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Calculation of Virtual Number of Teeth of Helical Gear
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The helix angle $\psi$, is always measured on the cylindrical pitch surface. $\psi$ value is not standardized. It ranges between $15^{\circ}$ and $45^{\circ} .$ Commonly used values are 15,23, 30 or $45^{\circ} .$ Lower values give less end thrust. Higher values result in smoother operation and more end thrust. Above $45^{\circ}$ is not recommended.

The circular pitch $(\mathrm{p})$ and pressure angle $(\alpha)$ are measured in the plane of rotation, as in spur gears. These quantities in normal plane are denoted by suffix $\mathrm{n}\left(p_{n}, \alpha_{n}\right)$ as shown in fig below. It may be observed,

$p_{n}=\mathrm{p} \cos \varphi,$ Normal module $m_{n}$ is $=\mathrm{m}$ cos $\varphi .$ Normal module is used for hob selection. The pitch diameter (d) of the helical gear is $\mathrm{d}=\mathrm{Z} \mathrm{m}=\mathrm{Z} m_{n} / \cos \varphi$

The axial pitch $(\mathrm{pa})$ is $(\mathrm{pa}) = \mathrm{p} / \tan \varphi$

In the case of a helical gear, the resultant load between mating teeth is always perpendicular to the tooth surface. Hence bending stresses are computed in the normal plane, and the strength of the tooth as a cantilever beam depends on its profile in the normal plane. Fig. 14 shows the view of a helical gear in the normal and transverse plane.

The table shows the difference between a transverse module and a normal module.

Parameter Transverse Module $m_t \text{ or } m$ Normal Module $m_n$
Definition It is the module of tooth datum orthogonal to the centre axis of gear It is the module of tooth datum orthogonal to the thread helix
Advantages Replaces spur gears having the same module, number of teeth, and centre distance. Modifications of spur gears are made by gear cutting or grinding machines, even if they have different helix angles.
Disadvantages Special gear cutting or grinding machines are required for processing each helix angle. Have a centre distance value different from that of a spur gear, although they have the same module size and the same number of gear teeth. The centre distance value is rarely an integral number.

The first figure shows the pitch cylinder and one tooth of a helical gear. The normal plane intersects the pitch cylinder in an ellipse. If ‘d’ is the pitch diameter of the helical gear, the major and minor axes of the ellipse will be $\frac{d}{\cos \varphi}$ and $\mathrm{d} .$ The radius of curvature '$R_{e}$' at the extremes of the minor axis from coordinate geometry is found to be $\frac{d}{2 \cos \varphi^{2}}$

The shape of the tooth in the normal plane is nearly the same as the shape of a spur gear tooth having a pitch radius equal to radius $R_e$ of the ellipse.

$\operatorname{Re}=\frac{d}{2 \cos \varphi^{2}}$

The equivalent number of teeth (also called virtual number of teeth), $Z_{v},$ is defined as the number of teeth in a gear of radius Re:

$$Z_{v}=\text { diameter } \times \text{ module }=\left(2 R_{e}\right)\left(m_{n}\right)$$

Substituting $R_{e}, m_{n}=\mathrm{m} \cos \varphi,$ and $\mathrm{d}=\mathrm{Z} \mathrm{m}$

$$Z_{v}=2 \frac{d}{2 \cos \varphi^{2}} \frac{m}{\cos \varphi}=\frac{Z}{\cos \varphi^{3}}$$