# Dynamic Stiffness Effect of Mechanical Components on Gear Mesh Misalignment
## 3 Gear Train Modeling Process with Dynamic Stiffness
Using the relations \(\mathbf{U}(x_{i},t)=\mathbf{u}e^{\pm jwt}\) and \(\mathbf{F}(x_{i},t)=\mathbf{f}e^{\pm jwt}\) with \(j^{2}=-1\), Equation (7) is rewritten as:  
\[\mathbf{K}_{DS}(\omega)\ \mathbf{u}=\mathbf{f},\mathbf{K}_{DS}(\omega)=\mathbf{K}- \omega^{2}\mathbf{M}\pm j\omega\mathbf{C} \tag{8}\]  
where \(\omega\) indicates the angular frequency (rad/s), and the dynamic stiffness, which contains the potential and kinetic energies, denoted as \(\mathbf{K}_{DS}(\omega)\) to distinguish it from the static stiffness \(\mathbf{K}\). The angular frequency component, \(\omega\), of the dynamic stiffness is an unknown variable, and depends on the critical speed of the gear drivetrain under the operation conditions. The housing structure is typically connected to the bearing center using a rigid body element (RBE), which is also known as a multipoint constraint [18; 19; 20; 21]. Here, we use a conventional RBE formulation with the displacement constraints between master and slave DOFs determined by the master-slave, Lagrange multiplier, and penalty methods. In the master-slave method, DOFs are separated among internal, slave, and master nodes, as shown in Figure 4. In this case, the DOFs at the bearing centers are defined as masters, the DOFs constrained to the master DOF are slaves, and the other DOFs are internal. The components of Equation (8), except for the damping matrix, can then be rewritten as  
\[\mathbf{M}=\left[ \begin{array}{ccc}\mathbf{M}_{II}&\mathbf{M}_{IS}&\mathbf{ M}_{IF}\\ \mathbf{M}_{SI}&\mathbf{M}_{SS}&\mathbf{M}_{SF}\\ \mathbf{M}_{FI}&\mathbf{M}_{FS}&\mathbf{M}_{FF}\end{array} \right],\mathbf{K}= \left[ \begin{array}{ccc}\mathbf{K}_{II}&\mathbf{K}_{IS}&\mathbf{K}_{IF}\\ \mathbf{K}_{SI}&\mathbf{K}_{SS}&\mathbf{K}_{SF}\\ \mathbf{K}_{FI}&\mathbf{K}_{FS}&\mathbf{K}_{FF}\end{array} \right],\mathbf{u}= \left( \begin{array}{c}\mathbf{u}_{I}\\ \mathbf{u}_{S}\\ \mathbf{u}_{F}\end{array} \right),\mathbf{f}=\left( \begin{array}{c}\mathbf{ f}_{I}\\ \mathbf{f}_{S}\\ \mathbf{f}_{F}\end{array} \right) \tag{9}\]  
in which the internal, slave, and master DOFs are denoted by the subscripts \(I\), \(S\), and \(F\), respectively. Here, we define a certain transformation matrix \(\mathbf{A}_{ij}^{T}\) as \(\mathbf{A}_{ji}\) (\(\mathbf{A}_{ij}^{T}=\mathbf{A}_{ji}\)). When we define the number of DOFs of the model as \(N\), the numbers of the internal, slave, and master DOFs can be denoted as \(N_{I}\), \(N_{S}\), and \(N_{F}\), respectively, and these typically have the following relation: \(N_{I}>>N_{S}>N_{F}\) and \(N=N_{I}+N_{S}+N_{F}\). The master DOFs are typically independent, and the slave DOFs are physically dependent on the master DOFs. Considering these constraints, the slave DOFs can be presented using the Boolean matrix \(\mathbf{B}\) as:  
\[\mathbf{u}_{S}=\mathbf{B}\mathbf{u}_{F} \tag{10}\]  
The Boolean matrix, \(\mathbf{B}\), is a \(N_{S}\) by \(N_{F}\) rectangular matrix. Using Equation (10), the displacement vector, \(\mathbf{u}\), is then transformed into the modified vector, \(\hat{\mathbf{u}}\), by eliminating the slave DOFs:  
Figure 4: One rigid body element (RBE), where the red dot represents a master node, yellow dots represent slave nodes, and blue dots represent internal nodes. \[\mathbf{u}=\mathbf{T}_{S}\mathbf{\hat{u}},\mathbf{T}_{S}=\left[ \begin{array}{cc} \mathbf{I}&\mathbf{0}\\ \mathbf{0}&\mathbf{B}\\ \mathbf{0}&\mathbf{I}\end{array} \right],\mathbf{\hat{u}}=\left( \begin{array}{ cc}\mathbf{u}_{I}\\ \mathbf{u}_{F}\end{array} \right) \tag{11}\]  
where \(\mathbf{T}_{S}\) is a transformation matrix, and \(\mathbf{I}\) is an identity matrix. Using Equations (9) and (11) in Equation (8), we obtain the following modified matrices:  
\[\mathbf{\hat{M}}=\mathbf{T}_{S}^{T}\mathbf{M}\mathbf{T}_{S}= \begin{bmatrix} \mathbf{M}_{II}&\mathbf{\hat{M}}_{IF}\\ \mathbf{\hat{M}}_{FI}&\mathbf{\hat{M}}_{FF}\end{bmatrix} ,\ \mathbf{\hat{K}}=\mathbf{T}_{S}^{T}\mathbf{K}\mathbf{T}_{S}= \begin{bmatrix}\mathbf{K}_{II}&\mathbf{\hat{K}}_{IF}\\ \mathbf{\hat{K}}_{FI}&\mathbf{\hat{K}}_{FF}\end{bmatrix}