# Dynamic Analysis

The discrete form of the equation of motion given in Equation of Motion for Translational Velocities represents a system of linear differential equations of second order and, in principal the solution to the equations can be obtained by standard procedures for the solution of differential equations with constant coefficients. However, in practical finite element analysis, a few effective methods are used. The procedures are generally divided into two methods of solution: direct integration method and mode superposition. Although the two techniques may at first look to be quite different, in fact they are closely related, and the choice for one method or the other is determined only by their numerical effectiveness.

In direct integration the equations of motion are directly integrated using a numerical step-by-step procedure. In this method no transformation of the equations into another basis is carried out. The dynamic equilibrium equation written at discrete time points includes the effect of inertia and damping forces. The variation of displacements, velocities and accelerations is assumed with each time interval $\text{\Delta}t$ . As the solution is obtained by a step-by-step procedure, the diverse system nonlinearities as geometric, material, contact and large deformation nonlinearity are taken into account in a natural way even if the resolution in each step remains linear.

The mode superposition method generally consists of transforming the equilibrium equation into the generalized displacement modes. An eigen value problem is resolved. The eigen vectors are the free vibration mode shapes of the finite element assemblage. The superposition of the response of each eigen vector leads to the global response. As the method is based on the superposition rule, the linear response of dynamically loaded of the structure is generally developed.

In the following, first the resolution procedure in direct integration method when using an explicit time discretization scheme is described. Then, the procedures of modal analysis are briefly presented. The implicit method will be detailed in Radioss Parallelization.