Understanding Cutter Selection for Helical Gears

When cutting gears, especially helical gears, selecting the correct cutter is crucial for achieving the desired tooth profile. Standard gear cutters are typically designed for spur gears and come in sets, with each cutter covering a specific range of teeth. For example, a No. 4 cutter is suitable for cutting spur gears with 26 to 34 teeth.

The Challenge with Helical Gears

Helical gears have teeth that are cut at an angle (α, the helix angle) to the gear's axis. Because of this angle, the shape of the tooth profile in the plane perpendicular to the tooth (the normal plane) is different from the shape in the plane of rotation (the transverse plane). Standard cutters are ground to produce the correct involute profile in the transverse plane for a spur gear.

To use standard spur gear cutters for cutting helical gears, we need to consider how the helix angle affects the tooth profile that the cutter sees. Imagine slicing the helical gear perpendicular to the tooth helix. The gear blank appears elliptical in this plane. The tooth profile in this normal plane corresponds to a spur gear having a larger number of teeth than the actual helical gear.

Introducing the Virtual or Equivalent Number of Teeth

To select the correct standard spur gear cutter for a helical gear, we calculate a 'virtual' or 'equivalent' number of teeth. This virtual number represents the number of teeth a spur gear would need to have in order to have a tooth profile in its transverse plane equivalent to the tooth profile of the helical gear in its normal plane. The standard cutter is then chosen based on this virtual number of teeth.

The formula for the virtual or equivalent number of teeth ($T_e$) for a helical gear is given by:

Where:

• \(T\) is the actual number of teeth on the helical gear.
• \(\alpha\) is the helix angle.
• \(T_e\) is the virtual or equivalent number of teeth.

The standard spur gear cutter is then selected based on this calculated virtual number of teeth, \(T_e\). The cutter number corresponds to the range of teeth that includes \(T_e\).

Applying the Concept to the Question

The question asks for the size of the cutter, which is determined by the virtual number of teeth calculated using the actual number of teeth and the helix angle. The options provide different formulas combining the number of teeth and the helix angle.

Let's look at the options:

1. \((number of teeth)/\alpha\): This formula does not represent the virtual number of teeth.
2. \((number of teeth)/\cos \alpha\): This formula relates to the tooth pitch in the normal plane, not the virtual number of teeth used for cutter selection.
3. \((number of teeth)/(\cos \alpha)^2\): This formula is also not the standard expression for the virtual number of teeth used for cutter selection.
4. \((number of teeth)/(\cos \alpha)^3\): This formula matches the standard formula for the virtual or equivalent number of teeth ($T_e = T / \cos^3 \alpha$). The cutter is selected based on this calculated value.

Therefore, the size of the cutter (or more precisely, the basis for selecting the cutter number from a standard set of spur gear cutters) is determined by the virtual number of teeth, which is calculated using the formula \((number of teeth)/(\cos \alpha)^3\).

In the given example, if a 30 T gear is cut with a helix angle \(\alpha\), the equivalent number of teeth would be \(30 / \cos^3 \alpha\). The cutter would be selected based on this calculated value, choosing a cutter number whose range includes \(30 / \cos^3 \alpha\).

Thus, the expression \((number of teeth)/(\cos \alpha)^3\) is what determines the correct cutter size from a standard set of spur gear cutters when cutting a helical gear.