Analysis of spur gear teeth for static deflections

The problem of the deflection of spur gear teeth has been previously investigated analytically by several authors. Two of the papers dealing with this topic were written by Timoshenko and Baud (18)t and Marin and Shenk (11). Both papers employ the cantilever beam approach in which the spur gear tooth is represented as a rigidly supported, cantilever beam of uniform thickness tapered in the plane of load application. Deflection is considered to be caused by bending moment, shearing force, and Hertzian pressure at the point of load application.

Two additional papers were written by Weber (25) and Attia (3) in which the actual involute gear tooth was considered to be an elastically supported cantilever beam rather than rigidly clamped at the root of the tooth. Likewise, circumferential deformation of the gear body was considered. However, the derived equations for the total deflection due to these effects are exceedingly complicated and require the use of numerical methods for their solution since some of the equations are in integral form.