# # Harmonic Analysis

Harmonic analysis is used to calculate the elastic structure responses to harmonic (sinusoidally varying) loads at steady state. This is done using modal superposition.

The dynamic equation of motion is:

Where

From the mode shape results of a modal dynamic analysis, the nodal displacements, velocities, and accelerations can be expressed as

where

Substituting these in the original equation gives

Pre-multiplying each term in this equation by the transpose of the mode shape gives

According to the orthogonality relationship of the mode shapes to the
mass matrix and the stiffness matrix and also assuming the mode shapes
are also orthogonal to the damping matrix (e.g. Rayleigh damping), this
equation can be replaced by a set of

Setting

Then the uncoupled equations can be expressed in a general form as follows

where all the terms are scalars. Solving this equation is equivalent to solving a single degree of freedom problem.

For the single degree of freedom problem subjected to harmonic load, the
dynamic magnification factors

where

and

The maximum displacement, velocity & acceleration of mode

Substituting gives the maximum actual nodal displacements, velocities & accelerations at the steady state of the forced vibration as

After obtaining the maximum nodal displacements, the element forces and moments etc can be calculated as in static analysis.