One Pendulum Less A few weeks ago we looked at how to balance an inverted pendulum as part of a 3 part series on balancing inverted pendulums. Part 2 is still being worked on, but here’s part 3, an exploration of how shaking an inverted pendulum up and down really fast can make it stable.…
Springs, Except More Fun In any introductory physics or differential equation’s class we are introduced to the simple spring mass system. Because of Hooks law it’s nice, linear, and easy to analyze. But not all springs follow Hooks Law. Some, like the duffing spring we’ll analyze in this post, are nonlinear and chaotic. What’s a…
Everything Orbits About Something In our solar system, Moons orbit around planets, which themselves orbit around the sun. Well, actual they orbit about the system barycenter, the combined center of mass between the two bodies. But can we get an object to orbit around a region without a barycenter? Yes! (Kinda) There are actually a…
One Pendulum Less We’ve looked at the chaotic dynamics of double pendulums, but let’s take a moment to step back at the double pendulum’s simpler sibling, the single pendulum. This is the first post in a 3-part series looking at balancing an inverted pendulum Balancing an inverted pendulum – Part 1 Balancing an inverted pendulum…
Intro/Motivation One of the most common tasks in dynamics and controls is how do we get an object from point A to point B. Sometimes, like in the brachistochrone problem, we get lucky and there is a nice analytical solution. Unfortunately, in most situations, there is no closed form answer, and we need to turn…
Recap of Lagrange Points In the circular restricted three body problem, there are a set of 5 points that if we place our spacecraft there, it’ll never move relative to the two bodies. These are called the Lagrange points and are the only equilibrium points for the system. Whenever we find an equilibrium point in…
Jacobi and Lagrange So far we have explored the Jacobi Integral and the Lagrange points. Now, we are ready to combine the two into one of the most useful concepts in the Circular restricted 3-Body Problem (CR3BP), forbidden regions. You’ve seen them in my visualizations before, but now we’ll finally go over what they are.…
Balancing a Marble Imagine you had a marble and you wanted to put it at the point between the Earth and the Moon where the force of gravity was equal from both. The two forces would cancel out, and the marble would remain in that location forever. If the sun and the earth were not…
Recap In the last two posts on what is the 3-body problem and the dynamics of the three body problem I’ve shown you the following GIF with a spacecrafts trajectory in the earth moon system. The trajectory is bouncing around what I’ve labeled a ‘forbidden region’. In this post we’ll go over the number behind…
