In mechanical transmission systems, the cylindrical gear is one of the most widely used components, and its quality and lifespan directly impact the operation of entire equipment. With economic development, the demand for large and heavy-duty cylindrical gears is increasing, and improving their quality and longevity can bring significant economic benefits to the heavy machinery industry. The failure modes of cylindrical gears primarily include tooth breakage due to bending stress and tooth surface damage such as fatigue pitting, spalling, plastic deformation, and wear caused by contact stress. Moreover, a substantial portion of failures in large cylindrical gears results from tooth surface damage under contact stress, with wear on working surfaces leading to disrupted geometric meshing relationships and reduced lifespan. Heat treatment is a critical process in cylindrical gear manufacturing, and local induction heating strengthening can specifically target areas prone to failure. To enhance the quality of induction heating treatment, numerous researchers worldwide have conducted finite element simulations of the induction heating process. This article employs a novel induction heating device to simulate the local induction heating of large cylindrical gears, analyzing temperature and stress fields to optimize heating uniformity and efficiency.
The induction heating process is highly nonlinear, characterized by rapid temperature changes and varying material properties with temperature. Therefore, the transient temperature field \( T(x, y, z, t) \) can be described by a three-dimensional nonlinear transient heat conduction equation. The governing equation is as follows:
$$ \frac{\partial}{\partial x} \left( \lambda \frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y} \left( \lambda \frac{\partial T}{\partial y} \right) + \frac{\partial}{\partial z} \left( \lambda \frac{\partial T}{\partial z} \right) + Q = \rho c \frac{\partial T}{\partial t} $$
Here, \( \lambda \) represents the temperature-dependent thermal conductivity, \( Q \) is the internal heat source intensity, \( \rho \) is the material density, and \( c \) is the temperature-dependent specific heat capacity. To solve this differential equation, initial and boundary conditions are required in addition to geometric, material, and time conditions. For a cylindrical gear, the initial condition is the set initial temperature, and the boundary condition on the tooth profile surface involves convection and radiation, governed by:
$$ \lambda \frac{\partial T}{\partial n} = -h (T – T_B) – \varepsilon \sigma (T^4 – T_B^4) $$
where \( T_B \) is the ambient temperature, \( h \) is the convective heat transfer coefficient, \( \varepsilon \) is the emissivity, \( \sigma \) is the Stefan-Boltzmann constant, and \( n \) is the outward normal to the surface.
The thermal stress analysis during induction heating of a metallic cylindrical gear falls under the category of thermo-elastoplastic problems. We adopt the incremental theory based on the Prandtl-Reuss theory, assuming isotropic hardening and adherence to the Mises yield criterion. Since material properties change with temperature, the yield stress is also temperature-dependent, and the Mises hardening criterion can be expressed as:
$$ \sigma = H \left( \int d\varepsilon^P, T \right) $$
In differential form:
$$ \frac{\partial \sigma}{\partial \sigma} d\sigma = \frac{\partial H}{\partial \varepsilon^P} d\varepsilon^P + \frac{\partial H}{\partial T} dT $$
In the elastic region, the total strain increment includes elastic and thermal strain increments, with the incremental stress-strain relationship given by:
$$ d\sigma = [D] (d\varepsilon – d\varepsilon^T) $$
In the plastic region, the total strain increment includes elastic, plastic, and thermal strain increments, leading to:
$$ d\sigma = [D]_{ep} (d\varepsilon – d\varepsilon^T) + d\sigma^T $$
Here, \( [D]_{ep} = [D] – [D]_p \), where \( [D]_p \) is the plastic matrix. Due to the nonlinear stress-strain relationship in thermo-elastoplastic problems, we simplify calculations using the “incremental load method,” linearizing the relationship in segments. For each loading step, increments \( \Delta \sigma \), \( \Delta \varepsilon \), and \( \Delta T \) replace differentials, resulting in incremental stress-strain relations for elastic and plastic regions, respectively:
$$ \Delta \sigma = [D] (\Delta \varepsilon – \Delta \varepsilon^T) $$
$$ \Delta \sigma = [D]_{ep} (\Delta \varepsilon – \Delta \varepsilon^T) + \Delta \sigma^T $$
where \( \Delta \sigma \) represents the instantaneous stress during induction heating.
To assess the quality of induction heating for cylindrical gears, temperature and stress uniformity are critical metrics. We define standard deviation and uniformity as follows. The standard deviation \( \sigma \) is calculated as:
$$ \sigma = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} (X_i – \bar{X})^2 } $$
where \( X_i \) are the data points (temperature or stress values), \( \bar{X} \) is the mean value, and \( N \) is the number of data points. Uniformity is then defined as:
$$ \text{Uniformity} = \left(1 – \frac{\sigma}{\bar{X}}\right) \times 100\% $$
This formulation allows us to quantitatively evaluate the homogeneity of temperature and stress distributions in the cylindrical gear during induction heating.
We conducted a preliminary experiment using a steel plate to validate our simulation model. The material was 45 steel, and we employed a novel induction heating device with an infrared thermometer to measure surface temperatures and a clamp power meter to collect electrical data. The simulation was performed using ANSYS APDL, incorporating material parameters from literature. The comparison between experimental and simulated temperature data showed good agreement, with errors within 10%, confirming the reliability of our finite element model for cylindrical gear applications.
For the cylindrical gear simulation, we considered a large spur cylindrical gear with a module of 40, 35 teeth, and a face width of 80 mm, made of 45 steel. To reduce computational cost, we analyzed a single tooth, given the symmetry of the cylindrical gear. The finite element model was constructed with careful mesh refinement: the tooth profile surface layer (4 mm depth) was meshed uniformly with dense elements, while the core region used non-uniform meshing; the tooth tip employed mapped meshing, and other parts used swept meshing. We used SOLID236 elements for harmonic magnetic field calculations, SOLID90 for temperature field analysis, and SOLID186 for structural stress analysis. The induction coil and magnetizer were included in the model, with two placement schemes: Scheme 1 and Scheme 2, differentiated by the orientation of the induction coil relative to the cylindrical gear tooth axis. In both schemes, the coil was positioned centrally above the tooth width in the axial direction.
The material properties for 45 steel are temperature-dependent and crucial for accurate simulation. Below is a summary of key parameters used in our model:
| Property | Value at 20°C | Temperature Dependence |
|---|---|---|
| Density (ρ) | 7850 kg/m³ | Assumed constant |
| Thermal Conductivity (λ) | 48 W/(m·K) | Decreases with temperature |
| Specific Heat Capacity (c) | 460 J/(kg·K) | Increases with temperature |
| Electrical Resistivity | 2.0e-7 Ω·m | Increases with temperature |
| Young’s Modulus (E) | 210 GPa | Decreases with temperature |
| Poisson’s Ratio (ν) | 0.3 | Assumed constant |
| Yield Strength | 355 MPa | Decreases with temperature |
The simulation solved for transient temperature and stress fields over a 5-second heating period. The temperature distribution on the tooth profile surface and mid-plane sections for both schemes revealed that the heat-affected zone was directly below the induction coil, with the highest temperature at the center. We selected observation points for detailed analysis: on the tooth surface, points were labeled O (center), A, B, C, D, and E along x or y directions; along the depth (z-direction), points at 1 mm, 2 mm, 3 mm, and 4 mm from the surface were considered. The temperature evolution showed rapid initial heating, with point O reaching the Curie temperature (approximately 768°C) at 1.7 seconds, causing a drop in magnetic permeability and reduced eddy current density, thus slowing the temperature rise. Other points exhibited lower heating rates with increasing distance from O, especially beyond the magnetizer-covered coil area.
The temperature uniformity analysis for both schemes is summarized in the table below, based on data from selected observation regions:
| Time (s) | Scheme 1 Temperature Uniformity (%) | Scheme 2 Temperature Uniformity (%) | Notes |
|---|---|---|---|
| 0 | 100.00 | 100.00 | Initial state |
| 1.0 | 85.34 | 82.15 | Rapid decrease due to differential heating |
| 2.0 | 88.92 | 86.40 | Begin recovery after Curie point |
| 3.0 | 91.50 | 89.22 | Stabilizing trend |
| 4.0 | 92.30 | 90.05 | Near steady-state |
| 5.0 | 92.62 | 90.28 | Final uniformity values |
Thermal stress evolution followed a different pattern: stresses initially increased rapidly due to temperature gradients, peaked, then decreased as material yield strength dropped with rising temperature, and eventually stabilized. Point O reached maximum stress around 0.4 seconds, approximately 354.16 MPa for Scheme 1 and 355.11 MPa for Scheme 2, both near the yield limit at that temperature. Other points showed lower peak stresses with increasing distance from O. The stress uniformity analysis revealed complex trends, as shown in the table below:
| Time (s) | Scheme 1 Stress Uniformity (%) | Scheme 2 Stress Uniformity (%) | Observations |
|---|---|---|---|
| 0 | 100.00 | 100.00 | Initial state, no stress |
| 1.0 | 88.45 | 90.12 | Decrease due to differential stress rates |
| 2.0 | 85.67 | 92.34 | Mixed trends from yielding and relaxation |
| 3.0 | 89.23 | 95.01 | Recovery as stresses equilibrate |
| 4.0 | 91.00 | 97.20 | High uniformity approached |
| 5.0 | 91.78 | 97.90 | Final stress uniformity values |
The relationship between temperature and stress uniformity can be expressed through a correlation analysis. We define a uniformity index \( U \) for both temperature \( T \) and stress \( S \), and compute their time-dependent behavior. The differential change in uniformity for temperature is influenced by heat conduction and eddy current distribution, while for stress, it depends on thermal expansion and material plasticity. A simplified model for uniformity evolution can be given by:
$$ \frac{dU_T}{dt} = -\alpha_T \cdot \nabla^2 T + \beta_T \cdot Q_{eddy} $$
$$ \frac{dU_S}{dt} = -\alpha_S \cdot \nabla \sigma + \beta_S \cdot \left( \frac{d\sigma_y}{dT} \right) $$
where \( U_T \) and \( U_S \) are temperature and stress uniformities, \( \alpha_T, \beta_T, \alpha_S, \beta_S \) are coefficients, \( \nabla^2 T \) is the Laplacian of temperature (indicating spatial variation), \( Q_{eddy} \) is eddy current heat source, \( \nabla \sigma \) is stress gradient, and \( \frac{d\sigma_y}{dT} \) is the temperature derivative of yield stress. These equations highlight that temperature uniformity improves with heat diffusion and stable eddy currents, while stress uniformity is affected by stress redistribution and yield strength changes.
For large cylindrical gears, the induction heating process must balance efficiency and uniformity. Our simulations show that Scheme 1 offers slightly better temperature uniformity (92.62% at 5 s) compared to Scheme 2 (90.28%), but Scheme 2 achieves superior stress uniformity (97.9% vs. 91.78%). This discrepancy arises because temperature uniformity is driven by thermal diffusion and electromagnetic effects, whereas stress uniformity involves mechanical responses like plasticity and relaxation. In practical applications for cylindrical gear hardening, optimizing both metrics is essential to prevent defects such as cracking or uneven hardness. The novel induction heating device demonstrated high efficiency, with rapid heating to the Curie point and controlled temperature profiles. However, the asynchronous trends between temperature and stress uniformity imply that process parameters (e.g., coil placement, power, frequency) must be tailored based on the desired outcome—whether prioritizing thermal homogeneity or mechanical integrity.
Further analysis involves sensitivity studies of key parameters. We derived formulas for uniformity as functions of material properties and process variables. For temperature uniformity in a cylindrical gear, considering radial and axial heat flow, we can approximate:
$$ U_T \approx 1 – \frac{\Delta T_{\text{max}}}{T_{\text{avg}}} = 1 – \frac{k \cdot P \cdot f(L, r)}{\rho c \cdot v \cdot A} $$
where \( \Delta T_{\text{max}} \) is the maximum temperature difference, \( T_{\text{avg}} \) is average temperature, \( k \) is a geometric factor, \( P \) is input power, \( f(L, r) \) is a function of coil length \( L \) and gear radius \( r \), \( v \) is heating speed, and \( A \) is area. For stress uniformity, incorporating thermal strain and elastic modulus:
$$ U_S \approx 1 – \frac{E \cdot \alpha \cdot \Delta T_{\text{max}}}{\sigma_y(T)} $$
with \( E \) as Young’s modulus, \( \alpha \) as thermal expansion coefficient, and \( \sigma_y(T) \) as temperature-dependent yield strength. These formulas emphasize that increasing power or decreasing heating speed can reduce temperature uniformity, while lower yield strength at high temperatures may improve stress uniformity.
In conclusion, the numerical simulation of local induction heating for large cylindrical gears provides valuable insights into temperature and stress field dynamics. The cylindrical gear, as a critical transmission component, benefits from targeted strengthening through induction heating. Our study confirms that the novel induction heating device is effective, with Scheme 1 and Scheme 2 offering trade-offs between temperature and stress uniformity. The cylindrical gear’s performance under induction heating hinges on complex interactions between electromagnetic, thermal, and mechanical phenomena. Future work should explore multi-objective optimization to simultaneously enhance both uniformities, potentially using advanced control algorithms or coil designs. This research contributes to improving the durability and reliability of cylindrical gears in heavy machinery, aligning with industrial demands for efficiency and cost-effectiveness.