# Parallel Direct Solution of Flexible Multibody Systems Based on Block Gaussian Elimination
## 2 Flexible Multibody Formulation with the Craig-Bampton Method
### Flexible Body Formulation with the Craig-Bampton Method
#### 2.2.2 Rigid Modes Removing and Modes Orthogonalization  
However, there are two defects in the modes set (24). Firstly, rigid modes with zero frequency exist in the subspace generated by the boundary modes \(\mathbf{\Phi}_{i}^{\text{\tiny{b}}\text{\tiny{b}}\text{\tiny{m}}}\); and secondly, the columns in \(\mathbf{\Phi}_{i}^{\text{\tiny{b}}\text{\tiny{m}}\text{\tiny{m}}}\) are not orthogonal to those in \(\mathbf{\Phi}_{i}^{\text{\tiny{v}}\text{\tiny{h}}\text{\tiny{m}}}\). Rigid modes are redundant because they have been accounted for by the floating frame of reference (FFR) coordinates, while modal orthogonality is essential for producing diagonal modal matrices of stiffness and mass.  
To address the two issues, reorthogonalization can be applied. First, we project Equation (19) onto the subspace of \(\overline{\mathbf{\Phi}}_{i}\) by letting \(\overline{f}_{i}=0\), substituting \(\mathbf{u}_{i}=\overline{\mathbf{\Phi}}_{i}\overline{\mathbf{q}}_{i}\), and multiplying \(\overline{\mathbf{\Phi}}_{i}^{\text{\tiny{T}}}\) on the left side, resulting in  
\[\Big{(}\overline{\mathbf{\Phi}}_{i}^{\text{\tiny{T}}}\overline{\mathbf{K}}_{i} \overline{\mathbf{\Phi}}_{i}\Big{)}\overline{\mathbf{q}}_{i}+\Big{(}\overline{\mathbf{ \Phi}}_{i}^{\text{\tiny{T}}}\overline{M}\overline{\mathbf{\Phi}}_{i}\Big{)} \overline{\overline{q}}_{i}=\mathbf{0}, \tag{25}\]  
wherein \(\overline{\mathbf{q}}_{i}\) represents the redundant modal coordinates corresponding to modes \(\overline{\mathbf{\Phi}}_{i}\). Secondly, we conduct a mode analysis on the projected vibration Equation (25) to derive the orthogonal modes [32]. By discarding zero-frequency modes and arranging the frequencies in descending order, the reorthogonalized non-zero-frequency modal matrix \(\overline{\overline{\mathbf{\Phi}}}_{i}\) can fulfill the following unit orthogonal condition corresponding to the mass matrix:  
\[\overline{\overline{\mathbf{\Phi}}}_{i}^{\text{\tiny{T}}}\overline{\mathbf{\Phi}}_{ i}^{\text{\tiny{T}}}\overline{\mathbf{M}}_{i}\overline{\mathbf{\Phi}}_{i}\overline{\overline{ \mathbf{\Phi}}}_{i}=\mathbf{E}_{n_{i}},\overline{\overline{\mathbf{\Phi}}}_{i}^{\text{ \tiny{T}}}\overline{\mathbf{\Phi}}_{i}^{\text{\tiny{T}}}\overline{\mathbf{K}}_{i} \overline{\mathbf{\Phi}}_{i}\overline{\overline{\mathbf{\Phi}}}_{i}=\mathbf{\Omega}_{i}^{ 2}, \tag{26}\]  
wherein \(\mathbf{E}_{n_{i}}\) represents the unit matrix with dimension \(n_{i}=M_{i}+N_{i}-6\) and \(\mathbf{\Omega}_{i}^{2}=\text{diag}(\omega_{i,1}^{2},\cdots,\omega_{i,n_{i}}^{2})\) is the diagonal matrix with non-zero modal circle frequencies \(\omega_{i,1}\geq\cdots\geq\omega_{i,n_{i}}>0\) in descending order.  
Then, the orthogonal non-zero-frequency elastic modes matrix of the CB method is  
\[\mathbf{\Phi}_{i}=\overline{\mathbf{\Phi}}_{i}\overline{\overline{\mathbf{\Phi}}}_{i} \triangleq\Big{[}\mathbf{\varphi}_{i,1}\quad\cdots\quad\mathbf{\varphi}_{i,n_{i}} \Big{]}. \tag{27}\]  
The FE displacement is then expressed by the modal coordinates \(\mathbf{q}_{i}=[q_{i,1};q_{i,2},\cdots;q_{i,n_{i}}]\) as  
\[\mathbf{u}_{i}=\mathbf{\Phi}_{i}\mathbf{q}_{i}, \tag{28}\]  
and the elastic vibration equations of flexible body \(i\) are diagonal:  
\[\Omega_{i}^{2}\mathbf{q}_{i}+\ddot{\mathbf{q}}_{i}=\mathbf{f}_{i}, \tag{29}\]  
wherein \(\mathbf{f}_{i}=\mathbf{\Phi}_{i}^{\text{\tiny{T}}}\overline{f}_{i}\) is the modal force of the external forces acting on the flexible body \(i\).