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Participation Factor And Effective Mass

The modal mass for mode ii is defined as

m^i=φiTMφi{\widehat{m}}_{i} = {\boldsymbol{\varphi}_{i}}^{T}\mathbf{M}\boldsymbol{\varphi}_{i}

The direction information can be extracted using the participation factor. The participation factor for mode ii in the jj direction is given by

Γij=φiTMrjm^i\Gamma_{ij} = \frac{{\boldsymbol{\varphi}_{i}}^{T}\mathbf{M}\mathbf{r}_{j}}{{\widehat{m}}_{i}}

where the rj\mathbf{r}_{j} vector is a rigid body vector in the jj direction. The effective mass is similar but defined as

m~ij=(φiTMrj)2m^i=m^iΓij2{\widetilde{m}}_{ij} = \frac{\left( {\boldsymbol{\varphi}_{i}}^{T}\mathbf{M}\mathbf{r}_{j} \right)^{2}}{{\widehat{m}}_{i}} = {\widehat{m}}_{i}{\Gamma_{ij}}^{2}

The rigid body vectors are defined as

rx={10000010},ry={01000001},...r_{x} = \begin{Bmatrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ \vdots \\ \end{Bmatrix},r_{y} = \begin{Bmatrix} 0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \\ \vdots \\ \end{Bmatrix},...

So a rigid body vector for a rotation of θ\theta about global z can be defined as

rθ={cosθsinθ0000cosθsinθ}r_{\theta} = \begin{Bmatrix} \cos\theta \\ \sin\theta \\ 0 \\ 0 \\ 0 \\ 0 \\ \cos\theta \\ \sin\theta \\ \vdots \\ \end{Bmatrix}

The effective mass in a rotated axis system can be calculated from the participation factors and effective masses.

Γiθ=φiTMrθm^i=φiTMrxm^icosθ+φiTMrym^isinθ=Γixcosθ+Γiysinθ\begin{aligned}\Gamma_{i\theta} &= \frac{{\boldsymbol{\varphi}_{i}}^{T}\mathbf{M}\mathbf{r}_{\theta}}{{\widehat{m}}_{i}} \\ &= \frac{{\boldsymbol{\varphi}_{i}}^{T}\mathbf{M}\mathbf{r}_{x}}{{\widehat{m}}_{i}}\cos\theta + \frac{{\boldsymbol{\varphi}_{i}}^{T}\mathbf{M}\mathbf{r}_{y}}{{\widehat{m}}_{i}}\sin\theta \\ &= \Gamma_{ix}\cos\theta + \Gamma_{iy}\sin\theta\end{aligned}

So

m~iθ=m^Γiθ2=m^(Γixcosθ+Γiysinθ)2\begin{aligned}{\widetilde{m}}_{i\theta} &= \widehat{m}{\Gamma_{i\theta}}^{2} \\ &= \widehat{m}\left( \Gamma_{ix}\cos\theta + \Gamma_{iy}\sin\theta \right)^{2}\end{aligned}

The sum of the effective mass in any given direction over all the modes is the total unrestrained mass. Staring with the definition of effective mass

m~ij=(φiTMrj)2m^i{\widetilde{m}}_{ij} = \frac{\left( {\boldsymbol{\varphi}_{i}}^{T}\mathbf{M}\mathbf{r}_{j} \right)^{2}}{{\widehat{m}}_{i}}

The rigid body vector can be written as

rj=Φaj\mathbf{r}_{j} = \mathbf{\Phi}\mathbf{a}_{j}

So the term in the numerator of the effective mass becomes φiTMΦaj{\boldsymbol{\varphi}_{i}}^{T}\mathbf{M}\boldsymbol{\Phi}\mathbf{a}_{j} so

m~ij=(φiTMΦaj)2m^i=(m^iaij)2m^i=m^iaij2{\widetilde{m}}_{ij} = \frac{\left( {\boldsymbol{\varphi}_{i}}^{T}\mathbf{M}\boldsymbol{\Phi}\mathbf{a}_{j} \right)^{2}}{{\widehat{m}}_{i}} = \frac{\left( {\widehat{m}}_{i}\mathbf{a}i_{j} \right)^{2}}{{\widehat{m}}_{i}} = {\widehat{m}}_{i}{\mathbf{a}_{ij}}^{2}

Also the total mass rjTMrj=ajTΦTMΦaj{\mathbf{r}_{j}}^{T}\mathbf{M}\mathbf{r}_{j} = {\mathbf{a}_{j}}^{T}\boldsymbol{\Phi}^{T}\mathbf{M}\boldsymbol{\Phi}\mathbf{a}_{j} and ΦTMΦ=diag(m~i)\boldsymbol{\Phi}^{T}\mathbf{M}\boldsymbol{\Phi} = diag\left( {\widetilde{m}}_{i} \right) so

ajTΦTMΦaj=im~iaij2{\mathbf{a}_{j}}^{T}\mathbf{\Phi}^{T}\mathbf{M}\boldsymbol{\Phi}\mathbf{a}_{j} = \sum_{i}^{}{\widetilde{m}}_{i}{a_{ij}}^{2}

So the sum of the effective masses over all the modes is the total unrestrained mass.