# Flexible-Link Multibody System Eigenvalue Analysis Parameterized with Respect to Rigid-Body Motion
## 3 Polynomial Representation of the Eigenpairs of Flexible-Link Multibody Systems
### Eigensolution Expansion
They play the same role of the multi-index \(\sigma\) in Equation (19), i.e., they are the multi-indices related to the system parameters \(\mathbf{x}\), and include \(p\) non-negative integer entries:  
\[\alpha_{a}=(\alpha_{a_{1}},\alpha_{a_{2}}\ldots\alpha_{a_{p}}), \quad a=1,\ldots,4\] \[\beta_{b}=(\beta_{b_{1}},\beta_{b_{2}}\ldots\beta_{b_{p}}), \quad b=1,\ldots,3 \tag{23}\] \[\gamma_{c}=(\gamma_{c_{1}},\gamma_{c_{2}}\ldots\gamma_{c_{p}}), \quad c=1,\ldots,2\]  
If the terms in Equation (22) are now ordered by powers of \(\delta\mathbf{x}\), by defining  
\[\tau=(\tau_{1},\ldots,\tau_{p})=\] \[(\alpha_{1_{1}}+\alpha_{2_{1}}+\alpha_{3_{1}}+\alpha_{4_{1}}, \ldots,\alpha_{1_{p}}+\alpha_{2_{p}}+\alpha_{3_{p}}+\alpha_{4_{p}})=\] \[(\beta_{1_{1}}+\beta_{2_{1}}+\beta_{3_{1}},\ldots,\beta_{1_{p}}+ \beta_{2_{p}}+\beta_{3_{p}})=\] \[(\gamma_{1_{1}}+\gamma_{2_{1}},\ldots,\gamma_{1_{p}}+\gamma_{2_{ p}})=\] \[\mathbf{\phi}_{i}(\mathbf{x})\approx\sum_{|\sigma|=0}^{t}\mathbf{\phi}_{i}^{(\sigma)} \delta\mathbf{x}^{\sigma},\ \ \ \ \lambda_{i}(\mathbf{x})\approx\sum_{|\sigma|=0}^{t}\lambda_{i}^{(\sigma)}\delta \mathbf{x}^{\sigma},\ \ \ \ \ \ i=1,\ldots,2n_{dof} \tag{21}\]0  
Equation (22) can be rearranged as  
\[\left( \begin{array}{ccccc}\\ \sum_{|\alpha|=0}^{t}&\lambda_{i}^{(\alpha_{1})}\lambda_{i}^{(\alpha_{2})} \overline{\mathbf{M}}^{(\alpha_{3})}\boldsymbol{\phi}_{i}^{(\alpha_{4})}+& \sum_{|\beta|=0}^{t}&\lambda_{i}^{(\beta_{1})}\overline{\mathbf{C}}^{(\beta_ {2})}\boldsymbol{\phi}_{i}^{(\beta_{3})}&\sum_{|\gamma|=0}^{t}&\overline{ \mathbf{K}}^{(\gamma_{1})}\boldsymbol{\phi}_{i}^{(\gamma_{2})}\\ \par a_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}=\tau&\par&\par&\par&\par&\par i=1, \ldots,2n_{dof}\end{array} \right)\delta\mathbf{x}^{\tau}=\mathbf{0}\] \[i=1,\ldots,2n_{dof} \tag{25}\]  
It results in a sum of \(n_{p,\tau}\) terms for each value of \(\tau\) (see Section 3.1) made by two coefficients each, i.e., the term between brackets and \(\delta\mathbf{x}^{\tau}\). Equation (25) has to hold for all \(\delta\mathbf{x}\in\mathbb{R}^{p}\), meaning that for a given \(|\tau|\), all the \(n_{p,\tau}\) terms between brackets resulting from Equation (25) have to be equal to zero:  
\[\sum_{|\alpha|=0}^{t} \lambda_{i}^{(\alpha_{1})}\lambda_{i}^{(\alpha_{2})}\overline{ \mathbf{M}}^{(\alpha_{3})}\boldsymbol{\phi}_{i}^{(\alpha_{4})}+\sum_{|\beta|=0 }^{t} \lambda_{i}^{(\beta_{1})}\overline{\mathbf{C}}  
\[=-\left(\begin{array}{ccccc}2\lambda_{i}^{(t)}\overline{\mathbf{M}}^{(0)} \boldsymbol{\phi}_{i}^{(0)}+\lambda_{i}^{(0)}\lambda_{i}^{(0)}\overline{\mathbf{ M}}^{(0)}\boldsymbol{\phi}_{i}^{(t)}+\lambda_{i}^{(t)}\overline{\mathbf{C}}^{(0)} \boldsymbol{\phi}_{i}^{(0)}+\lambda_{i}^{(0)}\overline{\mathbf{C}}^{(0)} \boldsymbol{\phi}_{i}^{(t)}+\overline{\mathbf{K}}^{(0)}\boldsymbol{\phi}_{i}^ {(t)}=\\ \\ \underbrace{\sum\limits_{|a|=1}^{t}\lambda_{i}^{(a_{1})}\lambda_{i}^{(a_{2})} \overline{\mathbf{M}}^{(a_{3})}\boldsymbol{\phi}_{i}^{(a_{4})}+\sum\limits_{| \beta|=1}^{t}\lambda_{i}^{(\beta_{1})}\overline{\mathbf{C}}^{(\beta_{2})} \boldsymbol{\phi}_{i}^{(\beta_{3})}+\sum\limits_{|\gamma|=1}^{t}\overline{ \mathbf{K}}^{(\gamma_{1})}\boldsymbol{\phi}_{i}^{(\gamma_{2})}\\ \rho_{1}\tau|\beta|\)}_{ \begin{subarray}{c}\gamma_{2}\
eq|\gamma|\\ \rho_{2}\
eq|\gamma|\\ \rho_{3}\
eq|\beta|\\ \rho_{1}+\rho_{2}+\rho_{3}=\tau\end{subarray} }\right)} \tag{27}\]  
By solving Equation (27) for the highest derivatives of the eigenpairs it holds that  
\[\left\{ \begin{array}{c}\boldsymbol{\phi}_{i}^{(t)}\\ \lambda_{i}^{(t)}\end{array} \right\}=-\underbrace{\left[\lambda_{i}^{(0)} \lambda_{i}^{(0)}\overline{\mathbf{M}}^{(0)}+\lambda_{i}^{(0)}\overline{ \mathbf{C}}^{(0)}+\overline{\mathbf{K}}^{(0)}\right.}_{\mathbf{S}_{i}}\left. \left(2\lambda_{i}^{(0)}\overline{\mathbf{M}}^{(0)}+\overline{\mathbf{C}}^{( 0)}\right)\boldsymbol{\phi}_{i}^{(0)}\right]^{\mathsf{t}}\mathbf{r}_{\tau,i} \tag{28}\]  
\[i=1,\ldots,2n_{dof}\]  
where \(\mathbf{S}_{i}\) is a rectangular matrix with \(n_{dof}\) rows and \(n_{dof}+1\) columns. Symbol \" +\" indicates the pseudo-inverse matrix.