# 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
#### 2.2.3 Flexible Body Formulation Under Floating Frame of Reference
 \tag{40}\]  
wherein, for flexible body \(i\), \(m_{i,p}\) is the lumped mass at node \(p\), \(\mathbf{H}_{i,p}\) is the symmetric lumped inertia matrix at node \(p\), \(\mathbf{\Phi}_{i,p}^{\text{t}}\) is the translational components of the modes \(\mathbf{\Phi}_{i}\) at node \(p\), while \(\mathbf{\Phi}_{i,p}^{\text{r}}\) is the rotational components of the modes \(\mathbf{\Phi}_{i}\) at node \(p\), \(\mathbf{\varphi}_{i,p,j}^{\text{t}}\) is the \(j^{\text{th}}\) column of \(\mathbf{\Phi}_{i,p}^{\text{t}}\), the wave upon represents the cross-production-to-matrix operator as the same as in Formula (35), and \(\mathbf{s}_{i,p}\) is the original position of node \(p\) in the FFR as shown in Figure 1.  
The unit orthogonal condition corresponding to the mass matrix as defined in Equation (26) gives  
\[m_{i,p}(\mathbf{\Phi}_{i,p}^{\text{t}})^{\text{T}}\mathbf{\Phi}_{i,p}^{\text{t}}+( \mathbf{\Phi}_{i,p}^{\text{r}})^{\text{T}}\mathbf{H}_{i,p}\mathbf{\Phi}_{i,p}^{\text{r}}= \mathbf{E}_{n_{i}}, \tag{41}\]  
Then, we substitute Formula (40) into the kinetic energy (39) to obtain the generalized mass matrix of the flexible body under the coordinates \(\mathbf{\xi}_{i}\) as  
\[\mathbf{M}_{i}= \begin{bmatrix}M_{i}^{\text{ff}}&M_{i}^{\text{tt}}&M_{i}^{\text{ rr}}\\ M_{i}^{\text{ff}}&M_{i}^{\text{tt}}&M_{i}^{\text{rr}}\\ M_{i}^{\text{ff}}&M_{i}^{\text{tt}}&M_{i}^{\text{rr}}\end{bmatrix} = \begin{bmatrix} \mathbf{E}_{n_{i}}&\mathbf{I}_{i,3}^{\text{T}}\mathbf{A}_{i}^{\text{T}}&\mathbf{I}_{i,4}^{ \text{T}}\mathbf{D}_{i}\\ \text{sym}&l_{i,1}\mathbf{E}_{3}&-\mathbf{A}_{i}(\mathbf{I}_{i,2}+\mathbf{I}_{i,3}q_{i})\mathbf{D}_ {i}\\ \text{sym}&\text{sym}&\mathbf{D}_{i}^{\text{T}}[\mathbf{I}_{i,7}-\frac{1}{j}q_{i,j}( \mathbf{I}_{i,8j}+\mathbf{I}_{i,8j}^{\text{T}})]\mathbf{D}_{i}\end{bmatrix} . \tag{42}\]  
wherein the superscripts \({}^{\text{f},\text{t},\text{r}}\) refer to flexible, translational, and rotational components, respectively. The expressions and meanings of the modal invariant submatrices \(I_{i,1}\), \(\mathbf{I}_{i,2}\), \(\mathbf{I}_{i,3}\), \(\mathbf{I}_{i,4}\), \(\mathbf{I}_{i,7}\), and \(\mathbf{I}_{i,8j}\) in (42) are listed in Table 1. The invariant submatrices are defined identically to those in the modal neutral file (MNF) [33], where \(\mathbf{I}_{i,5}\) and \(\mathbf{I}_{i,6}\) are commonly omitted. For one-dimensional and two-dimensional elements, such as cables, beams, plates, and shells, the moment of inertia of the nodes \(\mathbf{H}_{i,p}\) can be neglected. The potential energy of elastic deformation is only a function of the elastic coordinates \(\mathbf{q}_{i}\). Because \(\mathbf{u}_{i}=\mathbf{\Phi}_{i}\mathbf{q}_{i}\), as shown in (28), \(\overline{\mathbf{K}}_{i}\) is the stiffness matrix corresponding to FE variable \(\mathbf{u}_{i}\), as shown in Equation (19); with the orthogonal condition (26), the potential energy of elastic deformation is then  
\[V_{i}=\frac{1}{2}\mathbf{u}_{i}^{\mathsf{T}}\overline{\mathbf{K}}_{i}\mathbf{u}_{i}=\frac{ 1}{2}\mathbf{q}_{i}^{\mathsf{T}}\mathbf{\Phi}_{i}^{\mathsf{T}}\overline{\mathbf{K}}_{i}\bm {\Phi}_{i}\mathbf{q}_{i}=\frac{1}{2}\mathbf{q}_{i}^{\mathsf{T}}\mathbf{\Omega}_{i}^{2}\bm {q}_{i}. \tag{43}\]  
The body's general coordinate is \(\xi_{i}=[\mathbf{q}_{i};\mathbf{x}_{i};\mathbf{\alpha}_{i}]\), and the system stiffness matrix given by Equation (13) is \(\mathbf{K}=\frac{\partial\mathcal{E}_{\text{fast}}}{\partial\mathbf{\xi}}+\frac{ \partial\mathcal{E}_{\text{fast}}}{\partial\mathbf{\xi}}\), which includes both the elastic and inertial parts. Then, the stiffness matrix for body \(i\) is  
\[\mathbf{K}_{i}= \begin{bmatrix}\mathbf{\Omega}_{i}^{2}&&\\ &\mathbf{0}&\\ &&\mathbf{0}\end{bmatrix} +\frac{\partial(\mathbf{M}_{i}\tilde{\mathbf{\xi}}_{i})}{\partial \xi_{i}}. \tag{44}\]