Calculation of static transmission errors associated with thermo-elastic coupling contacts of spur gears

2.1. Calculation of key parameters of thermo-elastic coupling contacts

The heat flux and convection heat transfer coefficient are key parameters of thermo-elastic coupling temperature field of spur gears.

According to the reference [9-12], the heat flux of gear tooth surface can be written in:

where $q_{M1}$ is the heat flux of meshing surface of the driving gear, $q_{M2}$ is the heat flux of meshing surface of driven gear. $b$ is the half degree of contact bandwidth at the gear meshing point, $q$ is the instantaneous heat flow input on the surface of gear meshing tooth, $\omega$ is the angular velocity of gear, $v_S$ is the tangential velocity at the gear meshing point. Differentiating the driving and driven gear with the subscript 1 and 2.

According to the reference [10], the convective heat transfer coefficient of gear meshing surface can be written in:

where $k_o$ is the thermal conductivity of lubricating oil, ${\rho }_o$ is the density of lubricating oil, $c_o$ is the specific heat capacity of lubricating oil, ${\delta }_o=k_o/{\rho }_o{c}_o$ is the temperature coefficient of lubricating oil, $v_{o1}$ is the kinematic viscosity of lubricating oil at the temperature of oil supply, $r_c$ the radius of the indexing circle of a gear, $H$ is the height of the gear tooth, $q_{tot}$ is the standardized cooling capacity.

The convection heat transfer coefficient of the gear face can be written in:

where $N{u}_{ao}$ is the Nusselt number of the mixture of air and lubricating oil. ${Pr}_{ao}$ is the Prandtl number of the mixture of air and lubricating oil. $k_{ao}$ is the thermal conductivity of mixture of air and lubricating oil. $r_k$ is the rotation radius of the meshing point. $v_{ao}$ is the kinematic viscosity of ai and lubricating oil. $m_s$ is a constant that represents the temperature variation along the radial direction of the disc surface.

2.2. Calculation method of static transmission errors

According to the reference [11], thermo-elastic coupling contacts analysis of gears can use indirect coupling method by finite element analysis. The deformation of the driving and driven gear in the direction of the meshing line is obtained through thermo-elastic finite element analysis. The elastic contact equation in gear transmission can be given in [11]:

where [$K$] is the stiffness matrix. $\left\{u\right\}$ is the node displacement vector. $\left\{P\right\}$ is the load vector. $\left\{R\right\}$ is the contact force vector.

According to the heat-transfer process, the thermal balance equations for gear transmission system can be written in:

where [$K_t$] is the temperature stiffness matrix. $\left\{t\right\}$ is the node temperature vector. $\left\{P_t\right\}$ is thermal flow vector of generalized node.

Elastic contact and thermal contact are solved by iterative coupling according to Eq. (4) and Eq. (5). The deformation of the meshing point can be obtained in the node displacement vector $\left\{u\right\}$. The STE formula associated with thermo-elastic coupling contacts spur gear pair can be expressed as:

where ${\delta }_a$ is the deformation along the direction of the meshing ling of thermo-elastic coupling contacts. Differentiating the driving and driven gear with the subscript 1 and 2. $\mathrm{\Delta }{f}_{\sum }$ is the system equivalent error which is caused by machining and assembly errors, and it can be derived by:

where $f_{pb}$ is pitch error. $f_f$ is tooth profile error. $f_b$ is tooth error, $f_a$ is center distance installation error. $\alpha$ is gear pressure angle.

2.3. Differences between static transmission errors with and without thermo-elastic coupling contacts

The static transmission errors without thermo-elasti coupling contacts is mainly calculated by the equivalent tooth profile method based on material mechanics. The equivalent tooth shape method simplifies the gear teeth to variable-section cantilever beam based on elastic mechanics. According to the reference [2], it is considered that the comprehensive elastic deformation of the meshing tooth consists of the bending deformation ${\delta }_B$ of the rectangular and the trapezoid parts, the shear deformation ${\delta }_S$, the additional deformation ${\delta }_G$ caused by the elastic deformation of the base, and the contact deformation ${\delta }_{pv}$ of the tooth surface meshed with the tooth surface.

The deformation of the gear tooth at the loading point without thermo-elastic coupling contacts can be derived by:

The deformation of the gear tooth associated with thermo-elastic coupling contacts can be derived by:

where subscript $T$ are expressed as the deformation associated with thermo-elastic coupled contacts. ${\delta }_{Tt}$ is the heat deformation caused by temperature field.

The deformation in traditional calculation method is elastic deformation based on material mechanics. The coupling method of elastic deformation and heat deformation is taken into account when considering the thermo-elastic coupling contact.