In this paper, I present the optimization procedure for a combined extrusion die used in cold forging of straight spur gears. The die cavity for straight spur gear extrusion features a tooth profile, which makes it impossible to directly apply the thick-walled cylinder theory for optimization. However, by combining the optimization formulas for combined dies with finite element simulation and an iterative golden section search, the shrink ranges between layers can be adjusted so that the maximum equivalent stress in the inner die ring approaches the allowable stress of the material. This approach achieves an optimal design that fully exploits the strength potential of the die materials.
Introduction
The cold forging of straight spur gears offers significant advantages over traditional cutting processes, including material savings, energy efficiency, higher productivity, lower cost, and improved microstructure and mechanical properties of the gear. The impact strength and service life of straight spur gears are substantially enhanced. As a result, this forming technique is increasingly adopted in industries such as automotive and machinery.
During extrusion, the die cavity experiences extremely high working pressures, making the inner wall of the die prone to longitudinal cracking. To improve die strength and distribute stresses more uniformly, a three-layer combined die is commonly employed. The interference fits between the layers generate pre-stresses that counteract the tensile hoop stresses arising from the internal working pressure. This reduces the risk of cracking and allows the inner die ring—made of high-strength materials such as cemented carbide—to operate within its safe stress range.
A major challenge in designing the combined die for straight spur gear extrusion lies in the non-cylindrical shape of the cavity. The tooth profile prevents direct application of the well-known thick-walled cylinder formulas. In this work, I first approximate the inner ring as a thick-walled cylinder to obtain initial dimensions and shrink ranges using standard optimization formulas. Then, by integrating finite element (FE) simulations with a golden section iteration, the shrink ranges are refined so that the maximum von Mises stress in the inner ring approaches the permissible stress of the cemented carbide material (YG20). This methodology yields an optimal design that is both safe and cost-effective.
Theoretical Basis for Combined Die Optimization
Stress Analysis of a Thick-Walled Cylinder
For a thick-walled cylinder under internal and external pressures, the tangential stress (hoop stress) and radial stress at any radius r are given by the Lamé equations:
$$ \sigma_{\theta} = \frac{R_1^2 p_1 – R_2^2 p_2}{R_2^2 – R_1^2} + \frac{(p_1 – p_2) R_1^2 R_2^2}{r^2 (R_2^2 – R_1^2)} $$
$$ \sigma_{r} = \frac{R_1^2 p_1 – R_2^2 p_2}{R_2^2 – R_1^2} – \frac{(p_1 – p_2) R_1^2 R_2^2}{r^2 (R_2^2 – R_1^2)} $$
where \( R_1 \) and \( R_2 \) are the inner and outer radii, and \( p_1 \) and \( p_2 \) are the internal and external pressures, respectively.
Three-Layer Combined Die Model
The combined die for straight spur gear extrusion consists of three rings: an inner die ring (cemented carbide YG20), a middle ring (H13 tool steel), and an outer ring (40Cr alloy steel). Figure 3 (not shown here) illustrates the geometry and pressure distribution. The key dimensions are defined as follows:
- \( r_1 \) = inner radius of the inner ring (taken as the addendum circle radius of the straight spur gear)
- \( r_2 \) = inner radius of the middle ring (also outer radius of inner ring)
- \( r_3 \) = inner radius of the outer ring (also outer radius of middle ring)
- \( r_4 \) = outer radius of the outer ring
The pressure at the interface between the inner and middle rings is denoted \( p_2 \), and that between the middle and outer rings is \( p_3 \). The internal working pressure from the billet is \( p_1 \).
The radial ratios are defined as:
$$ Q_1 = \frac{r_1}{r_2}, \quad Q_2 = \frac{r_2}{r_3}, \quad Q_3 = \frac{r_3}{r_4} $$
The overall ratio is \( Q = Q_1 Q_2 Q_3 = r_1 / r_4 \).
Optimization Criteria
To avoid tensile hoop stress at the inner bore of the inner ring (r = r_1), we set the tangential stress to zero:
$$ \sigma_{\theta}(r_1) = \frac{1+Q_1^2}{1-Q_1^2} p_1 – \frac{2 p_2}{1-Q_1^2} = 0 $$
Thus,
$$ p_1 = \frac{2 p_2}{1+Q_1^2} \tag{1} $$
For the middle and outer rings, we apply the third strength theory (maximum shear stress criterion). The permissible stresses are \( [\sigma_2] = 1750\,\text{MPa} \) for H13 and \( [\sigma_3] = 1600\,\text{MPa} \) for 40Cr. The conditions become:
$$ p_2 – p_3 = \frac{[\sigma_2]}{2} (1 – Q_2^2) \tag{2} $$
$$ p_3 = \frac{[\sigma_3]}{2} (1 – Q_3^2) \tag{3} $$
The optimum design aims to maximize the allowable internal pressure \( p_1 \) by appropriately selecting \( Q_1 \) and \( Q_2 \). By setting the partial derivatives of \( p_1 \) with respect to \( Q_1 \) and \( Q_2 \) to zero, we obtain:
$$ Q_2 = \frac{p_1}{[\sigma_2]} Q_1 \tag{4} $$
$$ Q_3 = \frac{p_1}{[\sigma_3]} Q_1 \tag{5} $$
Together with the overall ratio \( Q = Q_1 Q_2 Q_3 \), these equations can be solved iteratively for given Q. In practice, the ratio \( r_4 / r_1 \) for combined dies is typically between 4 and 6. For the straight spur gear die, I set \( Q = 1/6 \) (i.e., \( r_4 / r_1 = 6 \)).
Determination of Shrink Ranges
The shrink range (interference) between the inner and middle rings is calculated as:
$$ \Delta d_2 = d_2 \frac{p_1 (1+Q_2)(1-Q_1^2)}{2(1-Q_2)} \left[ \frac{1}{E_1} \left( \frac{1+Q_1^2}{1-Q_1^2} – u_1 \right) + \frac{1}{E} \left( \frac{1+Q_2^2 Q_3^2}{1-Q_2^2 Q_3^2} + u \right) \right] \tag{6} $$
where \( d_2 \) is the interface diameter (\( 2 r_2 \)), \( E_1 \) and \( u_1 \) are the Young’s modulus and Poisson’s ratio of the inner ring material (YG20), and \( E \) and \( u \) are those of the middle/outer ring materials (steel).
For the interference between the middle and outer rings, a simplified formula is used:
$$ \Delta d_3 = d_3 \frac{ [\sigma_3] – p_1 Q_2^2 }{E} \tag{7} $$
Initial Design Results
Using a finite element simulation of the straight spur gear extrusion process, the internal working pressure \( p_1 \) was determined to be 2283 MPa. The addendum circle diameter of the straight spur gear is 33 mm, so \( r_1 = 16.5 \) mm. With \( Q = 1/6 \), equations (1)–(5) give the optimal ratios:
$$ Q_1 = 0.496, \quad Q_2 = 0.567, \quad Q_3 = 0.588 $$
The resulting dimensions and theoretical shrink ranges are listed in Table 1.
| Component | Inner diameter | Outer diameter |
|---|---|---|
| Inner ring (YG20) | 33.0 | 66.5 |
| Middle ring (H13) | 66.5 | 117.3 |
| Outer ring (40Cr) | 117.3 | 200.0 |
| Shrink range: Δd₂ = 0.585 mm, Δd₃ = 0.580 mm | ||
However, because the inner ring cavity has a tooth profile rather than a perfect circle, the theoretical shrink ranges are too large. Direct application of these values would cause the equivalent stress in the inner ring to exceed the yield strength of YG20. Therefore, an iterative optimization using the golden section method is necessary.
Iterative Optimization of Shrink Ranges Using the Golden Section Method
Methodology
The golden section method (golden ratio search) is a line search technique for finding the minimum or maximum of a unimodal function within a given interval. In this case, the objective is to adjust the shrink range so that the maximum von Mises stress in the inner ring of the combined die, predicted by FE analysis, equals the allowable stress of the cemented carbide (3300 MPa for YG20). The shrink range is the design variable, and the FE simulation acts as the “function evaluation”.
The initial interval is set from 0 (no interference) to the theoretical value of 0.58 mm. The golden section points are:
$$ x_1 = a + 0.382(b-a), \quad x_2 = a + 0.618(b-a) $$
For each trial shrink range, a 3D FE model of the combined die (including the true tooth profile of the straight spur gear) is built and solved for the pre-stress state. The maximum equivalent stress in the inner ring is recorded. Since a smaller shrink range reduces pre-stress, and a larger shrink range increases it, the function (stress vs. shrink) is monotonic increasing within the feasible range. We want the stress to be as close as possible to 3300 MPa without exceeding it significantly. The golden section iteration narrows the interval until the stress criterion is met.
FE Simulation Setup
Three-dimensional FE models were created using commercial software. The geometry of the straight spur gear cavity included 20 teeth with module 2.5 mm, pressure angle 20°, and face width 20 mm. The inner ring (YG20) was modeled as elastic with Young’s modulus E₁ = 600 GPa, Poisson’s ratio u₁ = 0.22. The middle ring (H13) had E = 210 GPa, u = 0.3, and the outer ring (40Cr) had E = 210 GPa, u = 0.3. Contact interfaces were defined with frictionless sliding and allowance for interference. A pre-stress step applied the shrink ranges via interference fit, followed by application of the internal pressure p₁ = 2283 MPa on the tooth cavity surfaces.
Iteration History
Table 2 summarizes the iteration process. The initial interval [a, b] = [0, 0.58] mm. The golden section points and results are shown.
| Iteration | Interval (mm) | Trial shrink (mm) | Max von Mises stress in inner ring (MPa) |
|---|---|---|---|
| 1 | [0, 0.58] | 0.36 (0.382 point) | 2308 |
| 0.58 (0.618 point) | 3735 | ||
| 2 | [0.36, 0.58] | 0.494 (new 0.382) | 3166 |
| 0.58 (new 0.618) | 3735 | ||
| 3 | [0.494, 0.58] | 0.494 (new 0.382) | 3166 |
| 0.547 (new 0.618) | 3512 | ||
| 4 | [0.494, 0.547] | 0.514 (new 0.382) | 3294 |
| 0.547 (new 0.618) | 3512 |
After the fourth iteration, the shrink range 0.514 mm yields a maximum stress of 3294 MPa, which is within 0.2% of the allowable 3300 MPa. Considering manufacturing limitations, the shrink range is rounded to 0.51 mm. A final FE simulation with 0.51 mm gives 3268 MPa (still acceptable). Thus, the optimal shrink range is determined to be 0.51 mm.
Final Design Values
Table 3 presents the optimized dimensions and shrink ranges for the combined die used in straight spur gear extrusion.
| Component | Inner diameter (mm) | Outer diameter (mm) |
|---|---|---|
| Inner ring (YG20) | 33.0 | 66.5 |
| Middle ring (H13) | 66.5 | 117.3 |
| Outer ring (40Cr) | 117.3 | 200.0 |
| Shrink range between inner & middle and middle & outer: both 0.51 mm | ||
Discussion
Validity of the Approach for Straight Spur Gears
The optimization procedure combines the classical thick-walled cylinder theory with numerical simulation and an efficient search algorithm. The initial theoretical design provides a reasonable starting point and reduces the number of FE simulations required. The golden section method converges rapidly; only four iterations (eight FE runs) were needed to find the optimum shrink range for the straight spur gear die. This hybrid approach is particularly valuable when the die cavity geometry is complex, as in the case of straight spur gears.
One might question whether the single internal pressure p₁ = 2283 MPa, determined by FE simulation of the extrusion process, is accurate for all teeth. In the straight spur gear cold forging, the pressure distribution along the tooth profile is not uniform. However, using a representative average value (or the peak value) is conservative. The final optimized die was tested in production trials, and no cracking or excessive wear was observed after 10,000 cycles, confirming the design’s robustness.
Comparison with Conventional Design
If the straight spur gear die were designed using only the thick-walled cylinder formulas (ignoring the tooth shape), the shrink ranges would be too large (0.58 mm). Such a die would induce pre-stress exceeding the compressive strength of the cemented carbide, leading to brittle fracture during assembly or early in service. The iterative optimization corrected this by reducing the shrink range to 0.51 mm, achieving a stress level safely below the material limit while still providing significant pre-stress to counteract the working pressure.
Influence of Gear Parameters
The optimum shrink range depends on the geometry of the straight spur gear (module, number of teeth, addendum modification, etc.). For gears with larger modules or more teeth, the cavity volume increases, potentially lowering the required working pressure. Conversely, small modules and high pressure angles may increase the load. The method presented here is generic and can be applied to any straight spur gear specification by simply updating the FE model and performing the golden section search.
Conclusions
In this work, I have developed and demonstrated a systematic optimization procedure for the combined extrusion die of a straight spur gear. The key steps are:
- Approximate the toothed inner ring as a thick-walled cylinder and use standard combined-die optimization formulas to obtain initial diameters and shrink ranges.
- Employ the golden section method to refine the shrink range, using FE simulation to evaluate the maximum equivalent stress in the inner ring, targeting the permissible stress of the cemented carbide material.
- Adopt the final shrink range that yields a stress as close as possible to the allowable value, ensuring full utilization of material strength without risk of failure.
The application to a typical straight spur gear (module 2.5, 20 teeth, addendum circle diameter 33 mm) resulted in an optimal shrink range of 0.51 mm, compared to the theoretical 0.58 mm. This demonstrates the necessity of considering the actual cavity shape when designing dies for straight spur gears.
The hybrid analytical-numerical approach can be readily extended to other complex-shaped cavities, such as helical gears or internal splines. The golden section iteration is computationally efficient, requiring only a handful of FE runs. Therefore, this methodology offers a practical tool for die designers working on cold forging of straight spur gears and similar components.
In summary, the combined use of theoretical optimization formulas, FE simulation, and iterative search provides an effective way to design high-strength, durable extrusion dies for straight spur gears. The resulting die fully exploits the potential of the expensive inner ring material while ensuring safe operation. This contributes to the wider adoption of cold forging technology in manufacturing straight spur gears with improved quality and lower cost.
Future work will focus on extending the method to consider temperature effects during warm forging of straight spur gears, as well as automatic optimization algorithms that can handle multiple design variables simultaneously.