Since the motion trajectory of the clamping head in the three-dimensional stretch forming of aluminum profiles is generally curved, and the aluminum profile undergoes stretching deformation during the bending process, it is difficult to obtain the motion trajectory of the bending clamping head. Therefore, a method for determining the motion trajectory of the clamping head in the three-dimensional stretch forming of aluminum profiles is proposed.

In modern industrial manufacturing, the application of aluminum profiles is increasingly widespread, particularly in the field of vehicle lightweighting.

## Unlocking Precision in Aluminum Profile Stretch Forming

However, three-dimensional stretch forming of aluminum profiles faces numerous challenges, with one major issue being the determination of clamping head trajectories. In traditional methods, obtaining accurate clamping head trajectories is often difficult, directly impacting product quality and forming effectiveness. To address this challenge, researchers have proposed an innovative approach that utilizes discretization approximation to determine clamping head trajectories, enabling precise control over the three-dimensional stretch forming process of aluminum profiles.

This new method not only effectively resolves the difficulty in obtaining clamping head trajectories during aluminum profile stretch forming but also enhances product quality and production efficiency. This article will discuss the challenges of aluminum profile stretch forming, limitations of existing methods, and the principles and application of the proposed discretization approximation method. Through detailed analysis and example verification, we will demonstrate the superiority and potential application prospects of this new method, providing new ideas and approaches for further optimization of aluminum profile stretch forming processes.

### Method for determining the clamping head motion trajectory

- Divide the initial two ends of the aluminum profile into fixed end and clamping end, and establish a coordinate system at the fixed end.
- Extract the central axis of the product, discretize it into several spatial points, and take each discrete point as a tangent point to the central axis of the product.
- Cut off corresponding lengths of line segments along each tangent line direction (from the tangent point to the clamping end), and the other end of the line segment is the spatial point where the clamping head trajectory passes.
- Connect these spatial points in sequence to obtain the trajectory of the clamping head.
- Finally, verify the proposed method through examples.

The verification results show that within a certain range, as the number of discrete points increases, the rebound value decreases rapidly, and then the rebound value changes little with further increase in discrete points. When analyzing, taking the middle part as the rebound reference benchmark, the obtained rebound value is closer to the actual result.

### Vehicle lightweighting and aluminum profiles

Vehicle lightweighting inevitably requires the use of lightweight alloys, among which aluminum alloy is one of the most widely used lightweight alloys. The vehicle body frame generally adopts aluminum profiles, and the forming process of aluminum profiles mainly includes bending and stretch forming. For aluminum profiles with simple structures, they can be achieved through die stamping; however, for aluminum profiles with complex structures, especially three-dimensional twisted parts that cannot be achieved through die bending, stretch forming processes are required.

### Challenges of aluminum profile stretch forming

Aluminum profiles have a relatively soft material, and to enhance their structural strength, they often adopt non-circular cross-section structures. However, non-circular cross-section structures cannot use balls as core rods, making it difficult to achieve three-dimensional bending structures of aluminum profiles through CNC bending machines. Instead, stretch forming processes are used. Compared with stamping and CNC bending, stretch forming requires determining the motion trajectory of the clamping head, which directly affects the quality of the product. The trajectory of the clamping head is a curve that cannot be determined in advance. If obtained through experimental methods, the cost and cycle will be greatly increased.

**Solution:** To solve the problem of determining the motion trajectory of the clamping head in three-dimensional stretch forming of aluminum profiles, a discretization approximation method is proposed. This method converts three-dimensional stretch forming into a model where one end is fixed and the other end moves with the shape, and then intercepts the trajectory points of each clamping head based on the extended line of the discrete tangent point of the central line. This method has strong adaptability to aluminum profile structures, can quickly determine the motion trajectory of the clamping head, and effectively reduces the product development cycle and trial mold times.

## Equivalent Transformation of Aluminum Profile Stretch Bending Process

Aluminum profiles are typically obtained through extrusion molding, and compared to bending forming, the quality control of extrusion molding is relatively straightforward, with good stability in product quality [5-8]. Stretch bending of aluminum profiles involves gradually fitting the profile to the mold surface using clamping heads at both ends. This method requires determining the motion trajectories of the two clamping heads in advance. However, for three-dimensional stretch bending, it is difficult to accurately predict the amount of elongation and deformation of the profile during stretching. Additionally, there is currently no specialized analysis software available for aluminum profile stretch bending. During the bending process of aluminum profiles, collapse and wrinkling are inevitable. Conventional foundational analysis software may experience excessive distortion of mesh elements, leading to computational failure. Therefore, it is nearly impossible to pre-obtain bending trajectories through simulation algorithms. Hence, this paper proposes a method for obtaining bending trajectories based on the geometric characteristics of the product.

Due to the relativity of motion, the movement of two clamping heads can be equivalently transformed into the movement of a single clamping head. To this end, this paper fully considers the structural characteristics of aluminum profile bending parts and transforms the bending process into an equivalent model with one end fixed and the other end clamped by the clamping head. The process of gradually fitting the mold from the fixed end to the clamping end is illustrated in Figure 1.

## Determination of Aluminum Profile Bending Trajectories

Firstly, using the fixed end (Figure 1) as the origin, the corresponding coordinate system is established. Secondly, the central axis of the product is extracted and uniformly discretized into several points. During the fitting process of the aluminum profile to the mold, the product is divided into the fitting section L and the non-fitting section L’. Assuming the product length is L, then L = L’ + L’, as shown in Figure 2. Due to the deformation of the product during the fitting process, the length of the non-fitting section can be obtained through L’ = L – L’, ensuring consistency with the actual value.

Since the separated central line of the product is a spatial curve, the direction of the tangent line at a certain point on the spatial curve is determined. The tangent line can be directly obtained in CAD software, and the corresponding line segment length can be obtained. The obtained clamping head motion trajectory is used as the control parameter of the bending machine, where the control parameters of the moving end clamping head are in the format of (time, X coordinate, Y coordinate, Z coordinate), and the control parameters of the fixed end clamping head are in the format of (time, 0, 0, 0). For parts with significant spatial deformations: (1) The central layer of the product (the surface without elongation) can be determined geometrically, and the projection of the central axis line on the central layer is taken as the virtual product axis line. (2) For profile parts where the central layer is difficult to obtain (twisted parts), the surface where the central line is located can be unfolded along a major direction (approximate stamping forming analysis) to determine the motion trajectory of the clamping head. Firstly, the central axis line of the product is discretized into n spatial points (the larger the n value, the more accurate the trajectory). At each point, the tangent line of the central axis line is drawn. Assuming the fitting section length at the k-th point (k = 1, 2, …, n) is L(k), then the non-fitting section length is L'(k) = L – L(k). Starting from the k-th point, a line segment of length L(k) is intercepted along the tangent direction, and the endpoint of the line segment is the point where the clamping head passes at the k-th point, and so on, until the entire trajectory of the clamping head is obtained.

The obtained clamping head motion trajectory is used as the control parameter of the bending machine, where the control parameters of the clamping end clamping head are in the format of (time, X coordinate, Y coordinate, Z coordinate), and the control parameters of the fixed end clamping head are in the format of (time, 0, 0, 0), thereby realizing one end fixed and the other end movable.

## Simulation Analysis and Validation of Aluminum Profile Bending Trajectory

### Verification of Clamping Head Bending Trajectory Determination Method

To validate the method proposed in this paper, a lightweight vehicle body crossbeam is used as the carrier. The product model is shown in Figure 3, which is a three-dimensional bending structure made of 6016-T6 aluminum alloy.

Firstly, the central axis line of the product is extracted and discretized in CAD software. 50 cutting points are uniformly preset on the central axis line (the number of cutting points is determined according to the complexity of the product structure, and it is related to the accuracy of the obtained trajectory, generally more cutting points are placed in positions with large curvature changes). As shown in Figure 4, tangent lines are drawn from the fixed end to the clamping end on the central axis line; a certain length of line segment (the length of the product central line when the mold is not fitted) is intercepted in each tangent direction, and the endpoint of the line segment is the point where the clamping head moves through. Finally, these points are connected in sequence to obtain the final clamping head motion trajectory. In practical applications, the coordinates of these points are directly read and used as the control parameters of the clamping head.

During the first bending process, bending is carried out according to the predetermined trajectory. Since one end of the product is fixed, the product itself will not undergo rigid displacement; however, due to the three-dimensional curve of the clamping head’s motion trajectory, the product will move with the clamping head. Due to the significant uncertainty of aluminum profile springback, springback is not considered when determining the bending motion trajectory of the clamping head for the first time. The bending motion trajectory of the clamping head is adjusted based on the springback value of the first trial sample. By scanning the physical sample, the actual central axis line of the product is extracted, and a corresponding relationship between the predetermined bending motion trajectory and the actual product central axis line is established. Firstly, the actual central axis line of the product is divided into n discrete points, and the theoretical central axis line of the product is also divided into n discrete points, with equal spacing between these points; the spatial distance between the same-numbered discrete points is calculated, and this distance is directly mapped to the corrected bending motion trajectory. Based on the simulation results, corresponding adjustments are made to the mold.

### Product Formability

Product formability is one of the critical assessment indicators for aluminum profile bending. However, due to the typically irregular cross-sectional structure of aluminum profiles, the internal support bodies used in steel tube bending processes cannot be applied in aluminum profile bending. As a result, defects such as wrinkling and cracking often occur in aluminum profile bending.

For the product structure shown in Figure 3, a corresponding analysis model is built in CAE analysis software. The clamping head control mode is set to displacement control. The final analysis result of product formability is shown in Figure 5.

From Figure 5, it can be observed that the product has a depression on the inner side of the arc at both ends. This is mainly due to the collapse of the internal cavity of the product during bending, as it lacks mechanical support. However, there are no other defects such as cracking or stacking. By comparing with the physical prototype, the analysis results are consistent with the actual product.

### Product Springback Analysis

By comparing the mechanical properties of steel and aluminum, it is evident that aluminum alloy forming faces several challenges:

**Low elongation and small deformation range**: Aluminum alloy parts are difficult to form and prone to cracking due to their low elongation and small deformation range.**Lower elastic modulus than steel**: After forming, aluminum alloy parts exhibit greater springback compared to steel, making it difficult to control the dimensional accuracy of the product.

Springback value is also one of the critical assessment indicators for aluminum profile bending. Figures 6 and 7 respectively show the analysis results of springback in the X and Y directions for the product.

After analysis, it is found that the springback value in the Z direction of the product is within 0.5 m. Hence, the springback values of this product are mainly concentrated in the X and Y directions, with the springback value in the Y direction being greater than that in the X direction. The maximum springback values occur on the same side, with a maximum springback value of 35 mm in the X direction and 8.4 mm in the Y direction.

### Physical Validation

The method proposed in this paper was validated through trial production, and the final processed product is shown in Figure 8.

The concave at the inner side of the arc at both ends of the product and the bulge formed on the sidewall indicate the lack of mechanical support in the internal cavity of the aluminum profile during the bending process, leading to collapse. The defect form and position in the CAE analysis results are consistent with the physical defects. This indicates that the bending trajectory designed in this paper can meet the actual production requirements.

By scanning the product, the actual springback values were obtained: the maximum springback value in the Y direction is 32 mm (compared to the analysis value of 35 mm), and the maximum springback value in the Y direction is 7.8 mm (compared to the analysis value of 8.4 mm). The analysis results are consistent with the actual results. When comparing different parts of the product (fixed end, clamping end, and middle part) as springback reference benchmarks, it was found that using the middle part as the springback reference benchmark yielded results closer to the actual results.

In practical analysis, when the number of discrete points is less than 10, the product deviates from the mold cavity and cannot smoothly fit the mold. When the number of discrete points exceeds 10, it can be observed from Figure 9 that within a certain range, the springback value rapidly decreases with the increase in the number of discrete points. Subsequently, with a further increase in the number of discrete points, the change in the springback value is not significant.

### Conclusion

- According to the principle of relativity in motion, transforming the double-head bending of aluminum profiles into single-head bending can effectively simplify the calculation model of aluminum profile bending.
- When comparing and analyzing different parts of the product (fixed end, clamping end, and middle part) as springback reference benchmarks, it was found that using the middle part as the springback reference benchmark yielded springback values closer to the actual results.
- When the number of discrete points is less than 10, the product deviates from the mold cavity and cannot smoothly fit the mold. When the number of discrete points exceeds 10, within a certain range, the springback value rapidly decreases with the increase in the number of discrete points. Subsequently, with a further increase in the number of discrete points, the change in the springback value is not significant.