In the realm of mechanical engineering, gears stand as ubiquitous and critically demanded components, with the spur and pinion gear pair being fundamental to power transmission systems. The cold precision forging of spur and pinion gears has garnered significant attention due to its ability to produce parts with superior mechanical properties, minimal heat treatment-induced deformation, enhanced wear resistance, and prolonged service life. However, the conventional cold forging process for such gears, which typically involves integral loading and forming, suffers from excessively high forming loads. This leads to accelerated die wear, reduced tool life, and increased production costs. While various pressure-relief or分流分流 techniques have been proposed to mitigate these loads, the final forging force often remains prohibitively high. Some researchers have explored a two-step process involving global pre-forming followed by local loading for the final tooth shape, achieving notable load reduction. Nevertheless, this method often leaves residual flash or protrusions on the gear face, necessitating additional machining. In this work, we propose and investigate a novel partitioned local forming (PLF) strategy for the cold precision forging of cylindrical spur and pinion gears. Using DEFORM-3D finite element analysis (FEA) software as our primary tool, we simulate the entire forming process. Our results demonstrate that this PLF process not only ensures complete tooth cavity filling but also dramatically reduces the required forming load. This analysis provides a theoretical foundation for advancing the practical industrial application of cold forging for spur and pinion gears.
The core innovation of our approach lies in the partitioned local forming methodology. “Partitioning” refers to the conceptual division of the target spur and pinion gear into two or more symmetric regions. “Local forming” denotes the sequential application of force to only one of these regions at a time, as opposed to the traditional integral loading where the entire face is pressed simultaneously. This sequential, localized loading drastically reduces the instantaneous contact area in the main force direction, leading to a substantial decrease in the required press tonnage. The fundamental principle is illustrated in the following description. Initially, a cylindrical billet is partitioned into two equal regions, labeled Zone① and Zone②. The first forming stage involves locally pressing only the material corresponding to Zone① into the tooth cavities of the die. Subsequently, in the second stage, the material in Zone② is formed. This step-by-step approach completes the forging of the entire spur and pinion gear. The strategic division and sequential forming are key to managing metal flow and stress distribution.
To quantify the process, we established a detailed geometric model. The target spur gear parameters are: module m = 1.5 mm, number of teeth z = 18, pressure angle α = 20°, face width B = 10 mm, and bore diameter d = 11.5 mm. Adhering to the principle of volume constancy and aiming for a billet diameter close to the gear’s root circle diameter, the initial billet dimensions were selected as Ø23 mm × 15 mm. Since the PLF process requires a pre-formed billet with distinct zones, a pre-forging operation is essential. The pre-forged billet features a 5 mm upward protrusion in Zone① and a 5 mm downward protrusion in Zone②, creating the necessary volume distribution for subsequent local forming. The three-dimensional models for the dies—including the punch, upper pressure plate, die cavity, lower pressure plate, and lower bolster—were meticulously created using Pro/ENGINEER software. These components work in concert to facilitate the sequential local forming of the spur and pinion gear.
The finite element model was constructed with a focus on computational efficiency and accuracy. Given the symmetry of both the pre-formed billet (quarter symmetry) and the gear teeth (cyclic symmetry), only one-quarter of the workpiece was modeled for the final forming simulation. The die components were defined as rigid bodies, while the workpiece material was defined as a plastic body using AISI-1045 (cold) from the DEFORM-3D material library. The friction condition at the tool-workpiece interface was modeled using the shear friction model with a coefficient of 0.12. The process was simulated under isothermal conditions at 20°C. The workpiece was discretized using tetrahedral elements, resulting in a robust model capable of capturing complex deformation patterns inherent in spur and pinion gear forging.
The forming sequence within the PLF process is precisely controlled. The total die stroke is 10 mm. The process begins with the punch moving downwards alone, pressing the protrusion in Zone① into the die cavity to form the teeth in that zone over a 5 mm stroke. Following this, both the punch and the die cavity assembly move downwards together. During this second 5 mm stroke, the lower bolster acts on the protrusion in Zone②, forcing that material into the cavity to form the remaining teeth. This two-stage motion sequence is critical for implementing the local loading strategy for the spur and pinion gear.
The deformation mechanics of the billet during the PLF process are complex and insightful. As the punch descends to form Zone①, metal flows primarily radially outward and into the tooth cavities of that zone. The initial contact and filling create distinct deformation patterns. Upon completion of Zone① and the commencement of Zone② forming, the already-formed teeth in Zone① act as a constraint, influencing the flow of material from Zone②. The final stage involves the complete filling of Zone②’s cavities, culminating in a fully formed spur and pinion gear profile. To analyze this in depth, we examine the equivalent strain and stress distributions, as well as the forming load evolution.
The equivalent strain distribution provides a window into the severity of deformation throughout the process. At 25% of the total stroke (during Zone① forming), strain concentration is observed in the mid-upper regions of the tooth roots in Zone① and at the interface between the Zone① protrusion and Zone② material. The maximum equivalent strain value at this stage is approximately 5.91. As Zone① forming completes at 50% stroke, the strained region expands. When Zone② forming begins, a new strain concentration zone appears in the mid-lower regions of the tooth roots in Zone②. By the end of the process (100% stroke), the strain distributes more uniformly across the entire gear, with a final maximum value around 5.19. This evolution suggests that the PLF process subjects different regions of the spur and pinion gear to high deformation at different times, rather than all at once, which is beneficial for load management. The equivalent strain $\epsilon_{eq}$ is a scalar measure of cumulative deformation, often calculated in plasticity theory. For a simple interpretation, the effective strain increment $d\bar{\epsilon}$ in metal forming can be related to the strain tensor components $d\epsilon_{ij}$:
$$ d\bar{\epsilon} = \sqrt{\frac{2}{3} d\epsilon_{ij} d\epsilon_{ij}} $$
The total equivalent strain is the integral of this increment along the deformation path.
| Forming Stage (Stroke %) | Primary Strain Concentration Zone | Approx. Max. Equivalent Strain |
|---|---|---|
| 25% (Zone① forming) | Tooth roots (Zone①), Interface (Zone①/②) | 5.91 |
| 50% (Zone① complete) | Expanded in Zone① teeth, Interface | 5.86 |
| 75% (Zone② forming) | Zone① teeth, Interface, Tooth roots (Zone②) | 4.91 |
| 100% (Process complete) | Uniformly distributed across gear | 5.19 |
The equivalent stress distribution mirrors the strain patterns but reveals the instantaneous resistance to deformation. During the initial phase of Zone① forming, high stress (up to ~133 MPa) is localized in the tooth root areas of Zone① and the inter-zone interface. As forming progresses, these stress concentrations diffuse as the material yields and flows. When Zone② forming is active, stress builds up in the tooth roots of that zone. Notably, once Zone① teeth are fully formed, the stress in that region stabilizes or even slightly decreases as the material work-hardens but experiences less additional deformation. In the final stage, the stress state becomes more homogeneous throughout the spur and pinion gear workpiece. The equivalent or von Mises stress $\sigma_{eq}$ is a key parameter for assessing yield and is given by:
$$ \sigma_{eq} = \sqrt{\frac{1}{2}\left[ (\sigma_{11}-\sigma_{22})^2 + (\sigma_{22}-\sigma_{33})^2 + (\sigma_{33}-\sigma_{11})^2 + 6(\sigma_{12}^2+\sigma_{23}^2+\sigma_{31}^2) \right] } $$
where $\sigma_{ij}$ are the components of the stress tensor. The material’s flow stress $\bar{\sigma}$ is often a function of this equivalent stress and is dependent on strain, strain rate, and temperature: $\bar{\sigma} = f(\bar{\epsilon}, \dot{\bar{\epsilon}}, T)$. For the cold forging of spur and pinion gears, the strain hardening effect is predominant.
| Material Constant | Symbol | Value | Unit |
|---|---|---|---|
| Strength Coefficient | K | 850 | MPa |
| Strain Hardening Exponent | n | 0.20 | – |
| Strain Rate Sensitivity (approx. for cold) | m | ~0.01 | – |
A simplified power-law model can be expressed as: $$ \bar{\sigma} = K \bar{\epsilon}^n $$ This relation helps explain the rising forming load as deformation progresses in each stage of forging the spur and pinion gear.
The most significant advantage of the PLF process is its dramatic reduction in forming load compared to conventional closed-die forging. The load-stroke curve for the PLF process exhibits two distinct peaks corresponding to the two forming stages. The first peak, representing the completion of Zone①, reaches a maximum load of approximately 16 kN. The second peak, corresponding to the completion of Zone②, reaches about 21.3 kN. It is noteworthy that the load for Zone② forming is roughly 30% higher than for Zone①. This difference arises because during Zone① forming, the material in Zone② is largely unconstrained and can flow relatively freely, offering less resistance. In contrast, during Zone② forming, the already-formed teeth in Zone① significantly constrain the material flow from Zone②, increasing the required pressure. To put this in perspective, a simulation of traditional single-stage closed-die forging for the same spur and pinion gear geometry yields a maximum forming load of approximately 71.6 kN. Therefore, the PLF process achieves a load reduction of about 78% for the first stage and 70% for the second stage. The average load throughout the PLF process is estimated to be around 40% lower than the average load during conventional forging. This reduction is directly attributable to the reduced instantaneous projected contact area in the loading direction during each local stage. The forming load $F$ can be conceptually related to the flow stress $\bar{\sigma}$ and the instantaneous contact area $A_c$:
$$ F \approx \bar{\sigma} \cdot A_c $$
In integral forging, $A_c$ is the entire gear face area from the start. In PLF, $A_c$ is initially only the area of Zone①, and later only the area of Zone②, which is roughly half the total area each time. This simple relation underscores the potency of the local loading approach for spur and pinion gear manufacturing.
| Process Type | Stage / Description | Maximum Load (kN) | Load Reduction vs. Conventional |
|---|---|---|---|
| Partitioned Local Forming (PLF) | Stage 1: Forming Zone① | 16.0 | ~78% |
| Stage 2: Forming Zone② | 21.3 | ~70% | |
| Conventional Closed-Die Forging | Single Stroke (Integral Forming) | 71.6 | Baseline (0%) |
The implications of this load reduction are profound for the production of spur and pinion gears. Lower forming loads translate directly into several practical benefits. Firstly, the service life of the forging dies is expected to increase significantly. Lower stresses on the tooling reduce the risk of fatigue failure, chipping, and wear, especially in the intricate tooth profiles of the spur and pinion gear cavities. Secondly, smaller, less powerful, and therefore less expensive forging presses can be employed for production. This lowers the capital investment barrier for adopting cold precision forging technology. Thirdly, the reduced load may also contribute to improved dimensional accuracy and surface finish of the forged spur and pinion gear by minimizing elastic deflections in the die system. Furthermore, the PLF process, as simulated, produces a gear with fully filled teeth and no significant flash or extra material on the end faces that would require subsequent machining. This aligns with the net-shape or near-net-shape ideal for precision forging of spur and pinion gears.
To generalize the findings, we can consider the scaling effects. For larger spur and pinion gears with more teeth or a larger module, the load reduction factor offered by PLF is expected to remain significant or even increase. The fundamental principle of reducing the instantaneous load-bearing area is independent of scale. However, the optimal partition strategy (e.g., dividing into more than two zones) may need to be investigated for very large gears to prevent excessive lateral flow or other defects. The partition ratio and the geometry of the pre-form are critical process parameters. The pre-form must ensure adequate volume distribution to each zone to enable complete filling without causing overlaps or voids at the interface between zones in the final spur and pinion gear. The interface region, as seen in the strain analysis, is a critical zone requiring careful design to ensure sound metallurgical bonding and mechanical integrity.
From a materials perspective, the sequential deformation in PLF may also influence the final microstructure and properties of the spur and pinion gear. Since different regions undergo their primary deformation at different times, the strain history is not uniform. This could lead to gradients in grain refinement and work hardening across the gear. For most applications, the final heat treatment would homogenize these properties, but for applications relying on the as-forged state, this is an area for further study. The cold working imparts high strength, and the controlled flow in PLF may help in achieving a more favorable fiber flow pattern around the tooth root, potentially enhancing fatigue resistance—a critical property for spur and pinion gears in high-duty cycles.
In conclusion, our finite element simulation-based investigation into the partitioned local forming process for cylindrical spur and pinion gears yields highly promising results. The key findings are systematically summarized as follows. Firstly, the PLF process is fully capable of producing a spur and pinion gear with completely filled tooth profiles without the need for significant post-forging machining, achieving the goal of net-shape forming. Secondly, the process fundamentally alters the loading paradigm by sequentially forming partitioned zones, leading to a drastic reduction in the required forming force. Specifically, the maximum load in the first forming stage is reduced by approximately 78%, and in the second stage by about 70%, compared to traditional integral forging. The average forming load throughout the PLF process is roughly 40% lower. These reductions are quantitatively substantial and have direct, positive implications for die life, press capacity requirements, and overall process economy. The successful numerical demonstration of this concept provides a strong theoretical and practical reference for the development and industrialization of advanced cold precision forging techniques for spur and pinion gears. Future work should focus on experimental validation, optimization of pre-form and partition design for various gear specifications, and a detailed analysis of the mechanical properties of gears produced via this novel method.
The mathematical modeling of such processes can be extended. For instance, an analytical estimate of the load for each stage can be attempted using upper-bound methods or slab analysis, considering the complex geometry of a spur and pinion gear tooth. The total power $J^*$ in an upper-bound approach might integrate the internal deformation power and the shear power losses along velocity discontinuity surfaces $S_d$:
$$ J^* = \frac{2}{\sqrt{3}} \bar{\sigma} \int_V \sqrt{\frac{1}{2} \dot{\epsilon}_{ij} \dot{\epsilon}_{ij}} dV + \frac{\bar{\sigma}}{\sqrt{3}} \int_{S_d} |\Delta v| dS $$
Applying such a model to the PLF process would be challenging but insightful for understanding the contribution of different deformation zones in the spur and pinion gear forging. Ultimately, the synergy between advanced simulation tools like DEFORM-3D and innovative process strategies like partitioned local forming paves the way for more efficient and cost-effective manufacturing of high-performance spur and pinion gears, which are the workhorses of countless mechanical systems.