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