Abstract
Helical gears, compared to spur gears, offer stronger bearing capacity and smoother transmission, making them indispensable components in high-power and high-precision transmission systems, especially in high-speed and heavy-load applications. The higher rotational speeds and loads lead to elevated body temperatures, a crucial parameter indicating potential gear scuffing failure. This paper investigates a pair of helical gears by establishing a contact model using MATLAB software, based on Hertz contact theory and heat transfer theory. We calculate and study the contact stress, friction heat flux, and convective heat transfer conditions at different positions. A method for calculating the steady-state temperature field of helical gears is proposed, and the friction heat flux under varying torques and speeds is analyzed. Finally, the results are integrated into ANSYS Workbench for simulation analysis to study the steady-state temperature field distribution under different torque conditions. The findings provide valuable insights into helical gear scuffing failure.
1. Introduction
Helical gears are widely used in high-power and high-precision transmission systems due to their superior load-bearing capacity and smooth transmission characteristics. Applications include wind turbines, ships, and aerospace equipment. However, as transmission power and speeds increase, thermal effects become more pronounced, leading to potential gear deformation and scuffing issues. Understanding the thermal behavior of helical gears is crucial for ensuring their reliability and longevity.
1.1 Background and Motivation
Previous studies have focused primarily on the thermal effects of spur gears, while research on helical gears, which operate under higher loads and speeds, is relatively scarce. Helical gears’ unique geometry and operational conditions necessitate a dedicated analysis framework.
1.2 Objectives
The primary objectives of this study are:
- To establish a contact model for helical gears using MATLAB based on Hertz contact theory.
- To analyze the friction heat flux and convective heat transfer conditions during gear meshing.
- To propose a method for calculating the steady-state temperature field of helical gears.
- To simulate and analyze the steady-state temperature field distribution under varying torque conditions using ANSYS Workbench.
1.3 Paper Organization
This paper is organized as follows: Section 2 details the contact analysis of helical gears, including the mathematical modeling and stress distribution. Section 3 focuses on the calculation of friction heat flux under different operational conditions. Section 4 presents the method for calculating the steady-state temperature field and its validation using ANSYS Workbench. Section 5 discusses the results and findings, followed by conclusions in Section 6.
2. Contact Analysis of Helical Gears
The unique geometry of helical gears introduces complexities in their contact behavior compared to spur gears. This section outlines the mathematical modeling and stress distribution analysis of helical gear pairs.
2.1 Geometric Modeling
The helical gear meshing model is illustrated , where Op and Op′ (respectively, Og and Og′) represent the rotation axes of the pinion (driving gear) and gear (driven gear). The base circle radii are denoted as rbp and rbg, and the rotational speeds are w1 and w2, respectively. The theoretical and actual meshing zones are indicated by N1,N2,N2′,N1′ and B1,B2,B2′,B1′, respectively.
The gear parameters used in this study are summarized in Table 1.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of Teeth (z) | 17 | 26 |
| Helix Angle (β) | 15° | 15° |
| Normal Module (mn) | 4.5 mm | 4.5 mm |
| Pressure Angle (α) | 20° | 20° |
| Face Width (b) | 20 mm | 20 mm |
Table 1: Gear Parameters
2.2 Contact Line Length
Due to the helix angle, the contact line length during helical gear meshing is time-varying. Based on geometric considerations, the contact line length L(t) can be expressed as:
L(t) = \begin{cases} b \sin\beta_b \left( \varepsilon_{\alpha_b} – \frac{t}{T_m} + \varepsilon_{\beta_b} \right) & 0 \leq t \leq T_m(\varepsilon_{\alpha_b} – \varepsilon_{\beta_b}) \\ b \sin\beta_b \left[ \cos\beta_b \left( \frac{t}{T_m} – \varepsilon_{\alpha_b} \right) p_{bt} + \varepsilon_{\beta_b} \right] & T_m(\varepsilon_{\alpha_b} – \varepsilon_{\beta_b}) < t \leq T_m\varepsilon_{\beta_b} end{cases}
where:
- εαb, εβb, and εγ represent the transverse, axial, and total contact ratios, respectively.
- Tm is the meshing period.
- βb is the helix angle.
- pbt is the tooth pitch.
- b is the face width.
The total contact line length Lz(t) for multiple contact lines can be calculated as:
Lz(t)=i=1∑NLi(t)
where N is the number of contact lines and Li(t) is the length of the i-th contact line at time t.
2.3 Hertz Contact Theory
The average contact stress σ and Hertzian contact half-width a can be calculated using Hertz contact theory:
sigma=2LER1R2/(R1+R2)πFn
a=πER1R2/(R1+R2)4FnL
where:
- Fn is the normal load.
- L is the contact line length.
- E is the equivalent elastic modulus.
- R1 and R2 are the radii of curvature of the contacting surfaces.
The contact stress distribution along the contact line .
3. Friction Heat Flux Analysis
The friction heat generated during gear meshing significantly contributes to the thermal behavior of helical gears. This section investigates the friction heat flux under varying operational conditions.
3.1 Friction Heat Flux Calculation
The friction heat flux q is determined by the contact stress, relative sliding velocity, and friction coefficient:
q=σfγV
where:
- σ is the contact stress.
- f is the friction coefficient.
- γ is the thermal conversion factor.
- V is the relative sliding velocity.
For the pinion, the friction heat flux q1 can be expressed as:
q1=b2aqβv1{1−(ax)2}1/2
where:
- a is the Hertzian contact half-width.
- β is the heat partition coefficient.
- v1 is the tangential velocity of the pinion.
- x is the position along the contact line.
The distribution of friction heat flux on the pinion.
3.2 Influence of Operational Parameters
The friction heat flux is significantly influenced by the gear’s rotational speed and applied torque. The variations in friction heat flux under different speeds and torques, respectively.
At a constant torque of 260 N·m, the peak friction heat flux increases from 1.14×105 W/m² to 3.44×105 W/m² as the speed increases from 800 rpm to 2400 rpm. Similarly, that at a constant speed of 1200 rpm, the peak friction heat flux increases from 1.05×105 W/m² to 2.35×105 W/m² as the torque increases from 160 N·m to 360 N·m.
4. Steady-State Temperature Field Analysis
This section proposes a method for calculating the steady-state temperature field of helical gears and validates the results using ANSYS Workbench.
4.1 Convective Heat Transfer Coefficients
The convective heat transfer coefficients for different gear surfaces are calculated based on their geometric characteristics and operational conditions. The relevant formulas are:
- For tooth flanks (M, N regions):
hmn=0.228Pr1/3dpλRe2/3
- For tooth end faces (D, E regions):
hde=0.6Prλ(0.56+0.26Pr1/2+Pr)2/3(vrw)1/2
- For tooth roots and tips (T, F regions):
htf=0.664Pr1/3λ(vrw)1/2
where:
- λ is the thermal conductivity.
- Re is the Reynolds number.
- Pr is the Prandtl number.
- dp is the pitch diameter.
- w is the angular velocity.
- vr is the kinematic viscosity of the lubricant.
The lubricant properties used in this study are listed in Table 2.
| Property | Value |
|---|---|
| Density (ρ) | 8600 kg/m³ |
| Thermal Conductivity (λ) | 0.13 W/m·K |
| Viscosity (μ) | 46.05 mm²/s |
| Prandtl Number (Pr) | 587.3 |
Table 2: Lubricant Properties
4.2 Steady-State Temperature Field Calculation
The steady-state temperature field is calculated by solving the heat transfer equation in conjunction with the boundary conditions determined from the convection heat transfer coefficients. The calculation is performed in MATLAB, and the results are then imported into ANSYS Workbench for validation.
4.3 ANSYS Workbench Simulation
The helical gear model is created in SolidWorks and imported into ANSYS Workbench for meshing and boundary condition application. The friction heat flux and convective heat transfer coefficients are applied as boundary conditions. The steady-state temperature field distributions under different torques.
The maximum temperatures occur at the tooth roots and tips, consistent with the high friction heat flux regions. The temperature peaks increase with torque, highlighting the need for thermal management strategies, especially at higher loads.
5. Results and Discussion
The results of the contact stress, friction heat flux, and steady-state temperature field analyses provide valuable insights into the thermal behavior of helical gears.
5.1 Contact Stress Distribution
The Hertzian contact stress distribution indicates high stress concentrations near the pitch line, emphasizing the need for adequate lubrication and material selection to mitigate wear and scuffing.
5.2 Influence of Operational Parameters
The friction heat flux analysis reveals that both speed and torque significantly impact the thermal load on helical gears. As speed and torque increase, so does the friction heat flux, leading to elevated gear temperatures .
5.3 Thermal Management Strategies
The steady-state temperature field simulations indicate that high temperatures concentrate at the tooth roots and tips. Therefore, thermal management strategies, such as optimized gear geometry, improved lubrication systems, and active cooling mechanisms, could mitigate thermal issues and extend gear life.
6. Conclusion
This paper presents a comprehensive analysis of the contact behavior and thermal effects in helical gears. The following key findings emerge:
- Contact Stress Distribution: The Hertzian contact stress distribution highlights the need for adequate lubrication and material selection to minimize wear and scuffing.
- Friction Heat Flux: Both speed and torque significantly influence the friction heat flux generated during gear meshing, with higher speeds and torques leading to increased thermal loads.
- Steady-State Temperature Field: Simulations show that maximum temperatures occur at the tooth roots and tips, emphasizing the importance of thermal management strategies to mitigate excessive heating.
Future work could focus on experimental validation of the simulation results and further optimization of gear geometry and lubrication systems to improve thermal performance and extend gear life.