A mathematical model of parametric tooth profiles for elliptical gears
Journal
Mechanism and Machine Theory
Journal Volume
221
Start Page
106386
ISSN
0094114X
Date Issued
2026-05
Author(s)
Abstract
This study presents a general mathematical model for any parametric tooth profiles of elliptical gears, incorporating three major parameters that relate the pitch point and the contact point. A first-order differential equation is derived to constrain the relationship between these parameters, ensuring the tooth profiles are conjugated. Analytical expressions for the profiles of the rack cutter and a pair of elliptical gears are provided, but the manufacturing tools and methods are not limited, allowing high customization. These expressions are presented using the same parameters without the need for coordinate transformation. To illustrate the applicability of this model, two types of teeth, involute and cycloid, are used as examples. Their analytical expressions are presented with geometric significance and are notably concise. Additionally, the analytical expressions for the base curves of involute teeth are derived. Some geometric properties are also derived to analyze the durability. Finally, two methods for designing tooth profiles are proposed, allowing researchers to either design new profiles or analyze and adjust existing profiles. Three numerical design examples demonstrate the improved geometric properties.
Subjects
Elliptical gear
Mathematical model
Non-circular gear
Pitch curve
Publisher
Elsevier Ltd
Type
journal article