# Dynamic Stiffness Effect of Mechanical Components on Gear Mesh Misalignment
## 2 Summary of the Gearbox Design Process  
An optimal gearbox design process requires various advanced methods and iteration procedures, as can be seen in Figure 2. In the first step, a preliminary gear sizing is performed to meet the specified design requirements for the gearbox. Design parameters relevant to the size of the gearbox, such as the gear ratio, center distance, and gear face width, can be determined in this step.  
In the second step, the gear macro-geometry is determined, including the normal module, number of gear teeth, pressure angle, helix angle, and profile shift coefficient, all of which are based on gear rating standards [10; 11; 12; 13; 14]. Bearings that can support the required gear loads are then selected, and shaft and housing are designed based on the gear macro-geometry.  
In the third step, the gear micro-geometry is determined, including the tip and root relief, lead slope, and crowning; these are based on the load distribution and transmission error analysis. Gear mesh misalignment between the meshed gears is a leading factor in the gear strength analysis, and the misalignment is iteratively computed using the shaft deflection and bearing deformation. The misalignment is then converted to displacement along the line of action on the gear mesh, as shown in Figure 3. A smaller mesh misalignment represents a more uniform load distribution on the gear mesh. The gear mesh misalignment in the line of action direction (Figure 3) is defined as:  
\[F_{\beta X}=b\ \tan\varphi,\ \varphi=\theta_{skew}\cos\alpha_{wt}+\theta_{slope} \tag{1}\]  
in which \(F_{\beta X}\) is the gear mesh misalignment; \(b\) is the face width; \(\varphi\) is the angle of gear mesh misalignment; \(\alpha_{wt}\) is the working transverse pressure angle; and \(\theta_{skew}\) and \(\theta_{slope}\) are the angles of gear skew and slope, respectively.  
From this gear mesh misalignment, we can compute the face load factor for contact stress which accounts for the effect of the load distribution over the face width on the gear contact stress. The definition of the face load factor is:  
Figure 2: Typical gearbox design procedure ([PERSON], 1951).  
\[K_{H\beta}=\frac{\text{Maximum load per unit face width}}{\text{Average load per unit face width}}=\frac{\text{max}(F/b)}{F_{m}/b} \tag{2}\]  
where \(K_{H\beta}\) is the face load factor, and \(F\) and \(F_{m}\) are the load at an arbitrary position of the tooth flank and the mean (average) load at the reference circle, respectively [10]. In engineering practice, the face load factor is computed as:  
\[K_{H\beta}=\left\{\begin{array}{c}\sqrt{\frac{2F_{\beta Y}c_{\beta\beta}}{F_{ m}/b}},\text{if}\frac{F_{\beta Y}c_{\beta\beta}}{2F_{m}/b}\geq 1,\\ 1+\frac{F_{\beta Y}c_{\beta\beta}}{2F_{m}/b},\text{if else},\end{array}\right. \tag{3}\]  
and then the effective gear mesh misalignment, denoted as \(F_{\beta Y}\), is defined as:  
\[F_{\beta Y}=F_{\beta X}-y_{\beta} \tag{4}\]  
where \(c_{\gamma\beta}\) is the mean value of the mesh stiffness per unit face width, and \(y_{\beta}\) is the running-in allowance for a gear pair.  
The gear mesh misalignment causes non-uniform load distribution along the face width in the plane of action, possibly resulting from gear elastic deflection, housing flexibility, and bearing deformation. As the most important factor in the gear strength rating, the face load factor must be equal to or greater than 1.0. A value of 1.0 is assumed to indicate that loads acting on gear tooth flanks are evenly distributed. As the value increases above 1.0, the load distribution on the gear tooth flanks becomes increasingly uneven [1; 2]. This iteration procedure corresponds to _Iteration_ #2 in the overall gear design process shown in Figure 2.  
Figure 3: Schematic of gearbox with transverse line of action between the meshed gears.