# A Rapid and Inexpensive Method for Finding the Basic Parameters of Involute Helical Gears
## 2 Materials and Methods
### Helical Gears  
In order to define the profile of a helical gear, a base cylinder is set on a plane \((P)\), as shown in Figure 5. In the \((P)\) plane, \((\Delta)\) is considered the straight line that makes the angle \((\beta_{b})\) with the generatrix of the cylinder tangent to the plane. By rolling the plane \((P)\) without slipping, a ruled surface is generated which can be used as a tooth flank. During the process of flank creation, the points of the straight line \((\Delta)\) will sit on the base cylinder as a helix with constant pitch \((E_{b})\), named the base helix. Each point of the straight line \((\Delta)\) describes an involute generated using the circles of radius \((r_{b})\) (the radius of the base cylinder). As an example, in the foreground plane, the involute \((e)\) is generated and in the background plane, the involute \((e^{\prime})\) is generated. All involute curves commence from the base cylinder and, due to their origin at the base helix, preserve a uniform angular discrepancy across the length of the cylinder's generatrix. The generation process of the flank's surface yields the conclusion that it constitutes a helical involute ruled surface. The intersection of the flank with an arbitrary cylinder of  
Figure 5: Defining the flank for a helical spur gear with involute teeth. radius \(r_{y}\) coaxial to the base cylinder is a helix \((E_{y})\). Based on Figure 6, where the unfolded cylinders are also shown, the inclination angle \(\beta_{y}\) of the helix on the cylinder of radius \(r_{y}\) is found. It must be noted that both helices share the same axial pitch. \[tan(\beta_{y})=\frac{2\pi r_{y}}{p_{z}}\Rightarrow tan(\beta_{y})=tan(\beta_{ b})\frac{r_{y}}{r_{b}} \tag{19}\]  
We introduce the convention that the parameters from a frontal section (normal to the axis of the wheel) are denoted with the index \"\(t\)\". Thus, the relation (19) becomes  
\[tan(\beta_{y})=\frac{tan(\beta_{y})}{cos(a_{yt})} \tag{20}\]  
where \(a_{yt}\) represents the pressure angle of the involute \((e)\) on the circle of radius \(r_{y}\), as in Figure 6. Given that the gearing interaction between the profiles in any section perpendicular to the axes of the wheels mirrors that of two spur gears, it can be asserted that the rack's profile will be linear across all frontal sections. Additionally, with helical teeth, the interaction between the teeth of the wheel and the rack occurs along a straight line \((\Delta)\). It is noteworthy that this line no longer aligns perpendicularly to the direction of displacement. As a result, it becomes apparent that the flank of the helical rack is planar, with the orientation of the teeth not being perpendicular to the direction of displacement. Instead, the teeth's direction forms an angle \(\beta_{0}\) with the direction of displacement. In a section perpendicular to the tooth, the size and the design of the reference helical rack must have standard dimensions, identical to that observed in a rack with spur teeth. Figure 7 shows the basic rack with helical teeth. The base plane \(P\) of the rack is presented in Figure 7. The intersection between the reference plane to any frontal plane is a straight line along which the tooth thickness is identical to the space width. With respect to the reference plane, the following parameters are defined:  
* The addendum of the tooth \(h_{a_{0}}\);  
Figure 6: Establishing the angle of the tooth helix on an arbitrary cylinder. Figure 7: The basic rack with helical teeth. * The dedendum of the tooth \(h_{f_{0}}\).