Profile shifted gear addendum and tip diameter

 I worked in R&D for transmission development. I used to study by myself to prove any formular for better understanding. Sometimes I look back my documents and surprised to myself how I did this all.

Below is to find the reason why profile shifted gear addendum and tip diameter is decided by center distance correction coefficient and profile shift coefficient of partner gear. I was wondering why some gear specs is decided by partner gear's specification so I calculated to see it.

Addendum and Dedendum based on gear 1's reference circle is as below.

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

Dedendum = $(1.25-x_2)m$

Total depth = Addendum + Dedendum

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

The clearance between 2 gears are same as length of AB.

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

​This proves that even after profile shift is applied, to meet the clearance (0.25m) between 2 gears, Addendum value should be decided by center distance correction coefficient and partner gear's profile shilft coefficient. For same reason, tip diameter is decided also by partner gear's profile shift coefficient.