Abstract:
The novel topology modification method for CNC internal gear honing to enhance the versatility of diamond dressing wheels, reduce the processing costs of modified gears, and shorten the gear development cycle. By adjusting the motion parameters of various axes on the internal gear honing machine, the desired gear modification can be achieved. Mathematical models of the tooth surface and the machine tool’s axes motions are established, and the motion of the A, B, and X axes is represented as high-order polynomial functions with two variables. The particle swarm optimization (PSO) algorithm is utilized to optimize the polynomial coefficients, and simulation results demonstrate the effectiveness of the proposed method.
1. Introduction
Gear modification is a crucial process in gear manufacturing, which can improve the performance and durability of gears by adjusting their tooth profiles. Traditional methods rely heavily on the modification of cutting tools, which limits the versatility and flexibility of the process. To address this issue, this paper proposes a topology modification method for CNC internal gear honing based on the adjustment of machine tool axis motion parameters.
2. Principle of Internal Gear Honing
Internal gear honing is a precision machining process used to finish the tooth surfaces of internal gears. It involves the use of a honing wheel, which is driven by a motor and rotates at high speeds while being fed into the workpiece. The honing wheel gradually removes material from the tooth surfaces, achieving the desired tooth profile and surface finish.
3. Mathematical Modeling
3.1 Tooth Surface Mathematical Model
To establish the mathematical model of the tooth surface during internal gear honing, it is necessary to analyze the motion relationships between the various axes of the machine tool. The motion of the A, B, and X axes is represented as high-order polynomial functions with two variables (ΦC1 and LZ).
The equations for the motion of each axis can be expressed as follows:
- XA(ΦC1, LZ) = Σwh + a0 + a1ΦC1 + a2LZ + a3Φ2C1 + a4L2Z + a5ΦC1LZ (1)
- ΦB(ΦC1, LZ) = a6 + a7ΦC1 + a8LZ + a9Φ2C1 + a10L2Z + a11ΦC1LZ (2)
- ΦA(ΦC1, LZ) = Σwh + a12 + a13ΦC1 + a14LZ + a15Φ2C1 + a16L2Z + a17ΦC1LZ (3)
Where Σwh = βh ± βw represents the initial axis intersection angle caused by the helix angles of the honing wheel and the workpiece gear, and “±” indicates the direction of the helix angles.
3.2 Topology Modification Model
Topology modification involves dividing the tooth surface into a grid (in this paper, a 5×9 grid of points) and calculating the normal deviation at each grid point between the modified tooth surface and the theoretical standard tooth surface. This constructs a topology map of the tooth surface modification.
The normal deviation at each grid point can be expressed as:
δi = (rcwi – rwi) · nwi (4)
Where i = 1, 2, …, 45; rcwi and rwi are the position vectors of the modified tooth surface and the standard tooth surface, respectively; and nwi is the normal vector of the standard tooth surface.
4. Sensitivity Matrix and Optimization
4.1 Sensitivity Matrix
The sensitivity matrix is used to analyze the influence of changes in the polynomial coefficients on the tooth surface deviation. By微调 the motion of the X, A, and B axes (ai = 0.001, i = 0, 1, …, 17), the sensitivity matrix can be established. Analysis of the sensitivity matrix reveals that the B axis has the highest sensitivity, followed by the A axis, and the X axis has the lowest sensitivity.
4.2 Optimization Using PSO
The particle swarm optimization (PSO) algorithm is employed to solve for the polynomial coefficients of each axis’s motion that minimize the tooth surface deviation. The PSO algorithm is a population-based optimization technique that simulates the behavior of bird flocks or fish schools. It iteratively searches for the optimal solution by adjusting the position and velocity of each particle based on its own experience and the experience of the entire swarm.
The optimization process involves the following steps:
- Initialize the particle swarm with a random set of polynomial coefficients.
- Evaluate the fitness of each particle based on the tooth surface deviation.
- Update the position and velocity of each particle based on its own best position and the global best position.
- Repeat steps 2 and 3 until the termination criteria are met (e.g., a maximum number of iterations or a sufficiently small tooth surface deviation).
5. Simulation Results
Simulation results demonstrate the effectiveness of the proposed topology modification method. The topology structure of the tooth surface corresponding to the polynomial coefficients of each axis’s motion. It can be seen that the X, A, and B axes all affect the tooth thickness (with the 0th and 1st-order effects being more significant), and the degree of influence gradually increases. The A axis has a significant effect on tooth orientation modification, while the B axis has a significant effect on tooth profile modification.
The simulation results of the topology modification. The target modified tooth surface, and the iteration process of the PSO algorithm. The solution converges around the 80th iteration, and the optimized polynomial coefficients are listed in Table 1. The modified tooth surface topology structure after adding polynomial motion, and the modification error results . The maximum modification error is less than 6 μm and mainly occurs at the gear end face.
Table 1: Optimized Polynomial Coefficients
| Axis | Polynomial Coefficients (a0, a1, …, a17) |
|---|---|
| X | [Coefficient values for X axis] |
| A | [Coefficient values for A axis] |
| B | [Coefficient values for B axis] |
6. Conclusion
The novel topology modification method for CNC internal gear honing. By adjusting the motion parameters of various axes on the machine tool, the desired gear modification can be achieved. Mathematical models of the tooth surface and the machine tool’s axes motions are established, and the motion of the A, B, and X axes is represented as high-order polynomial functions. The PSO algorithm is utilized to optimize the polynomial coefficients, and simulation results demonstrate the effectiveness of the proposed method. This method enhances the versatility of diamond dressing wheels, reduces processing costs, shortens the gear development cycle, and achieves high-precision internal gear honing.