Abstract
To obtain the steady-state temperature field and thermal deformation field of electrically driven helical gears, this paper derives the calculation methods for the average friction heat flux density on the meshing surface and the convective heat transfer coefficients of other surfaces based on the meshing principle of helical gears, heat transfer theory, and tribology. A parametric thermal analysis model of helical gears is established using the APDL language, and the steady-state temperature field and thermal deformation field of the helical gears are obtained. The influence rules of different gear models on the steady-state temperature field and thermal deformation field are revealed. The research results show that the steady-state temperature field varies gradually, with the highest temperature on the meshing surface and the lowest temperature on the hub. Two distinct high-temperature regions exist on the meshing surface, located in the two double-tooth meshing areas. In the tooth height direction, the tooth surface temperature exhibits an “M”-shaped distribution. The maximum temperature in the double-tooth meshing area at the meshing-in end is close to the tooth root, while the maximum temperature in the double-tooth meshing area at the meshing-out end is close to the tooth tip. In the tooth width direction, the tooth surface temperature is distributed asymmetrically. For a clockwise rotating right-handed driving gear, the high-temperature zone in the double-tooth meshing area at the meshing-in end is close to the front face of the gear, while the high-temperature zone in the double-tooth meshing area at the meshing-out end is close to the rear face of the gear. Due to the accumulation of deformation, the overall thermal deformation of the helical gears are largest at the tooth tip and lowest at the hub, with the maximum deformation occurring near the two end faces of the gear. When considering the deformation along the meshing line direction, the thermal deformation on the rear face is greater than that on the front face, consistent with the temperature distribution along the tooth width. To save computational cost, the single-tooth model can be used to replace the full-tooth model with high accuracy in calculating the steady-state temperature field, but only the full-tooth model is accurate in calculating the thermal deformation field, and other models have large errors.
1. Introduction
Helical gears possess excellent transmission performance and load-bearing capacity, making them widely used in electrically driven transmission systems. However, during the transmission process of helical gears, due to sliding friction in both axial and radial directions between the meshing tooth surfaces, a significant amount of heat is generated under high-speed and heavy-load conditions, leading to adhesive failure and thermal deformation of the gears. Thermal deformation alters the involute tooth profile of the gears, causing changes in the transmission meshing characteristics and generating vibration and noise. For electrically driven gears, the higher the rotational speed, the more heat is generated during gear transmission, thus making the study of the temperature field and thermal deformation field of electrically driven helical gears significant. Currently, the main methods for studying the temperature field and thermal deformation field of gears include numerical calculation methods [1][2], experimental analysis methods [3], and finite element methods [4][10].
Li Guihua et al. [11] proposed an approximate method for calculating the temperature field of meshing gears based on mechanical drawing principles and heat transfer theory, using Saint-Venant’s principle for correction. The results were very close to those obtained by the finite element method. Sütter et al. [12] introduced an experimental method for measuring flash temperatures on sliding surfaces, utilizing an enhanced CCD camera and visible light temperature measurement to measure the temperature field at the sliding surface. Guo Heng et al. [13] established a finite element analysis model for gear temperature fields and transmission errors using Workbench and explored the influence of changes in rotational speed and torque on gear temperature and the impact of changes in rotational speed and temperature on gear transmission errors. Li Wei et al. [14] derived formulas for calculating the friction heat flux density and convective heat transfer coefficient of gears and established a single-tooth parametric model of spur gears using the APDL language to obtain the single-tooth temperature field of spur gears, analyzing the influence of different parameters on gear temperature. Xue Jianhua et al. [15] analyzed the meshing surface of helical gears, derived the distribution laws of contact lines, loads, and friction heat fluxes, and established the bulk temperature field of helical gears considering temperature effects, conducting thermomechanical coupling analysis.
In summary, most finite element analyses of the temperature field and thermal deformation field of helical gears currently utilize the Workbench or APDL language to establish finite element analysis models of gears and study the influence of key factors such as rotational speed, load, and material on the results. However, there is no published literature on the impact of the model itself on the accuracy of the analysis results. This paper establishes a parametric model of helical gears using the APDL language to investigate the influence of different models, such as partial teeth and full teeth, on the calculation accuracy of the steady-state temperature field and thermal deformation field. The solution process.
2. Heat Balance Equation and Boundary Conditions of Helical Gears
During the meshing process of helical gears, heat is generated due to friction between the two contacting meshing surfaces. Part of this heat is transmitted into the gear bulk, while the rest is carried away through convective heat transfer between the gear surfaces and lubricating oil. After a period of operation, the helical gears system reaches a thermal equilibrium state, at which point the bulk temperature field of the helical gears are steady-state temperature field. The heat conduction differential equation for the steady-state temperature field of the gear transmission system is derived based on the law of conservation of energy and Fourier’s law of heat conduction, as shown in Equation (1).
lambda(∂x2∂2t+∂y2∂2t+∂z2∂2t)=0(1)
Where: λ is the thermal conductivity, and t is the temperature.
To solve for the steady-state temperature field of the gear, it is necessary to first determine the boundary conditions for each surface. The schematic diagram of each gear surface.
The boundary conditions for each gear surface can be obtained using Newton’s cooling formula and Fourier’s law. The meshing surface is a combination of the second and third types of boundary conditions (S and T), as shown in Equation (2), while the non-meshing surface, tooth tip, tooth roots on both sides, and end faces are the third type of boundary condition (T), as shown in Equation (3).
−λ(∂n∂t)=s1(tc−to)−qw(2)
−λ(∂n∂t)=s2(tc−to)(3)
Where: n is the normal vector of the heat transfer surface, s1 is the convective heat transfer coefficient of the meshing surface, s2 is the convective heat transfer coefficient of the tooth tip, non-meshing surfaces, end faces, and tooth roots on both sides, tc and to are the initial temperatures of the gear and lubricating oil, respectively, and qw is the average friction heat flux density.
3. Average Friction Heat Flux Density and Convective Heat Transfer Coefficients
3.1 Dimensionless Coordinates
To represent the position of any meshing point on the gear meshing line, this paper introduces the dimensionless coordinate Γ. the schematic diagram of the dimensionless coordinates on the meshing line of an external meshing gear.
Where NI is the theoretical meshing-in point, NA is the actual meshing-in point, NB and ND are the two boundaries between the single-tooth and double-tooth meshing areas, NC is the node, NE is the actual meshing-out point, and NO is the theoretical meshing-out point. Taking the meshing line as the coordinate axis, the dimensionless coordinate Γ of any point NK on the meshing line is defined as the ratio of the distance from this point to the node NC to the length of the line segment NINC, with the positive direction from point NI to point NO. The calculation formula for the dimensionless coordinate Γ of point NK can be obtained from the geometric as shown in Equation (4).
Gamma=rb1tanαncrb1tanαnk−rb1tanαnc=tanαnctanαnk−1(4)
Where: αnc is the pitch circle pressure angle, and αnk is the active gear pressure angle corresponding to the meshing point NK.
3.2 Average Friction Heat Flux Density
To solve for the steady-state temperature field of the helical gears, the average friction heat flux density needs to be applied to the meshing surface. The value of the average friction heat flux density is determined by Equation (5).
qw=2T1πkfμPMfτ0(v1−v2)(5)
Where: kf is the tooth-to-tooth distribution coefficient of friction heat flux density, μ is the friction coefficient, PM is the maximum Hertzian contact pressure, f is the thermal energy conversion coefficient (taken as 0.95 in this paper), T1 is the meshing period of the driving gear, τ0 is the half-bandwidth of time-domain contact, and v1 and v2 are the tangential velocities of the meshing points of the driving and driven gears, respectively, determined by Equation (6).
tau0=v1b0,b0=πEPM8wR=2πRv1wE,v1=30R1πn1,v2=30R2πn2(6)
Where: E is the equivalent elastic modulus, R is the combined radius of curvature, w is the unit line load, determined by Equations (7) and (8).
w=Lrb1n1cosβb9549PηK(7)
R=R1+R2R1R2,E1=E11−u12+E21−u22(8)
Where: P is the input power, η is the transmission efficiency, L is the contact line length, βb is the base helix angle, R1 and R2 are the radius of curvature of the driving and driven gears, respectively, u1 and u2 are the Poisson’s ratios of the driving and driven gears, respectively, and E1 and E2 are the elastic moduli of the driving and driven gears, respectively.
3.3 Convective Heat Transfer Coefficients
(1) Tooth Tip Convective Heat Transfer Coefficient
The tooth tip convective heat transfer coefficient is determined by Equation (9).
hd=0.664λoPo0.333(νoω)0.5(9)
Where: λo is the thermal conductivity of the lubricating oil, Po is the Prandtl number of the lubricating oil (Po=ρocoνo/λo, where ρo is the density of the lubricating oil, co is the specific heat capacity of the lubricating oil, and νo is the kinematic viscosity of the lubricating oil), and ω is the angular velocity of the gear.
(2) Tooth Surface and Tooth Root Transition Surface Convective Heat Transfer Coefficient
The convective heat transfer coefficient of the tooth surface and tooth root transition surface is determined by Equation (10).
ha=0.228Re0.731Po0.333Ldλo(10)
Where: Ld is the pitch circle radius of the driving gear, Re is the Reynolds number of the lubricating oil, determined by Equation (11).
Re=νoωrnk2(11)
Where: rnk is the radius of any meshing point on the gear.
(3) End Face Convective Heat Transfer Coefficient
End face convective heat transfer is similar to that of a rotating disk and can be divided into laminar, transitional, and turbulent flow states based on the Reynolds number. The convective heat transfer coefficients for the three states are shown in Equation (12).
h_t = \begin{cases} 0.308 \lambda_{mix} \left( m_z + 2 \right)^{0.5} \times P_{mix}^{0.5} \left( \frac{\omega}{\nu_{mix}} \right)^{0.5}, & R_e \leq 2 \times 10^5 \\ 10^{-19} \lambda_{mix} \left( \frac{\omega}{\nu_{mix}} \right)^4 r_{nk}^7, & 2 \times 10^5 \leq R_e \leq 2.5 \times 10^5 \\ 0.0197 \lambda_{mix} \left( m_z + 2.6 \right)^{0.2} \times \left( \frac{\omega}{\nu_{mix}} \right)^{0.8} r_{nk}^{0.6}, & R_e \geq 2.5 \times 10^5 end{cases} \quad (12)
Where: mz has a value of 2, in an oil-gas mixture, Pmix is the Prandtl number, νmix is the kinematic viscosity, and λmix is the thermal conductivity. The characteristic parameters of the oil-gas mixture can be constructed by linearly combining the characteristic parameters of the oil and gas media in a certain proportion αmix, as shown in Equation (13).
numix,λmix,Pmix=νa,λa,Pa⋅(1−αmix)+νo,λo,Po⋅αmix(13)
Where: νa, λa, and Pa are the viscosity coefficient, thermal conductivity, and Prandtl number of the air, respectively. d is the correction coefficient for the impact of actual operating conditions on the mixed flow, determined by d=rnk/ra, where ra is the tooth tip circle radius.
4. Finite Element Parametric Model
The parameters of the helical gears and lubricating oil are shown in Tables 1 and 2.
Table 1: Helical gears parameters
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Driving gear speed n1 (rpm) | 2000 | Elastic modulus E (GPa) | 206 |
| Number of teeth z1,z2 | 23, 30 | Poisson’s ratio υ | 0.3 |
| Module m (mm) | 3 | Specific heat capacity c (J/kg·K) | 465 |
| Pressure angle α (°) | 20 | Thermal conductivity λ (W/m·K) | 46 |
| Tooth width b (mm) | 20 | Coefficient of thermal expansion k (10−5/K) | 11.3 |
| Helix angle β (°) | 8 | Density ρ (kg/m³) | 7850 |
| Input power P (kW) | 50 |
Table 2: Lubricating Oil Parameters
| Parameter | Value |
|---|---|
| Lubricating oil type | SCH632 |
| Density (ρ_o) (kg/m³) | 870 |
| Kinematic viscosity (ν_o) (cSt) | 320 (at 40°C), 38.5 (at 100°C) |
| Thermal conductivity (λ_o) (W/(m·K)) | 2000 |
| Specific heat capacity (c_o) (J/(kg·K)) | 0.14 |
These parameters provide detailed information about the lubricating oil used in the study, including its type, density, kinematic viscosity at different temperatures, thermal conductivity, and specific heat capacity. These properties are crucial for accurately modeling the heat transfer and friction processes within the gear system.