Fractal Model of Double Involute Gears Time-Varying Meshing Stiffness Considering Friction Factors

Most conventional time-varying gear mesh stiffness calculations do not take into account the effect of tooth profile on the time-varying gear mesh stiffness. Due to the machining characteristics of gears, there is usually a certain amount of roughness on the tooth surface, similar to the distribution of many micro-convexities on the tooth surface. This tooth surface profile is characterised by a random distribution of micro-convexities in a small area, but a self-similar distribution over the whole range. In order to solve more accurately for the time-varying mesh stiffness of double involute gears, this paper takes into account the microscopic morphology and friction of the tooth surfaces and combines a pair of inverse stepped tapered roller contact models to “segment” the gear contact line at each moment of engagement, with the contact on each segment being equivalent to a cylindrical contact. And based on M-B fractal theory and Hertzian contact theory, a time-varying meshing stiffness model for double involute gears considering fractal tooth surface roughness is developed. Analysis of the effect of tooth profile parameters and friction factors on the time-varying meshing stiffness of gears. By comparing the results of the gear mesh stiffness fractal model, finite element model and ISO 6336-1:2006 (E), the calculation results of the fractal model proposed in this paper are closer to the ISO calculation results, and the error between them is within 9.4%, which verifies the reasonableness of the proposed fractal model. The effects of friction factor, gear material characteristics, fractal dimension and roughness amplitude on the time-varying mesh stiffness of double involute gears were analysed using Matlab software based on the proposed fractal model of double involute gear time-varying mesh stiffness. The results showed that under the same load conditions, the increase in the friction factor leaded to an increase in the critical area for the elastic-plastic deformation of the micro-convex body on the tooth surface, i.e. an increase in the percentage of the elastic contact area of the gear teeth during meshing. Therefore, the gear meshing stiffness was increased. When the value of the friction factor was in the range of 0.1~0.3, the gear mesh stiffness was less affected by the change in the friction factor. When the value of the friction factor was in the range of 0.3~0.9, the gear mesh stiffness was more influenced by the variation of the friction factor. An increase in the material properties of the gear, i.e. an increase in the yield limited of the gear material, increased the elastic contact area of the gear teeth during meshing and increased the meshing stiffness of the double involute gear. The increase in fractal dimension leaded to a reduction in tooth surface roughness, the tooth surface became smoother and the contact area increases, resulting in an increase in the meshing stiffness of the double involute gear. When the fractal dimension was less than 1.4, there was no significant effect on the mesh stiffness of double involute gears. A reduction in roughness amplitude increased the mesh stiffness of double involute gears. As the fractal dimension increased, the magnitude of the change in double involute gear mesh stiffness with roughness amplitude became more pronounced. This study provided a theoretical basis for the estimation of the time-varying mesh stiffness of double involute gears, the development of machining processes, the selection of machining materials and the analysis of gear friction dynamics in conjunction with tooth surface profiles.