Abstract:
Addressing the simulation issues related to gear hobbing based on solid modeling technology, this paper develops a simulation method using a three-dimensional (3D) Dexel model for cylindrical gear hobbing. The kinematic model of cylindrical gear hobbing is studied, and triangular mesh models are created based on the geometric parameters of the hob and gear blank, which are then converted into Dexel models within the simulation engine. Efficient cutting width evaluation (CWE) calculations and geometric simulations of undeformed chips are carried out with the aid of Delaunay triangulation and Alpha shape reconstruction. An oblique cutting model is employed to calculate the force distribution at all engagement nodes at each time step.
1. Introduction
In numerical control (NC) machining simulation, the 3D Dexel discrete geometry technique achieves a good balance between computational speed and prediction accuracy. For instance, Li et al. [7] utilized a 3D Dexel model to represent both the workpiece and the tool, achieving faster simulation speeds compared to unidirectional Dexel models while maintaining high NC simulation accuracy. Therefore, we propose a simulation method for gear hobbing based on the 3D Dexel model. Unlike simulation methods that solely calculate CWE, the proposed method can extract geometric characteristics of the workpiece throughout the entire process, enabling the simulation and measurement of machining errors caused by residual stresses and elastic deformations in a virtual environment. Additionally, when cutting conditions change, the proposed simulation method can still successfully predict cutting forces in multiple directions.
2. Simulation of Gear Hobbing Based on the 3D Dexel Model
2.1 Kinematic Analysis of Gear Hobbing
Taking spur gears as an example, a kinematic analysis of gear hobbing is conducted.
The hob (tool) and gear (workpiece) rotate synchronously around their axes with angular velocities ωc and ωg, respectively. The hob feeds along the gear width with an axial feed rate vf, resulting in an axial displacement da. The hob axis is inclined at an angle γ to effectively engage the tool with the gear. For tools and gears with the same rotational direction, γ is defined as:
γ = β – η (1)
Where β is the gear helix angle, and η is the hob thread helix angle.
The angular velocity ωg of the workpiece gear is selected based on the number of teeth on the workpiece and the number of threads on the hob:
ωg = (Ng / Nc) · (1 – σvfsinβ / (πNgmn)) · ωc (2)
Where Ng is the number of teeth on the workpiece, Nc is the number of threads on the hob, mn is the normal module of the gear, σ = sgnβ · sgnη · sgnvf is a correction parameter, and rpc is the pitch radius of the hob. Typically, the cutting depth dc in gear hobbing is set to dc = 2.25mn. At the start of the machining operation, the hob center distance dr is set as follows:
dr = rtip + rag – dc (3)
Where rtip is the radius of the hob tip, and rag is the tooth tip radius.
In the tool coordinate system (TCS), a point along the hob cutting edge can be represented by the vector rc(t). According to the requirements of the 3D Dexel model, the same point can also be represented as rw(t) in the workpiece coordinate system (WCS). Additionally, two extra coordinate systems are defined: the machine coordinate system (MCS) and the auxiliary coordinate system (ACS). The MCS is stationary and shares a common origin and Z-axis with the WCS.
2.2 CWE Calculation and Extraction of Undeformed Chips
The 3D Dexel modeling engine used in this paper is ModuleWorks. To establish the simulation, triangular mesh models are created based on the geometric parameters of the hob and gear blank and converted into Dexel models within the engine. To reduce computational complexity, each tooth profile on the hob is modeled as an infinitely thin layer extruded from its rake face. The gear blank is modeled as a cylinder. The kinematic model uses process parameters to simulate the relative motion of the hob relative to the gear. At each time step, the hob adopts interpolation for cutting at different trajectory points. The visualization of gear hobbing simulation based on 3D Dexel in the WCS.
2.3 Cutting Force Calculation
The cutting velocity vector vc is calculated as follows:
vc = ωc × rPi,c – ωg × rPi,g + vf (8)
Where Pi is the ith engagement node on the cutting edge, ωc is the tool angular velocity vector, ωg is the gear angular velocity vector, rPi,c is the coordinate of point Pi in TCS, rPi,g is the coordinate of point Pi in WCS, and vf is the vector representation of the axial feed rate.
3. Simulation Results and Verification
In the hobbing process of spur cylindrical gears, the Z-axis of the WCS mainly coincides with the tangential cutting direction, capturing the most significant force component. The transverse X/Y axes reflect the influence of local feed and radial cutting forces. When building the simulation, the Kienzle model [14] is adopted to individually update the local cutting force coefficients Ktc, Kfc, Krc at the engagement nodes based on the rake angle and inclination angle of the engagement nodes, referencing the method in [13].
The hob minimum/maximum envelope, the influence of cut-in and cut-out strokes, and the engagement motion are well captured. The root mean square (RMS) value and standard deviation of force prediction errors throughout the process are calculated by subtracting the measured cutting forces from the simulated cutting forces for each sample. Additionally, the error values are compared to the peak cutting forces in the three corresponding directions (X: 520 N, Y: 521 N, Z: 2363 N) in terms of percentage.
4. Conclusion
A novel simulation method for cylindrical gear hobbing is proposed. This simulation method utilizes a 3D Dexel model for efficient CWE calculations and extraction of workpiece geometric characteristics, enabling predictions of kinematics, undeformed chips, and cutting forces. Experimental results demonstrate that the measured results align well with the simulation predictions, with prediction errors (RMS and standard deviation) ranging between 4% and 12%, further improving the efficiency and accuracy of gear hobbing simulations based on solid modeling. Future work will focus on further refining the geometric representation of the hob and workpiece, modeling surface positioning errors, and predicting hobbing vibrations.