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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:

Mu¨+Cu˙+Ku=psin⁡(ωt)\mathbf{M}\ddot{\mathbf{u}} + \mathbf{C}\dot{\mathbf{u}} + \mathbf{Ku} = \mathbf{p}\sin(\omega t)

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

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

u=Φqu˙=Φq˙u¨=Φq¨\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 q,q˙,q¨\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}} are the displacement, velocity, and acceleration in modal (generalized) coordinates, for the mm modes analysed.

Substituting these in the original equation gives

MΦq¨+CΦq˙+KΦq=psin⁡(ωt)\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

ΦTMΦq¨+ΦTCΦq˙+ΦTKΦq= ΦTpsin⁡(ωt)\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 mmuncoupled dynamic equations of motion as shown below.

φiTMφiq¨i+φiTCφiq˙i+φiTKφiqi=φiTpsin⁡(ωit){\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

m^i=φiTMφik^i=φiTKφic^i=φiTCφip^i=φiTp\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

m^iq¨i+c^q˙i+k^qi=p^isin⁡(ωit){\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 μ\mu of the displacement for mode ii in complex number notation is

μi=[(1−(ωωi)2+i2ξ(ωωi))]−1=μi,R−iμi,I\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

μR=AA2+B2,μI=BA2+B2\mu_{\mathfrak{R}} = \frac{A}{A^2+B^2},\quad \mu_{\mathfrak{I}} = \frac{B}{A^2+B^2}
A=1−(ωωi)2,B=2ξ(ωωi)A = 1 - \left( \frac{\omega}{\omega_{i}} \right)^{2},\quad B = 2\xi\left( \frac{\omega}{\omega_{i}} \right)

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

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

qi=μp^ik^i=μp^im^iωi2q˙i=qiω=μp^iωk^i=μp^iωm^iωi2q¨i=qiω2=μp^iω2k^i=μp^iω2m^iωi2\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

u=∑i=1mφiqiu˙=∑i=1mφiq˙iu¨=∑i=1mφiq¨i\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.