Abstract
Aiming at the current problems of low gear shaping efficiency and large cutting force fluctuations in non-circular gears, and based on the gear shaping cutting force modeling and the relationship established between the cutting force and the cutting area, this article proposes a circular feed rate optimization method based on constant cutting force constraint. To improve the processing efficiency of the workpiece and reduce the fluctuation of the cutting force, with the machine parameters and the constant cutting area as the constraints, and taking the curvature radius of the pitch curve as the segmented basis, an optimization model of variable circumferential feed is established. Through the analysis of the third-order elliptic gear slotting example, the effectiveness of the variable circle feed optimization model is verified. The segmented optimization avoids frequent and severe changes in the speed of each axis of motion, and realizes the optimization of the processing parameters of non-circular gears.
1. Introduction
At present, the main processing methods for non-circular gears are numerical control gear shaping and numerical control hobbing. Compared with numerical control hobbing, numerical control gear shaping has a wider processing range and does not produce the problems of undercutting that are prone to occur in non-circular gears with large curvature changes or a small number of teeth during hobbing. Therefore, gear shaping processing should be more versatile. However, due to issues with processing technology, processing accuracy, and processing efficiency, the development of non-circular gears has been restricted to a certain extent. Reasonably optimizing processing parameters has a significant impact on the rational use of machine tool equipment and reducing manufacturing costs. Among the many process parameters, the feed rate is the most important one affecting processing efficiency and quality.
2. Literature Review
Table 1 below summarizes some key research on process parameter optimization in the field of machining:
| Author | Research Focus | Main Findings |
|---|---|---|
| Ding Hongjian et al. | Milling experiments on thin-walled parts | By comparing cutting forces at different cutting positions on the same tool path, process parameters were optimized to obtain process parameters that vary with the cutting path, thereby controlling cutting forces and tool wear by changing process parameters. |
| Lo | Feed rate optimization for complex surface machining | An optimization model for feed rate adjustment based on surface and curve interpolation was established. Verified through actual processing, the model can maintain uniform fluctuations in cutting forces. However, this optimization method has the disadvantages of complex technology and poor operability. |
| Cai et al. | Feed rate optimization model based on cutting force model | A feed rate optimization model was established to calculate the feed rate for each tool position data, and the calculated feed rate values were written into the updated tool position file to optimize the feed rate. |
3. Cutting Force Modeling
During gear shaping of non-circular gears, the main cutting force Fz accounts for the largest proportion of the cutting force components, and the main cutting force is the basis for calculating machine tool power, tool and fixture strength, as well as for selecting the geometric angle and cutting amount of the tool. Therefore, in this paper, the main cutting force is taken as the main component, and the constant main cutting force is taken as the constraint condition for the subsequent variable circumferential feed optimization. The gear shaper cutter is a multi-edged tool with a small variation range in the tool rake angle, so the gear shaping cutting process can be simplified as an orthogonal cutting model. The basic relationship of the gear shaping force is established, and a simplified diagram of the cutting force is obtained based on the force translation method.
4. Simulation of Non-circular Gear Shaping
4.1 Non-circular Gear Shaping Linkage Model
Let the pitch curve of the non-circular gear be c: r = r(φ), where φ is the polar angle. The unit tangent vector at point m is t, and the unit normal vector is n. A segment m1m = Δd is intercepted on the normal line, where m1 is the center of the shaper cutter, r0 is the radius of the shaper cutter, β is the angle between the polar radius vector om and om1, μ is the angle between the tangent and the polar radius, and γ is the angle between m1m and om1. The differential geometric relationships can be used to solve μ = arctan(dr/dφ) (0 ≤ μ < π).
4.2 Gear Shaping Motion Simulation Based on SolidWorks
According to the non-circular gear shaping relationship and linkage model, the processing parameters are determined based on the parameters of the workpiece and the shaper cutter. Matlab is used to program and simulate the gear shaping process to obtain the coordinate points of the tool tooth profile trajectory. The processing data is imported into SolidWorks, and the solid removal function of SolidWorks is used to realize the calculation of the geometric entities of the tool tooth and the instantaneous tooth space. Finally, the undeformed chip shape produced by each stroke of cutting and the shape of the workpiece after cutting can be obtained. By establishing an intersectable plane with the undeformed chip and extracting the intersection data points, the chip entity can be reconstructed in Matlab. The undeformed chip area can be obtained through Matlab’s powerful computing capabilities, which reflects the variation rule of the cutting force during the processing.
Table 2. Simulation Process Parameters
| Feed Time | Feed Amount s /mm | Radial Feed Rate st /(mm/s) | Circumferential Feed per Stroke sr /mm | Stroke Number n |
|---|---|---|---|---|
| First Feed | 30.06 | 72.67 | 200 | – |
| Second Feed | 20.06 | 31.8 | – | 180 |
| Third Feed | 1.75 | 0.048 | 318 | 180 |
5. Optimization Model of Variable Circumferential Feed Based on Constant Cutting Force Constraint
5.1 Optimization Objectives
Due to the variable curvature radius of the pitch curve of non-circular gears, cutting force fluctuations are more pronounced during the machining process. To address this issue and improve both workpiece quality and processing efficiency, the optimization of the circumferential feed rate, with constant cutting force serving as a constraint, is proposed for the gear shaping process of non-circular gears. The primary objective of this optimization is to minimize the fluctuations in cutting force and enhance processing efficiency. Specifically, this translates to minimizing the variations in both the processing time (t) and the cutting area (A).
5.2 Constraint Conditions
To ensure the feasibility and effectiveness of the optimization, several constraint conditions must be considered:
- Machine Parameters Constraint: The stroke number (n) of the spindle must fall within the maximum and minimum stroke numbers allowed by the machine tool to ensure cutting stability and comply with the machine’s performance limitations. Similarly, to increase the cutting area where the cutting depth is smaller, the circumferential feed rate (s_k) needs to be increased, but it must also remain within the machine’s operational range.
- Constant Cutting Area Control Constraint: Due to the continuous variation of the pitch curve during the actual non-circular gear shaping process, the cutting area also changes. Therefore, a constant cutting area constraint is imposed to keep the variations in the cutting area within a predetermined range. Experimental data has shown that maintaining the cutting area fluctuations within 5% of the target cutting area (A_obj) can optimize processing efficiency and quality while protecting the machine tool and cutting tool.
- Target Cutting Area Constraint: The selection of A_obj depends on the processing stage. During rough machining, the goal is to quickly remove material, so the maximum cutting area during the cutting process should be chosen as A_obj. In contrast, during semi-finishing machining, the objective is to achieve the designed dimensions and accuracy requirements of the workpiece, and thus, the average area within one cutting area cycle should be selected as A_obj.
5.3 Optimization Algorithm
The optimization algorithm involves several steps:
- Fitting and Prediction: The cumulative cutting area corresponding to each processing position (T) is obtained by fitting the cumulative cutting area from conventional processing strokes to a polynomial equation. With the target cutting area (A_obj) selected based on the constraint conditions, the approximate circumferential feed rate (s_k’) for the k-th stroke can be calculated using the fitted equation and the target cutting area.
- Iterative Calculation: Using the calculated approximate circumferential feed rate (s_k’) as the initial value for that stroke, an iterative calculation method is employed to determine the circumferential feed rate (s_k) for each stroke that satisfies the target cutting area and machine parameter constraints.
- Simulation and Verification: Each calculated circumferential feed rate at each processing position needs to be verified using the gear shaping simulation method to ensure that the cutting area meets the constraint conditions of the optimization model. If not, the optimization algorithm is used to adjust the data for that processing position until the constraints are satisfied, and then the optimization continues for the next processing position.
5.4 Segmented Optimization of Variable Circumferential Feed Rate
Due to the diverse shapes of non-circular gears and their pitch curves, the material removal rate varies during the machining process depending on the workpiece contour. To address this, the pitch curve is segmented based on its curvature radius, which reflects the concavity and convexity of the workpiece contour. This segmentation allows for the optimization of the circumferential feed rate for different sections of the pitch curve.
After segmenting the pitch curve, the variable circumferential feed rate optimization algorithm is applied to each segment. The range of variation of the circumferential feed rate must satisfy the machine parameter constraints. By limiting the difference in circumferential feed rate between adjacent strokes and adjusting and uniformizing the initially optimized feed rates, the optimized feed rates can change uniformly within the desired range, ensuring a stable cutting process and high processing quality.
In summary, the optimization model of variable circumferential feed based on constant cutting force constraint aims to minimize cutting force fluctuations and improve processing efficiency by considering machine parameters, constant cutting area control, and target cutting area constraints. Through segmented optimization and iterative calculation, the model achieves stable cutting and high-quality processing of non-circular gears.