# Flexible-Link Multibody System Eigenvalue Analysis Parameterized with Respect to Rigid-Body Motion
## 2 Modeling of Flexible-Link Multibody Systems  
### Motion Equations  
The total motion of a FMBS undergoing large rigid body motion and small elastic deformations can be modeled as an elastic motion, due to the link flexibility superposed onto a rigid body motion. Different strategies exist to model both the elastic and the rigid motions. The widespread approach to model link flexibility is by means of finite element model. Such a model is then embedded into a floating frame [20], in a corotational frame [32], or in a moving reference configuration [33], which represents the rigid motion. Depending on the selected model strategy and the type of coordinates used (i.e., relative, absolute, natural, minimal, or nodal coordinates) the motion equations result in a nonlinear set of DAEs or ODEs. Despite the large use of the DAE formulation, in this paper dynamic models formulated by means of ODEs will be adopted, as we are interested in the system modal parameters. Indeed, models employing ODEs lead to the exact system spectrum, unlike DAE formulation that may lead to approximate eigenvalues or introduce spurious ones [27].  
The motion equations of a FMBS represented by means of a minimum set of second-order ODEs expressed in terms of the system degrees of freedom (i.e., independent coordinates) take the following matrix form,  
\[\mathbf{M}(\mathbf{q}_{r})\ddot{\mathbf{q}}+\mathbf{C}(\mathbf{q}_{r},\dot{ \mathbf{q}}_{r})\dot{\mathbf{q}}+\mathbf{K}(\mathbf{q}_{r})\mathbf{q}=\mathbf{ f}(\mathbf{q}_{r},\mathbf{u}) \tag{1}\]  
where \(\mathbf{q}=\left\{\mathbf{q}_{f}^{\mathrm{T}}\quad\mathbf{q}_{r}^{\mathrm{T}}\right\} ^{\mathrm{T}}\) is the vector of the independent coordinates, including the vector \(\mathbf{q}_{f}\in\mathbb{R}^{n_{f}}\) of the elastic coordinates of the flexible links and the vector \(\mathbf{q}_{r}\in\mathbb{R}^{n_{r}}\) of the rigid body variables. The number of rigid coordinates, \(n_{r}\), is equal to number of rigid body motions of the system, whereas the number of the elastic coordinates, \(n_{f}\), depends on the number of finite elements employed to discretize the flexible links; their sum gives the total number of the system dofs, \(n_{dof}=n_{f}+n_{r}\). In Equation (1), \(\mathbf{M}\) is the symmetric, positive definite mass matrix; \(\mathbf{C}\) is the matrix containing the damping, centrifugal, and Coriolis terms; \(\mathbf{K}\) is the system stiffness matrix, which is semi-positive definite as rigid-body motion is allowed. According to the model formulation employed, the \(\mathbf{K}\) matrix can depend or not on the rigid body coordinates \(\mathbf{q}_{r}\). Indeed, e.g., if the floating frame of reference formulation [20] is employed, \(\mathbf{K}\) is constant, whereas in the case of the Equivalent Rigid Link System (ERLS) formulation [34], it depends nonlinearly on \(\mathbf{q}_{r}\). The term on the right-hand side of Equation (1) represents the vector of the gravity, friction, and generalized external forces, it depends on the system coordinates, \(\mathbf{q}_{r}\), and on the external inputs, \(\mathbf{u}\).