# 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  
The dynamic model of a FMBS depends nonlinearly on the values assumed by the rigid motion coordinates and their derivatives. Therefore, the solution of the eigenvalue problem obtained by means of a linearized model has only a local validity, i.e., it holds for the expansion point \(\mathbf{P}=\mathbf{P}_{0}\). The validity of such a solution can be extended by deriving a polynomial representation of the eigenpairs in the system rigid motion coordinates and velocities. To this end, the Taylor series for multivariate functions (see Section 3.1) and the eigenvalue problem formulation of Equation (12) are here exploited. Let \(\mathbf{x}\in\mathbb{R}^{p\geq 2n_{t}}\) be the vector of model parameters including the rigid motion coordinates \(\mathbf{q}_{r}\); their velocities \(\mathbf{q}_{r}\); and, possibly, the parameters which the system model can depend on, such as payload mass, stiffness varying with the operative conditions, design parameters, and so on. Let us approximate, by means of Taylor series (see Equation (19)), the linearized system matrices, which are functions of \(\mathbf{x}\), about the operative point \(\mathbf{x}_{0}\):  
\[\overline{\mathbf{M}}(\mathbf{x})\approx\sum_{|\sigma|=0}^{t}\overline{ \mathbf{M}}^{(\sigma)}\delta\mathbf{x}^{\sigma},\ \ \ \ \overline{\mathbf{C}}(\mathbf{x})\approx\sum_{|\sigma|=0}^{t}\overline{ \mathbf{C}}^{(\sigma)}\delta\mathbf{x}^{\sigma},\ \ \ \ \overline{\mathbf{K}}(\mathbf{x})\approx\sum_{| \sigma|=0}^{t}\overline{\mathbf{K}}^{(\sigma)}\delta\mathbf{x}^{\sigma} \tag{20}\]  
Now, let us assume that also the eigenpairs can be approximated by means of a Taylor expansion:  
\[\mathbf{\phi}_{i}(\mathbf{x})\approx\sum_{|\sigma|=0}^{t}\mathbf{\phi}_{i}^{(\sigma)} \delta\mathbf{x}^{\sigma},\ \ \ \ \lambda_{i}(\mathbf{x})\approx\sum_{|\sigma|=0}^{t}\lambda_{i}^{(\sigma)}\delta \mathbf{x}^{\sigma},\ \ \ \ \ \ i=1,\ldots,2n_{dof} \tag{21}\]  
Note that having assumed to analytically know the system matrices, the Taylor series in Equation (20) are known expressions; conversely, the derivatives of the eigensolutions (\(\lambda_{i}^{(\sigma)}\), \(\mathbf{\phi}_{i}^{(\sigma)}\))with respect to \(\mathbf{x}\) in Equation (21) are unknown quantities and must be determined. To this end, let us rewrite the eigenvalue problem of Equation (12) as a Taylor series expansion truncated to the \(t\)-th order through Equations (20) and (21):  
\[\sum_{|\alpha|=0}^{t}\lambda_{i}^{(\alpha_{1})}\lambda_{i}^{(\alpha _{2})}\overline{\mathbf{M}}^{(\alpha_{3})}\boldsymbol{\phi}_{i}^{(\alpha_{4})} \delta\mathbf{x}^{\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}}+\sum_{|\beta|=0 }^{t}\lambda_{i}^{(\beta_{1})}\overline{\mathbf{C}}^{(\beta_{2})}\boldsymbol{ \phi}_{i}^{(\beta_{3})}\delta\mathbf{x}^{\beta_{1}+\beta_{2}+\beta_{3}}+\sum_{| \gamma|=0}^{t}\overline{\mathbf{K}}^{(\gamma_{1})}\boldsymbol{\phi}_{i}^{( \gamma_{2})}\delta\mathbf{x}^{\gamma_{1}+\gamma_{2}}=\mathbf{0}\\ i=1,\ldots,2n_{dof} \tag{22}\]  
where \(\alpha=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})\), \(\beta=(\beta_{1},\beta_{2},\beta_{3})\) and \(\gamma=(\gamma_{1},\gamma_{2})\) are three multi-indices. The components of such multi-indices set the order of each factor of the three addends in Equation (22), so that their whole order is equal to \(|\alpha|=|\beta|=|\gamma|\). Note that the entries of \(\alpha\), \(\beta\) and \(\gamma\) are multi-indices too.