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 represents the spatial distribution of load and the time variation.

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

where are the displacement, velocity, and acceleration in modal (generalized) coordinates, for the modes analysed.

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 uncoupled dynamic equations of motion as shown below.

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 of the displacement for mode in complex number notation is

where

and is the natural frequency of mode .

The maximum displacement, velocity & acceleration of mode in the modal coordinates are

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.