# Static Simulation

Static simulation is useful for finding the equilibrium configurations for models having one or more degrees of freedom. The equilibrium is defined as that configuration where all forces and moments acting on all parts in the model at rest equal zero. MotionSolve provides two different algorithms for static simulation.

## Maximum Kinetic Energy Attrition Method (MKEAM)

This method, outlined below, is based on numerical integration of a special dynamic formulation of the static problem.
1. The model is formulated as a dynamics problem from which all damping has been removed. The result is a conservative system whose total energy, defined as the sum of kinetic and potential energies, remains invariant with time.
2. Numerical integration is started. As integration, the kinetic energy of the system is monitored. When a peak (maximum) is detected, integration stops and backtracks as necessary to locate the peak in time, within some precision. Since the system is conservative, this instant also corresponds to a valley (minimum) for potential energy.
3. At this point, all velocities and accelerations in the model are set to zero, leading to zero kinetic energy. Then, the integration is restarted.
4. Steps 1-3 constitute one iteration. If the peak is located perfectly, then the model will already be at the equilibrium configuration. However, due to the discrete nature of the integrator, this is usually not the case. Thus, it is necessary to repeat steps 1-3 until the process converges as defined by three convergence parameters:
• Maximum Kinetic Energy Tolerance (Max K.E. Tol.) - Denotes the change in maximum kinetic energy from one iteration to the next.
• Maximum Delta q Tolerance (Max Delta q Tol.) - Denotes the maximum change in generalized coordinates from one iteration to the next.
• Maximum Iterations (Max Iterations) - Denotes the maximum number of iterations before the simulation stops.
The MKEAM method has following benefits:
1. It only finds the stable equilibriums.
2. It is suitable for problems where the equilibrium configuration is far from the model configuration. Such situations are problematic for the Newton-Raphson algorithm employed in the Force Imbalance Method (FIM). A simple example of such a model is a spring loaded pendulum that must make several complete revolutions to reach the equilibrium configuration.

## Force Imbalance Method (FIM)

This method sets all the velocity and acceleration terms in the equations of motion to zero to obtain a system of nonlinear algebraic equations. The generalized coordinates and constraint forces are the unknowns. This system is then solved using a variant of Newton-Raphson to find the equilibrium configuration.

The FIM method makes no distinction among equilibriums based on stability. In other words, it is equally capable of finding both stable and unstable equilibriums.

The FIM method has difficulty in cases where the equilibrium configuration is far from the model configuration and for models dominated by discontinuous effects such as contact. In that sense, the MKEAM and FIM methods are complementary.

There are two variations of FIM available in MotionSolve:
1. FIM_S is a modified implementation which supports all MotionSolve elements except Force_Contact.
2. FIM_D is a time integration based approach to a quasi-static solution. It is not applicable for a purely static solution. In addition to the parameters specified in the Param_Static element, FIM_D also uses the parameters specified in the Param_Transient element to control the DAE integration. FIM_D defaults to FIM_S when a pure static solution is required.