# 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:

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

Where $\mathbf{p}$ represents the spatial distribution of load and $\omega$ the time variation.

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

\begin{aligned} {\mathbf{u} = \mathbf{\Phi q}}\\{\dot{\mathbf{u}} = \mathbf{\Phi}\dot{\mathbf{q}} }\\{\ddot{\mathbf{u}} = \mathbf{\Phi}\ddot{\mathbf{q}}} \end{aligned}

where $\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}}$ are the displacement, velocity, and acceleration in modal (generalized) coordinates, for the $m$ modes analysed.

Substituting these in the original equation gives

$\mathbf{M\Phi}\ddot{\mathbf{q}} + \mathbf{C\Phi}\dot{\mathbf{q}} + \mathbf{K\Phi q} = \mathbf{p}\sin(\omega t)$

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

$\mathbf{\Phi}^{T}\mathbf{M\Phi}\ddot{\mathbf{q}} + \mathbf{\Phi}^{T}\mathbf{C\Phi}\dot{\mathbf{q}} + \mathbf{\Phi}^{T}\mathbf{K\Phi q} = \ \mathbf{\Phi}^{T}\mathbf{p}\sin(\omega t)$

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 $m$uncoupled dynamic equations of motion as shown below.

${\boldsymbol{\varphi}_{i}}^{T}\mathbf{M}\boldsymbol{\varphi}_{i}{\ddot{q}}_{i} + {\boldsymbol{\varphi}_{i}}^{T}\mathbf{C}\boldsymbol{\varphi}_{i}{\dot{q}}_{i} + {\boldsymbol{\varphi}_{i}}^{T}\mathbf{K}\boldsymbol{\varphi}_{i}q_{i} = {\boldsymbol{\varphi}_{i}}^{T}\mathbf{p}\sin\left( \omega_{i}t \right)$

Setting

\begin{aligned}{\widehat{m}}_{i} &= {\boldsymbol{\varphi}_{i}}^{T}\mathbf{M}\boldsymbol{\varphi}_{i}\\ {\widehat{k}}_{i} &= {\boldsymbol{\varphi}_{i}}^{T}\mathbf{K}\boldsymbol{\varphi}_{i}\\ {\widehat{c}}_{i} &= {\boldsymbol{\varphi}_{i}}^{T}\mathbf{C}\boldsymbol{\varphi}_{i}\\ {\widehat{p}}_{i} &= {\boldsymbol{\varphi}_{i}}^{T}\mathbf{p}\end{aligned}

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

${\widehat{m}}_{i}{\ddot{q}}_{i} + \widehat{c}{\dot{q}}_{i} + \widehat{k}q_{i} = {\widehat{p}}_{i}\sin\left( \omega_{i}t \right)$

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 $\mu$ of the displacement for mode $i$ in complex number notation is

$\mu_{i} = \left\lbrack \left( 1 - \left( \frac{\omega}{\omega_{i}} \right)^{2} + i2\xi\left( \frac{\omega}{\omega_{i}} \right) \right) \right\rbrack^{- 1} = \mu_{i,\mathfrak{R}} - i\mu_{i,\mathfrak{I}}$

where

$\mu_{\mathfrak{R}} = \frac{A}{A^2+B^2},\quad \mu_{\mathfrak{I}} = \frac{B}{A^2+B^2}$
$A = 1 - \left( \frac{\omega}{\omega_{i}} \right)^{2},\quad B = 2\xi\left( \frac{\omega}{\omega_{i}} \right)$

and $\omega_{i}$ is the natural frequency of mode $i$.

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

\begin{aligned} q_{i} &= \mu\frac{{\widehat{p}}_{i}}{{\widehat{k}}_{i}} &= \mu\frac{{\widehat{p}}_{i}}{{\widehat{m}}_{i}{\omega_{i}}^{2}} \\ {\dot{q}}_{i} = q_{i}\omega &= \mu\frac{{\widehat{p}}_{i}\omega}{{\widehat{k}}_{i}} &= \mu\frac{{\widehat{p}}_{i}\omega}{{\widehat{m}}_{i}{\omega_{i}}^{2}} \\ {\ddot{q}}_{i} = q_{i}\omega^{2} &= \mu\frac{{\widehat{p}}_{i}\omega^{2}}{{\widehat{k}}_{i}} &= \mu\frac{{\widehat{p}}_{i}\omega^{2}}{{\widehat{m}}_{i}{\omega_{i}}^{2}} \end{aligned}

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

$\mathbf{u} = \sum_{i = 1}^{m}\boldsymbol{\varphi}_{i}\mathbf{q}_{i}\qquad\dot{\mathbf{u}} = \sum_{i = 1}^{m}\boldsymbol{\varphi}_{i}{\dot{\mathbf{q}}}_{i}\qquad\ddot{\mathbf{u}} = \sum_{i = 1}^{m}\boldsymbol{\varphi}_{i}{\ddot{\mathbf{q}}}_{i}$

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