Equation 5–25 (open loop) and Equation 5–39 (closed loop) provide a relation between generalized accelerations and generalized velocities {q}.

Equation 5–7 provides a relation between generalized velocities {q} and the time derivatives of the geometry variables

These two sets of equations form a system of first order explicit ordinary differential equations (ODE).

(5–46)

This system is integrated using the explicit Runge-Kutta method RK4.

RK4 Method

The fourth order method is based on four estimations. Given an initial value y at time value t, and a time step value dt, the following four estimations are formed:

(5–47)

(5–48)

(5–49)

(5–50)

A fourth order approximation of y(t+dt) is given by:

(5–51)

Adaptive Time Stepping (for explicit time integration)

Time step dt must be chosen carefully for the integration of the ODEs to ensure that it is stable (that is, not becoming exponentially large), and accurate (that is, the difference between the approximation of the solution and the exact solution is controlled).

RK4 is conditionally stable, meaning that stability can be guaranteed if the time step is small enough. While the algorithm is accurate when it is stable, the time step chosen must be large enough to maintain computational efficiency.

For both integration schemes, quantifying the amount of kinetic energy contained in the highest order term of the polynomial approximation can give a good indication of whether the time step should be reduced or increased.

If the energy in the high order term is too large, it is likely that the approximation is inaccurate, and the time step should smaller.

If this energy is significant and controlled, the time step can be accepted, but the time step used will be smaller.

If the energy is low, then the next time step can be increased.

Rigid body systems usually have relatively slow motion, but the following factors can lead to smaller time steps:

These factors imply that the optimal time step varies with the system velocities and configuration, and thus cannot be determined before running the solution. As a consequence, automatic time stepping generally should not be turned off.

When automatic time stepping is used, the energy balance of the system is maintained within the tolerance that is requested. Note that impacts and shock can be non-conservative, and will affect the energy balance. This loss during impact is detailed in Contact and Stops.