# Dynamic Stiffness Effect of Mechanical Components on Gear Mesh Misalignment
## 3 Gear Train Modeling Process with Dynamic Stiffness  
This section presents the reduced-order modeling procedure for the gear train structure and introduces the definition of dynamic stiffness. Among the gear train components, the gears and bearings have been simply modeled using analytical and empirical techniques. However, the shaft and housing structural components are modeled using the finite element (FE) method. In particular, the FE method is the most popular approach to modeling housing structures with arbitrary and complicated geometries, and the approach is applied here via Romax-DESIGNER (R17) and KISSsoft (2017). This is the most popular gearbox design and analysis software used to consider the elastic behavior of the gearbox components in the design process [5,18,19]. Before modeling the FE housing, the initial positions of bearing center are first determined within the rigid body assumption (Figure 3) as:  
\[\mathbf{q}_{0}=\left[\left(\mathbf{q}_{0}^{1}\right)^{T}\left(\mathbf{q}_{0}^ {2}\right)^{T}\cdots\left(\mathbf{q}_{0}^{j}\right)^{T}\cdots\left(\mathbf{q}_ {0}^{N_{B}}\right)^{T}\right]^{T}\!\!,\ \mathbf{q}_{0}^{j}=\left( \begin{array}{c}x_{0}^{j}\\ y_{0}^{j}\\ z_{0}^{j}\end{array} \right) \tag{5}\]  
where \(\mathbf{q}_{0}\) is the initial position vector of the bearings, \(\mathbf{q}_{0}^{j}\) is the initial position of the \(j\)th bearing, and \(N_{B}\) denotes the number of bearings (variables associated with bearings are denoted by the subscript \(B\)). The bearing center plays an important role in connecting the gearbox components, including the shaft, bearings, gears, and housing. Using the initial position \(\mathbf{q}_{0}\) within the quasi-static condition, the following initial reaction forces of the bearings are computed as:  
\[\mathbf{f}_{0}=\left[\left(\mathbf{f}_{0}^{1}\right)^{T}\left(\mathbf{f}_{0}^ {2}\right)^{T}\cdots\left(\mathbf{f}_{0}^{j}\right)^{T}\cdots\left(\mathbf{f} _{0}^{N_{B}}\right)^{T}\right]^{T}\!\!,\ \mathbf{f}_{0}^{j}=\left(f_{x_{0}}^{j}\ f_{y_{0}}^{j}\ f_{x_{0}}^{j}\ T_{y_{0}}^{j}\ T_{z_{0}}^{j}\right)^{T} \tag{6}\]  
where \(\mathbf{f}_{0}\) is the initial reaction force vector of the bearings; \(\mathbf{f}_{0}^{i}\) is the initial reaction force of the \(j\)th bearing; and \(f\) and \(T\) are the force and torque (moment) components, respectively. The initial reaction force, \(\mathbf{f}_{0}\), is then loaded into the gear train model at the initial bearing center, \(\mathbf{q}_{0}\), and the gear train deformation can be computed in the quasi-static condition. The bearing center is then updated according to the gear train deformation, which means that the bearing reaction force also changes. Therefore, the reaction force and bearing center location should be iteratively computed until these updated values are the same as the previous values of \(\mathbf{q}_{i+i}=\mathbf{q}_{i}\) and \(\mathbf{f}_{i+i}=\mathbf{f}_{i}\). This information is required to more accurately predict and compute gear mesh misalignment. The linear equations of motion for the housing structure are then defined as:  
\[\mathbf{M}\ddot{\mathbf{U}}(x,t)+\mathbf{C}\dot{\mathbf{U}}(x,t)+\mathbf{K} \mathbf{U}(x,t)=\mathbf{F}(x,t) \tag{7}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are the mass, damping, and stiffness matrices of the FE housing structure, respectively; \(\ddot{\mathbf{U}}(x,t)\), \(\dot{\mathbf{U}}(x,t)\), \(\mathbf{U}(x,t)\), and \(\mathbf{F}(x,t)\) are the acceleration, velocity, displacement, and force vectors, respectively; and \(x\) and \(t\) are the space and time variables.