# 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
Note that on the right-hand side of Equation (28) only the eigenpair derivatives up to order \(t-1\) appear. Therefore, by proceeding iteratively from the first order (that requires the only knowledge of the eigenpairs computed at the expansion point \(\lambda_{i}^{(0)},\quad\boldsymbol{\phi}_{i}^{(0)}\)) up to the desired Taylor order, it is possible to compute all the coefficients of the Taylor series expansion in Equation (22). In such a way, the eigenvectors and eigenvalues of a FMBS are expressed as polynomial functions of the system rigid-body motion and of other possible variables of interest. Although the method has been derived for the right eigenvectors (\(\boldsymbol{\phi}\)), its extension to the left (\(\boldsymbol{\psi}\)) ones is trivial: the same procedure has to be repeated by substituting the linearized system matrices with their transposes, i.e., the starting eigenvalue problem is the following one,  
\[\left(\lambda^{*}\lambda^{*}\overline{\mathbf{M}}^{\mathrm{T}}+ \lambda^{*}\overline{\mathbf{C}}^{\mathrm{T}}+\overline{\mathbf{K}}^{\mathrm{ T}}\right)\boldsymbol{\psi} = \boldsymbol{0} \tag{29}\]  
Note that the derivatives of each eigenpair (\(\lambda_{i}\), \(\boldsymbol{\phi}_{i}\)) are independent from the others \(2n_{dof}-1\) pairs; therefore, the problem can be solved for just the eigenpairs of interest, as well as if the eigenpairs are all complex conjugate, it is enough to solve the problem for \(n_{dof}\) eigenpairs. The order \(t\) at which to truncate the Taylor series must be chosen as a trade-off between accuracy and complexity of the approximated eigensolutions. Indeed, a higher truncation order better approximates the real solutions, but the number of addends of the polynomial functions representing the eigensolutions rapidly increases with it, compromising the computational benefits of the method. An evidence of this is provided in Table 1. It shows the number of addends of the approximated functions expanded up to a certain order \(t\) and parameterized on the rigid-body motion of systems having \(n_{r}\) rigid dofs. The parameterization about dynamic equilibrium configurations is computationally more demanding with respect to the one made about static equilibrium configurations, as the first uses twice the number of parameters. Therefore, a computationally efficient approximation of the eigenpairs about dynamic equilibrium configurations can be obtained only for systems with few rigid dofs. Conversely, if, as it is common in practice, the eigenpairs are computed about static equilibrium configurations, they can be efficiently approximated with the proposed method also if they depend on several rigid dofs. Regardless the computational efficiency and hence the possibility to employ the approximatedeigenpairs in real-time applications, the method provides a function that puts in correlation the modal parameters with the system configurations; these could be used in offline methods such as optimization techniques.