## 2 Equations of Motion for Flexible Multibody Systems
### Floating Frame of Reference Formulation  
The Lagrange equation for the \(i\)-th body in the FMBD system can be written as:  
\[\frac{d}{dt}\left(\frac{\partial T^{i}}{\partial\dot{\mathbf{q}}^{i}}\right)^{T} -\left(\frac{\partial T^{i}}{\partial\mathbf{q}^{i}}\right)^{T}+\mathbf{C}_{ \mathbf{q}^{i}}^{T}\mathbf{\lambda}^{i}=\mathbf{Q}^{i}, \tag{1}\]  
where \(\mathbf{q}^{i}\) is the generalized coordinate vector including the reference (rigid) and flexible coordinates. \(T^{i}\) is the kinetic energy. \(\mathbf{C}_{\mathbf{q}^{i}}^{T}\) is the constraint Jacobian matrix. \(\mathbf{\lambda}^{i}\) is the vector of Lagrange multipliers. \(\mathbf{Q}^{i}\) is the vector of generalized forces. The generalized forces \(\mathbf{Q}^{i}\) acting on the \(i\)-th body can be derived using the virtual work \(\delta W_{d}^{i}\) of the elastic force and \(\delta W_{e}^{i}\) of the external forces as:  
\[\delta W^{i}=\delta W_{d}^{i}+\delta W_{e}^{i}, \tag{2}\]  
where \(\delta W_{d}^{i}\) and \(\delta W_{e}^{i}\) are defined as:  
\[\delta W_{d}^{i}=-\mathbf{q}^{i^{T}}\mathbf{K}^{i}\delta\mathbf{q}^{i},\quad \delta W_{e}^{i}=\mathbf{Q}_{e}^{i^{T}}\delta\mathbf{q}^{i}, \tag{3}\]  
where \(\mathbf{K}^{i}\) is the stiffness matrix and \(\mathbf{Q}_{e}^{i}\) is the external force vector. Substituting Equation (3) into Equation (2), the generalized force \(\mathbf{Q}^{i}\) can be expressed as:  
\[\mathbf{Q}^{i}=-\mathbf{K}^{i}\mathbf{q}^{i}+\mathbf{Q}_{e}^{i}. \tag{4}\]  
The terms on the left side in Equation (1) without the third term can be written as [1]:  
\[\frac{d}{dt}\left(\frac{\partial T^{i}}{\partial\dot{\mathbf{q}}^{i}}\right)^ {T}-\left(\frac{\partial T^{i}}{\partial\mathbf{q}^{i}}\right)^{T}=\mathbf{M} ^{i}\ddot{\mathbf{q}}^{i}-\mathbf{Q}_{v}^{i}, \tag{5}\]  
where \(\mathbf{M}^{i}\) is the mass matrix and \(\mathbf{Q}_{v}^{i}\) is the quadratic velocity vector defined as:  
\[\mathbf{Q}_{v}^{i}=-\mathbf{M}^{i}\dot{\mathbf{q}}^{i}+\left[\frac{\partial} {\partial\mathbf{q}^{i}}\left(\frac{1}{2}\dot{\mathbf{q}}^{i^{T}}\mathbf{M}^{ i}\dot{\mathbf{q}}^{i}\right)\right]^{T}. \tag{6}\]  
Using Equations (4) and (5) in Equation (1), the following EOMs neglecting the damping are obtained:  
\[\mathbf{M}^{i}\ddot{\mathbf{q}}^{i}+\mathbf{K}^{i}\mathbf{q}^{i}+\mathbf{C}_ {\mathbf{q}^{i}}^{T}\mathbf{\lambda}^{i}=\mathbf{Q}_{e}^{i}+\mathbf{Q}_{v}^{ i}, \tag{7}\]  
and it can be specifically written in a partitioned form of the reference and flexible coordinates as:  
\[\begin{bmatrix}\mathbf{M}_{R}^{i}&\mathbf{M}_{FR}^{i}\\ \mathbf{M}_{RF}^{i}&\mathbf{M}_{F}^{i}\end{bmatrix}\begin{bmatrix}\ddot{ \mathbf{q}}_{R}^{i}\\ \ddot{\mathbf{q}}_{F}^{i}\end{bmatrix}+\begin{bmatrix}\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{K}_{F}^{i}\end{bmatrix}\begin{bmatrix}\mathbf{q}_{R}^{i}\\ \mathbf{q}_{F}^{i}\end{bmatrix}+\begin{bmatrix}\mathbf{C}_{\mathbf{q}_{R}^{i} }^{T}\\ \mathbf{C}_{\mathbf{q}_{F}^{i}}^{T}\end{bmatrix}\mathbf{\lambda}^{i}=\begin{bmatrix} \begin{bmatrix}\mathbf{Q}_{e}^{i}\\ \mathbf{Q}_{e}^{i}\end{bmatrix}_{F}\end{bmatrix}+\begin{bmatrix}\begin{bmatrix} \mathbf{Q}_{v}^{i}\\ \mathbf{Q}_{v}^{i}\end{bmatrix}_{F}\end{bmatrix}, \tag{8}\]  
where subscripts \(R\) and \(F\) denote the quantities with respect to the reference and flexible coordinates, respectively.