# Flexible-Link Multibody System Eigenvalue Analysis Parameterized with Respect to Rigid-Body Motion
## 4 Results  
The effectiveness of the method in approximating the variability of the eigenpairs of FMBS due to system configuration change has been validated by means of the test case shown in Figure 1. It is an open-chain, planar mechanism with two flexible links and two revolute joints, and therefore two rigid dofs. The system has been modeled by means of the ERLS formulation [34], which leads to a dynamic model consisting of a minimum set of nonlinear ODEs (the interested reader can find the analytical nonlinear model of the studied system in the Appendix A.1). The link flexibility has been modeled through finite elements, in particular, uniform two-node and six-dof beam elements (see Figure 1) have been employed with the properties shown in Table 2. The system dynamic model has 23 dofs, of which two are the rigid motion coordinates and 21 are elastic dofs. The two rigid motion coordinates have been chosen as the absolute angular position of the shoulder and the elbow joints, respectively. They have been denoted \(q_{1}\) and \(q_{2}\) in Figure 1.  
To test the correctness of the method in approximating the eigenpairs under motion condition, the system is forced to follow a given trajectory. In particular, the end-effector (point E in Figure 1) moves along a horizontal linear path of 0.3 m in 0.5 s. The corresponding trajectory in the joint coordinates is shown in Figure 2.  
To apply the method under investigation, the system dynamic model has been linearized about a dynamic equilibrium configuration: \(\mathbf{P}_{0}=(\mathbf{q}_{0}=\mathbf{q}(T/2),\dot{\mathbf{q}}_{0}=\dot{ \mathbf{q}}(T/2),\ddot{\mathbf{q}}_{0}=\ddot{\mathbf{q}}(T/2),\mathbf{u}_{0}= \mathbf{u}(T/2))\), where \(T\) is the time period of the trajectory, the numerical values of the dynamic equilibrium configuration are stated in Appendix A.2. Then, the linearized system has been parameterized on the positions and velocities of the rigid motion coordinates, i.e., \(\mathbf{x}=\left\{q_{1}\quad q_{2}\quad q_{1}\quad q_{2}\right\}^{\text{T}}\). Starting from the solution of the eigenvalue problem computed at the expansion point  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Static Equilibrium**} & \multicolumn{4}{c}{**Dynamic Equilibrium**} \\  & & \multicolumn{4}{c}{\((\mathbf{x}=\mathbf{q}_{r})\)} & \multicolumn{4}{c}{\((\mathbf{x}^{\text{T}}=\{\mathbf{q}_{r}^{\text{T}}\}\)} & \multicolumn{4}{c}{\((\mathbf{x}^{\text{T}}=\{\mathbf{q}_{r}^{\text{T}}\}\)} & \multicolumn{4}{c}{\(\mathbf{\dot{q}}_{r}^{\text{T}})\)} \\ \(\mathbf{n}_{r}\) & \(\mathbf{t}=\mathbf{1}\) & \(\mathbf{t}=\mathbf{2}\) & \(\mathbf{t}=\mathbf{3}\) & \(\mathbf{t}=\mathbf{4}\) & \(\mathbf{t}=\mathbf{5}\) & \(\mathbf{t}=\mathbf{1}\) & \(\mathbf{t}=\mathbf{2}\) & \(\mathbf{t}=\mathbf{3}\) & \(\mathbf{t}=\mathbf{4}\)\(\left\{q_{1}(T/2)\quad q_{2}(T/2)\quad\dot{q}_{1}(T/2)\quad\dot{q}_{2}(T/2)\right\} ^{\text{T}}\), the eigenpair derivatives up to the fourth Taylor series expansion have been computed by means of Equation (28). Finally, fourth-grade polynomials, representing the eigenvalues and eigenvectors as function of the angular positions and velocities of the shoulder and the elbow joints, have been obtained. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Property** & **Symbol** & **Value** \\ \hline Length first link & \(L_{1}\) & \(0.6\) m \\ Length second link & \(L_{2}\) & \(0.45\) m \\ Bending moment of inertia & J & \(3.97\cdot 10^{-8}\) m\({}^{4}\) \\ Circular cross-sectional area & A & \(7.06\cdot 10^{-4}\) m\({}^{2}\) \\ Mass density & \(\rho\) & \(2700\) kg/m\({}^{3}\) \\ Linear mass density & \(\rho_{l}\) & \(1.906\) kg/m \\ Young’s modulus & E & \(69\cdot 10^{9}\) Pa \\ Mass proportional damping coefficients & \(\alpha\) & \(8.7\cdot 10^{-1}\) m/s \\ Stiffness proportional damping coefficients & \(\beta\) & \(4.5\cdot 10^{-6}\) m\(\cdot\)s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Finite element model parameters. Figure 2: Joint trajectory.