Abstract
The mathematical framework used to address the geometrically non-conjugate gear tooth contact problem in three-dimensional space represents a sophisticated task that requires substantial computational resources. The conventional approach to tooth contact analysis (TCA), involving five non-linear equations with five unknown parameters, uses an implicit model where convergence cannot be guaranteed. In recent years, researchers have proposed several new methods for analyzing gear tooth contact, offering more efficient alternatives to the conventional TCA model. However, most of these methods rely on a discretized approach, resulting in approximate solutions and the use of additional optimization algorithms, such as particle swarm optimization to find the initial contact or grid representation of the tooth surface composed of nodal points. This additional manipulation complicates the process of determining the contact trace and increases the computational load. Furthermore, most of these methods cannot be applied to all spiral gear tooth engagement cases involving complex misalignments and tooth surface modifications; they are limited to specific cases, such as pure-rolling gear drives, parallel axis contacts, or situations without axial misalignment. Moreover, the majority of these methods, like the conventional model, suffer from the challenge of determining proper starting values to run the iterative process required to solve the TCA non-linear equations, where a random choice of starting values might lead to the divergence of the numerical algorithm. To overcome this limitation and enhance the convergence properties, a novel TCA method has been introduced. This investigation contrasts the conventional and proposed novel model in terms of their convergence probability, accuracy, and computational speed, thereby contributing to a comprehensive understanding of the strengths and limitations inherent in these approaches. Compared to conventional and other TCA models suggested in the past few years, the proposed novel TCA method uses fundamental geometric principles to analytically reduce the non-linear meshing equations from five to two and decrease the number of unknown variables from five to two. This approach does not require the application of discretization methods or additional optimization algorithms. The iterative numerical calculation process can be initiated with any values, and the system will find all contact points exactly at once. Furthermore, the proposed model is applied to simulate the involute tooth contact of spiral bevel gears within a spherical coordinate system. This application aims to examine the impact of misalignment or profile modifications on the transmission error, path of contact, and stress.