Picture This Humans are a visual species, we look for patterns in life and we are great at finding them when they are visual. We can tell two people apart just by looking at their faces. Sometime’s we’re too good at seeing patterns and we see faces in a piece of burnt toast. If we’re…
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…
Lissajous curves From orbits around Lagrange Points, to double pendulums, we often run into a family of loopy, beautiful, curves. These are called Lissajous curves, and describe complex harmonic motion. In layman terms, Lissajous curves appear when an object’s motion’s have two independent frequencies. Today, we’ll explore another system that produces Lissajous curves, a double…
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.…
