# Flexible-Link Multibody System Eigenvalue Analysis Parameterized with Respect to Rigid-Body Motion
## 2 Modeling of Flexible-Link Multibody Systems
### Modal Analysis for Systems with Nonsymmetric Matrices  
The linearized damping matrix \(\overline{\mathbf{C}}\) in Equation (11) is, in general, neither proportional to the mass and stiffness matrices nor symmetric (due to the velocity-dependent terms); therefore, the eigenvalue problem associated to the model (11) takes the form of a quadratic eigenvalue problem:  
\[\left(\lambda^{2}\overline{\mathbf{M}}+\lambda\overline{\mathbf{C} }+\overline{\mathbf{K}}\right)\boldsymbol{\phi} = \mathbf{0} \tag{12}\] \[\boldsymbol{\psi}^{*}\left(\lambda^{2}\overline{\mathbf{M}}+ \lambda\overline{\mathbf{C}}+\overline{\mathbf{K}}\right) = \mathbf{0} \tag{13}\]  
where \(\lambda\) is a system eigenvalue, and \(\boldsymbol{\phi}\in\mathbb{R}^{n_{dof}}\) and \(\boldsymbol{\psi}\in\mathbb{R}^{n_{dof}}\) are the corresponding right and left eigenvectors, respectively. In Equation (13), symbol \(*\) denotes the conjugate transpose, as \(\lambda\), \(\boldsymbol{\phi}\), and \(\boldsymbol{\psi}\) may be complex valued. The quadratic eigenvalue problem in Equations (12) and (13) has \(2n_{dof}\) eigenvalues, which are symmetric with respect to the real axis of the complex plane, being the system matrices real [35]. This means that the system eigenvalues can be either real or complex, but in the last case, they occur in complex conjugate pairs as well as the corresponding eigenvectors: if \(\lambda_{1}=\lambda_{r}+i\lambda_{i}\) is a system eigenvalue, then \(\lambda_{2}=\lambda_{r}-i\lambda_{i}\) is a system eigenvalue too, and the corresponding eigenvectors are \(\boldsymbol{\phi}_{1}=\boldsymbol{\phi}_{r}+i\boldsymbol{\phi}_{i}\) and \(\boldsymbol{\phi}_{2}=\boldsymbol{\phi}_{r}-i\boldsymbol{\phi}_{i}\), respectively.  
Although the quadratic eigenvalue problem in Equations (12) and (13) could be directly solved, it is usually transformed in a generalized eigenvalue problem  
\[\mathbf{Aw} = \lambda\mathbf{Bw} \tag{14}\] \[\mathbf{z}^{*}\mathbf{A} = \lambda\mathbf{z}^{*}\mathbf{B} \tag{15}\]  
by reducing the \(n_{dof}\)-dimensional set of second-order homogeneous differential equations in Equation (11) to a \(2n_{dof}\)-dimensional first-order set of equations:  
\[\underbrace{\begin{bmatrix}\overline{\mathbf{C}}&\overline{\mathbf{M}}\\ \mathbf{N}&\mathbf{0}\end{bmatrix}}_{\mathbf{B}}\underbrace{\begin{matrix} \delta\dot{\mathbf{q}}\\ \delta\dot{\mathbf{q}}\end{matrix}}_{\hat{\boldsymbol{y}}}-\underbrace{ \begin{bmatrix}-\overline{\mathbf{K}}&\mathbf{0}\\ \mathbf{0}&\mathbf{N}\end{bmatrix}}_{\mathbf{A}}\underbrace{\begin{matrix} \delta\mathbf{q}\\ \delta\dot{\mathbf{q}}\end{matrix}}_{\mathbf{y}}=\begin{bmatrix}\mathbf{0}\\ \mathbf{0}\end{bmatrix} \tag{16}\]  
where \(\mathbf{N}\in\mathbb{R}^{n_{dof}\times n_{dof}}\) can be any nonsingular matrix; here it is considered \(\mathbf{N}=\overline{\mathbf{M}}\). The quadratic (Equations (12) and (13)) and the generalized (Equations (14) and (15)) eigenvalue problems have the same \(2n_{dof}\) eigenvalues, whereas the corresponding eigenvectors are correlated by the following relations,  
\[\mathbf{w}=\begin{cases}\boldsymbol{\phi}\\ \lambda\boldsymbol{\phi}\end{cases}\mathbf{z}=\begin{cases}\boldsymbol{\psi} \\ \lambda^{*}\boldsymbol{\psi}\end{cases} \tag{17}\]  
If all the system eigenvalues are distinct and the left and the right eigenvectors are normalized so that \(\mathbf{w}_{i}^{*}\mathbf{B}\mathbf{z}_{i}=1\), the following biorthonormality relations hold,  
\[\mathbf{z}_{i}^{*}\mathbf{Aw}_{j}=\left\{\begin{array}{lcl}\lambda_{i}& \text{if}&i=j\\ 0&\text{if}&i\
eq j\end{array}\right.;&\mathbf{z}_{i}^{*}\mathbf{Bw}_{j}=1= \left\{\begin{array}{lcl}1&\text{if}&i=j\\ 0&\text{if}&i\
eq j\end{array}\right. \tag{18}\]