# Parallel Direct Solution of Flexible Multibody Systems Based on Block Gaussian Elimination
## 2 Flexible Multibody Formulation with the Craig-Bampton Method
### Time Integration Procedures of Flexible Multibody Systems  
#### 2.1.1 System Governing Equations  
The Lagrange equations of the first kind are utilized to express the governing equations of the flexible multibody system [2] as  
\[\left\{\begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L (\tilde{\xi},\tilde{\xi})}{\partial\xi}\right)-\frac{\partial L(\tilde{\xi}, \tilde{\xi})}{\partial\xi}+\left(\frac{\partial\xi}{\partial\xi}\right)^{ \mathrm{T}}\boldsymbol{\lambda}=\boldsymbol{\tau},\\ \boldsymbol{c}(\boldsymbol{\xi})=\boldsymbol{0},\end{array}\right. \tag{1}\]  
wherein \(\boldsymbol{\xi}\) is the general displacement of the bodies, \(L=T-V\) denotes the Lagrange function with the system's kinetic energy \(T\) and potential energy \(V\), \(\boldsymbol{\tau}\) is the general force vector of the external loads, \(\boldsymbol{c}\) is a set of complete constraint equations that are solely dependent on the general displacement \(\boldsymbol{\xi}\), and \(\boldsymbol{\lambda}\) corresponds to the Lagrangian multipliers of the constrains.  
The constraints between bodies are explicitly expressed by the algebraic equations \(\boldsymbol{c}(\boldsymbol{\xi})=\boldsymbol{0}\) in Equation (1). This facilitates convenient computer coding. However, the variable \(\boldsymbol{\lambda}\) has the same magnitude as the constraint forces and often introduces numerical problems due to its large difference in magnitude compared to the general displacement variable \(\boldsymbol{\xi}\). This issue can be effectively resolved by multiplying the constraint equations \(\boldsymbol{c}(\boldsymbol{\xi})=\boldsymbol{0}\) by a scale factor \(a\) to balance the magnitudes between \(\boldsymbol{\xi}\) and \(\boldsymbol{\lambda}\), as follows:  
\[\left\{\begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L (\tilde{\xi},\tilde{\xi})}{\partial\xi}\right)-\frac{\partial L(\tilde{\xi}, \tilde{\xi})}{\partial\xi}+\left(a\cdot\frac{\partial\xi}{\partial\xi}\right)^ {\mathrm{T}}\boldsymbol{\lambda}=\boldsymbol{\tau},\\ a\cdot\boldsymbol{c}(\boldsymbol{\xi})=\boldsymbol{0}.\end{array}\right. \tag{2}\]  
A method for calculating the scale factor \(a\) has been detailed in the authors' previous work [31], and readers are encouraged to consult it for further information. Since Equation (2) does not differ fundamentally from Equation (1), we still use Equation (1) instead of Equation (2) by taking \(\boldsymbol{c}\) as \(a\cdot\boldsymbol{c}\) in Equation (1).  
In Equation (1), the Lagrange function relating terms are actually the general inertial forces \(\boldsymbol{F}_{\mathrm{iner}}\) and general elastic forces \(\boldsymbol{F}_{\mathrm{elas}}\), and they are the internal forces within the bodies. By separating them from the external forces \(\boldsymbol{\tau}\) and constraint forces \(\boldsymbol{c}_{\boldsymbol{\xi}}^{\mathrm{T}}\boldsymbol{\lambda}\), the system governing equation can be rewritten as  
\[\left\{\begin{array}{l}\boldsymbol{F}_{\mathrm{iner}}(\boldsymbol{\xi}, \dot{\boldsymbol{\xi}},\ddot{\boldsymbol{\xi}})+\boldsymbol{F}_{\mathrm{elas}} (\boldsymbol{\xi})=\boldsymbol{\tau}(\boldsymbol{\xi},\dot{\boldsymbol{\xi} })-\boldsymbol{c}_{\boldsymbol{\xi}}^{\mathrm{T}}(\boldsymbol{\xi})\boldsymbol {\lambda}\,\\ \boldsymbol{c}(\boldsymbol{\xi})=\boldsymbol{0}\,\end{array}\right. \tag{3}\]  
wherein \(\boldsymbol{c}_{\boldsymbol{\xi}}^{\mathrm{T}}\) is the abbreviation of \((\partial\boldsymbol{c}/\partial\boldsymbol{\xi})^{\mathrm{T}}\).