# Flexible-Link Multibody System Eigenvalue Analysis Parameterized with Respect to Rigid-Body Motion
## 1 Introduction  
The dynamic behavior of a mechanical system can be easily studied by means of modal analysis, which provides the system modal parameters or characteristics (i.e., natural frequencies, mode shapes, and damping ratios). The knowledge of the modal characteristics is a fundamental requirement for the implementation of several advanced model-based engineering techniques, such as motion planning [1], control design [2; 3], stability analysis [4; 5], model reduction [6; 7; 8; 9; 10], model updating [11; 12; 13], and structural modification [14; 15; 16]. Modal analysis relies on the solution of an eigenvalue problem, which seeks the eigenvalues and eigenvectors associated to a linear system of equations. However, the adoption of such an analysis may provide a useful insight also for the study of mechanical systems whose dynamics is not governed by linear time-invariant equations. A significant class of mechanical systems that falls in this folder is the one of the flexible-link multibody systems (FMBSs). Such systems are robots or mechanisms that can deflect due to external loads or internal body forces, whose motion is described by means of kineto-elastodynamic models, hereafter referred to as dynamic models [17]. Several contributionscan be found in the literature on the modeling of such systems as well as survey papers [18; 19] and books [20; 21]. The system elastic behavior is represented by continuous ordinary and partial differential equations. Such equations, so as to be simplified and solved, are discretized by means of lumped parameters, assumed modes, or finite element methods [19]. The common approach for modeling FMBSs consists in the use of the finite element method to discretize the flexible links and to represent their elastic deformations and in superposing such deformations to a known rigid body motion. Based on the set of coordinates chosen to model the rigid body motion, i.e., a minimum set of independent coordinates (representing the system degrees of freedom) or a redundant set of coordinates (including both the independent and dependent coordinates), the equations of motion are formulated as a set of ordinary differential equations (ODEs) or a coupled set of differential and algebraic equations (DAEs) to be solved simultaneously [20], respectively. In both the aforementioned cases, due to the large displacements to which FMBSs are subjected, their dynamic models are highly nonlinear and depend on the system configuration. Therefore, the related mass, stiffness, and damping matrices are in general nonconstant, as well as the resulting modal parameters. A common practice used in the literature to apply the modal analysis to FLMBSs consists of linearizing the nonlinear dynamic model about a selected configuration so that the modal parameters can be computed. In particular, if the system model is formulated by means of ODEs, the eigenvalue analysis can be straightforwardly applied to the linearized system equations [12; 22]. Conversely, if the system model is formulated by means of DAEs, two different strategies are possible for computing the eingensolutions: (a) transform the motion equations from DAEs to ODEs, linearize the resulting model, and then compute the eigenpairs [5; 23; 24]; (b) perform a direct eigenanalysis, i.e., the eigenanalysis for the system of equations resulting from the direct linearization of the DAEs [25; 26]. The main difference between the eigensolutions obtained from linearized ODEs or DAEs is that the linearized ODEs allow to obtain the exact problem spectrum, while the linearized DAEs are affected by the linearization method and introduce spurious eigenvalues in the spectrum [27]. Due to such an evidence, this paper will focus on FMBSs modeled by means of a minimum set of ODEs. Although the adoption of linearized models, on the one hand, allows the computation of the eigenpairs of FMBSs, on the other hand, it does not take directly into account the variability of the modal parameters due to the system configuration change.