# Flexible-Link Multibody System Eigenvalue Analysis Parameterized with Respect to Rigid-Body Motion
## 2 Modeling of Flexible-Link Multibody Systems
### Linearization of the Equations of Motion  
Due to the strong coupling between the gross rigid body motion and the fine motion (vibration) of the flexible links, the system motion equations are highly nonlinear differential equations in the coordinates and velocities.  
To compute the modal characteristics of the system, the motion equations are to be linearized. Let us rewrite Equation (1) as follows.  
\[\mathbf{\Gamma}(\mathbf{P})=\mathbf{\Gamma}(\mathbf{q},\dot{\mathbf{q}},\ddot{ \mathbf{q}},\mathbf{u})=\mathbf{M}\ddot{\mathbf{q}}+\mathbf{C}\dot{\mathbf{q}} +\mathbf{K}\mathbf{q}-\mathbf{f} \tag{2}\]  
Equation (2) can be linearized about an operating point \(\mathbf{P}_{0}=(\mathbf{q}_{0},\dot{\mathbf{q}}_{0},\ddot{\mathbf{q}}_{0}, \mathbf{u}_{0})\) as follows.  
\[\mathbf{\Gamma}(\mathbf{P}_{0}+\delta\mathbf{P})\simeq\left.\frac{\partial \mathbf{\Gamma}}{\partial\mathbf{q}}\right|_{\mathbf{P}=\mathbf{P}_{0}}\delta \mathbf{q}+\left.\frac{\partial\mathbf{\Gamma}}{\partial\dot{\mathbf{q}}} \right|_{\mathbf{P}=\mathbf{P}_{0}}\delta\dot{\mathbf{q}}+\left.\frac{ \partial\mathbf{\Gamma}}{\partial\ddot{\mathbf{q}}}\right|_{\mathbf{P}= \mathbf{P}_{0}}\delta\ddot{\mathbf{q}}+\left.\frac{\partial\mathbf{\Gamma}}{ \partial\mathbf{u}}\right|_{\mathbf{P}=\mathbf{P}_{0}}\delta\mathbf{u} \tag{3}\]  
where  
\[\frac{\partial\mathbf{\Gamma}}{\partial\mathbf{q}} =\overline{\mathbf{K}}= \frac{\partial\mathbf{M}}{\partial\mathbf{q}}\otimes\ddot{ \mathbf{q}}+\frac{\partial\mathbf{C}}{\partial\mathbf{q}}\otimes\dot{\mathbf{ q}}+\frac{\partial\mathbf{K}}{\partial\mathbf{q}}\otimes\mathbf{q}+\mathbf{K}+ \frac{\partial\mathbf{f}}{\partial\mathbf{q}} \tag{4}\] \[\frac{\partial\mathbf{\Gamma}}{\partial\dot{\mathbf{q}}} =\overline{\mathbf{C}}= \frac{\partial\mathbf{C}}{\partial\dot{\mathbf{q}}}\otimes\dot{ \mathbf{q}}+\mathbf{C}\] (5) \[\frac{\partial\mathbf{\Gamma}}{\partial\dot{\mathbf{q}}} =\overline{\mathbf{M}}= \mathbf{M}\] (6) \[\frac{\partial\mathbf{\Gamma}}{\partial\mathbf{u}} =-\overline{\mathbf{F}}= -\frac{\partial\mathbf{f}}{\partial\mathbf{u}} \tag{7}\]  
In Equations (4) and (5), the symbol \(\otimes\) indicates the inner product between the partial derivative of a matrix \(\mathbf{A}\in\mathbb{R}^{n_{def}\times n_{def}}\) with respect to vector \(\mathbf{s}\in\mathbb{R}^{n_{def}}\) by vector \(\mathbf{b}\in\mathbb{R}^{n_{def}}\):  
\[\frac{\partial\mathbf{A}}{\partial\mathbf{s}}\otimes\mathbf{b}=\left[\frac{ \partial\mathbf{A}}{\partial\mathbf{s}_{1}}\mathbf{b}\quad\dots\quad\frac{ \partial\mathbf{A}}{\partial\mathbf{s}_{n_{def}}}\mathbf{b}\right] \tag{8}\]  
Equations (4)-(7) represent the most general formulation of the linearized system matrices, i.e., when the operating point is chosen as a dynamic equilibrium state of the system. As it is typically the case, whenever the system model is linearized about a static equilibrium position, i.e., \(\mathbf{P}_{0}=(\mathbf{q}_{0},\mathbf{0},\mathbf{u}_{0})\), some of the terms in the linearized stiffness \(\overline{\mathbf{K}}\) and damping \(\overline{\mathbf{C}}\) matrices vanish. Then, it holds that  
\[\frac{\partial\mathbf{\Gamma}}{\partial\mathbf{q}} =\overline{\mathbf{K}}= \frac{\partial\mathbf{K}}{\partial\mathbf{q}}\otimes\mathbf{q}+ \mathbf{K}+\frac{\partial\mathbf{f}}{\partial\mathbf{q}} \tag{9}\] \[\frac{\partial\mathbf{\Gamma}}{\partial\dot{\mathbf{q}}} =\overline{\mathbf{C}}= \mathbf{C} \tag{10}\]  
Finally, the system motion equations linearized about an operating point can be written as  
\[\overline{\mathbf{M}}\delta\dot{\mathbf{q}}+\overline{\mathbf{C}}\delta\dot{ \mathbf{q}}+\overline{\mathbf{K}}\delta\mathbf{q}=\overline{\mathbf{F}}\delta \mathbf{u} \tag{11}\]