Transient Loads are only present in the system during a limited time window. Examples for transient excitations are:

  • Explosions
  • Shock or crash
  • Impact
  • Hammers
  • Jumping

When analytical methods become infeasible, time step procedures are used to determine the solution of a system under a transient load. The basis of all procedures is the concept to determine the temporal progression of the displacement w, the velocity \dot{w} and the acceleration \ddot{w} step-wise. The time domain is discretized into finite time intervals and the equations of motion are then only fulfilled in an approximate manner. Thus, only discrete points in time t_i are considered and, usually, a time increment \Delta t is chosen. The values at the time step t_{i+1} = t_i + \Delta t are determined from the values at time step t_i and possibly t_{i-1}, t_{i-2}, \ldots.


Numerical time step procedures

Time step methods can be used to solve the general equation of motion for displacement \mathbf{u}, velocity \dot{\mathbf{u}}, and acceleration \ddot{\mathbf{u}}:

(1) \begin{equation} \mathbf{M} \ddot{\mathbf{u}} (t) + \mathbf{C} \dot{\mathbf{u}} (t) + \mathbf{K} \mathbf{u} (t) = \mathbf{f} (t). \label{eq:MDOF_eq_motion_general} \end{equation}

All methods can be classified as explicit or implicit methods. An explicit method calculates the response values only on the basis of the known values of the preceding time step. An implicit method uses the preceding time step as well as the following (predicted) time step to obtain a solution for the current time step.

The choice of the time discretization has a large influence on the quality of the solution. Choosing a too large time step results in errors in the solution, as wave motion can not be depicted properly. Also numerical errors can introduce artificial (positive or negative) damping, which must be considered when using a time step method.

Examples for time step methods