Due to the complexity of spiral bevel gear machining, the cutting tools can be a significant cost of gear manufacturing. Unlike the price of the material, which is fixed by the market, the cost of tooling and subsequent re-grinding can be lowered through reducing tool wear and increasing tool life. Current methods for making these improvements include costly trial-and-error techniques. These can be expensive, time consuming, and may require use of production equipment, disrupting work-flow and product delivery.
This work details an alternative approach to physical testing for predicting and reducing tool wear using the finite element method. This virtual design approach uses real-world cutting tool geometry, automatically generated gear blanks, and known process kinematics to simulate the cutting process. The cutting geometry was scanned with a combination of laser and MRI techniques and then converted into a 3D CAD model for use in modeling. Gear blanks were automatically generated using parametric inputs and material removed to begin simulation mid-process to reduce simulation time. The kinematics for spiral bevel machining were implemented for the non-generating motion and are able to model both face-milling and face-hobbing machining methods. The simulation results included tool temperature and tool stress, which have been shown to be the main drivers in tool wear.
To validate the modeling approach, experimental tests were run for a variety of edge preparations, feeds, and speeds. The experimental results collected included forces, tool wear pictures, and chips. The comparison of the simulated and experimental results found good correlation for forces and between tool wear marks and the temperature and stress contours on the tool. The chip shape was also found to be similar. Based on this comparison, suggestions for reducing tool wear with the modeling approach are made. Additionally, the lessons learned, potential benefits, and pitfalls of this approach to tool wear reduction and future work are summarized.
1 Introduction
The custom nature of spiral bevel gears and the high cost of cutting tools have continued to put pressure on manufacturers to make timely improvements for reducing the cost of gear machining. However, improvement usually comes at a cost of interruption of production processes to allow for expensive and time-consuming trial-and-error implementations. It is highly desirable for tooling designers and manufacturing engineers to have a virtual machining simulation tool to study the spiral bevel gear machining process including understanding the tool wear, chip formation, and the effect of machining on the workpiece surface.
In this paper, a physics-based finite element model is developed for simulating the spiral bevel gear non-generating process (Formate® process of Gleason Works) for face milling (single indexing) and face hobbing (continuous indexing). The model requires inputs for defining the uncut gear blank, the tool, process parameters, and materials. Then it creates in-cut gear geometry, explicitly meshes gear and cutter, and simulates the complicated kinematic motion between the tool and workpiece. The model is capable of predicting the forces, temperature, chip shape, and stresses during the machining process. In addition, the model allows for setting up the simulation at the beginning, middle, or end of the cutting process, which enables capturing the specific area of interest such as studying a single rotation of the cutting tool or single cutting tooth pattern in high detail while still receiving a solution in a reasonable amount of time.
To ensure the accuracy of simulation solutions, a set of validation experiments were conducted. The experiment data of forces and chips are collected and compared with their simulation counterparts. The experimental tool wear marks are compared with simulation temperature and stress fields to identify a correlation. Suggestions on how to use this finite element model to reduce tool wear are provided.
2 Modeling Methodology
2.1 Finite Element formulations
In this study, the finite element solution is obtained by using AdvantEdge software. It is an explicit dynamic, thermo-mechanically coupled finite element modeling package designed specifically for metal cutting [1]. It employs an explicit Lagrangian finite element formulation equipped with advanced material constitutive model, robust contact algorithm, and continuous adaptive meshing capability. The model accounts for dynamic effects, heat conduction, full thermal-mechanical coupling, plasticity in finite deformation and large strain rate, and frictional contact between deformable meshes. A brief description of the finite element formulation is given as follows:
The finite element formulation is derived based on the Lagrangian description of a dynamic equilibrium equation using the principal of virtual work, which can be expressed by the following weak form:
where P is the first Piola-Kirchhoff stress field, the subscript n + 1 refers to (n + 1)th time step, f, a, and t are the body forces, accelerations, and boundary tractions, respectively, ρ0 is the mass density on the reference configuration B0, η is an admissible virtual displacement field, and ∇0 denotes the deformation gradient.
Upon discretization with finite elements, the governing equation of (1) can be re-written in a matrix form as
where M is the mass matrix,
Rextn+1 is the external force array,
and Rintn+1 represents the internal force array,
Here Na is the shape function.
The explicit time integration method and central difference scheme are used for time discretization
where d and v are displacement and velocity vectors, respectively, and Δt is the time increase from (n)th time step to (n + 1)th time step.
Substantial amount of heat is generated during the machining process and has a considerable influence on the mechanical responses. The primary sources of heat are the plastic work in the workpiece and the friction at the tool-chip interface. To get the finite element solution of the thermal effects, the weak form of the first law of thermodynamics in the current configuration Bt (with the current Neumann boundary Btq) is given by
where ρ is the current mass density, c is the heat capacity, T is the temperature field, η is an admissible virtual temperature field, and q is the heat flux; s is the heat rate generated by the plastic work, which can be estimated as]
where W. is the plastic power density and β is the Taylor-Quinney coefficient [2].
ℎ is the heat rate generated by the tool-chip interface friction, which can be calculated as
where t is the contact traction vector and [[v]] is the jump in velocity vector across the contact interface.
Applying the finite element discretization and forward Euler algorithm, the thermal time stepping equation is obtained and expressed in a matrix form as
where T is the temperature vector, and K and C are the conductivity and heat capacity matrix, respectively. Q is the vector of sources of heat.
The coupled thermo-mechanical equations are solved by a staggered procedure. The mechanical step is assumed to be isothermal, i.e. temperature is assumed to be constant. The thermal step only deals with the heat generation and temperature change.
Machining involves complicated contact conditions, including the tool-workpiece contact on the tool rake face and relief face, the chip-workpiece contact, which occurs when the chip curls over and touches the workpiece, and the chip-chip contact, which occurs during chip segmentation. A contact algorithm developed by Taylor and Flanagan [3] is employed to model the contact conditions. Two surfaces that come into contact are treated as a pair of master surface (rigid) and slave surface (deformable); the impenetrability of two surfaces is achieved following an explicit predictor-corrector scheme.
To capture fine-scale features developed at the tool-chip interface such as the localized shear bands and to alleviate element distortion for ensuring accuracy of solution, an adaptive meshing technique is used in solving the FEM equations. The adaptive meshing is triggered once the element distortion reaches a critical value. Mesh refinement, improvement, and coarsening are automatically applied. In regions where materials experience plastic deformation, workpiece mesh is refined. On the other hand, the mesh coarsening procedure is applied in regions where plastic deformation is inactive in order to reduce the computation cost.
Since high-speed machining is a high strain, high strain rate, and high temperature operation, AdvantEdge describes the material constitutive behaviors using a power law model in which the material’s flow stress is governed by the following equation:
where σ is the flow stress, g is the strain hardening function, Γ is the strain rate sensitivity function, and Θ is the thermal softening function. εp, ε., and T denote the equivalent plastic strain, plastic strain rate, and temperature, respectively. Flow stress models other than the default power law model, such as the Johnson-Cook model [4], can also be implemented and conveniently interfaced with AdvantEdge using a User Defined Yield Surface (UDYS) capability.
2.2 Kinematics of Spiral Bevel Gear Machining
The kinematics for spiral bevel gear machining, including face milling and face hobbing non-generating processes were implemented in AdvantEdge to define the relative positions and motions of the cutter and gear. The method of determining the relative positions of cutter and gear is explained in detail by Stadtfeld [5], which requires input of multiple geometric parameters such as mean cone distance, mean spiral angle, and nominal cutter radius. For face-milling non-generating processes, the gear is fixed during machining. For face-hobbing non-generating motion, the angular velocity of the gear ωg is determined as follows
where ωcu is the angular velocity of cutter, Ns is the number of cutter blade groups, and Ng is the number of gear teeth. Additionally, the feed and starting depth of cut are also required for determining the kinematics of non-generating motion.
The machining kinematics are first used by the AdvantEdge mesh engine in the steps of placing the cutter and creating the in-cut gear geometry. Then, the meshing of the gear and cutter is applied automatically, followed by adding boundary conditions. The simulation is run from the in-cut gear geometry while following the kinematics.
It is worth mentioning that the handedness of cutter and gear was taken into account while determining the machining kinematics in AdvantEdge. Any combinations of cutter and gear handedness can be correctly handled by modifying the positions and rotating directions accordingly.
Figure 1 shows two examples of machining kinematics for different processes and different handedness of the cutter and gear.
3 Experimental Validation
To validate the kinematics and accuracy of the FEM solution for spiral bevel gear machining model, a series of face-milling non-generating tests were carried out. Concurrently with the machining tests, simulations were set up and run for comparison. The validated parameters include force, chip, and tool wear.
3.1 Experiment setup
The validation tests were carried out on a Mori-Seiki NH6300 DCG 5-axis horizontal machining center. A Kistler 9255B table mounted piezoelectric dynamometer was used for measuring forces. Images of chips and tool wear were taken using a Flexbar Optiflex digital microscope. Due to the limitations of the machining center and to ensure high quality data collection, a single-insert cutter is used. Figure 2 shows the setup of the machine test.
The cutting tool material used in the machining test was high speed steel. Two workpiece materials that are commonly used in the gear manufacturing industry were machined — steel X53 and 9310. During the machining test, a flood coolant of a mixture of oil and limited amount of water was used for lubrication, controlling cutting temperature and flushing out chips.
Table 1 shows the feeds and surface speeds used for the validation tests. Nine sets of cutting conditions were studied according to common industry practices, covering high and low surface speeds and feeds.
3.2 FEM Simulation setup
FEM simulations were set up according to the experiments. Figure 3 shows the whole cutter and a single tooth used in the simulation to reduce the meshing and running time.
The single tooth is imported into AdvantEdge as a STEP file. The simulation was set up through a graphic user interface (GUI) designed specifically for spiral bevel gear machining. Parameters defining the workpiece geometry and machining process were input directly. Tool and workpiece materials were selected from a large material library integrated in AdvantEdge. Parameters relating the initial meshing and adaptive meshing were set according to the geometry size and chip load, respectively.
After the simulation setup, the initial meshes of workpiece and tool at a full-cutting position were automatically generated though the AdvantEdge mesh engine. An example of initial mesh is shown in Figure 4. Meshes on the tooth cutting edge and on the workpiece cutting surfaces are refined to ensure high simulation accuracy. Meshes on other parts of tool and workpiece are coarsened to speed up the computation. The FEM problem was solved using multi-threaded parallel computing methodology. The output results include contours of temperature, stresses, and plastic strain on the workpiece and the tool throughout the simulation, and time histories of global force, torque, and maximum tool temperature. Figure 4 shows a representative simulation of results on the right.
3.3 Comparison of experiments and simulations
The experimental tangential and radial forces were measured by dynamometer to compare with their simulation counterparts. The chips from experiments and simulations were also compared in terms of the chip shape and size. In addition, the tool wear was visually inspected using a digital microscope and compared with the simulation temperature and stress contours to find correlations.
3.3.1 Cutting forces
Figure 5 shows the comparison of forces between experiments and AdvantEdge simulations for steel X53. The simulation was found to accurately capture the expected force variation trends with respect to feed and speed, i.e. cutting forces increase while increasing feed and decrease while increasing speed. Higher feed leads to larger chip load, which causes higher cutting forces. On the other hand, higher cutting speed causes higher cutting temperature, which softens the workpiece material and reduces the cutting forces. The magnitudes also correlate quite well between simulation and experiment. On average, the simulation tangential force deviated from the experimental data by 17.8 percent, while the simulated forces in the radial direction differed from the experiment by 16.4 percent.
It is worth noting that the low-speed tests show a larger discrepancy between experiment and simulation. By analyzing the video and post-machined workpiece, it is believed that this was caused by some sticking condition that is occurring on the machine between the tool and workpiece. This condition is not currently modeled in AdvantEdge.
In addition to the steel X53, another material steel 9310 was also tested. Figure 6 shows the validation of forces with different materials. Apparently, use of different materials does not change the accuracy of the validation test.
3.3.2 Chip formation
The chips generated during machining test were collected and analyzed. The chip shape between experiments and simulations was compared, and a good similarity was found between the two, as shown in Figure 7.
In addition, the simulation result of plastic strain contour on the chip was checked to identify the locations where significant plastic strain occurs. The large plastic strain can indicate impending material failure and possible chip separation. Figure 8 shows the comparison of high plastic strain area on the chip in experiment and simulation. The simulation was found to accurately predict the plastic strain distribution and the high plastic strain area in the chip.
3.3.3 Tool wear
After the machining test, the cutter tooth was examined using a digital microscope and found to have noticeable tool wear marks on the rake face (crater wear) and flank face (flank wear). The tool wear pictures were then compared with simulation contour results to identify a correlation between tool wear and simulation available outputs. Figure 9 shows the comparison between experimental tool wear marks and simulation temperature and pressure contours. It can be seen that the high temperature area matches the tool wear marks very well. This close correlation between tool wear, high temperature, and high pressure is consistent with the common understanding of tool wear process and is used in building multiple tool wear predictive models [6].
Combining with a tool wear predictive model which is usually a function of tool temperature and tool stress, the finite element simulation tool can be used to predict tool wear during spiral bevel gear machining, which would be beneficial for both tool design and process optimization. For instance, a set of simulations can be set up and run with the design of experiments (DOE) method to study the influence of rake angle, edge preparation, and coating material on the tool stress and temperature fields, and thus the tool wear to get an optimal tool design.
4 Concluding remarks
In this study, a physics-based finite element model for simulating spiral bevel gear face-milling and face- hobbing non-generating processes is presented. The model can be used to predict the forces, temperature, chip shape, and stresses during the machining process. These data enable more informed decisions and offer the ability to test a wide range of ideas prior to investing in new cutters or making process changes on the floor. For example, instead of guessing why a cutter fractured, virtual machining simulation allows users to simulate their process conditions to see the areas of high stress and high temperature that indicate tool failure. The accuracy of finite element solutions is validated by a series of simple machining tests. The simulated forces are found to accurately predict trends, and magnitudes are on average within 20 percent of experiment data. Chip shapes are found to be similar between experiments and simulations. The simulation results of temperature and stress fields also are found to be closely correlated to experimental tool wear marks. It is demonstrated that this finite element model will be beneficial for the study of tool wear reduction in spiral bevel gear machining. Users can accelerate the optimization of tool design and machining process for reducing tool wear by coupling the virtual-machining simulation with DOE approach.
In summary, the ability to virtually model the complex cutting process of spiral bevel gear machining has the potential to reduce the cutting cycle time and improve the tool life, which will lower the cost to manufacture gears.
Currently the finite element model focuses on smaller diameter gears that are commonly used in the automotive industry. There are still some improvements necessary to expand the modeling capability to larger size gears and cutters. Other future developments for additional features include the simulation of the generating process in spiral bevel gear machining and adding more gear specific materials.
5 Acknowledgements
This study was partially funded by NAVAIR SBIR program award and supported by Meritor, General Motors, Ford, Rolls-Royce and Aero Gear.
Bibliography
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- Johnson, G. R. and Cook, W. H., 1983, “A Constitutive Model and Data for Metals Subjected to Large Strains, High Strain Rates and High Temperatures.” Proceedings of the 7th International Symposium on Ballistics: pp.541-546.
- Taylor, G. I. and Quinney, H., 1931, “The Plastic Distortion of Metals.” Philo. Trans. Roy. Soc. A., London 230, pp.323-362.
- Taylor, L. M. and Flanagan, D. P., 1987, PRONTO 2D: A Two-Dimensional Transient Solid Dynamics Program. SAND86-0594.
- Stadtfeld, H. J., 2014. Gleason Bevel Gear Technology: Basics of Gear Engineering and Modern Manufacturing Methods for Angular Transmissions. Rochester, N.Y.
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Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented September 2018 at the AGMA Fall Technical Meeting in Oakbrook, Illinois. 18FTM18
