Dynamical Simulation - Symmetric/Torque Model

Symmetric/Torque Model

The two types of symmetric objects that will simplify Euler's equations are “symmetric tops” and “symmetric spheres.” The first assumes one degree of symmetry, this makes two of the I terms equal. These objects, like cylinders and tops, can be expressed with one very simple equation and two slightly simpler equations. This does not do us much good, because with one more symmetry we can get a large jump in speed with almost no change in appearance. The symmetric sphere makes all of the I terms equal (the Moment of inertia scalar), which makes all of these equations simple:

• The N terms are applied torques about the principal axes
• The terms are angular velocities about the principal axes
• The I term is the scalar Moment of inertia:

where

• • V is the volume region of the object,
  • r is the distance from the axis of rotation,
  • m is mass,
  • v is volume,
  • ρ is the pointwise density function of the object,
  • x, y, z are the Cartesian coordinates.

These equations allow us to simulate the behavior of an object that can spin in a way very close to the method simulate motion without spin. This is a simple model but it is accurate enough to produce realistic output in real-time Dynamical simulations. It also allows a Physics engine to focus on the changing forces and torques rather than varying inertia.