# Flexible-Link Multibody System Eigenvalue Analysis Parameterized with Respect to Rigid-Body Motion
## 1 Introduction
This last point is typically addressed by discretizing a given system motion/task in a certain number of operative points about which the nonlinear model is linearized and a new eigenvalue problem solved [5; 6]. Following such an approach, several eigenvalue problems are to be solved, which is typically a computational expensive operation; additionally, unless interpolation techniques are used, the eigenpairs computed at different operating points are not related among them. An approach that could help in overcoming such an issue has been proposed by [PERSON] et al. [28]. In such a paper, a method to get a parametric representation of the eigenvalues and eigenvectors of an undamped second-order mechanical system, whose model is analytically known, has been presented. In [29], [PERSON] et al. extended the method to proportionally damped systems. In this approach, the parametric representation of the eigenpairs is inferred from an iterative Taylor series expansion of the eigenvalue problem associated to the linearized system matrices about a parametric operating point. A first extension and application of the approach proposed by [PERSON] et al. to FMBSs characterized by small deformation and negligible damping and velocity-dependent terms has been proposed by the authors in [30; 31]. In particular, in [31], the authors performed a preliminary investigation on the method capability to approximate the modal content over a wide range of the FMBS parameters, including not only the rigid motion coordinates, but also the payload handled by a two-degree-of-freedom (dof) planar robot carrying out a pick-and-place trajectory. As promising results have been obtained in these preliminary studies, this paper aims at providing a comprehensive dissertation on the extension of such a method to FMBSs characterized by small deformations and non negligible damping and velocity-dependent terms. Indeed, the knowledge of an analytical relationship between the system motion condition and the natural frequencies, damping factors and modal shapes of, at least, the main vibrational modes, can be fruitfully exploited for, e.g., the development and implementation of more efficient model-based motion planning and control strategies towards to vibration minimization or the set-up of optimization problems based on the system modal characteristics. Additionally, a similar analytical expression allows for an easy sensitivity analysis of the system model to parameter uncertainties. The paper is set out as follows. Starting from a nonlinear dynamic model formulated by means of a minimum set of ODEs, Section 2 derives the linearized dynamic model of a FMBS about a dynamic equilibrium configuration and discusses the eigenvalue problem for systems having nonsymmetric matrices, as it is the case of the linearized models of FMBSs. Section 3 outlines the method to derive the polynomial expressions for the eigenpairs of a FMBS in its generalized coordinates. In Section 4, the effectiveness of the method is proved by applying it to a flexible planar robot following a predefined trajectory. Finally, Section 5 gives concluding remarks.