Swinging Sticks Mathematics

Double Pendulums

How Swinging Sticks Works - Double Pendulum
  • x = Horizontal Coordinate
  • y = Vertical Coordinate
  • θ = Angle of Pendulum
  • L = Length
  • m = mass
  • g = gravity (9.81 m/s2)

A double pendulum is a pendulum that has another one attached to the end of it.

For the most part, the motion is seemingly chaotic however there is actually mathematic beauty behind the Swinging Sticks

The Swinging Sticks is considered a simple or compound double pendulum because it operates solely in a 2D plain.

Swinging Sticks Physics

See Swinging Sticks Physics

The Swinging Sticks equations for double pendulums are solved using the Runge-Kutta methods of numerical analysis. This technique was developed in the early 1900's by mathematicians C. Runge and M.W. Kutta

Calculate all the following equations for

x1, x2, y1, y2

1) Calculate the position of masses

x = L sin θ

2) Next take the derivative to find the velocity

x1' = θ1' L1 cos θ1

x2' = x1' + θ2' L2 cos θ2

3) Take another derivative to find the acceleration

x1'' = −θ1'2 L1 sin θ1 + θ1'' L1 cos θ1

x2'' = x1'' − θ2'2 L2 sin θ2 + θ2'' L2 cos θ2

0 = vertical downwards

Counter-clockwise = positive

Click here to learn more about the physics behind the Swinging Sticks Kinetic Energy Sculpture.