher order could handle much larger time steps than necessary in animation. I then showed how to apply Parker-Sochacki to a Driven Pendulum and what the simulation looks like. Now I am adding damping.
Like last time, I wrote the equations in Latex and have an image of the pdf file here:
I didn’t mention this in the previous post. One price of using Parker-Sochacki is having to solve for more equations. For all three types of pendulums there are only two first order Ordinary Differential Equations to solve using other numerical methods. For the Simple Pendulum, Parker-Sochacki converts the system to having 4 First Order Differential Equations. For the Driven Pendulum and the Damped Driven Pendulum, Parker-Sochacki converts the system to having 6 First Order Differential Equations. There are more equations but the result is being able to take larger time steps.
Also, a note on higher time steps. Rigid Body simulations I have done like the projectile with gravity and drag or these pendulums appear to work fine at a lower order. When it comes to simulations like this in animation, it depends on what the problem is to see if Parker-Sochacki will be a benefit. Most likely there will be at least a little speed up.
Simulations like the 2-body problem are where I see an advantage. This is where you have the norm in the denominator, which I have seen in many simulations.
Here is a video of the simulation rendered in Houdini. Notice that the pendulum does not swing as much later in the simulation as it does in the beginning.
This shows that Parker-Sochacki is applicable to forces such as drive, damping, gravity, and drag.