# Dynamic Stiffness Effect of Mechanical Components on Gear Mesh Misalignment
## 3 Gear Train Modeling Process with Dynamic Stiffness
, \tag{12}\]  
and these component matrices are then defined as:  
\[\mathbf{\hat{M}}_{IF}=\mathbf{M}_{IS}\mathbf{B}+\mathbf{M}_{IF},\ \mathbf{\hat{M}}_{IF}^{T}=\mathbf{\hat{M}}_{FI},\ \mathbf{\hat{M}}_{FF}=\mathbf{M}_{FF}+\mathbf{M}_{FS}\mathbf{B}+\mathbf{B}^{T} \mathbf{M}_{SF}+\mathbf{B}^{T}\mathbf{M}_{SS}\mathbf{B} \tag{13}\]  
\[\mathbf{\hat{K}}_{IF}=\mathbf{K}_{IS}\mathbf{B}+\mathbf{K}_{IF},\ \mathbf{\hat{K}}_{IF}^{T}=\mathbf{\hat{K}}_{FI},\ \mathbf{\hat{K}}_{FF}=\mathbf{K}_{FF}+\mathbf{K}_{FS}\mathbf{B}+\mathbf{B}^{T} \mathbf{K}_{SF}+\mathbf{B}^{T}\mathbf{K}_{SS}\mathbf{B} \tag{14}\]  
This is a typical modeling result for an FE housing with RBEs. Projecting the internal DOFs into the master DOFs at the bearing centers is the next step towards considering the housing effect in the gearbox design process. Using Equations (12) to (14), the linear static equations, which include \(\mathbf{\hat{K}}\mathbf{\hat{u}}=\mathbf{\hat{f}}\), can be specifically presented as:  
\[\left[\begin{array}{cc}\mathbf{K}_{II}&\mathbf{\hat{K}}_{IF}\\ \mathbf{\hat{K}}_{FI}&\mathbf{\hat{K}}_{FF}\end{bmatrix}\right]\left( \begin{array} []{cc}\mathbf{u}_{I}\\ \mathbf{u}_{F}\end{array} \right)=\left( \begin{array}{cc}\mathbf{f}_{I}\\ \mathbf{f}_{F}+\mathbf{B}^{T}\mathbf{f}_{S}\end{array} \right) \tag{15}\]  
With the assumption of \(\mathbf{f}_{I}=\mathbf{0}\), the well-known static constraint modes can be derived as:  
\[\mathbf{u}_{I}=\mathbf{\Psi}\mathbf{u}_{F},\mathbf{\Psi}=-\mathbf{K}_{II}^{- 1}\ \mathbf{\hat{K}}_{IF} \tag{16}\]  
where \(\mathbf{\Psi}\) is a matrix of the static constraint modes that connects the internal DOFs (\(\mathbf{u}_{I}\)) and the master DOFs (\(\mathbf{u}_{F}\)) [22; 23]. This connection is valid only for \(\mathbf{f}_{F}\), which implies that the external forces at the bearing center are dominant. Using Equation (16), the modified displacement vector, \(\mathbf{\hat{u}}\), can be presented simply by the master DOFs as:  
\[\mathbf{\hat{u}}=\mathbf{T}_{I}\mathbf{\overline{u}},\mathbf{T}_{I}=\left[ \begin{array}{cc}\mathbf{\Psi}\\ \mathbf{I}\end{array} \right],\mathbf{\overline{u}}=\mathbf{u}_{F} \tag{17}\]  
Therefore, the mass and stiffness matrices in Equation (12) can be projected using \(\mathbf{T}_{I}\), as:  
\[\mathbf{\overline{M}}=\mathbf{T}_{I}^{T}\mathbf{\hat{M}}\mathbf{T}_{I}, \mathbf{\overline{K}}=\mathbf{T}_{I}^{T}\mathbf{K}\mathbf{T}_{I},\mathbf{ \overline{f}}=\mathbf{T}_{I}^{T}\mathbf{\hat{f}} \tag{18}\]  
and can be detailed as:  
\[\mathbf{\overline{M}}=\mathbf{\hat{M}}_{FF}+\mathbf{\hat{K}}_{FI}\mathbf{ \Psi}+\mathbf{\Psi}^{T}\ \mathbf{\hat{K}}_{IF}+\mathbf{\hat{K}}_{FI}\mathbf{K}_{II}^{-1}\mathbf{M}_{II} \mathbf{K}_{II}^{-1}\ \mathbf{\hat{K}}_{IF},\ \mathbf{\overline{K}}=\mathbf{\hat{K}}_{FF}-\mathbf{\hat{K}}_{FI}\mathbf{K}_{II}^ {-1}\ \mathbf{\hat{K}}_{IF} \tag{19}\]  
The condensed matrices, \(\mathbf{\overline{M}}\) and \(\mathbf{\overline{K}}\), are \(N_{F}\) by \(N_{F}\) matrices. The master DOFs, \(\mathbf{u}_{F}\), represent the bearing DOFs, which generally consist of six DOFs with three translations (\(u_{x},u_{y},u_{z}\)) and three rotations (\(\theta_{x},\theta_{y},\theta_{z}\)). Therefore, \(N_{F}\) is defined as \(6N_{B}\) (\(N_{f}=6N_{B}\)). The reduced matrices in Equation (19) are then represented as:\[\overline{\mathbf{M}}= \begin{bmatrix}\overline{\mathbf{M}}_{B_{1}B_{1}}&\overline{ \mathbf{M}}_{B_{1}B_{j}}\\ \overline{\mathbf{M}}_{B_{1}B_{1}}&\overline{\mathbf{M}}_{B_{1}B_{j}}\end{bmatrix} ,\ \overline{\mathbf{K}}= \begin{bmatrix}\overline{\mathbf{K}}_{B_{1}B_{1}}& \overline{\mathbf{K}}_{B_{1}B_{j}}\\ \overline{\mathbf{K}}_{B_{1}B_{1}}&\overline{\mathbf{K}}_{B_{1}B_{j}}\end{bmatrix}