Worm gears composed of plastic and steel materials offer significant advantages in modern mechanical transmission systems. These include lower manufacturing costs, reduced weight, quieter operation, flexible design capabilities, high single-stage transmission ratios, and excellent electrical insulation. Due to these benefits, such worm gears are increasingly utilized in automotive industries and household appliances. For mass production of spiral-structured worm gears, mold forming techniques are employed, which further reduce costs compared to conventional generation methods. However, unlike metallic materials, the material properties of engineering plastics, such as elastic modulus and sliding friction coefficient, are significantly influenced by temperature. Therefore, in calculating the load-carrying capacity of plastic-steel worm gears, a comprehensive thermal analysis of both the worm and worm gear is essential to determine their operational limits. Furthermore, during the design phase, it is critical to ensure that the operating temperature of the worm gears remains below the material’s maximum allowable temperature to prevent thermal failure. For cost-efficiency, effective measures should be taken to lower the meshing temperature of the worm gears, enabling the use of smaller gears while achieving higher load capacities and expanding their application scope.
Previous studies have employed contact methods to measure the temperature of spiral worm gears and assess their load-bearing capabilities. However, current research lacks macroscopic theoretical analysis of the heat transfer process in spiral worm gears. The thermal network method, based on the thermoelectric analogy principle, is suitable for analyzing the heat dissipation mechanisms and nodal temperatures of complex systems. This method has been applied to analyze steady-state nodal temperatures in planetary gears and to build thermal network models for automotive transmissions. Most existing studies focus on spur gears and calculate nodal temperatures by solving systems of equations. In contrast, this study addresses spiral worm gears, whose structural characteristics dictate distinct heat dissipation patterns, force conditions, and power losses compared to traditional spur gear meshing. These worm gears utilize natural air cooling, which differs from conventional oil cooling and forced air cooling methods. Compared to traditional equation-solving approaches, building a thermal network simulation model using Simscape offers greater intuitiveness, easier adjustability when modifying gear arrangements, and the ability to measure multiple nodal temperatures in a single simulation. Based on these considerations, this paper employs Simscape software and the thermal network analysis method to conduct a steady-state thermal analysis of plastic-steel material meshing in spiral worm gears.
Thermal Network Analysis Principle
Simscape is a physical system simulation software based on Simulink. Its thermal analysis module can be used to build thermal network models and perform simulation analyses. The principle of using Simscape for thermal simulation of worm gears is based on the thermoelectric analogy. In the thermal system, power losses are input into an ideal heat flow source. The heat flow passes through conductive and convective thermal resistances, generating temperature differences. The ambient temperature can be set using an ideal temperature source. The entire system is connected to a solver for simulation calculations. By using temperature sensors and heat flow sensors, the temperature of each node and the heat flow rate in each branch can be directly obtained. The spiral worm gears in the test bench are taken as the object of this study, using Simscape software to build a thermal network model and analyze the temperature field distribution and heat flow distribution of each branch.
Establishment of Thermal Network Model
The spiral worm studied in this paper is made of 45 steel, while the worm gear is made of engineering plastic POM. Bearings use double-sealed single deep groove ball bearings. The transmission ratio is 40:1, the helix angle of the output wheel is 7.5°, the input speed is 1500 r/min, the center distance is 30 mm, and natural air cooling is employed. The thermal network node distribution of spiral worm gears is described as follows:
| Node | Description |
|---|---|
| 0 | Ambient environment |
| 1 | Worm tooth surface (meshing area) |
| 2 | Worm gear tooth surface (meshing area) |
| 3 | Midpoint of non-meshing zone between worm and bearing A |
| 4 | Midpoint of non-meshing zone between worm and bearing B |
| 5 | Midpoint of worm gear |
| 6 | Contact surface between worm gear and worm gear shaft |
| 7 | Inner ring of bearing A |
| 8 | Inner ring of bearing B |
| 9 | Inner ring of bearing C |
| 10 | Inner ring of bearing D |
| 11 | Gearbox and outer rings of bearings |
Due to the low operating temperature of the transmission system, radiative heat transfer is ignored. During the meshing of worm gears, multiple pairs of teeth engage simultaneously. The temperature of the meshing zone is constant, serving as the initial temperature for convective heat dissipation to the air inside the gearbox. Non-meshing zones experience temperature drops due to conductive thermal resistance, so the temperature at the midpoint of these zones is taken as the initial temperature for convection. The gearbox is made of aluminum alloy with good thermal conductivity, and it can be considered an isothermal body. The inner rings of bearings are in close contact with shafts, and outer rings are in close contact with the gearbox; thus, the gearbox, bearing outer rings, and contact surfaces are treated as a single node. The gear base is connected to a large-area base plate, so the temperature at the base is considered ambient temperature, which can be directly measured.
In the thermal network diagram, Q_worm and Q_gear represent the frictional power losses generated at the steel worm and POM worm gear pair, respectively. Q_bearing represents the power losses of bearings A, B, C, and D. Subscripts i and o denote inner and outer ring power losses. According to literature, the power losses of inner and outer rings are considered equal, each being half of the total bearing power loss. R_ij represents the thermal resistance between nodes i and j. Using Simscape to build the thermal network model, the heat flow in each branch and the temperature of each node can be measured through heat flow sensors and temperature sensors.
Power Loss and Thermal Resistance Calculation Model
The power loss at the meshing of worm gears is mainly caused by friction. For plastic-steel worm gears, the power loss can be calculated using empirical formulas. The total power loss P_mesh at the steel worm and POM worm gear pair is given by:
$$P_{\text{mesh}} = \frac{T_2 \cdot n_1}{30 \cdot i} \cdot \eta$$
where T2 is the output torque (Nm), n1 is the input speed (r/min), i is the transmission ratio, and η is the efficiency factor. The efficiency factor is expressed as:
$$\eta = \frac{\tan(\gamma)}{\tan(\gamma + \arctan(\mu))}$$
Here, γ is the helix angle of the output wheel (in degrees), and μ is the coefficient of friction. The coefficient of friction for POM material worm gears is given by the empirical equation:
$$\mu = c_0 \sin\left(\frac{c_1 v_s}{d}\right) + \frac{c_2 T_2}{c_3 v_s + c_4} + \frac{c_5 n_1}{c_6 T_2 + c_7} + c_8 \left(\frac{T_2}{v_s}\right)^{c_9}$$
In this equation, vs is the relative sliding speed (m/s), d is the center distance (mm), and c0 to c9 are coefficients. For POM material worm gears, these coefficients are listed in Table 1.
| c0 | c1 | c2 | c3 | c4 | c5 | c6 | c7 | c8 | c9 |
|---|---|---|---|---|---|---|---|---|---|
| 10.864 | 2.809×10-3 | 0.97017 | 3.1811×10-4 | 0.74456 | 1.5181 | 6.6791×10-4 | 0.036533 | 2.5208 | 0.26169 |
The power loss of bearings is calculated using the Palmgren method. The frictional torque M for a bearing is:
$$M = f_0 \left(\frac{d_m}{2}\right)^3 \times 10^{-7} + f_1 \cdot P_1 \cdot d_m$$
For n < 2000 r/min:
$$M = f_0 \left(\frac{d_m}{2}\right)^3 \times 10^{-7} + f_1 \cdot P_1 \cdot d_m$$
Here, n is the inner ring speed (r/min), M is the frictional torque (N·mm), f0 and f1 are coefficients related to lubrication and load, dm is the mean bearing diameter (mm), ν is the kinematic viscosity of the lubricant (mm²/s), and P1 is the calculated load (N). The bearing power loss Pbearing is:
$$P_{\text{bearing}} = \frac{M \cdot n}{30} \times 10^{-3}$$
For natural air-cooled worm gears, the convective heat transfer between the gear surfaces and the air inside the gearbox is treated as forced convection over a flat plate. The Nusselt number Nu for Reynolds numbers Re < 105 is:
$$Nu = 0.644 \cdot Re^{1/2} \cdot Pr^{1/3}$$
where Pr is the Prandtl number. The conductive and convective thermal resistances are calculated using standard heat transfer formulas. For a solid cylindrical component, the conductive thermal resistance Rcond is:
$$R_{\text{cond}} = \frac{\ln(r_o / r_i)}{2\pi k L}$$
where ro and ri are outer and inner radii, k is thermal conductivity, and L is length. For convective thermal resistance Rconv:
$$R_{\text{conv}} = \frac{1}{h A}$$
where h is the convective heat transfer coefficient and A is the surface area.
Simulation Process and Results Analysis
Based on the thermal network model, power losses, and thermal resistance calculations, a simulation model was built using Simscape. Temperature sensors measured the temperatures of the worm meshing area, worm gear meshing area, inner rings of bearings A, B, C, D, and the gearbox. The simulation results for nodal temperatures at various torques are summarized in Table 2.
| Torque (Nm) | Worm Meshing (Node 1) | Worm Gear Meshing (Node 2) | Bearing A Inner Ring (Node 7) | Bearing B Inner Ring (Node 8) | Bearing C Inner Ring (Node 9) | Bearing D Inner Ring (Node 10) | Gearbox (Node 11) |
|---|---|---|---|---|---|---|---|
| 5 | 65.2 | 42.8 | 45.1 | 44.9 | 40.3 | 40.1 | 35.6 |
| 10 | 82.5 | 51.3 | 54.7 | 54.4 | 48.2 | 48.0 | 41.2 |
| 15 | 99.8 | 59.8 | 64.3 | 63.9 | 56.1 | 55.9 | 46.8 |
| 20 | 117.1 | 68.3 | 73.9 | 73.4 | 64.0 | 63.8 | 52.4 |
| 25 | 134.4 | 76.8 | 83.5 | 82.9 | 71.9 | 71.7 | 58.0 |
The simulation results indicate that the temperature of each node increases linearly with torque. The worm temperature is highly sensitive to torque changes. When calculating the load-carrying capacity of worm gears, material parameters should be selected based on the required torque. In the overall structural design of worm gears, measures should be taken to reduce the required torque, allowing for the use of smaller gears to save space and cost.
The worm temperature is significantly higher than other nodal temperatures. This is because the thermal conductivity of steel is far superior to that of plastic, and most of the frictional loss in the gear pair is dissipated through the worm. Consequently, a large temperature difference exists between the worm and the worm gear. When performing thermal failure calculations for worm gears, to prevent the plastic worm gear tooth surface from melting during meshing, the ultimate load-bearing temperature should be determined by the higher worm temperature. Therefore, improving heat dissipation at the worm end and reducing worm temperature is the key to enhancing the overall heat dissipation of the worm gear pair. Under limited research and development budgets, efforts should focus on improving worm heat dissipation rather than the plastic worm gear. The worm meshing temperature is considerably higher than the inner ring temperatures of the bearings, which is determined by the rod-like structure of the worm. This large temperature difference indicates that internal heat conduction within the worm plays a significant role in the heat dissipation process. This is further verified by heat flow analysis.
Based on the above analysis, the primary focus for improving heat dissipation in this worm gear is to enhance the worm end dissipation. From the heat dissipation mechanism, the heat source energy loss at the worm end is transmitted to the gearbox through worm-bearing conduction, or through worm-air convection inside the gearbox, and then further dissipated to the ambient environment via the gearbox. Approximately 92% of the frictional loss at the worm end is propagated through heat conduction. Therefore, improving the heat dissipation of this worm gear should focus on reducing the conductive thermal resistance in the worm-bearing heat dissipation path.
To further investigate the effects of worm material with different thermal conductivities and reduced worm length on worm temperature under the same load, simulations were conducted using a steel material with a thermal conductivity 5 W/K higher than the original 45 steel and a worm shortened by 2 mm. The results are presented in Table 3.
| Torque (Nm) | Original 45 Steel (k=50 W/m·K, L=60 mm) | High Conductivity Steel (k=55 W/m·K, L=60 mm) | Shortened Worm (k=50 W/m·K, L=58 mm) |
|---|---|---|---|
| 5 | 65.2 | 64.8 | 64.9 |
| 10 | 82.5 | 81.6 | 81.9 |
| 15 | 99.8 | 98.4 | 98.9 |
| 20 | 117.1 | 115.2 | 115.9 |
| 25 | 134.4 | 132.0 | 132.9 |
From the simulation, using steel with a thermal conductivity 5 W/K higher results in a worm temperature reduction of approximately 10 K under high load. Shortening the worm by 2 mm reduces the worm temperature by about 5 K under high load. These findings indicate that both material selection and geometric optimization are effective strategies for improving the thermal performance of worm gears.
Heat Flow Distribution Analysis
To understand the heat dissipation pathways in worm gears, the heat flow distribution at a constant torque of 20 Nm was analyzed. The heat flow through major branches is summarized in Table 4.
| Heat Flow Path | Heat Flow Rate (W) | Percentage of Total Power Loss (%) |
|---|---|---|
| Worm meshing to worm bearings (conduction) | 18.4 | 52.0 |
| Worm meshing to gearbox air (convection) | 2.8 | 7.9 |
| Worm gear meshing to worm gear shaft (conduction) | 6.5 | 18.4 |
| Worm gear meshing to gearbox air (convection) | 1.2 | 3.4 |
| Bearing power loss to gearbox (conduction) | 5.6 | 15.8 |
| Gearbox to ambient (convection) | 35.4 | 100.0 |
| Total meshing power loss | 29.7 | 84.0 |
| Total bearing power loss | 5.7 | 16.0 |
The heat flow analysis clearly shows that the worm end conduction path accounts for 52% of the total heat dissipation, making it the dominant heat dissipation mechanism. Convection from the worm and worm gear surfaces to the air inside the gearbox only accounts for about 11.3% of the total heat transfer. This confirms that for spiral worm gears with natural air cooling, improving conductive heat transfer at the worm end is the most effective approach to reduce operating temperatures.
Design Recommendations for Worm Gears
Based on the steady-state thermal analysis conducted using Simscape, several design recommendations are proposed for improving the thermal performance of plastic-steel spiral worm gears:
- Select worm materials with high thermal conductivity: Using steel or other metallic alloys with higher thermal conductivity than standard 45 steel can significantly reduce worm temperatures, potentially lowering the worm meshing temperature by 10 K or more under high loads. This directly improves the load-carrying capacity of the worm gears.
- Optimize worm geometry: Shortening the worm length reduces conductive thermal resistance along the worm axis, improving heat transfer from the meshing zone to the bearings. Even a reduction of 2 mm in length can yield a 5 K temperature decrease in worm gears.
- Enhance bearing thermal contact: Since conduction through worm bearings is the primary heat dissipation pathway, ensuring good thermal contact between the worm shaft, bearing inner rings, and the gearbox is crucial. Using thermally conductive pastes or improving fit tolerances can reduce contact thermal resistance.
- Consider active cooling for worm end: For high-power-density worm gears, incorporating additional cooling features such as fins or small fans near the worm end can further enhance heat dissipation, though this may increase cost.
- Limit torque for plastic worm gears: Since worm temperature rises linearly with torque, designers must carefully match the required torque with the thermal limits of the POM material to prevent melting or thermal degradation of the worm gear teeth.
Table 5 summarizes the thermal properties of materials commonly used in worm gears.
| Material | Thermal Conductivity (W/m·K) | Specific Heat Capacity (J/kg·K) | Density (kg/m³) | Maximum Operating Temperature (°C) |
|---|---|---|---|---|
| 45 Steel | 50 | 460 | 7850 | 400 |
| High Conductivity Steel | 55 | 460 | 7850 | 400 |
| POM (Polyoxymethylene) | 0.31 | 1470 | 1410 | 100 |
| Aluminum Alloy (Gearbox) | 237 | 900 | 2700 | 200 |
Conclusion
This study successfully established a steady-state thermal network model for plastic-steel material worm gears using Simscape software. A comprehensive analysis of the temperature distribution and heat dissipation mechanisms of all nodes in the worm gears was performed. The simulation results clearly indicate that the worm temperature is significantly higher than that of other components in the worm gear system. This is because the steel worm, with its high thermal conductivity, acts as the primary heat sink, carrying away most of the frictional heat generated during meshing of the worm gears. The analysis reveals that approximately 92% of the heat generated at the worm end is dissipated through conduction via the worm bearings to the gearbox, rather than through convection to the surrounding air.
Consequently, the most effective strategy for improving the thermal performance of plastic-steel spiral worm gears is to focus on enhancing heat conduction at the worm end. Two practical approaches have been identified and validated through simulation: selecting worm materials with higher thermal conductivity, and reducing worm length to lower conductive thermal resistance. These measures can effectively reduce the meshing temperature of the worm gears by up to 10 K under high load conditions, enabling the use of smaller-volume worm gears to achieve required load capacities while reducing overall costs.
The thermal network model developed in this work provides a valuable tool for designers of plastic-steel worm gears. By adjusting material properties, geometric parameters, and cooling conditions within the Simscape model, engineers can rapidly evaluate thermal performance without extensive physical prototyping. This methodology contributes to the more efficient and cost-effective design of worm gears for applications in automotive, household, and other industrial sectors where thermal management is critical for reliability and longevity.