# Transient Simulation

Transient simulation can be performed on systems with zero or greater degrees of freedom. For systems with zero degrees of freedom, kinematic simulation is used. For systems with more than zero degrees of freedom, the dynamic simulation is used.

## Kinematic Simulation

A model with zero degrees of freedom is defined as kinematic. The algebraic equations that define the constraints imposed by various joints and applied motions completely specify the motion of the system. The constraints and their time derivatives are used to compute the system displacement, velocity, and accelerations. Then, the equations of force balance are used to algebraically calculate the constraint reaction forces. The solution is therefore algebraic in nature.

To obtain accurate reaction forces, you must provide realistic mass, inertia, and center of gravity information for the bodies. Incorrect values for these parameters produce correct motion, but incorrect reaction forces. The solver does not warn you because these parameters do not directly affect motion results, which is the primary purpose of a kinematic simulation.

## Dynamic Simulation

Dynamic simulation refers to the numerical integration of the ordinary differential or differential algebraic equations. The algebraic equations result from constraints in your model. It is applicable to models with one or more degrees of freedom.

The dynamic simulation accounts for all the accelerations (linear, angular, centrifugal, and coriolis), forces, and constraints. In other words, it solves the equations of motion in their most general form, including nonlinear effects. This enables you to develop accurate system level simulations of complex mechanical systems.

MotionSolve supports three different equation formulations for dynamic simulations:
The ODE Formulation
The ODE (Ordinary Differential Equations) formulation supports both stiff and non-stiff integrators. MotionSolve transforms the DAE form equations of motion into ODE form equations using coordinate partitioning and then solves the resulting equation using an ODE integrator. Both stiff and non-stiff integrators are supported. The stiff integrators supported by this formulation are (a) VSTIFF and (b) MSTIFF. The integrator ABAM is used to integrate non-stiff solutions.
The Index-3 (I3) Formulation
The I3 formulation provides the DAE (Differential Algebraic Equations) form of the equations of motion to an integrator capable of solving this form of the equations of motion. DASPK is the only integrator in MotionSolve capable of solving the I3 form of the equations of motion. In the I3 formulation, the integrator does not monitor the integration local error in the velocity states. Consequently, I3 solutions typically tend to be very fast, though slightly inaccurate in velocities sometimes.
The Stabilized Index-1 (SI1) Formulation
The SI1 formulation provides a "stabilized" index-1 DAE form of the equations of motion to an integrator capable of solving this form of the equations of motion. DASPK is the only integrator in MotionSolve capable of solving the SI1 form of the equations of motion. In the SI1 formulation, the integrator monitors the integration local error in both the displacement and velocity states. Consequently, SI1 solutions typically tend to be very accurate. The typical speed of SI1 solutions, compared to I3 solutions, is somewhat slower.

For more detailed description of various integrators, see Parameters: Transient Solver in the XML Format Reference Guide.