Straight spur gears are fundamental mechanical components widely used in automotive, machine tool, and agricultural industries. Traditionally, these gears are produced by cutting methods, which suffer from low material utilization, low efficiency, and interruption of the forging flow lines that degrade mechanical properties. To address these issues, cold extrusion precision forming has been extensively studied. However, conventional one-way cold extrusion of straight spur gears often encounters high forming loads and poor filling at tooth corners due to unfavorable friction conditions. In this work, we propose a controlled movable die forming process for straight spur gears. By synchronizing the die movement with the punch, we aim to reduce the forming load and improve the filling uniformity at the tooth corners. This paper presents a comprehensive numerical simulation using Deform-3D, an optimization of the die speed ratio, and experimental validation. The results demonstrate that when the die speed is half of the punch speed, the tooth profile is uniformly filled, the corner filling is complete, and the forming load is reduced by approximately 20% compared to conventional one-way extrusion. The study provides a practical reference for the design of straight spur gear cold extrusion processes and dies.
Introduction
The forming of straight spur gears by cold extrusion is a near-net-shape process that can significantly improve material utilization and mechanical performance. However, the typical flat geometry (height-to-diameter ratio less than 1/2 to 1/10) makes it difficult to fill the narrow tooth cavities, especially at the bottom corners. In conventional closed-die extrusion, the punch pushes the billet downward into a stationary die cavity. The friction between the billet and the die wall opposes the radial flow, causing non-uniform filling: the upper part of the tooth fills faster than the lower part, leading to incomplete filling and high required forming loads. To overcome this, we adopt the floating die principle, where the die moves axially in the same direction as the punch. This converts the friction force from an obstacle into a beneficial force that assists the metal flow toward the bottom of the die cavity. The concept is referred to as a controlled movable die. In this study, we systematically investigate the effect of die speed relative to punch speed on the forming behavior of a straight spur gear with modulus m = 3, number of teeth z = 18, and pressure angle α = 20°. The billet material is 10 steel (similar to AISI 1010), and the forming temperature is room temperature (20 °C). The punch speed is fixed at 6 mm·s⁻¹, while the die speed varies among 0, 3, and 6 mm·s⁻¹. The friction coefficient is 0.12, and the step displacement per simulation increment is 0.08 mm. The billet is a solid cylindrical bar.
Process Principle of Controlled Movable Die
In conventional one-way extrusion (die speed Vm = 0), the friction force τ between the billet and the die wall opposes the downward flow of the metal, as shown schematically in the original study. This counteracting friction impedes the radial filling of the tooth cavities, especially at the lower end, resulting in a high forming load and potential underfill. In contrast, with a movable die moving downward at a controlled speed Vm, the relative motion between the die and the billet can be adjusted. If the die moves slower than the bulk metal flow, the friction still opposes flow; but if the die moves faster than the local metal flow, the friction direction reverses and becomes a “positive” friction that assists the downward movement. The key is to achieve a die speed that maximizes this beneficial effect. Based on the fundamental friction theory, the axial friction force per unit area is given by:
$$\tau = \mu \sigma_n$$
where μ is the Coulomb friction coefficient and σn is the normal contact stress. The direction of τ depends on the relative velocity between the billet and the die. In the movable die case, when Vm > Vmetal, the friction acts downward, pulling the metal into the lower cavity. This is particularly helpful during the final stage of filling when metal flow slows down. The optimal velocity ratio is determined by the condition that the die speed equals the average metal flow speed at the die wall interface. In practice, we found through numerical experiments that a ratio of Vm / Vpunch = 1/2 yields the best result for the straight spur gear geometry investigated.
Numerical Simulation Setup
The finite element software Deform-3D was employed to simulate the cold extrusion process. The geometric parameters of the straight spur gear are listed in the following table.
| Parameter | Value |
|---|---|
| Modulus (m) | 3 |
| Number of teeth (z) | 18 |
| Pressure angle (α) | 20° |
| Billet material | 10 steel |
| Forming temperature | 20 °C |
| Punch speed Vd | 6 mm·s⁻¹ |
| Die speed Vm | 0, 3, 6 mm·s⁻¹ |
| Friction coefficient μ | 0.12 |
| Step displacement | 0.08 mm |
| Billet type | Solid cylindrical bar |
The simulation was performed for three cases: Vm = 0 (conventional one-way), Vm = 3 mm·s⁻¹ (half of punch speed), and Vm = 6 mm·s⁻¹ (equal to punch speed). The forming load, metal flow pattern, and filling quality were recorded at each step. The total stroke was sufficient to completely form the gear tooth profile.
Results and Discussion
Conventional One-Way Extrusion (Vm = 0)
In the conventional case, the die is stationary. At step 90 of the simulation (near the final stage), the upper part of the tooth cavity is already filled, while the lower part remains incompletely filled. This is due to the upward frictional resistance that hinders downward material flow. The predicted forming load curve shows a sharp increase at the end, reaching 9920 kN. The large load is required to deform the metal into the remaining unfilled corners. The non-uniform filling leads to defects such as underfill at the bottom tooth flank. This observation confirms that conventional one-way extrusion is inefficient for straight spur gears with a flat geometry.
Movable Die Extrusion (Vm = 3 mm·s⁻¹)
When the die moves downward at 3 mm·s⁻¹ (half the punch speed), the friction force becomes beneficial. At the same step 90, the tooth profile is uniformly filled from top to bottom. The material flows more smoothly into the lower cavity because the die motion reduces the relative velocity and reverses the friction direction. The predicted forming load curve exhibits a lower maximum load of 8130 kN, which is about 18% lower than the conventional case. The filling is complete with no underfill or folding defects. The benefit is attributed to the “positive friction” that assists the material flow into the die cavity. This can be expressed by the effective shear stress condition:
$$\tau_{\text{effective}} = \mu \sigma_n \cdot \text{sign}(V_{\text{relative}})$$
where sign(Vrelative) is positive when the die moves faster than the local metal flow. In our optimal case, the relative velocity is small and positive, maximizing the assisting component.
Movable Die Extrusion (Vm = 6 mm·s⁻¹)
When the die speed equals the punch speed (6 mm·s⁻¹), the die moves at the same velocity as the punch. In this scenario, the lower part of the tooth fills faster than the upper part, causing an inverse non-uniformity. The forming load is similar to the conventional case (around 9800 kN), and the filling quality is not improved. This indicates that too high a die speed over-assists the bottom flow while starving the top, and the overall benefit is lost. Therefore, there exists an optimal ratio between die speed and punch speed for the straight spur gear.
Summary of Simulated Loads
The following table summarizes the maximum forming loads for the three die speeds investigated.
| Die Speed Vm (mm·s⁻¹) | Punch Speed Vd (mm·s⁻¹) | Maximum Load (kN) | Load Reduction vs. Conventional |
|---|---|---|---|
| 0 | 6 | 9920 | — |
| 3 | 6 | 8130 | 18.0% |
| 6 | 6 | 9800 | 1.2% |
The results clearly show that the optimal condition is Vm = Vd/2. The reduction in forming load is consistent with the improved metal flow. The underlying mechanism can be explained by considering the force balance on an infinitesimal element of the billet. The axial equilibrium includes the punch force, the die wall friction, and the backpressure from the lower punch. The movable die reduces the frictional drag and even provides a driving component, thereby lowering the required punch force. Mathematically, the punch force F can be approximated by:
$$F = A \sigma_y + \int_{S} \tau \, dS$$
where A is the cross-sectional area, σy is the yield stress of the material, and the integral is over the die–billet contact surface S. With the movable die, the integral term becomes smaller or even negative (if friction assists), leading to a smaller total force.
Experimental Validation
To confirm the numerical predictions, cold extrusion experiments were conducted on a four-column hydraulic press. The material was 10 steel billets. The die motion was controlled hydraulically using a quantitative pump and a proportional speed control valve. Both conventional (stationary die) and movable die (Vm = 3 mm·s⁻¹) processes were tested. The forming load was measured with a load cell. The experimental results are summarized in the table below.
| Process | Measured Max Load (kN) | Simulated Max Load (kN) | Difference (%) |
|---|---|---|---|
| Conventional (Vm=0) | 9850 | 9920 | 0.7 |
| Movable die (Vm=3 mm·s⁻¹) | 8080 | 8130 | 0.6 |
The experimental loads are in good agreement with the simulation. The load reduction achieved with the movable die is about 20%, consistent with the simulation. More importantly, the formed straight spur gears from the movable die process exhibited complete tooth profiles with no corner underfill or collapse. The surface quality was satisfactory, and no folding defects were observed. A photograph of an experimental gear is shown in the following figure, which demonstrates the excellent filling of the tooth cavity.
The experimental results validate the numerical analysis and confirm the effectiveness of the controlled movable die process for straight spur gears. The die speed ratio of 1/2 (relative to punch speed) is recommended as a robust design rule for similar cold extrusion processes.
Conclusions
In this work, we have developed and demonstrated a controlled movable die cold extrusion process for straight spur gears. Through numerical simulation and experimental validation, the following conclusions are drawn:
- The conventional one-way extrusion of straight spur gears suffers from non-uniform filling and high forming loads due to unfavorable friction.
- By moving the die downward at an appropriate speed, the friction force can be transformed into a beneficial force that assists material flow into the lower tooth cavities.
- For the straight spur gear with modulus 3 and 18 teeth, the optimal die speed is half of the punch speed (3 mm·s⁻¹ when punch speed is 6 mm·s⁻¹). At this ratio, the tooth profile is uniformly filled, corners are fully saturated, and the forming load is reduced by approximately 20% compared to conventional extrusion.
- Both too low (0) and too high (equal to punch speed) die speeds degrade the filling uniformity and load reduction.
- Experimental results confirm the numerical predictions, with the formed straight spur gears exhibiting complete filling and no defects.
The proposed controlled movable die technique provides an effective way to enhance the formability of straight spur gears in cold extrusion, offering potential for industrial applications where low forming loads and high quality are required. Future work may explore alternative gear geometries and material types to extend the applicability of this method.