Improved Time-Varying Tooth Stiffness Calculation in Cracked Spur Gear Using Modified Limiting Line

Defects and cyclic loads often lead to tooth-root cracks in spur gear transmission systems, affecting system stiffness, vibration patterns, and lifespan. Traditional methods using straight limiting lines and parabolic curves to assess reduced load-bearing areas due to cracks have limitations, including substantial errors with deep cracks and incompatibility with semi-analytical techniques.

This paper introduces a novel approach: a modified limiting line for calculating gear mesh stiffness over a broader range of crack depths. Gear body is treated as rigid to avoid error in gear-body deflection estimates. The modified limiting line is defined by minimizing the difference between mesh stiffness obtained using analytical and finite element methods at a particular mesh position. Moreover, the orientation is used to derive mesh stiffness at additional mesh sites for a given crack configuration. Also, an optimization problem involving a compatibility condition is proposed to determine the load-sharing ratios during double tooth pair engagement.

The optimization problems, featuring nonlinear constraints, are solved using sequential quadratic programming. The mesh stiffness and load-sharing ratios are obtained for various crack configurations and are verified using the finite element method. Moreover, the dynamic responses at different crack levels are obtained.

The current approach demonstrates better accuracy at higher crack levels than the existing analytical methods and is computationally less expensive than finite element methods.