Theory

This is a simple double pendulum simulation experiment. Key pendulum parameters are outlined below. The simulation numerically integrates the differential equations of motion for α₁ and α₂ in order to determine the pendulum mass positions θ₁ and θ₂. These equations can be derived from the Euler-Lagrange differential equations for θ₁ and θ₂, which can be formulated analytically from the double pendulum model as described here. The pendulum is rendered based on the calculated pendulum positions.

m₁: upper pendulum mass
m₂: lower pendulum mass
l₁: length of the link connecting m₁ to the pendulum anchor
l₂: length of the link connecting m₂ to m₁
θ₁: the angular position of m₁ about the pendulum anchor
θ₂: the angular position of m₂ about m₁
ω₁: the angular velocity of m₁ about the pendulum anchor
ω₂: the angular velocity of m₂ about m₁
α₁: the angular acceleration of m₁ about the pendulum anchor
α₂: the angular acceleration of m₂ about m₁

Note: The default integration scheme generally tends to loose energy over time; however, certain pendulum configurations will cause the opposite effect to occur.

For this simulation, the equations for α₁ and α₂ were sourced from here.