Cylindrical gears are fundamental components prized for their high transmission efficiency, broad applicability, and strong load-bearing capacity, making them indispensable in sectors like aerospace, automotive, and machine tools. Conventional manufacturing methods, such as milling, hobbing, and shaping, are increasingly revealing limitations including inferior mechanical properties, significant material waste, and high production costs. As an advanced near-net-shape forming technology, cross-rolling of gears offers a promising alternative with advantages like high production efficiency, minimal material usage, and excellent surface quality. The forming force during this process, influenced by material properties, roller parameters, and process conditions, is critical for final workpiece quality and tool life. Accurately calculating this force is essential for optimal roll design and equipment selection.
Understanding the material behavior is the first step. We begin by establishing a constitutive model for a typical gear steel, 40CrNiMo, under relevant conditions.
Establishment of the Constitutive Model for 40CrNiMo Steel
The flow stress of a material during plastic deformation is governed by strain, strain rate, and temperature. To model this for 40CrNiMo, isothermal compression tests were conducted. Specimens were heated to temperatures of 20°C, 80°C, 140°C, and 200°C and compressed at strain rates of 0.005 s⁻¹, 0.05 s⁻¹, and 0.5 s⁻¹. The obtained true stress-true strain curves, shown in the figure below, indicate that flow stress decreases with increasing temperature and increases with increasing strain rate—a behavior typical of many metals.
The Johnson-Cook (J-C) model is widely used to describe such thermo-viscoplastic behavior. The original J-C model is expressed as:
$$ \sigma = (A + B\varepsilon^n)\left(1 + C\ln{\dot{\varepsilon}^*}\right)\left[1 – \left(\frac{T – T_0}{T_m – T_0}\right)^m\right] $$
where \( \sigma \) is the equivalent stress, \( \varepsilon \) is the equivalent plastic strain, \( \dot{\varepsilon}^* = \dot{\varepsilon}/\dot{\varepsilon}_0 \) is the dimensionless strain rate, \( T \) is the deformation temperature, \( T_0 \) is the reference temperature, \( T_m \) is the melting temperature, and \( A, B, C, n, m \) are material constants. Using 20°C and 0.005 s⁻¹ as reference conditions, the constants were determined via linear regression from the experimental data. However, predictions from this original model showed significant deviations, especially at higher temperatures.
To improve accuracy, a modified Johnson-Cook model was employed. This model replaces the power-law hardening term with a cubic polynomial:
$$ \sigma = (A_1 + B_1\varepsilon + B_2\varepsilon^2 + B_3\varepsilon^3)\left(1 + C_1\ln{\dot{\varepsilon}^*}\right)\left[1 – \left(\frac{T – T_0}{T_m – T_0}\right)^m\right] $$
The material constants \( A_1, B_1, B_2, B_3, C_1, \) and \( m \) were fitted from the experimental data. The final modified constitutive model for the cylindrical gear steel 40CrNiMo is:
$$ \sigma = (-221 + 17519\varepsilon – 64889\varepsilon^2 + 82894\varepsilon^3)(1 + 0.017\ln{\dot{\varepsilon}^*})\left[1 – \left(\frac{T – 20}{1510 – 20}\right)^{0.7744}\right] $$
The Root Mean Square Error (RMSE) for this modified model was calculated to be 9.575, confirming its significantly better predictive capability compared to the original model for the conditions relevant to cylindrical gear forming.
| Element | C | Si | Mn | Cr | Ni | Mo | S | P | Fe |
|---|---|---|---|---|---|---|---|---|---|
| Content | 0.39 | 0.20 | 0.67 | 0.78 | 1.34 | 0.199 | 0.003 | 0.016 | Bal. |
Theoretical Modeling of Cross-Rolling Force for Cylindrical Gears
The cross-rolling process involves complex three-dimensional forces. For analysis, we consider the radial forming force \( F_{\tau} \), which is primary for equipment design. Its calculation requires determining the average contact pressure and the instantaneous contact area between the tooth-profile roll and the cylindrical gear workpiece.
Tooth Height Evolution Model
During cross-rolling, material is displaced from the workpiece blank to form the gear teeth. Assuming volume constancy (plane strain condition in the central region), the growth in tooth height is directly related to the roll’s indentation depth. The process of generating the roll profile is based on the principle of gear generation using a rack tool. By varying the center distance (simulated by the rack’s offset coefficient), the evolving tooth profile of the workpiece during forming can be derived, and subsequently, the conjugate roll profile.
The coordinate transformation from the rack tool to the workpiece and then to the roll is key. The workpiece tooth profile in its moving coordinate system \( O_1X_1Y_1 \) is given by:
$$
\begin{bmatrix} x_{gi} \\ y_{gi} \\ 1 \end{bmatrix} = \begin{bmatrix} x_{ti}\cos\gamma – y_{ti}\sin\gamma + r_2\cos\gamma + S\sin\gamma \\ x_{ti}\sin\gamma + y_{ti}\cos\gamma + r_2\sin\gamma – S\cos\gamma \\ 1 \end{bmatrix}, \quad i=1,2,3
$$
where \( (x_{ti}, y_{ti}) \) are coordinates of the rack tool profile (representing tooth tip line, tip fillet, and tooth flank), \( \gamma \) is the workpiece rotation angle, \( r_2 \) is the workpiece pitch radius, and \( S \) is the rack translation distance.
The forming of a single tooth slot is analyzed in two stages:
Stage 1: The roll’s tip arc and tip fillet penetrate the workpiece. The indented area \( S_{BCDG} \) on the blank and the grown tooth area \( S_{AEFB} \) are approximated as polygons and trapezoids. Equating these areas yields the tooth height growth \( H_1 \).
Stage 2: The roll’s involute flank also contacts the workpiece. The indented area \( S_{KLMNG} \) and the more complex grown tooth area \( S_{HIJK} \), involving circular and involute segments, are calculated. Equating these gives the tooth height growth \( H_2 \) for this stage.
For a cylindrical gear with module \( m = 1.75 \) mm, pressure angle \( \alpha = 20^\circ \), and 46 teeth, the theoretical relationship between roll indentation depth and tooth height growth was calculated, providing a crucial input for contact area determination.
Contact Area Model
The contact area is dynamic and highly complex. We simplify by considering the instantaneous interference region when the roll is pressed into the pre-formed workpiece tooth profile at a specific indentation depth, corresponding to a roll rotation angle where contact arc length is maximal. The total contact area \( A_c \) is obtained by integrating the contact arc length \( l_c(z) \) along the tooth width (axial direction, z):
$$ A_c = \int_{0}^{L} l_c(z) \, dz $$
where \( L \) is the face width of the cylindrical gear. The contact arc length \( l_c \) itself is an integral of the differential arc length along the roll profile in contact:
$$ l_c = 2 \int_{x_0}^{x_i} \sqrt{1 + (y’)^2} \, dx $$
For the two-stage model, the total contact area is the sum of areas from Stage 1 (\( S_1 \)) and Stage 2 (\( S_2 \)):
$$ S_1 = \int_{0}^{\frac{H_{D_i}}{\sin\alpha_0}} \frac{l_{r1}}{2} \, dz, \quad S_2 = \int_{\frac{H_{D_i}}{\sin\alpha_0}}^{\frac{H_{N_i}}{\sin\alpha_0}} \frac{l_{r1} + l_{r2}}{2} \, dz $$
$$ A_c = S_1 + S_2 $$
Here, \( l_{r1} \) and \( l_{r2} \) are the contact arc lengths for Stage 1 and Stage 2 profiles, \( H_{D_i} \) and \( H_{N_i} \) are indentation depths, and \( \alpha_0 \) is the pressure angle. This method, combining the “thin slice” and integral approaches, allows for a theoretical estimation of the contact area during cylindrical gear cross-rolling.
Average Stress and Forming Force Model
To find the average contact pressure \( \bar{\sigma} \), the slab method (or principal stress method) is applied. Considering an infinitesimal element in the deformation zone and assuming plane strain conditions (minimal axial flow in the central region of the cylindrical gear), the force equilibrium in the radial direction leads to:
$$ \sum F_r = \sigma_r h l – (\sigma_r + d\sigma_r) h l – 2 \tau l \, dr = 0 $$
where \( h \) is the elemental height, \( l \) is the axial length, and \( \tau \) is the shear stress at the interface. Using the Tresca yield criterion \( \sigma_{\theta} – \sigma_r = \sigma_s \) (where \( \sigma_s \) is the yield stress) and a constant friction condition \( \tau = \mu k = \mu \sigma_s / \sqrt{3} \), the differential equation can be solved. Applying boundary conditions gives the normal stress distribution on the contact surface. Integrating this stress over the contact area yields the total forming force \( F_0 \). The average stress is then:
$$ \bar{\sigma} = \frac{F_0}{A_c} = \frac{F_0}{S_1 + S_2} $$
Finally, the theoretical radial forming force \( F \) for the cylindrical gear cross-rolling process is:
$$ F = \bar{\sigma} \cdot A_c $$
This model can be evaluated for both single-tooth engagement (simplified case) and double-tooth engagement (more realistic) scenarios by using the corresponding contact areas \( A_c^{\text{(one)}} \) and \( A_c^{\text{(two)}} \).
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Workpiece Teeth Number | \( z_2 \) | 46 | – |
| Module | \( m \) | 1.75 | mm |
| Pressure Angle | \( \alpha \) | 20 | ° |
| Addendum Coefficient | \( h_a^* \) | 1.0 | – |
| Dedendum Coefficient | \( c^* \) | 0.25 | – |
| Initial Workpiece Radius | \( r_0 \) | 40.11 | mm |
| Friction Factor | \( \mu \) | 0.15 | – |
Finite Element Simulation of Cylindrical Gear Cross-Rolling
To validate the theoretical models, 3D finite element (FE) simulations were performed using DEFORM software. Both single-tooth and double-tooth models of the cylindrical gear forming process were set up.
Model Setup
The roll, mandrel, and axial baffles were defined as rigid bodies. The cylindrical gear workpiece was modeled as a plastic body with the modified Johnson-Cook constitutive model for 40CrNiMo assigned. A shear friction model with a factor of 0.15 was applied at the roll-workpiece interface. The process parameters were set to replicate cold rolling: temperature = 20°C, roll rotational speed = 1.05 rad/s, and axial feed rate = 0.7 mm/s. The initial workpiece diameter was Φ80.8 mm.
Simulation Results and Analysis
The forming process of the cylindrical gear can be divided into three stages: dividing, forming, and finishing. In the dividing stage, the roll marks the tooth slots on the blank. In the main forming stage, the tooth profile is fully developed, and material flows to form the involute shape, often creating “flash” or ears. In the finishing stage, the profile is calibrated and smoothed.
1. Tooth Height Evolution: The growth of tooth height from the FE simulation was measured at different cross-sections. The results showed good agreement with the theoretical predictions from the tooth height model, confirming the area constancy principle underlying the model for the cylindrical gear.
2. Contact Area: The instantaneous contact area was extracted from the simulation history. For the single-tooth model, the theoretical and simulated contact areas showed close correlation. For the double-tooth model, the simulated area was often lower than the theoretical prediction because, in practice, material flow sometimes prevented perfect simultaneous engagement of both roll teeth with the workpiece—a condition assumed in the theoretical double-tooth model.
3. Forming Force: The radial force history from FE simulation exhibited a single-peak trend: increasing during initial engagement and forming, then decreasing during finishing as the contact area reduces.
- For the single-tooth model, the theoretical maximum force was 258.49 kN, and the simulated maximum was 255.36 kN. The relative error was only 1.23%.
- For the double-tooth model, the theoretical maximum force was 347.79 kN, and the simulated maximum was 315.95 kN. The relative error was 10.08%. The higher error is attributed to the ideal double-contact condition in the theory versus the occasional single-contact in the simulation due to material flow.
These results validate the accuracy of the theoretical forming force model, especially for the single-tooth case which is a fundamental building block for understanding the process.
| Model Type | Theoretical Max Force (kN) | FE Simulated Max Force (kN) | Relative Error |
|---|---|---|---|
| Single-Tooth Model | 258.49 | 255.36 | 1.23% |
| Double-Tooth Model | 347.79 | 315.95 | 10.08% |
Experimental Verification
Physical cross-rolling experiments were conducted on a dedicated gear rolling machine to further validate the models. The cylindrical gear workpiece (40CrNiMo, Φ80.8 mm) was rolled using a designed tooth-profile roll under the same process parameters used in the simulation.
Experimental Results and Comparison
1. Tooth Height: The formed cylindrical gears were sectioned, and the tooth profiles were measured using a coordinate measuring system. The measured tooth height growth at different indentation depths was compared with theoretical and FE results. The maximum relative error between theory and experiment was 4.48%, demonstrating the practical applicability of the tooth height evolution model.
2. Forming Force: The radial forming force was recorded during the experiment. The force-time curve also showed a characteristic single-peak shape. The experimental maximum force was 331.25 kN.
Comparing all three values for the double-tooth engagement scenario:
$$ F_{\text{theory}}^{\text{max}} (347.79\ \text{kN}) > F_{\text{exp}}^{\text{max}} (331.25\ \text{kN}) > F_{\text{FE}}^{\text{max}} (315.95\ \text{kN}) $$
The theoretical model gives a slightly higher, and therefore conservative, estimate—which is safe for equipment design. The experimental value is higher than the FE value, likely due to factors like strain hardening and dry friction conditions not fully captured in the simulation. The 4.48% relative error between the theoretical and experimental maximum force confirms the high accuracy and practical utility of the developed forming force model for cylindrical gear cross-rolling.
| Source | Maximum Forming Force (kN) | Notes |
|---|---|---|
| Theoretical Model | 347.79 | Conservative, based on ideal double-contact. |
| Finite Element Simulation | 315.95 | Reflects dynamic material flow. |
| Experimental Measurement | 331.25 | Includes real-world effects like dry friction. |
Conclusion
- A modified Johnson-Cook constitutive model was successfully established for 40CrNiMo cylindrical gear steel, providing an accurate description of its flow stress under forming-relevant temperatures and strain rates, with an RMSE of 9.575.
- A theoretical model for calculating the forming force in cylindrical gear cross-rolling was developed. It integrates a tooth height evolution model based on area constancy, a contact area model derived using the thin-slice and integral method, and an average stress solution from the slab method under plane strain assumption.
- Finite element simulations and physical experiments were conducted for a cylindrical gear with module 1.75 mm and 46 teeth. The results validated the models:
- The theoretical maximum forming force showed excellent agreement with the FE result for the single-tooth model (1.23% error).
- For the more practical double-tooth engagement, the theoretical model provided a conservative estimate with a 10.08% error compared to simulation and a 4.48% error compared to the experimental maximum force.
The developed theoretical forming force model demonstrates sufficient accuracy for industrial applications in the design of rolling tools and the selection of processing equipment for cylindrical gear cross-rolling, contributing to the advancement of this efficient near-net-shape manufacturing technology.