# Periodic Load Analysis

GSA periodic load analysis is to calculate the maximum elastic structure responses to generic periodic loads at steady state. Modal superposition method is used in GSA periodic load analysis.

The dynamic equation of motion subjected to periodic loads is

$\mathbf{M}\ddot{\mathbf{u}} + \mathbf{C}\dot{\mathbf{u}} + \mathbf{Ku} = \mathbf{p}f(t)$

Where $f(t)$ is a harmonic load function. Using a Fourier Series, the periodic function of time can be expressed as a number of sine functions

$f(t) = \sum_{h = 1}^{H}{r_{h}\sin\left( \frac{2\pi h}{T}t \right)}$

where $r_{h}$ are the Fourier coefficients (or dynamic load factor) defined by the user and $T$ is the period of the periodic load frequency and $H$ is the number of Fourier (harmonic) terms to be considered.

Substituting in the first equation we can rewrite as a number of dynamic equations of motion subjected to harmonic loads:

$\mathbf{M}\ddot{\mathbf{u}} + \mathbf{C}\dot{\mathbf{u}} + \mathbf{Ku} = \mathbf{p}r_{h}\sin\left( \frac{2\pi h}{T}t \right)$

The maximum responses of this can be solved using harmonic analysis for each of the harmonic loads $(h = 1,2,\ldots)$ then the maximum responses from the periodic loads can be calculated using square root sum of the squares (SRSS)

${R=\sqrt{\sum_{h = 1}^{H}{{R}_{h,max}}^2}}$