How profile shift affects gear addendum and tip diameter

During my work in transmission research and development, I often studied engineering formulas to understand how they were derived rather than simply memorizing them.

When I encountered a formula in a technical reference book, I liked to work through the derivation myself whenever possible. This approach helped me understand the underlying principles, remember the relationships more effectively, and apply them to different engineering problems.

While studying profile-shifted gears, I became curious about why the addendum and tip diameter of one gear could depend on the profile shift coefficient of its mating gear, as well as the center distance correction coefficient.

Instead of accepting these relationships as given formulas, I used basic gear geometry to investigate why they occur.

This post documents that derivation and the geometric relationship behind the result.

1. Understanding the Gear Geometry

A profile-shifted gear is generated with a profile shift that changes the position of the basic rack relative to the reference position.

When two gears mesh, their tooth geometry must be considered together. The resulting geometry depends not only on the profile shift coefficients of the individual gears but also on the center distance correction used for the gear pair.

The attached sketches illustrate the relationship between the reference circles, base circles, addendum circles, and the tooth geometry of the two mating gears.

For the following derivation, the notation is:

• (m): Gear module

• (x_1): Profile shift coefficient of gear 1

• (x_2): Profile shift coefficient of gear 2

• (y): Center distance correction coefficient

The equations below use the notation and geometric relationships from my original calculation.

2. Addendum and Dedendum

For the gear geometry considered in this example, the relevant addendum and dedendum expressions are

Addendum = $(1 + y - x_2)\,m$

Dedendum = $(1.25-x_1)m$

Total depth = Addendum + Dedendum

$$<span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">=\; (</span></span>\mathrm{<span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">2.25</span></span>}<span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">+</span></span>\mathrm{<span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">y</span></span>}<span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">-</span></span><span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">(</span></span>{\mathrm{<span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">x</span></span>}}_{\mathrm{<span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">1</span></span>}}<span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">+</span></span>{\mathrm{<span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">x</span></span>}}_{\mathrm{<span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">2</span></span>}}<span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">)</span></span><span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">)</span></span>\text{<span style="font-family: arial;"><span style="font-size: small;">\hspace{0.17em}</span></span>}\mathrm{<span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">m</span></span>}$$

These expressions show that the tooth dimensions cannot always be considered independently of the mating gear and the corrected center distance.

The next step is to examine the clearance between the two gears.

3. Deriving the Clearance Between the Gears

In the geometric construction shown in the diagram, the clearance between the two gears is represented by the length AB.

Using the addendum and dedendum relationships above, the clearance is expressed as

$$A\!B = (1.25 - x_2)m - \{(1 + y - x_2)m - y m\}$$
$$= \boxed{0.25\,m}$$

The result shows that the clearance in this geometric construction is equal to one quarter of the module.

Notice that the terms containing the profile shift coefficient (x_2) cancel out. The center distance correction term (y) also cancels through the subtraction shown above.

This cancellation is the key mathematical observation in the derivation.

4. Why Does the Mating Gear Affect Addendum and Tip Diameter?

The derivation illustrates why the addendum of a profile-shifted gear cannot always be determined by considering that gear in isolation.

The tooth geometry of a mating gear pair must satisfy the required geometric relationships and clearance. When the profile shift and center distance correction are introduced, the relevant dimensions must remain consistent with those constraints.

Consequently, the addendum expression for the gear considered in this construction includes the center distance correction coefficient (y) and the mating gear's profile shift coefficient (x_2).

The tip diameter is also related to the addendum. For an external spur gear, the tip diameter is calculated from the reference diameter and the addendum:

d_a=d+2h_a

where:

• (d_a): Tip diameter

• (d): Reference diameter

• (h_a): Addendum

For a standard external spur gear, the reference diameter is

d=mz

where (z) is the number of teeth.

Combining these relationships gives

d_a=mz+2h_a

Thus, when the addendum expression depends on the relevant profile shift and center distance correction terms, the resulting tip diameter depends on those terms as well.

5. Conclusion

The purpose of this calculation was not simply to obtain an addendum or tip diameter formula. I wanted to understand why the formula contained terms associated with the mating gear and the center distance correction.

By working through the geometry, I could see how the clearance relationship led to the cancellation of certain terms and why the dimensions of a mating gear pair had to be considered together.

This is one of the approaches I found useful during engineering development: instead of memorizing a formula from a reference book, I tried to derive it from the underlying geometric relationships.

Understanding the derivation made the formula easier to remember and gave me a better basis for applying it to other gear design problems.