# Analytical Model of Gear Thermal Stiffness Based on the Potential Energy Method
## 1 Introduction  
Gear stiffness is the basis for studying various characteristics of gear transmission systems. References [1; 2; 3; 4; 5; 6; 7; 8; 9; 10] present a variety of calculation methods for gear mesh stiffness and includes analytical, finite element and experimental research methods. The potential energy method is the most widely used analytical method. Numerous studies on factors affecting gear stiffness have been conducted by many scholars at home and abroad to obtain more accurate gear mesh stiffness calculations [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22]. For example, the influence of tooth root cracks [11; 12], tooth profile errors [13], tooth surface peeling [14; 15], gear tooth modifications [16] and temperature [17; 18; 19; 20; 21; 22; 23; 24] on gear stiffness. In terms of the influence of temperature on gear stiffness, [PERSON] et al. [17] established a mathematical expression of spur gear tooth profile deformation caused by the change in gear surface contact temperature. The influence of contact temperature on spur gear mesh stiffness has been studied based on the Hertz contact theory. The gear thermal stiffness is considered a simple superposition of elastic stiffness and temperature stiffness (formed by thermal deformation caused by temperature rise). Their results showed that the gear thermal stiffness is greater than the gear elastic stiffness. [PERSON] et al. [18] proposed an analytical algorithm for the mesh stiffness of helical cylindrical gears considering the effect of temperature. The tooth profile deformation caused by the contact temperature deformation of the tooth surface was considered in the stiffness model. Thus, the effect of temperature is introduced in the gear stiffness. Their research results showed that after considering the influence of temperature rise, the thermal expansion of the tooth profile leads to a reduction in gear backlash, an increase in contact area of the gear tooth and a decrease in average deformation of the contact area. The mesh stiffness of the helical gear pair increased during the whole meshing process. [PERSON] et al. [19] obtained the base circle radius of a gear in the free expansion state based on the plane strain/stress model. Theactual involute profile was obtained after being affected by temperature. The mathematical expression of the time-varying thermal mesh stiffness of a gear pair with temperature was established by using the potential energy method. The effect of temperature on gear mesh stiffness has been studied, and the results have shown that gear stiffness increases after considering the influence of temperature. However, many scholarly research results conflict with these results. For example, [PERSON] et al. [20] proposed a thermal expansion model and a gear mesh stiffness model based on linear thermal expansion theory, involute profile theory and the energy method. Their research showed that an increase in temperature reduces gear mesh stiffness. [PERSON] et al. [21; 22; 23] crafted a semi-analytical formula of gear thermal mesh stiffness based on the finite element method. The influence mechanism of temperature on the gear mesh stiffness was revealed [21]. Then, the gear thermal stiffness was introduced into a system dynamic equation to study the influence of temperature on the dynamic characteristics of a gear transmission system [22]. Thereafter, the correctness of the thermal stiffness model was indirectly verified by an experimental method through the change of the natural frequency of the system dynamic response with temperature [23]. According to [PERSON]'s research, the gear thermal stiffness was equal to the ratio of the external load on the gear to the thermoelastic coupling deformation of the gear. The thermoelastic coupling deformation of the gear was equal to the difference between the gear elastic deformation under the joint action of the external load, thermal load and thermal elastic coupling load and the gear thermal expansion in the free state.