Analysis of the Effect of Three-Dimensional Topology Modification on Temperature Field and Thermal Deformation of Internal Helical Gears Pair

Abstract

The transmission accuracy and meshing performance of the gearbox is determined by the internal helical gears pair. Thermal deformation of internal helical gears pair is derived from sliding friction between the contacting teeth surface, resulting in shock, vibration, and misalignments. The purpose of this paper is to compare the influence of a modified gear and an unmodified gear on the temperature field and transmission characteristics of a planetary gear system under the same working conditions. This study presents an innovative temperature field model for gear pairs utilizing Surf152 elements, integrating Hertzian contact theory, tribological principles, and finite element analysis. For the first time, we quantitatively demonstrate the enhancement of thermo-mechanical performance through topological modification in helical gears. Under light-load conditions (200 rpm), the modified gear configuration exhibits a 6.38% reduction in tooth surface temperature and a 46.5% decrease in thermal deformation compared to conventional designs. Experimental validation confirms these improvements, showing an average 62.35% reduction in transmission error. These findings establish a novel methodology for high-precision gear design while providing critical theoretical foundations for planetary gear systems, ultimately leading to significant improvements in both transmission accuracy and operational lifespan.

1. Introduction

The internal helical planetary gear train is usually used in high-speed rail, wind turbine, ship, high-speed machine tools, etc., due to its advantages of stable operation, high bearing capacity and wear resistance. However, in the transmission process, the relative sliding between the tooth surfaces will inevitably produce friction heat. Because the lubricating oil cannot completely take away the friction heat, the temperature of the gear will gradually increase and a corresponding thermal deformation will occur. Especially under poor lubrication or extreme conditions, the friction heat effect is more prominent, which may cause failure phenomena such as pitting and scuffing on the tooth surface, thus affecting the meshing performance, transmission efficiency, and service life of the gear. Finally, the transmission accuracy of the reducer will be affected [1]. Tooth surface modification is one of the important means to improve the meshing performance of gears. Through micro-geometric correction, the edge contact stress and friction heat concentration can be effectively reduced, so as to optimize the temperature field distribution and suppress the thermal deformation. However, at present, research on the steady-state heat effect of the modification of the internal helical gear pair under thermo-elastic coupling is still limited. Therefore, it is of great academic value and engineering significance to improve the performance and reliability of the reducer by studying the temperature rise effect of the modification on the internal helical planetary transmission system during the working process and its influence on the transmission accuracy.

A large amount of research on the temperature field and thermal deformation of gears has been carried out by scholars at home and abroad. The effects of thermoelastic deformation, contact deformation and oil film thickness of the meshing point on backlash under thermoelastic coupling conditions were considered by Y. B. He et al. [2]. The temperature distribution within the gear is examined in conjunction with the calculated thermal input, the penetration depth of the lubricant, and the thermal conductivity coefficient. The contact stress of gear teeth in the process of gear transmission is analyzed by Hao Lianjing et al. [3] using the finite element method, and the influence of gear modification on the gear transmission system is studied. The thermoelastic coupling finite element model of the gear pair before and after reshaping is established by Song H et al. [4]. Through the comparison of stress analysis, it is observed that the maximum equivalent stress and contact stress of the gear pair after reshaping are significantly reduced, which improves the bearing condition of the gear meshing process. The thermal-elastic coupling analysis was conducted by Taburdagitan M et al. [5], by considering the elastic deformation of the gears, the forces between the contacting tooth pairs, and the heat generation during contact. The finite element method was used by Fan Zhimin et al. [6] to study the steady-state thermal characteristics of the double involute gear transmission system. By constructing a multi-parameter stepped simulation model, the influence of different tooth profile characteristic parameters on the peak temperature distribution in the meshing area was systematically analyzed. The simplified boundary conditions handling method of a straight-toothed cylindrical gear model was proposed by Huang Hua liang et al. [7], based on the boundary element method, to analyze the bulk temperature of the gear. The steady-state temperature distribution of the gear was obtained by Zhang Yueming and others [8], who divide the engaging tooth surface into several strip areas along the width direction to apply frictional heat flux at different engaging positions, and they analyze the impact of the number of strip areas on the temperature field. Thermal coupling analysis was conducted by Li Run fang et al. [9] to solve the thermal deformation of gears resulting from transmission temperatures. The heat transfer parameters of gears was determined by Luo Biao et al. [10], to analyze the temperature field and thermal deformation. The temperature variation in the tooth surface of both the modified gear and the unmodified gear during transmission was compared by Yang Long et al. [11], to evaluate the impact of tooth modification on the temperature field of the gears. The pressure distribution, friction coefficient, and stress distribution within the contact zone of meshing surfaces were determined by Dai Ling et al. [12], considering factors such as tooth width, wedge gap, and surface roughness.

The operating conditions and geometric characteristics of the gear are also the main factors affecting the tooth surface temperature and transmission error. Therefore, the study of these factors is also an indispensable part. In Wang YZ et al. [13], the finite element model of a single tooth of the spiral bevel gear is established, and the temperature field and thermal deformation at different speeds are analyzed. The results show that the thermal deformation and steady-state temperature will gradually increase with the increase in speed. For Li J et al. [14], the finite element method is used as a design tool to predict the temperature and distribution of gears and thermal stress and thermal deformation of gears, while the heat production of different surfaces during gear meshing is analyzed. The results show that the error of this method is very small compared with the ISO calculation results. Pan et al. [15] used the finite element method to solve the load distribution of the gear. Combining it with the heat transfer theory, the thermal analysis model of the gear is established. The oil–gas mixing parameters are introduced, and the effects of oil–gas mixing materials and loads on the temperature field are compared. Li W et al. [16] used the numerical calculation and finite element method to compare the modified gear and the unmodified gear. The results show that the surface temperature and thermal deformation of the modified gear are significantly reduced. Liu MY et al. [17] applied a modification method considering meshing deformation to suppress the temperature rise of plastic gears to improve the position of the maximum contact pressure and the relative sliding speed. In Hou T et al. [18], a new design of a curved small modulus gear was proposed and the temperature rise of the dry meshing was studied. It was found that a high speed and torque lead to a large temperature rise, and the highest temperature is in the middle of the tooth of the initial contact trajectory of the gear. Wang C [19] calculated the thermal deformation of the double helical gear considering the installation error and machining error; this provides a theoretical basis for the modification of a double helical gear under the influence of thermal deformation. Liu SY et al. [20] established a thermoelastic coupling finite element model to analyze the friction heat generation of spiral bevel gears. Based on this model, the steady-state temperature field of a single tooth is calculated by the frictional heat flux, which is used as the initial condition to predict the flash temperature of the meshing surface at different rotational speeds. Luo B et al. [21] analyzed the reasons for the influence of modification and pressure angle change on the gear temperature field and studied the influence mechanism of each factor. Fatourehchi E et al. [22] combined a tribological model with three-dimensional thermal fluid analysis and predicted the heat generation and dissipation rate of the oil jet in the meshing contact of lubricating gears in an air–oil mist environment.

Most of the studies mentioned above focused on the investigation of thermal-elastic deformation through gear modification, without considering the selection of optimal modification parameters analysis of a gear transmission error under the action of the temperature field was seldom carried out; thus, the study of the transmission error characteristics of modified gears under a thermal effect becomes increasingly significant. In this study, we assume that the viscous shear heating and hydrodynamic cooling effects of lubricants are not considered in the gear meshing process. These assumptions enable us to focus on the thermal behavior analysis of dry tooth meshing and provide a basis for subsequent research.

The study employs finite element contact analysis to identify the optimal modification parameters through systematic parameter selection. A comprehensive numerical investigation is conducted to analyze the heat transfer characteristics of both modified and unmodified gear pairs under various operational conditions, incorporating calculations of convective heat transfer coefficients and frictional heat flux distributions. Through advanced finite element simulations, we systematically characterize the distinct thermal behaviors of modified and unmodified gears, particularly focusing on differences in tooth surface temperature fields and thermal deformation patterns. Furthermore, an indirect coupling methodology is implemented to examine temperature-dependent transmission errors, enabling comparative analysis of modified versus conventional gear performance under identical rotational speeds but varying thermal conditions.

2. Steady-State Thermal Analysis of Internal Helical Gears Pair

During the meshing of the gears pair, the interaction between the lubricant and frictional heat is initiated per cycle, due to the tooth root region remaining continuously immersed by the lubricant fluid. As a heat transfer medium, the lubricant can efficiently absorb and dissipate heat from the gear surface. In order to simplify the analysis, this study assumes that the heat generation and cooling caused by jet flow or immersion in the lubricant oil pool are not considered in the gear meshing process. When the gear pair reaches the thermal equilibrium state, although the meshing process will still lead to a slight fluctuation in the temperature field, the fluctuation is very small and the influence on the overall temperature field distribution is not significant. Therefore, it is reasonable to use the steady-state temperature field model to analyze the thermodynamic characteristics under various working conditions.

Heat flux density refers to the amount of heat transferred per unit area per unit time; the computational formula can be expressed as:

$$\Phi =-\lambda A{\displaystyle \frac{dT}{dx}}$$

(1)

$$q=-\lambda {\displaystyle \frac{dT}{dx}}$$

(2)

Here, $\Phi$ represents the frictional heat flux, $q$ denotes the heat flux density, $\lambda$ is the thermal conductivity, and A is the cross-sectional area perpendicular to the direction of heat transfer. The negative sign indicates that heat transfer occurs in the opposite direction to the rising temperature.

Heat transfer inherently results in the temperature gradients, the temperature field stands for the distribution of temperatures across meshing nodes at any given moment, and can be expressed as follows:

$$T=f\left(x,y,z,t\right)$$

(3)

In three-dimensional unsteady heat transfer theory, Equation (4) represents the governing partial differential equations for the temperature field. The heat balance equation can be expressed as follows:

$${\displaystyle \frac{\partial T}{\partial t}}=a\left({\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial z^2}}\right)+{\displaystyle \frac{\stackrel{•}{\Phi }}{\rho c}}$$

(4)

where $\rho$ denotes the medium’s density, $c$ is the specific heat capacity, $T$ is the temperature field function, $t$ represents the time variable, $\lambda$ is the thermal conductivity, $\stackrel{•}{\Phi }$ is the intensity of the volumetric heat source, and $\alpha$ is the thermal diffusivity $\alpha =\lambda /\rho c$.

By analyzing the meshing characteristics of the internal helical gear pair, it is found that the contact time between the tooth surfaces is extremely short during meshing and driving and the temperature response time can be neglected; consequently, the temperature distribution across each tooth surface can be assumed to be uniform. This simplification enables the complex temperature field of the gears pair to be represented by a single-tooth steady-state temperature model. During transmission, heat generation primarily results from the frictional work at the meshing surfaces, while heat dissipation occurs through convection from the cooling medium and heat dissipation from the gear body, reaching a dynamic balance between heat input and output. Due to the constant physical properties of the temperature field without an internal heat source, the thermal conductivity differential equation can be expressed as:

$${\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial z^2}}=0$$

(5)

2.1. Boundary Condition Analysis of Steady-State Temperature Field

During the meshing of an internal helical gears pair, the transmission efficiency of the gears is determined by the heat generated from friction. The temperature gradient gradually decreases with the heat transferred from the tooth surface to the depth of the gear profile. Once the gear reaches thermal equilibrium, its temperature distribution may be stable. Given that the frictional heat generated by each tooth remains constant over one revolution, the temperature distribution of the entire gear can be inferred by analyzing the temperature of a single tooth. The digital tooth of the helical gear is demonstrated in Figure 1.

Based on the results from the tooth profile calculation in Figure 1, the matching of working conditions with boundary conditions can be analyzed, and the tooth profile can be categorized according to its operating conditions as follows:

In the tooth contact region, the M-zone, as the primary meshing surface, generates significant thermal effects due to friction, which satisfy the superposition of the second and third types of boundary conditions. The heat flow density $q$ reflects the energy transfer characteristics of the meshing process, indicating the dynamic balance between heat flow input and output, as well as heat transfer with the surrounding environment. Therefore, the boundary condition is expressed as follows:

$$-\lambda ({\displaystyle \frac{\partial T}{\partial t}})=h_m(T-T_0)-q$$

(6)

During the heat transfer analysis, the tooth root, tooth top, and non-meshing working tooth surfaces are referred to as the T-zone, transferring heat through the lubricating medium via convective heat transfer, except for the meshing region. The meshing surface satisfies the third type of boundary conditions, and the corresponding mathematical expression is expressed as follows:

$$-\lambda ({\displaystyle \frac{\partial T}{\partial n}})=h_t(T-T_0)$$

(7)

The convective heat exchange occurs between the gear surface such as end faces, the cross-section of the gear tooth, and shaft bore regions, and cooling media such as lubricating oil and air. These surfaces also satisfy Type III boundary conditions.

End face of gear tooth: S zone

$$-\lambda ({\displaystyle \frac{\partial T}{\partial n}})=h_s(T-T_0)$$

(8)

Bore section of gear shaft: D zone

$${\displaystyle \frac{\partial T}{\partial n}}=0$$

(9)

Cross-section of gear tooth: P, Q zone

Heat transfer also occurs within the gear cross-section, and it can be assumed that the heat transferred is uniform across both surfaces.

$$T{|}_p=T{|}_q$$

(10)

$${\displaystyle \frac{\partial T}{\partial n}}{|}_p={\displaystyle \frac{\partial T}{\partial n}}{|}_q$$

(11)

The above includes: $\lambda$-Thermal conductivity of the gear material, W/(m·K); $h_m,h_t,h_s$-heat transfer coefficient at each working zone, W/(m·K); $T_0$-ambient temperature, °C; $T$-gear body temperature, °C; $n$-external normal direction of the heat exchange surface.

2.2. Finite Element Analysis of Gear Body Temperature Field

The finite element method is applied to express the continuous temperature field with a finite number of discrete nodal temperatures, and is equivalent to solving the extremum problem of the corresponding generalized function. This can be approximated using the variational principle of generalized functions to obtain an approximate solution, as outlined in the literature [23]:

The function space mapping for the steady-state temperature field under the second type of boundary conditions is expressed as follows:

$$J\left[T\left(x,y\right)\right]={\displaystyle \frac{\lambda }{2}}{\displaystyle \underset{\Omega }{\iint }\left[\left(\frac{\partial T}{\partial x}\right)+\left(\frac{\partial T}{\partial y}\right)\right]}dxdy+{\displaystyle \underset{{\Gamma }_1}{\int }qTds}$$

(12)

where the thermal conductivity $\lambda$ and the heat flow density $q$ at the boundary are known quantities.

The function space mapping for the steady-state temperature field under the third type of boundary conditions is expressed as follows:

$$J\left[T\left(x,y\right)\right]={\displaystyle \frac{\lambda }{2}}{\displaystyle \underset{\Omega }{\iint }\left[\left(\frac{\partial T}{\partial x}\right)+\left(\frac{\partial T}{\partial y}\right)\right]}dxdy+{\displaystyle \underset{\Gamma 2}{\int }\left(\frac{1}{2}h{T}^2-h{T}_{0}T\right)ds}$$

(13)

The temperature of the medium $T_f$, the heat transfer coefficient $h$, and the thermal conductivity of the solid $\lambda$ are known quantities.

The function space mapping for the temperature field of the internal helical gear pair is expressed as follows

$$J\left[T\left(x,y\right)\right]={\displaystyle \frac{\lambda }{2}}{\displaystyle \underset{\Omega }{\iint }\left[\left(\frac{\partial T}{\partial x}\right)+\left(\frac{\partial T}{\partial y}\right)\right]}dxdy+{\displaystyle \underset{\Gamma 2}{\int }\left(\frac{1}{2}h{T}^2-\left(h{T}_0-q\right)T\right)ds}$$

(14)

Select any cell and the temperature at a point on the boundary line can be expressed as a linear interpolation of the temperatures at the two end nodes:

$$T=\left(1-{\displaystyle \frac{s}{s_k}}\right)T_i+{\displaystyle \frac{s}{s_k}}{T}_j$$

(15)

The temperature interpolation function is expressed as follows:

$$T\left(x,y\right)=N_i{T}_i+N_j{T}_j+N_k{T}_k$$

(16)

In the formula above, $N_i={\displaystyle \frac{1}{2A}}\left(a+b_{i}x+c_{i}y\right),i,j,k$.

The interpolating function of the temperature field indicates that solving the temperature field involves determining the temperature values at each node, a process involving the application of multivariate function operators. Thus, the solution to the temperature field can be viewed as an extremum problem for a multivariate function.

Assuming the computational region consists of n nodes, the expression for $J\left[T\left(x,y\right)\right]$ is converted into the form of $J\left[T_1,T_1\cdots T_n\right]$. The extremum condition is:

$${\displaystyle \frac{\partial J}{\partial T_m}}={\displaystyle \sum \frac{\partial J^e}{\partial T_m}}=0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\left(m=1,2,\dots n\right)$$

(17)

By substituting the temperature interpolation function into the spatial mapping of the temperature field function for derivation, the resulting extremum conditions are:

$${\displaystyle \frac{\partial J^e}{\partial T_i}}={\displaystyle \underset{e}{\iint }\lambda \left[\frac{\partial T}{\partial x}\frac{\partial }{\partial T_i}\left(\frac{\partial T}{\partial x}\right)+\frac{\partial T}{\partial y}\frac{\partial }{\partial T_i}\left(\frac{\partial T}{\partial y}\right)\right]}dxdy+{\displaystyle \underset{jk}{\int }h\left(T-T\right)}{\displaystyle \frac{\partial T}{\partial T_i}}=0$$

(18)

In Equation (18), the first term of the equation represents the temperature of the internal cell, which is characterized by a rigid array of temperatures.

$${\displaystyle \frac{\partial J^e}{\partial T_i}}={\displaystyle \frac{\lambda }{4A}}\left[\left({b_i}^2+{c_i}^2\right)T_i+\left(b_i{b}_j+c_i{c}_j\right)T_j+\left(b_i{b}_k+c_i{c}_k\right)T_k\right]$$

(19)

Its matrix form is notated as:

$$\left[\begin{array}{c}{\displaystyle \frac{\partial J^e}{\partial T_i}}\\ {\displaystyle \frac{\partial J^e}{\partial T_j}}\\ {\displaystyle \frac{\partial J^e}{\partial T_k}}\end{array}\right]={\displaystyle \frac{\lambda }{4A}}\left[\begin{array}{ccc}{b_i}^2+{c_i}^2& b_i{b}_j+c_i{c}_j& b_i{b}_k+c_i{c}_k\\ & {b_j}^2+{c_j}^2& b_j{b}_k+c_j{c}_k\\ & & {b_k}^2+{c_k}^2\end{array}\right]\left[\begin{array}{c}{T}_i\\ T_j\\ T_k\end{array}\right]={\left[H\right]}^e{\left[T\right]}^e=0$$

(20)

The rigid array of temperatures in the second term of the temperature field function space mapping in Equation:

$$\left[\begin{array}{ccc}0& 0& 0\\ & {\displaystyle \frac{\lambda s_i}{3}}& {\displaystyle \frac{\lambda s_i}{6}}\\ & & {\displaystyle \frac{\lambda s_i}{3}}\end{array}\right]\left[\begin{array}{c}{T}_i\\ T_j\\ T_k\end{array}\right]-\left[\begin{array}{c}0\\ {\displaystyle \frac{\lambda s_i}{2}}{T}_a\\ {\displaystyle \frac{\lambda s_i}{2}}T\end{array}\right]={\left[H^1\right]}^e{\left[T\right]}^e-{\left[p\right]}^e$$

(21)

The temperature rigid matrix of the boundary cell is obtained by combining the two components of the temperature rigid matrix together; the overall temperature field equation is expressed as follow:

$${\left[\overline{H}\right]}^e{\left[T\right]}^e={\left[p\right]}^e$$

(22)

The steady-state temperature field equations involve several known parameters, closely relevant to the boundary conditions of the thermal conductivity differential equation. The boundary conditions of the steady-state temperature field of the internal helical gears pair includes meshing working surfaces, the tooth top, the non-working tooth surface at the tooth root, the end face, the cross-section, and portions of the shaft bore, all of which are influenced by the frictional heat flux of the meshing surfaces and the convective heat transfer coefficients of the other surfaces.

2.3. Calculation of Frictional Heat Flow in the Working Face

The magnitude of the frictional heat flow $q_k$ on the meshing surface is determined by the contact pressure $F_k$, relative sliding speed $u_c$, friction coefficient $f$ and the ratio $\gamma$ of frictional work converted into heat, on the meshing surface of the internal helical gear pair. Consequently, the frictional heat flux is calculated by the following formula:

$$q_k=F_k{u}_{c}f\gamma$$

(23)

The tooth contact pressure in an internal helical gear pair varies with the location of the contact point. To mitigate the influence of end face effects, the average contact stress is selected as a crucial parameter for calculating the frictional heat flow density. Based on Hertzian contact theory, the transmission process can be approximated as the contact between two cylinders; the radius of cylinders is regarded as the equivalent radius of curvature at the meshing point.

$$p_{\mathit{max}}=\sqrt{{\displaystyle \frac{F_{\mathit{max}}E}{2\pi {\rho }_{\sum k}\left(1-v^2\right)}}}$$

(24)

According to the literature, the average contact stress of the internal helical gear pair is equal to π/4 times the maximum contact stress [24]. Hence, the average contact stress of the internal helical gear pair is:

$$F_k={\displaystyle \frac{\pi }{4}}\sqrt{{\displaystyle \frac{w_{k}E}{2\pi {\rho }_{\sum k}(1-v^2)}}}$$

(25)

where $E$ is the modulus of elasticity of the two cylindrical materials, $v$ is the Poisson’s ratio of the materials of the two cylindrical, and ${\rho }_{\sum k}$ is the radius of curvature of any meshing point.

As shown in Figure 2, Point O is the center of the internal helical gear and θ is the angle of the velocity direction. The internal helical gear rotates counterclockwise to drive the external helical gear to rotate in the same direction. OQ is the pitch circle radius R₀ of the inner helical gear, V₁ is the absolute velocity of the Q point on the pitch circle, V₂ is the velocity of the outer helical gear at this point, and V₃ is the absolute velocity of the P point on the inner helical gear in the vertical direction of the pitch circle; the angular velocity of the outer helical gear is W₁ and the angular velocity of the inner helical gear is W₂. When the inner helical gear acts, there is the following kinematic relationship:

The absolute speed of the external helical gear at point Q is:

$$V_1=L_{OQ}\overrightarrow{w_1}=R_0\overrightarrow{w_1}$$

(26)

The absolute velocity of the internal helical gear at point P is:

$$V_3=L_{op}\overrightarrow{w_1}=(R_0+l_{PQ})\overrightarrow{w_1}$$

(27)

The speed of the internal helical gear at point Q is:

$$V_2=V_3+l_{PQ}\overrightarrow{{\omega }_2}=(R_0+l_{PQ})\overrightarrow{{\omega }_1}+l_{PQ}\overrightarrow{{\omega }_2}$$

(28)

Combining Formulas (26) and (28), the relative sliding velocity of the internal helical gear relative to the external helical gear at point Q is obtained:

$$\Delta V=V_2-V_1=l_{PQ}(\overrightarrow{{\omega }_1}+\overrightarrow{{\omega }_2})$$

(29)

Then, the relative sliding velocity at any contact point M of the two gears can be approximately solved by Equation (29). At this time, the rotation radius at M on the external helical gear is:

$$L_{OM}=L_{OQ}\cos \theta$$

(30)

The coefficient of friction is determined experimentally, influenced by the material characteristics and lubrication conditions. The friction coefficient of the internal helical gears pair is expressed as follows:

$$f=0.12{\left({10}^{-3}{\displaystyle \frac{w{R}_a}{{\eta }_f\Delta V{\rho }_{\sum }}}\right)}^{0.25}$$

(31)

In the equation above, $w$ denotes the average line load per unit length along the tooth contact, $R_a$ represents the roughness of the two gear tooth surfaces, ${\eta }_f$ is the lubricant viscosity at the body temperature, and ${\rho }_{\sum }$ is the synthesis radius of curvature at the meshing points. As a general guideline, the coefficient of friction can be approximated as 0.06.

The frictional energy generated during the meshing process of the internal helical gear pair is not entirely converted into thermal energy. The energy conversion coefficient is typically in the range of 0.9 to 0.95 [25].

2.4. Analysis and Calculation of Convective Heat Transfer Coefficient

(1)

Convection heat transfer coefficient of addendum

The convective heat transfer between the cylindrical surface of the addendum and the lubricant is approximated as the convective heat transfer of the lubricant flowing over a slender plate [26]. The convective heat transfer coefficient is given by the following equation:

$$h_c=0.664{P_c}^{0.333}\lambda {\left({\displaystyle \frac{\omega }{\eta }}\right)}^{0.5}$$

(32)

$p_c$ represents the Prandtl number of the lubricant, $P_c={\rho }_1{c}_1{v}_1/\lambda$, $p_1$, $c_1$ $v_1$$\lambda$ are the specific gravity, heat capacity, dynamic viscosity and thermal conductivity of the lubricant, respectively. $\omega$ is the angular velocity of the gears and $\eta$ is the dynamic viscosity of the lubricant.

(2)

Convective heat transfer coefficient of meshing tooth surface

The convective heat transfer coefficient between the meshing tooth surface and the lubricating oil is determined by the following equation:

$$h_d=0.228{R_e}^{0.731}{P_c}^{0.333}\lambda /L_d$$

(33)

In the above equation, $L_d$ represents the pitch circle diameter of the gear, $R_e$ is the Reynolds number, as determined by Equation $R_e=\omega r_k^2/v_0$, and $r_k$ is the radius of any meshing point on the tooth surface.

(3)

Flow heat transfer coefficient end face

The end-face flow heat transfer in an internal helical gears pair can be analyzed analogously to the heat transfer problem of a rotating disk. The flow of lubricating oil over the surface of the disk can be classified into three states: laminar flow, transitional flow, and turbulent flow. The calculation of the heat transfer coefficient is strongly dependent on the lubricant’s flow state, which can be effectively characterized by the Reynolds number [27].

The flow state of the lubricant varies with the Reynolds number, which in turn affects the convective heat transfer coefficient, as given by:

$$\left\{\begin{array}{l}{h}_e=0.308\lambda {\left(m+2\right)}^{0.5}{P_r}^{0.5}{\left({\displaystyle \frac{\omega }{\eta }}\right)}^{0.5}, R_e\le 2\times {10}^5, \mathrm{laminar} \mathrm{flow}\\ h_e=10\times {10}^{-20}\lambda {\left({\displaystyle \frac{\omega }{\eta }}\right)}^4{R_c}^7, 2\times {10}^5\le R_e\le 2.5\times {10}^5, \mathrm{Transition} \mathrm{layer} \mathrm{flow}\\ h_e=0.0197\lambda {\left(m+2.6\right)}^{0.2}{\left({\displaystyle \frac{\omega }{\eta }}\right)}^{0.8}{R_c}^{0.6}, R_e>2.5\times {10}^5, \mathrm{Turbulent} \mathrm{flow}\end{array}\right.$$

(34)

In the equation above, $\lambda$ represents the thermal conductivity of the lubricant, $P_r$ is the Prandtl number of the lubricant–air mixture, $\eta$ is the kinematic viscosity of the lubricant–air mixture, and m is an exponential constant that characterizes the change in surface temperature of the disk in the radial direction. For the purposes of this study, a quadratic distribution is assumed, i.e., m = 2. $R_c$ denotes any radius on the surface of the disk.

3. Contact Analysis of Topologically Modified Internal Helical Gear Pair

3.1. Basic Parameters of Internal Helical Gears

Analysis of the forces acting on the tooth surfaces and the transmission error of internal helical gears pair is carried out in view of the multi-tooth contact during meshing. Finite element analysis is applied to construct the dynamic model to investigate the force mechanism in detail. The parameters of the modified gear are shown in

Table 1. Basic geometrical parameters of gears.

Gear Parameters	Driving Wheel	Driven Wheel
Helical direction	left-handed	left-handed
Number of teeth z	79	26
Normal modulus mn/(mm)	3.5	3.5
Face module mt/(mm)	3.5782	3.5782
Normal pressure angle α (°)	20	20
Helix angle β/(°)	12	12
Tooth width b/mm	60	60
Face pressure angle αt (°)	20.410
Transverse contact ratio	1.211
Overlap ratio	1.135
Total contact ratio	2.346

.

Firstly, based on the basic geometric parameters of the internal helical gear pair, combined with the principle of gear meshing kinematics and the theory of spatial surface equation, the corresponding calculation program is developed through a MATLAB numerical calculation platform to realize the accurate coordinate solution of the tooth surface points of the involute main and driven gears. The mathematical tooth surface model is shown in Figure 3a. It is necessary to export the calculated discrete point coordinate values of the tooth surface in DAT format, and then import the data into UG, and model through surface generation, Boolean operation and other operations, and finally generate the gear model. The assembly drawing is shown in Figure 3b.

Matlab2021 software is applied to solve the tooth profile equations generated based on the topological modification principle, and to calculate the tooth profile data. Tooth profile modification amount: $\Delta L_1=b_m{r}_b^2{(u_1-u_0)}^2$, Here, b_m is the tooth profile modification coefficient, r_b is the radius of the gear base circle, u₁ is the angle of the involute, and u₀ is the starting point of the modification area. Tooth modification amount: $\Delta L_2=b_h{(l_p-b_1/2)}^2$, b_h is the tooth modification coefficient, lp is the wheel longitudinal displacement, and b₁ is the tooth width. As shown in Figure 4, it is the mathematical model of the profile of the external helical gear. The relevant modification coefficients are shown in

Table 2. Gear profile coefficients.

Shaping	Tooth Profile Modification Factor bm	Toothedness Modification Factor bh
Case 0	0.00003	0.0001
Case 1	0.00001	0.0003
Case 2	0.000015	0.0006
Case 3	0.00007	0.000008

.

3.2. Finite Element Calculation Example Analysis

The inner helical gear is regarded as the driving gear, while the external helical gear is the driven gear. The material of both gears is 20CrMnTi, with a density of 7800 kg/m³. The Young’s modulus of elasticity is 1.4 × 10⁵ MPa, and the Poisson’s ratio is 0.3. The permissible contact stress is 745 MPa, and the permissible bending stress is 510 MPa. Frictional contact exists in the meshing of internal helical gears pair, with a friction coefficient of 0.25. The unit type is (secondary unit) C3D8R, the number of hexahedral meshes of the internal helical gear is 62,352, and the number of hexahedral meshes of the external helical gear is 53,821.

The contact analysis is carried out in three steps by the dynamic implicit method: the main analysis step is set to be 0.1/0.01. Firstly, the internal helical gear is fixed and the external helical gear is rotated to ensure contact and convergence. Secondly, the external helical gear is fixed and the internal helical gear is rotated to avoid excessive contact. Finally, the rotation degree of freedom of the two gears is released to simulate the rotation process of the external helical gear driven by the internal helical gear, in which the surface-to-surface contact is used to construct the connection relationship between the gears, the gear speed and torque are set in different analysis steps (taking the light load 50 N·m as an example), and the torque is applied to the internal helical gear.

A four-tooth simulation and analysis model, modeled in Abaqus, is shown in Figure 5. Python script files are embedded within the Abaqus2022 simulation software to extract and analyze the contact stresses during the meshing under a different speed and torque.

After adjusting the modification coefficient, four sets of simulation settings are set to be consistent, and the tooth surface contact analysis is carried out. From Figure 6, it can be seen that the contact area of Case 0 along the tooth profile direction is larger and the contact area in the middle area presents a slender ellipse, indicating that it is caused by the smaller coefficient b_h. After Case 1 increases b_h and decreases b_m, it is found that the contact area along the tooth profile direction is larger. Case 2 increases b_h and decreases b_m, and it is found that the contact area gradually moves closer to the middle. After re-adjustment, the contact area of Case 3 appears as an elliptical shape, and the contact area is mainly in the middle of the tooth surface. In the first three sets of simulation results, the contact area of the tooth surface is flat and long, causing edge contact, but. In Case 3, the tooth surface contact area is elliptical, and the meshing area appears in the middle of the tooth width. The two ends are sharp, the tooth surface contact stress is small, and the elliptical shape gradually decreases outward, so the bearing capacity is strong. Therefore, the tooth surface contact effect of Case 3 is the best. In addition, the red wire frame in the figure is the force area of the contact process, and the red dotted frame represents the main stress area.

3.3. Simulation Analysis of Transmission Error

In order to verify the smoothness of the internal meshing helical gear pair transmission, four sets of simulation results under different modification factors are analyzed; data on the transmission error are simulated respectively, as shown in Figure 7; the change amplitude in the transmission error of Case 3 is the smallest and has an excellent modification effect.

4. Thermoelastic Coupling Analysis of Topologically Modified and Non-Modified Internal Helical Gear Pairs

4.1. Single-Tooth Simulation Model Loading

Based on the previous calculation results, the frictional heat flux and convective heat transfer coefficient of the contact surface are specified. The corresponding convective heat transfer coefficients of the other surfaces are given. Finite element thermal analysis is performed by means of hexahedral unit Multizone mesh division; the Multizone mesh uses the block technology, in which different colors represent different block regions, aiming to improve the mesh quality and computational efficiency. By generating a mesh that is more adaptable to the geometric shape in each small block, it can capture key physical phenomena and save computing resources with a thermal conductivity of 46 W/(m·K) and a specific heat capacity of 556 J/(kg·°C). To ANSYS2022 thermal analysis, the convective heat transfer coefficient and frictional heat flow cannot be applied simultaneously to the meshing tooth surface. Thus, APDL language is applied to create Surf152 Surface Effect Elements on the meshing tooth surface. The loaded model, shown in Figure 8, is used for the steady-state temperature field and thermal deformation of the gear teeth under different working conditions.

4.2. Temperature Field and Thermal Deformation Simulation Analysis

From Section 3, it can be seen that Case 3 is the best group of modification effects. Case 3 is compared with the unmodified gears. The ambient temperature is set to 20 °C, the rotational speed of the driving wheel is 200 rpm, and the load- torque is 2 N·m. Under these conditions, the calculated friction heat flux and convective heat transfer coefficient are applied to the corresponding surface of the two meshing teeth surface to obtain the temperature field change diagram. Based on the steady temperature field, the thermal deformation calculation of a single tooth is carried out. The steady-state temperature field data are exerted to the model of the static analysis structure, and the deformation of the shaft hole area and the end face are restricted. The steady-state temperature field of the gear is shown in Figure 9.

From Figure 9a, it can be seen that the peak temperature of the tooth profile of the standard gear reaches 50.892 °C, while the minimum temperature is 33.408 °C; in contrast, the maximum temperature of the tooth profile of the topologically optimized gear in Figure 9b is 47.647 °C, and the minimum temperature is 32.462 °C. It can be seen that the maximum temperature decreased by 6.376% and the minimum temperature decreased by 2.832% The highest frictional heat flow density appears at the addendum region. The low overall convective heat transfer coefficient leads to less efficient heat dissipation and results in relatively high temperatures near the tooth top within the engagement region.

The highest temperature is concentrated at the center of the tooth width, with a significant decrease in temperature towards the shaft bore and tooth ends. The temperature distribution at the center r varies more significantly than that of the addendum. The temperature of the tooth surface with a three-dimensional topology modification is 3 °C lower than that of the tooth surface without a modification. Because the contact area of the tooth surface of the topology optimization is mainly concentrated in the middle of the tooth surface, the heat dissipation condition in the middle of the tooth width is poor, so the high temperature area of the meshing surface moves down obviously. According to the calculation formula of the friction heat flow, the friction heat flow generated by the gear pair during transmission is affected by the relative sliding speed, average contact stress, energy conversion coefficient, and friction coefficient. The formulas of average contact stress and relative sliding velocity demonstrate that, under the same working conditions, the two parameters of the gear with three-dimensional topology modification are lower than those of the gear without three-dimensional topology modification, while the friction coefficient and energy conversion coefficient are basically constant.

Therefore, the frictional heat flow of the gear with three-dimensional topology modification is lower than that of the gear without modification, resulting in a lower temperature of the meshing surface of the former. The convective heat transfer coefficient of the internal helical gear pair is affected by the characteristics of the lubricating oil and the working conditions, so there is little difference in the convective heat transfer coefficient between the two kinds of gear. Therefore, the temperature distribution trends of the two tooth profiles are similar, and are gradually reduced from the meshing surface to both ends.

As shown in Figure 10, the temperature distribution along the tooth width exhibits an upward convex pattern, initially increasing and subsequently decreasing. This behavior is primarily attributed to the enhanced convective heat transfer at the end faces, which provide superior heat dissipation conditions, while the convective heat transfer coefficient in the middle region of the tooth width is lower, resulting in relatively poorer heat dissipation conditions.

Figure 11 illustrates that, along the meshing line, the temperature is higher at the upper portion of the tooth. The temperature at the middle of the tooth width exceeds that at the ends, and the overall temperature variation diminishes progressively. From the addendum to the dedendum, the temperature decreases gradually, with the sequence of the distribution following: addendum, pitch circle, and then dedendum. At the pitch circle, the high temperature region shows a significant shrinkage phenomenon because the relative sliding velocity at the pitch circle is zero theoretically and the heat dissipation condition is relatively poor, so that a higher temperature level can be maintained. When the gear with a three-dimensional topology modification is engaged in meshing transmission, because the meshing surface clearance makes the lubricating oil contact the tooth surface and take away the heat, the temperature of the meshing area is reduced, indicating that the modification improves the heat dissipation effect, optimizes the load distribution of the tooth surface, and makes it more uniform. The non-uniformity of the tooth profile temperature may cause impact and vibration during the transmission of the internal helical gear pair, which in turn affects its transmission characteristics. Therefore, the temperature factor needs to be considered in the transmission analysis.

Through thermal–structural coupling simulation analysis, Figure 12 presents the thermal deformation results of the gears subjected to a steady-state temperature field. The analysis reveals that the deformation gradually decreases from the addendum to the dedendum, and the temperature of the addendum is higher due to the large friction heat flow. The maximum deformation of the gear without a three-dimensional topology modification and the gear with a three-dimensional topology correction is 0.43324 μm and 0.27498 μm, respectively. The topology optimization reduces the deformation by 36.5%. It can be seen that topology optimization can effectively reduce thermal deformation. Additionally, the fit of the gear bore to the shaft is subject to axial constraints, while the tooth top remains unconstrained. Along with the accumulation of radial thermal expansion effects, this results in the most significant deformation at the addendum.

The deformation data in the tooth width direction can be extracted by analyzing the results from each node on the meshing surface, as shown in Figure 13. On the meshing surface of the gear, the deformation at the addendum and pitch circle appears an upward bulge shape, while the dedendum shows more dramatic deformation in the middle region of the tooth width. Overall, the deformation at the addendum is generally more outstanding than that at the pitch circle and dedendum; especially, the central region appears as a particularly prominent deformation, while the sides appear as a relatively slight deformation. This phenomenon is primarily attributed to the higher temperature field in the central region. To the standard gear, the deformation at the middle of the tooth top is 0.433 μm, while the maximum deformation at the middle of the pitch circle and tooth root are 0.292 μm and 0.216 μm, respectively. Tooth modification and optimization enable the deformations at the three positions to be reduced to 0.274 μm, 0.181 μm, and 0.091 μm, respectively.

Figure 14 presents the results of converting the deformation from the tooth width direction to the meshing line direction. The maximum deformations at the addendum, pitch circle, and middle of the dedendum are 0.4 μm, 0.23 μm, and 0.17 μm, respectively. After three-dimensional topology modification optimization, the maximum deformations at the addendum, pitch circle, and middle of the dedendum are reduced to 0.24 μm, 0.18 μm, and 0.08 μm, respectively.

4.3. Thermoelastic Coupling Analysis of Three-Dimensional Topologically Modified Tooth Profile Under Different Operating Conditions

Figure 15 shows the temperature field distribution and thermal deformation of the gear under different working conditions after three-dimensional topology modification and optimization. It can be observed that the steady-state temperature increases in parallel with the working conditions during the meshing. It can be further observed that the steady-state temperature increases in parallel with the working conditions during the meshing. It can be seen from Figure 15a,b that the maximum temperature increases by 41.2%. while the maximum temperature of the contact tooth surface rises together, although the overall trend remains consistent. The temperature increase is attributed to the rise in frictional heat flux caused by the increased input conditions, which consequently elevate the steady-state temperature rising. As the convective environment remains unchanged, the convective heat transfer coefficient stays relatively constant, and heat dissipation remains stable. Consequently, the temperature distribution of the internal helical gear pair changes little, close to the actual transmission results.

5. Analysis of the Effect of Operating Temperature Rise on Transmission Error of Three-Dimensional Topological Modification of Internal Helical Gear Pair

Meshing of the simplified internal helical gears pair model is simulated by means of Hypermesh for hexahedral meshing. The temperature increase of the gears under working conditions is introduced into the Transient Structural module of Workbench as a load through the method of indirect coupling. The approach facilitates the simulation and analysis of gear transmission errors. To investigate the effect of a temperature rise on gear transmission error, the ambient temperature is set to 20 °C, with a revolving speed of 1000 rpm and a load of 3 N·m. The gear transmission is simulated under working conditions at temperatures of 30 °C, 80 °C, and 120 °C. The simulation data are employed to calculate the gear transmission error corresponding to the temperature conditions.

The simulation results are presented in Figure 16. As shown in Figure 16b, a rising leads to premature meshing of the gears due to thermal deformation, out of accordance with the scheduled meshing. Furthermore, thermal deformation intensifies with a rising temperature. The increase in gear temperature also results in an increase in the transmission error.

6. Experimental Validation

6.1. Transmission Error Test Principle and Test Scheme

In this section, the transmission errors of the unmodified and modified internal helical gear pairs in different time periods will be detected, and the results of the two transmission errors will be compared and analyzed to observe the effect of the modification. The speed is set to 1000 r/min, and the load is 3 N·m. The measurement is performed every 30 min. With the extension of time, the planetary gear reducer will generate friction heat during continuous operation, and the overall temperature will gradually increase. Therefore, it is of great significance to discuss the influence of the gear temperature on the transmission error under time-varying conditions.

The transmission error test principle is as follows: Figure 17. The test bench is composed of a drive motor, an input grating, an output grating, a measured reducer, a sensor, and a tooling base. The input gear shaft is fixed to the reducer shaft, and the actual rotation angle of the input end of the reducer is recorded by the input grating. The runout of the connecting rod of the output support disc is controlled within 0.03 mm, and then connected with the output grating angle sensor to measure the actual rotation angle of the reducer output. When the motor starts, the input gear shaft is driven to rotate, and the grating is input to record the rotation angle position at the moment. Due to the influence of the transmission error, the rotation angle position captured by the grating encoder at the output end at the same time does not conform to its theoretical position; that is, the angle signal frequencies of the two grating encoders are different. The rotation angle positions captured at both ends of the input and output are connected according to each instantaneous, and the bearing transmission error curve can be obtained.

6.2. Transmission Error Experimental Platform Construction and Experimental Results Comparison

According to the above test principle, the transmission error test platform is built. The test platform adopts a modular design, which is composed of a drive module, input measurement module, reducer module, output measurement module, supporting tooling module, and controller module, respectively. The grating support base can be replaced according to the model and size of the tested reducer. At the same time, the platform ensures the accuracy of the center hole of the reducer by the machining accuracy of the support base. The data of the input and output measurement modules are transmitted to the computer-based controller module by using the data acquisition card, and the test results of the transmission error are output in real time after processing by the computer. The test bench is shown in Figure 18. Figure 19 is the software measurement interface of the PC end of the transmission error.

In the transmission of the helical planetary reducer, most of the transmission error comes from the transmission of the internal helical gear pair. From Figure 20, it can be seen that compared with the transmission error curve before and after modification, when the working time is 30 min, the maximum transmission error after modification is 16.32″, and the transmission error of the unmodified gear system is 53.25″. The modification reduces the transmission error by about 68.35%. When the working time is 60 min, the modification reduces the transmission error by 58.38%. When the working time is 90 min, the modification reduces the transmission error by about 59.32%. The average value of the overall reduction in the transmission error after modification is 62.35%. Therefore, in the design of the transmission system, if the influence of the temperature rise on the transmission error is to be ensured, it is necessary to modify the gear to ensure the stability and reliability of the system.

6.3. Comparison of Simulation and Experimental Results

With the extension in the working time, the working temperature of the working gear system will gradually increase [28]. Therefore, we compare the simulation results of the fifth section with the experimental results. The working temperatures of the simulation results are 20 °C and 120 °C, and the working times of the modified gear in the experimental results are 30 min and 90 min. In Figure 21, through the a and b diagrams, we find that the change trend of the two is the same, and the change rule is to increase first and then decrease. The maximum difference is 10.11″ and 16.54″, respectively. The average values of the transmission error fluctuation between the simulation results and the experimental results are 17.7″ and 6.3″ and 24.9″ and 10.2″, respectively. It can be seen that the experimental results are higher than the simulation results, which is due to the neglect of the installation error and the transmission error of other gear trains.

7. Conclusions

(1) The frictional heat flux generated by the 3D topology modified internal helical gear pair during transmission is lower than that of the standard gear unmodification. The temperature of the modified gears is lower than the unmodified gears in the meshing area. Under the condition of a light load (200 r/min), the simulation analysis shows that the tooth surface temperature is reduced by 6.38% and the thermal deformation is reduced by 36.5% after modification. When the internal helical gears pair with a three-dimensional topology modification is engaged in the meshing transmission, there is a certain gap between the meshing tooth surfaces, and there are more contact areas between the lubricating oil and the tooth surface, which takes away more friction heat, thereby reducing the contact temperature of the tooth surface. Therefore, the gear after three-dimensional topology modification is helpful to improve the heat dissipation efficiency.

(2) Thermoelastic coupling simulation analysis of the topological design of the internal helical gears pair is carried out and reveals that both the temperature distribution and thermal deformation of the internal helical gear are significantly reduced under various revolving speeds and load conditions. To the modified gear, the meshing region within the high-temperature band tends to concentrate at the nodes, resulting in a more uniform temperature distribution across the addendum, pitch circle, and dedendum. Compared to the standard gear, modification can reduce the maximum deformation under the temperature field, lower the transmission error, and improve the transmission accuracy.

(3) The temperature of the internal helical gear pair under steady-state working conditions increases with the rising of the load during transmission. Simultaneously, the maximum temperature in the tooth contact area increases with the load. At 400 and 800 r/min, the tooth surface temperature increased by 41.2%. Although the specific temperature value will change, the overall trend in the temperature change is consistent with the trend in the load increase. Although the specific temperature values may vary, the overall temperature change trend remains consistent with the increasing load.

(4) Under the same working conditions, the transmission error comparison experiments of modified and unmodified gears was carried out through experiments. It was verified that the transmission error of the internal helical gear pair would gradually increase with the increase in the working temperature. Taking the unmodified gear as an example, the transmission error increased by 51.3% on average when working for 30 min and 90 min, which proved the reliability of the simulation part. It is also verified that the modification can reduce the transmission error of the planetary reducer, and the overall transmission error after modification is reduced by 62.35% on average, which provides a reference for the design and stability of different types of planetary gear reducers.