- Email: [email protected]
Modeling of Surface Location Errors in a Multi-scale Milling Simulation System Using a Tool Model Based on Triangle Meshes
Available online at www.sciencedirect.com
ScienceDirect Procedia CIRP 37 (2015) 188 – 192
CIRPe 2015 - Understanding the life cycle implications of manufacturing
Modeling of surface location errors in a multi-scale milling simulation system using a tool model based on triangle meshes T. Siebrechta,* , P. Kerstinga,b , D. Biermanna , S. Odendahla , J. Bergmanna b Institute ∗
a Institute of Machining Technology, TU Dortmund University, Baroper Str. 303, 44227 Dortmund, Germany of Manufacturing Technology and Quality Management, OvGU Magdeburg, Universit¨atsplatz 2, 39106 Magdeburg, Germany
Corresponding author. Tel.: +49-231-755-5819; fax: +49-231-755-5141. E-mail address: [email protected]
Abstract Simulation systems are used to optimize milling processes by analyzing surface location errors, which can be predicted by combining various modeling techniques in a multi-scale way. In order to visualize surface location errors, a multi-dexel representation of the workpiece can be used. The application of a triangle mesh representation of the tools to cut the workpiece model is presented, which results in a higher accuracy of the predicted surface location errors in comparison to a Constructive-Solid-Geometry-(CSG)-based approach. Both models are evaluated by simulating a milling process and the simulation results are validated by comparing them to experimental results. c 2015 2015 The The Authors. Authors. Published Published by by Elsevier Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license © Peer-review statement : Selection and peer-review under responsibility of the International Scientific Committee of the 4th CIRP Global Web (http://creativecommons.org/licenses/by-nc-nd/4.0/). Conference the responsibility person of the Conference Chaircommittee Dr. John Ahmet Erkoyuncu. under of the organizing of CIRPe 2015 - Understanding the life cycle implications of manufacturing Peer-review in Keywords: Milling, Geometric modeling, Surface
1. Introduction Simulations of milling processes are used in various industrial sectors for analyzing and optimizing the machining of different workpieces. Especially when using milling tools with high L/D ratios, vibrations are likely to occur and, thus, result in a bad surface quality. By modeling the dynamic behavior of the tools, vibrations can be predicted in order to optimize the process parameter values without carrying out extensive experiments. Different modeling techniques can be used to simulate milling processes while taking the dynamic behavior of the tool, the machine, and the workpiece into account [1]. Analytical methods can be applied to predict chatter for various process parameters [2]. The resulting process forces are calculated using empirical models, which are calibrated based on measurements. Alternatively, finite element analysis (FEA) [3,4] or molecular dynamics models, e.g, smoothed particle hydrodynamics (SPH) [5] or the discrete element method (DEM) [6], allow simulating the interaction between the tool and the workpiece. This can be used to model chip formation or thermal effects, for example. A convenient way to analyze the influence of vibrations of the milling tools on the process result is the visualization of surface location errors (SLE). Geometric-kinematic approaches with empirical force models [7] allow the simulation of the process
forces based on the geometric engagement situations between tool and workpiece by analyzing the undeformed chip thickness. Using these process forces and the resulting deflections of the tool, the local surface location errors can be predicted. The geometric-kinematic simulation system used in the presented work utilizes a multi-scale model of the workpiece, which is described in Section 2 in detail. The Constructive Solid Geometry (CSG) modeling technique [8] is used to calculate the undeformed chip thickness. An additional multi-dexel representation [9,10] is used to visualize the surface location errors on the machined workpiece surface. When cutting the multi-dexel workpiece model with the CSG representation of the tool, artifacts can occur as described in Section 3 in more detail. These artifacts are erroneous information in the visualization of the surface location errors. This occurrence of artifacts can be reduced by cutting the dexels with triangle mesh representations of the tools, which are deformed according to the simulated deflections. This method is described in Section 4. A comparison to the CSG-based approach and a validation based on experimental results are given in Section 5. Section 6 comprises conclusions from the presented work and an outlook on future research work.
2212-8271 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of CIRPe 2015 - Understanding the life cycle implications of manufacturing doi:10.1016/j.procir.2015.08.064
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Current tool position
CSG rays cutting edge 1
Tool shape
Initial workpiece shape CSG ray
Undeformed chip
Workpiece
Fig. 1: Representation of the machined workpiece shape using the CSG modeling technique.
2. Simulation of the machining process The utilized time-domain simulation system [7] is based on the interpretation of NC programs. By using NC programs in the simulation, a direct optimization of milling processes is possible. The calculated tool path is subdivided into discrete time steps, where each time step corresponds to the length of one feed per tooth. The material removal process is modeled once for each of these simulation steps while a detailed analysis of the engagement situation is carried out for multiple substeps. Both, workpiece and tool, are represented using the CSG technique to model the material removal process. Simple primitives, e.g., cylinders, spheres, or tori, are combined to model the boundary shapes of the rotating tools. Workpieces are represented as the difference between a cuboid, which corresponds to the initial stock, and instances of the CSG model of the tool at every previous simulation step as shown in Fig. 1. In every simulation step, the material removal is modeled by appending the new pose (position and orientation) of the tool to this list of subtracted shapes. Before the calculation of the process forces, the undeformed chip thicknesses are needed and therefore determined by scanning the chip shapes using cast rays. In order to take the variation of the chip thickness along the length of the tool into account, the cutting edges are subdivided into several slices which are treated separately. This allows to model the helix angle by rotating each slice accordingly. The rays of the model of an end-mill are shown in Fig. 2a. Process forces are calculated using an empirical, non-linear force model. These process forces are then used as excitation of a model of the tool dynamics, which is based on damped harmonic oscillators, resulting in a three-dimensional deflection of the tool for every substep of the simulation [11]. By taking the deflections into account during the calculation of the chip thickness values, the regenerative effect and, thus, chatter vibrations can be predicted. A schematic overview of the subsequent modeling steps of the simulation system is given in Fig. 3. The offsets of the CSG rays resulting from the deflections are shown in Fig. 2b. While the described CSG approach allows to model the material removal and to calculate process forces efficiently, it is difficult to visualize the shape of the machined workpiece and especially the surface location errors. Therefore, the simulation system comprises an additional multi-dexel model of the workpiece. As shown in Fig. 4, a dexel board is defined for each of the three spatial axes, representing points on the workpiece surface. The parts of the dexels intersecting the tool shape are removed in every simulation step to model the cutting
Height of cutting slice
Fig. 2: Simulation model of a milling tool with rays representing the two cutting edges without (a) and with (b) deflections.
Geometric model
Undef. chip shape
Undef. Analysis of undef. chip chip shape thickness
Empirical force model Process forces
Accounting for the regenerative effect Tool deflections
Dynamic model
Fig. 3: Schematic overview of the simulation steps (cf. [12]).
process. The resolution of the model can be adapted to the simulated process by choosing an appropriate distance between the dexels in the grids. In contrast to the CSG model, the multi-dexel workpiece model can be visualized by displaying the start- and endpoints of the dexels, which are directly representing points on the surface of the workpiece (cf. Fig. 4). The calculated surface location errors for these points can be directly visualized by coloring them accordingly. 3. Calculation of the surface location errors The dexel model is used to visualize the surface location errors, but their calculation is based on the CSG model. For every cut dexel, the deflection of the tool a at the time, when the cutting edge was cutting the dexel, is looked up from the history of deflections of the dynamics model. The surface location error s for this surface point is then calculated by projecting a onto the direction of the CSG ray as shown in Fig. 5. If Current tool position Initial workpiece shape
Workpiece Undeformed chip
Dexel board 2
Previous tool positions
CSG rays cutting edge 2
Dexel board 1
Fig. 4: Representation of the machined workpiece shape using a multi-dexel model.
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a)
Undeformed chip
CSG ray
d
a
b)
s
d : Dexel offset
Tool
d s Undeformed chip
Dexel
a : Tool deflection
CSG ray
a
Triangles
Cutting edge at beginning of tooth feed
Dexel
s : Surface location error
Fig. 5: Schematic overview of the calculation of negative (a) and positive (b) surface location errors.
the tool is deflected towards the workpiece surface, s becomes negative. This indicates that the material removal was too high at this location. In order to visualize the influence of the tool deflections on the topography of the workpiece surface, the cut dexel is displaced by an offset d which is calculated by projecting a onto the direction of the dexel. Especially if the deflection is much lower than the undeformed chip thickness, this approach yields sufficiently accurate results. However, in case of very large deflections, the calculated surface location errors can be inaccurate for the following reasons. The ideal, undeflected CSG tool shape is used to find the dexels of the multi-dexel boards that have to be cut in a simulation step. Especially for high deflections perpendicular to the direction of a dexel, it is possible that this selection contains dexels that would not have to be cut if the local deflection was taken into account. Accordingly, the selection would contain additional dexels when considering the local deflections. Despite the inaccurate selection of dexels for cutting, they are still displaced by d and colored according to the local surface location error s. The neglect of these effects leads to artifacts on the simulated workpiece surface, which are analyzed in Section 5 in more detail. Therefore, the varying local deflections are taken into account by using a triangle mesh model as described in Section 4. 4. Triangle mesh tool model In order to reduce the artifacts resulting from cutting the dexel boards with the CSG model of the tools, an alternative tool model based on triangle meshes can be used. By cutting the workpiece with triangle meshes, it is possible to take the varying deflections during each tooth feed step into account and, thereby, increase the accuracy of the simulation results at higher deflections. In every simulation substep, the cutting edges are modeled by connecting the tips of the corresponding CSG rays, resulting in a line strip for every cutting edge. By connecting the line strips of subsequent substeps using triangles, a triangle mesh representation of the tool in the current simulation step is generated. If the varying deflections of the tool are taken into account while generating the line strips, the resulting triangle mesh is deformed to model the movement of the cutting edges. In Fig. 6, a triangle mesh representation of a vibrating end-mill is shown schematically. Simulation parameters such as tool radius and height, the number of cutting slices, and the number of substeps per simulation tooth feed are usually set once per tool in a simulation
Fig. 6: Triangle mesh representation of a milling tool with deflections.
setup. Thus, the mesh only needs to be generated once for each tool in its undeflected state, reducing the required amount of calculations per simulation step. The deflections of the tool can then be modeled by moving the vertices. One problem of the new approach is the generation of the CSG rays in the centers of the cutting slices as shown in Fig. 2. Therefore, the generated mesh does not span the full height of the tool using the endpoints of these rays, which are located half a cutting slice height below or above the top and the bottom of the tool shape. In order to resolve this, the highest and the lowest rays are moved upwards and downwards along the cutting edges, respectively. Another problem arises if surface normal vectors are stored at the tips of the dexels in order to calculate lighting when visualizing the workpiece. When cutting with the CSG tool model, the inverted normals of the tool shape, e.g., a cylinder, can be used to calculate the workpiece normals. Accordingly, these normals can be calculated based on the faces of the triangle mesh. However, if the triangle mesh is strongly deformed due to high deflections, the normals of its faces differ from the original state, resulting in poor visualization quality. Therefore, the original normal vectors in the undeflected state are stored at the vertices of the mesh and used when calculating the normals of the workpiece during cutting. Using the CSG-based approach, deflections are modeled by changing the lengths of the scan rays according to the deflections as shown in Fig. 2. In case of slot or pocket milling, it is not possible to model the influence of tool vibrations on the surface quality of the floor of the workpiece because the deflectiondependent tilt of the tool is neglected. The application of triangle meshes has the advantage that three-dimensional deformations of the tool model are possible without further effort. 5. Application and experimental validation In order to analyze the influence of the proposed triangle mesh method in comparison to the CSG approach, a threeaxis milling process was simulated and validation experiments were conducted on a Deckel Maho DMU 50 eVolution five-axis machining center. The machined workpiece was a pocket with dimensions of 119 mm × 82 mm × 19.5 mm, which is shown in Fig. 7. An end mill with two flutes, a diameter of d = 12 mm, a corner radius of 2 mm, and a helix angle of 30◦ was used
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a)
m
0.4
m m
20 m
m
20
20 mm
100 mm Fig. 7: Shape of the test workpiece with the highlighted section that has been analyzed.
CSG
Surface location error
15
0
m
mm Artifacts
0.04 0.0 -0.04 -0.4
Low spatial frequency
b)
0.4 mm
Mesh
Surface location error
for the milling process. The workpiece (EN AW-7075) was manufactured using an axial depth of cut a p = 4.5 mm, a radial depth of cut ae = 4 mm, a spindle speed of n = 15 800 rpm, and a feed velocity of v f = 6320 mm min−1 , resulting in a feed per tooth of t f = 0.2 mm. Machining these kind of workpieces, chatter vibrations are especially likely to occur in the corners since the engagement angle becomes higher in comparison to the flank milling sections. Therefore, the analysis of the process is focused on the machining of a corner of the workpiece, which is highlighted in Fig. 7. This milling process was simulated using the CSG-based approach and the new triangle mesh model. Impact hammer tests were conducted to measure the dynamic modes of the milling tool to calibrate the dynamics model of the simulation. The coefficients of the cutting force model were estimated based on calibration experiments. The spatial resolution of the dexel model of the workpiece, i.e., the distance between adjacent dexels in the grids, was set to 0.1 mm. In order to show fine details and potentially large deflections in the corner, a logarithmic color scale was chosen to visualize the surface location errors. Fig. 8 shows the simulated surfaces using the CSG and the triangle mesh method, respectively. In both cases, chatter marks are clearly visible in the corner and the influence of the helix angle and the axial depth of cut is predicted in a similar way. However, the maximum negative surface location error is slightly higher when using the CSG approach. Furthermore, several artifacts are present as expected due to the described limitations of the model. These artifacts are completely avoided by using a triangle mesh to cut the dexels. The assumption that artifacts are most likely to occur if the deflections of the tool are high in comparison to the resolution of the dexel board is confirmed by the observation of equal results on the flat surface regions, where the tool deflections are significantly lower. By deforming the triangle mesh in a three-dimensional way according to the calculated deflection-dependent tilt of the tool, surface structures on the floor of the pocket can also be modeled using the triangle meshes, which can be seen in Fig. 8 as well. The resulting surface of the machined workpiece is shown in Fig. 9. As in the simulations, chatter vibrations occurred in the corner and the influence of the helix angle and the axial depth of cut can be observed as well. The spatial frequency of the chatter marks matches the triangle mesh simulation slightly more closely. This is especially noticeable in the lower section which was cut by the corner of the tool (Fig. 9a). Fig. 10 shows a comparison of the machined corner to the simulated surface contour. In order to depict the deviations caused by the deflections, the ideal surface according to the NC
0.04 0.0 -0.04 -0.4
Fig. 8: Comparison of surface location errors calculated using the CSG approach (a) and the triangle mesh model (b).
Fig. 9: Photo of the surface resulting from the experiments.
path is shown as well. When the tool enters the corner, the measured and simulated surface location error of approximately 0.3 mm matches and the contours of the surfaces are similar. The roughness of the simulated surface seems to be slightly higher, which can be explained by the neglect of effects like process damping in the simulation. Due to the higher complexity of calculating the intersections between the dexels and the triangle mesh representation in comparison to the CSG model of the tool, which comprises two cylinders and a torus only, the computation time is higher. In case of the analyzed corner, the simulations took approximately 231 s using the CSG model and 367 s using the triangle mesh model on a standard desktop computer, corresponding to an increase of the simulation time of 58.9 %. The computation time depends on different factors including the number of cutting slices and simulation substeps, which directly influence the number of triangles, and the resolution of the dexel boards, which influences the number of required ray intersections in each
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generated surfaces have to be known more precisely, the model based on a triangle mesh yields more accurate results. In the presented model, the triangle mesh is generated based on the local deflections at a fixed tool position for every cut. In future research work, the possibility of taking the feed velocity and the trochoidal movement of the cutting edges into account will be analyzed.
0.3 mm
Simulated surface
Ideal surface
Acknowledgements
2 mm
This project has received funding from the European Unions Seventh Framework Programme for research, technological development and demonstration under grant agreement no 609306.
Fig. 10: Comparison of the simulated and measured surface contours (top view).
References simulation step. Therefore, the given durations are only valid for the specific presented simulation setup, but the observed increase of simulation time can be interpreted as a rough estimation of the computational cost of the new modeling approach. 6. Conclusions and outlook The presented geometric-kinematic milling simulation system allows the prediction of surface location errors resulting from the dynamic behavior of the milling tools. By using a multi-scale workpiece model, the undeformed chip thicknesses and, thus, the process forces can be calculated accurately based on a CSG model and the surface location error can be visualized based on a multi-dexel model. The CSG modeling technique is used to model the tool shape and the material removal is simulated by removing the intersecting volume with the tool from both workpiece models. However, due to the assumption of an undeflected tool when selecting the dexels that are cut in a simulation step, the calculated surface location errors are not always accurate and artifacts can occur in the visualized workpiece shape. Analyzing an exemplary milling process, this effect could be shown in the corner of a pocket at high deflections of the milling tool due to chatter vibrations. An alternative method based on a triangle mesh model of the tool for cutting the dexel boards was presented. By displacing the vertices of the mesh, the local deflections of the tool can be taken into account when selecting the dexels that are cut in a simulation step. Using the new modeling approach to simulate the machining of the exemplary corner, the discussed artifacts are not present in the visualized surface anymore and face-milled surface location errors are predicted. Comparing the simulations to the surface resulting from a machining experiment, both approaches can be used to predict the occurrence of chatter-effected regions correctly, but the surface location errors predicted using the new method are more accurate when the deflections of the tool are high. However, a disadvantage of the approach based on a triangle mesh is the higher simulation time, which was increased by 58.9 % for the analyzed process. Therefore, the applicability of both techniques depends on the required accuracy and the application of the generated simulation results. If it is sufficient to locate chatter-effected regions, the CSG model should be preferred due to its higher efficiency. If the topographies of the
[1] Altintas, Y., Kersting, P., Biermann, D., Budak, E., Denkena, B., Lazoglu, I.. Virtual process systems for part machining operations. CIRP Annals – Manufacturing Technology 2014;63(2):585–605. [2] Altintas, Y., Weck, M.. Chatter stability of metal cutting and grinding. CIRP Annals – Manufacturing Technology 2004;53(2):619–642. [3] Mackerle, J.. Finite-element analysis and simulation of machining: a bibliography (1976–1996). Journal of Materials Processing Technology 1999;86(1):17–44. ¨ uven, H.N.. Prediction of workpiece [4] Budak, E., Tunc¸, L.T., Alan, S., Ozg¨ dynamics and its effects on chatter stability in milling. CIRP Annals – Manufacturing Technology 2012;61(1):339–342. [5] Limido, J., Espinosa, C., Sala¨un, M., Lacome, J.L.. SPH method applied to high speed cutting modelling. International journal of mechanical sciences 2007;49(7):898–908. [6] Eberhard, P., Gaugele, T.. Simulation of cutting processes using mesh-free lagrangian particle methods. Computational Mechanics 2013;51(3):261– 278. [7] Odendahl, S., Kersting, P.. Higher efficiency modeling of surface location errors by using a multi-scale milling simulation. Procedia CIRP 2013;9:18– 22. [8] Foley, J., Feiner, S., Hughes, J.. Introduction to computer graphics. Addison-Wesley, Reading, Mass; 1994. [9] Huang, Y., Oliver, J.H.. NC milling error assessment and tool path correction. In: Proceedings of the 21st annual conference on Computer graphics and interactive techniques. ACM; 1994, p. 287–294. [10] M¨uller, H., Surmann, T., Stautner, M., Albersmann, F., Weinert, K.. Online sculpting and visualization of multi-dexel volumes. In: Proceedings of the eighth ACM symposium on solid modeling and applications. ACM; 2003, p. 258–261. [11] Surmann, T., Enk, D.. Simulation of milling tool vibration trajectories along changing engagement conditions. International Journal of Machine Tools and Manufacture 2007;47(9):1442–1448. [12] Siebrecht, T., Odendahl, S., Hense, R., Kersting, P.. Interpolation method for the oscillator-based modeling of workpiece vibrations. In: Proceedings of the 3rd International Conference on Virtual Machining Process Technology. Calgary, Canada; 2014,.
ScienceDirect Procedia CIRP 37 (2015) 188 – 192
CIRPe 2015 - Understanding the life cycle implications of manufacturing
Modeling of surface location errors in a multi-scale milling simulation system using a tool model based on triangle meshes T. Siebrechta,* , P. Kerstinga,b , D. Biermanna , S. Odendahla , J. Bergmanna b Institute ∗
a Institute of Machining Technology, TU Dortmund University, Baroper Str. 303, 44227 Dortmund, Germany of Manufacturing Technology and Quality Management, OvGU Magdeburg, Universit¨atsplatz 2, 39106 Magdeburg, Germany
Corresponding author. Tel.: +49-231-755-5819; fax: +49-231-755-5141. E-mail address: [email protected]
Abstract Simulation systems are used to optimize milling processes by analyzing surface location errors, which can be predicted by combining various modeling techniques in a multi-scale way. In order to visualize surface location errors, a multi-dexel representation of the workpiece can be used. The application of a triangle mesh representation of the tools to cut the workpiece model is presented, which results in a higher accuracy of the predicted surface location errors in comparison to a Constructive-Solid-Geometry-(CSG)-based approach. Both models are evaluated by simulating a milling process and the simulation results are validated by comparing them to experimental results. c 2015 2015 The The Authors. Authors. Published Published by by Elsevier Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license © Peer-review statement : Selection and peer-review under responsibility of the International Scientific Committee of the 4th CIRP Global Web (http://creativecommons.org/licenses/by-nc-nd/4.0/). Conference the responsibility person of the Conference Chaircommittee Dr. John Ahmet Erkoyuncu. under of the organizing of CIRPe 2015 - Understanding the life cycle implications of manufacturing Peer-review in Keywords: Milling, Geometric modeling, Surface
1. Introduction Simulations of milling processes are used in various industrial sectors for analyzing and optimizing the machining of different workpieces. Especially when using milling tools with high L/D ratios, vibrations are likely to occur and, thus, result in a bad surface quality. By modeling the dynamic behavior of the tools, vibrations can be predicted in order to optimize the process parameter values without carrying out extensive experiments. Different modeling techniques can be used to simulate milling processes while taking the dynamic behavior of the tool, the machine, and the workpiece into account [1]. Analytical methods can be applied to predict chatter for various process parameters [2]. The resulting process forces are calculated using empirical models, which are calibrated based on measurements. Alternatively, finite element analysis (FEA) [3,4] or molecular dynamics models, e.g, smoothed particle hydrodynamics (SPH) [5] or the discrete element method (DEM) [6], allow simulating the interaction between the tool and the workpiece. This can be used to model chip formation or thermal effects, for example. A convenient way to analyze the influence of vibrations of the milling tools on the process result is the visualization of surface location errors (SLE). Geometric-kinematic approaches with empirical force models [7] allow the simulation of the process
forces based on the geometric engagement situations between tool and workpiece by analyzing the undeformed chip thickness. Using these process forces and the resulting deflections of the tool, the local surface location errors can be predicted. The geometric-kinematic simulation system used in the presented work utilizes a multi-scale model of the workpiece, which is described in Section 2 in detail. The Constructive Solid Geometry (CSG) modeling technique [8] is used to calculate the undeformed chip thickness. An additional multi-dexel representation [9,10] is used to visualize the surface location errors on the machined workpiece surface. When cutting the multi-dexel workpiece model with the CSG representation of the tool, artifacts can occur as described in Section 3 in more detail. These artifacts are erroneous information in the visualization of the surface location errors. This occurrence of artifacts can be reduced by cutting the dexels with triangle mesh representations of the tools, which are deformed according to the simulated deflections. This method is described in Section 4. A comparison to the CSG-based approach and a validation based on experimental results are given in Section 5. Section 6 comprises conclusions from the presented work and an outlook on future research work.
2212-8271 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of CIRPe 2015 - Understanding the life cycle implications of manufacturing doi:10.1016/j.procir.2015.08.064
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T. Siebrecht et al. / Procedia CIRP 37 (2015) 188 – 192
Current tool position
CSG rays cutting edge 1
Tool shape
Initial workpiece shape CSG ray
Undeformed chip
Workpiece
Fig. 1: Representation of the machined workpiece shape using the CSG modeling technique.
2. Simulation of the machining process The utilized time-domain simulation system [7] is based on the interpretation of NC programs. By using NC programs in the simulation, a direct optimization of milling processes is possible. The calculated tool path is subdivided into discrete time steps, where each time step corresponds to the length of one feed per tooth. The material removal process is modeled once for each of these simulation steps while a detailed analysis of the engagement situation is carried out for multiple substeps. Both, workpiece and tool, are represented using the CSG technique to model the material removal process. Simple primitives, e.g., cylinders, spheres, or tori, are combined to model the boundary shapes of the rotating tools. Workpieces are represented as the difference between a cuboid, which corresponds to the initial stock, and instances of the CSG model of the tool at every previous simulation step as shown in Fig. 1. In every simulation step, the material removal is modeled by appending the new pose (position and orientation) of the tool to this list of subtracted shapes. Before the calculation of the process forces, the undeformed chip thicknesses are needed and therefore determined by scanning the chip shapes using cast rays. In order to take the variation of the chip thickness along the length of the tool into account, the cutting edges are subdivided into several slices which are treated separately. This allows to model the helix angle by rotating each slice accordingly. The rays of the model of an end-mill are shown in Fig. 2a. Process forces are calculated using an empirical, non-linear force model. These process forces are then used as excitation of a model of the tool dynamics, which is based on damped harmonic oscillators, resulting in a three-dimensional deflection of the tool for every substep of the simulation [11]. By taking the deflections into account during the calculation of the chip thickness values, the regenerative effect and, thus, chatter vibrations can be predicted. A schematic overview of the subsequent modeling steps of the simulation system is given in Fig. 3. The offsets of the CSG rays resulting from the deflections are shown in Fig. 2b. While the described CSG approach allows to model the material removal and to calculate process forces efficiently, it is difficult to visualize the shape of the machined workpiece and especially the surface location errors. Therefore, the simulation system comprises an additional multi-dexel model of the workpiece. As shown in Fig. 4, a dexel board is defined for each of the three spatial axes, representing points on the workpiece surface. The parts of the dexels intersecting the tool shape are removed in every simulation step to model the cutting
Height of cutting slice
Fig. 2: Simulation model of a milling tool with rays representing the two cutting edges without (a) and with (b) deflections.
Geometric model
Undef. chip shape
Undef. Analysis of undef. chip chip shape thickness
Empirical force model Process forces
Accounting for the regenerative effect Tool deflections
Dynamic model
Fig. 3: Schematic overview of the simulation steps (cf. [12]).
process. The resolution of the model can be adapted to the simulated process by choosing an appropriate distance between the dexels in the grids. In contrast to the CSG model, the multi-dexel workpiece model can be visualized by displaying the start- and endpoints of the dexels, which are directly representing points on the surface of the workpiece (cf. Fig. 4). The calculated surface location errors for these points can be directly visualized by coloring them accordingly. 3. Calculation of the surface location errors The dexel model is used to visualize the surface location errors, but their calculation is based on the CSG model. For every cut dexel, the deflection of the tool a at the time, when the cutting edge was cutting the dexel, is looked up from the history of deflections of the dynamics model. The surface location error s for this surface point is then calculated by projecting a onto the direction of the CSG ray as shown in Fig. 5. If Current tool position Initial workpiece shape
Workpiece Undeformed chip
Dexel board 2
Previous tool positions
CSG rays cutting edge 2
Dexel board 1
Fig. 4: Representation of the machined workpiece shape using a multi-dexel model.
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a)
Undeformed chip
CSG ray
d
a
b)
s
d : Dexel offset
Tool
d s Undeformed chip
Dexel
a : Tool deflection
CSG ray
a
Triangles
Cutting edge at beginning of tooth feed
Dexel
s : Surface location error
Fig. 5: Schematic overview of the calculation of negative (a) and positive (b) surface location errors.
the tool is deflected towards the workpiece surface, s becomes negative. This indicates that the material removal was too high at this location. In order to visualize the influence of the tool deflections on the topography of the workpiece surface, the cut dexel is displaced by an offset d which is calculated by projecting a onto the direction of the dexel. Especially if the deflection is much lower than the undeformed chip thickness, this approach yields sufficiently accurate results. However, in case of very large deflections, the calculated surface location errors can be inaccurate for the following reasons. The ideal, undeflected CSG tool shape is used to find the dexels of the multi-dexel boards that have to be cut in a simulation step. Especially for high deflections perpendicular to the direction of a dexel, it is possible that this selection contains dexels that would not have to be cut if the local deflection was taken into account. Accordingly, the selection would contain additional dexels when considering the local deflections. Despite the inaccurate selection of dexels for cutting, they are still displaced by d and colored according to the local surface location error s. The neglect of these effects leads to artifacts on the simulated workpiece surface, which are analyzed in Section 5 in more detail. Therefore, the varying local deflections are taken into account by using a triangle mesh model as described in Section 4. 4. Triangle mesh tool model In order to reduce the artifacts resulting from cutting the dexel boards with the CSG model of the tools, an alternative tool model based on triangle meshes can be used. By cutting the workpiece with triangle meshes, it is possible to take the varying deflections during each tooth feed step into account and, thereby, increase the accuracy of the simulation results at higher deflections. In every simulation substep, the cutting edges are modeled by connecting the tips of the corresponding CSG rays, resulting in a line strip for every cutting edge. By connecting the line strips of subsequent substeps using triangles, a triangle mesh representation of the tool in the current simulation step is generated. If the varying deflections of the tool are taken into account while generating the line strips, the resulting triangle mesh is deformed to model the movement of the cutting edges. In Fig. 6, a triangle mesh representation of a vibrating end-mill is shown schematically. Simulation parameters such as tool radius and height, the number of cutting slices, and the number of substeps per simulation tooth feed are usually set once per tool in a simulation
Fig. 6: Triangle mesh representation of a milling tool with deflections.
setup. Thus, the mesh only needs to be generated once for each tool in its undeflected state, reducing the required amount of calculations per simulation step. The deflections of the tool can then be modeled by moving the vertices. One problem of the new approach is the generation of the CSG rays in the centers of the cutting slices as shown in Fig. 2. Therefore, the generated mesh does not span the full height of the tool using the endpoints of these rays, which are located half a cutting slice height below or above the top and the bottom of the tool shape. In order to resolve this, the highest and the lowest rays are moved upwards and downwards along the cutting edges, respectively. Another problem arises if surface normal vectors are stored at the tips of the dexels in order to calculate lighting when visualizing the workpiece. When cutting with the CSG tool model, the inverted normals of the tool shape, e.g., a cylinder, can be used to calculate the workpiece normals. Accordingly, these normals can be calculated based on the faces of the triangle mesh. However, if the triangle mesh is strongly deformed due to high deflections, the normals of its faces differ from the original state, resulting in poor visualization quality. Therefore, the original normal vectors in the undeflected state are stored at the vertices of the mesh and used when calculating the normals of the workpiece during cutting. Using the CSG-based approach, deflections are modeled by changing the lengths of the scan rays according to the deflections as shown in Fig. 2. In case of slot or pocket milling, it is not possible to model the influence of tool vibrations on the surface quality of the floor of the workpiece because the deflectiondependent tilt of the tool is neglected. The application of triangle meshes has the advantage that three-dimensional deformations of the tool model are possible without further effort. 5. Application and experimental validation In order to analyze the influence of the proposed triangle mesh method in comparison to the CSG approach, a threeaxis milling process was simulated and validation experiments were conducted on a Deckel Maho DMU 50 eVolution five-axis machining center. The machined workpiece was a pocket with dimensions of 119 mm × 82 mm × 19.5 mm, which is shown in Fig. 7. An end mill with two flutes, a diameter of d = 12 mm, a corner radius of 2 mm, and a helix angle of 30◦ was used
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a)
m
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100 mm Fig. 7: Shape of the test workpiece with the highlighted section that has been analyzed.
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for the milling process. The workpiece (EN AW-7075) was manufactured using an axial depth of cut a p = 4.5 mm, a radial depth of cut ae = 4 mm, a spindle speed of n = 15 800 rpm, and a feed velocity of v f = 6320 mm min−1 , resulting in a feed per tooth of t f = 0.2 mm. Machining these kind of workpieces, chatter vibrations are especially likely to occur in the corners since the engagement angle becomes higher in comparison to the flank milling sections. Therefore, the analysis of the process is focused on the machining of a corner of the workpiece, which is highlighted in Fig. 7. This milling process was simulated using the CSG-based approach and the new triangle mesh model. Impact hammer tests were conducted to measure the dynamic modes of the milling tool to calibrate the dynamics model of the simulation. The coefficients of the cutting force model were estimated based on calibration experiments. The spatial resolution of the dexel model of the workpiece, i.e., the distance between adjacent dexels in the grids, was set to 0.1 mm. In order to show fine details and potentially large deflections in the corner, a logarithmic color scale was chosen to visualize the surface location errors. Fig. 8 shows the simulated surfaces using the CSG and the triangle mesh method, respectively. In both cases, chatter marks are clearly visible in the corner and the influence of the helix angle and the axial depth of cut is predicted in a similar way. However, the maximum negative surface location error is slightly higher when using the CSG approach. Furthermore, several artifacts are present as expected due to the described limitations of the model. These artifacts are completely avoided by using a triangle mesh to cut the dexels. The assumption that artifacts are most likely to occur if the deflections of the tool are high in comparison to the resolution of the dexel board is confirmed by the observation of equal results on the flat surface regions, where the tool deflections are significantly lower. By deforming the triangle mesh in a three-dimensional way according to the calculated deflection-dependent tilt of the tool, surface structures on the floor of the pocket can also be modeled using the triangle meshes, which can be seen in Fig. 8 as well. The resulting surface of the machined workpiece is shown in Fig. 9. As in the simulations, chatter vibrations occurred in the corner and the influence of the helix angle and the axial depth of cut can be observed as well. The spatial frequency of the chatter marks matches the triangle mesh simulation slightly more closely. This is especially noticeable in the lower section which was cut by the corner of the tool (Fig. 9a). Fig. 10 shows a comparison of the machined corner to the simulated surface contour. In order to depict the deviations caused by the deflections, the ideal surface according to the NC
0.04 0.0 -0.04 -0.4
Fig. 8: Comparison of surface location errors calculated using the CSG approach (a) and the triangle mesh model (b).
Fig. 9: Photo of the surface resulting from the experiments.
path is shown as well. When the tool enters the corner, the measured and simulated surface location error of approximately 0.3 mm matches and the contours of the surfaces are similar. The roughness of the simulated surface seems to be slightly higher, which can be explained by the neglect of effects like process damping in the simulation. Due to the higher complexity of calculating the intersections between the dexels and the triangle mesh representation in comparison to the CSG model of the tool, which comprises two cylinders and a torus only, the computation time is higher. In case of the analyzed corner, the simulations took approximately 231 s using the CSG model and 367 s using the triangle mesh model on a standard desktop computer, corresponding to an increase of the simulation time of 58.9 %. The computation time depends on different factors including the number of cutting slices and simulation substeps, which directly influence the number of triangles, and the resolution of the dexel boards, which influences the number of required ray intersections in each
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generated surfaces have to be known more precisely, the model based on a triangle mesh yields more accurate results. In the presented model, the triangle mesh is generated based on the local deflections at a fixed tool position for every cut. In future research work, the possibility of taking the feed velocity and the trochoidal movement of the cutting edges into account will be analyzed.
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Acknowledgements
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This project has received funding from the European Unions Seventh Framework Programme for research, technological development and demonstration under grant agreement no 609306.
Fig. 10: Comparison of the simulated and measured surface contours (top view).
References simulation step. Therefore, the given durations are only valid for the specific presented simulation setup, but the observed increase of simulation time can be interpreted as a rough estimation of the computational cost of the new modeling approach. 6. Conclusions and outlook The presented geometric-kinematic milling simulation system allows the prediction of surface location errors resulting from the dynamic behavior of the milling tools. By using a multi-scale workpiece model, the undeformed chip thicknesses and, thus, the process forces can be calculated accurately based on a CSG model and the surface location error can be visualized based on a multi-dexel model. The CSG modeling technique is used to model the tool shape and the material removal is simulated by removing the intersecting volume with the tool from both workpiece models. However, due to the assumption of an undeflected tool when selecting the dexels that are cut in a simulation step, the calculated surface location errors are not always accurate and artifacts can occur in the visualized workpiece shape. Analyzing an exemplary milling process, this effect could be shown in the corner of a pocket at high deflections of the milling tool due to chatter vibrations. An alternative method based on a triangle mesh model of the tool for cutting the dexel boards was presented. By displacing the vertices of the mesh, the local deflections of the tool can be taken into account when selecting the dexels that are cut in a simulation step. Using the new modeling approach to simulate the machining of the exemplary corner, the discussed artifacts are not present in the visualized surface anymore and face-milled surface location errors are predicted. Comparing the simulations to the surface resulting from a machining experiment, both approaches can be used to predict the occurrence of chatter-effected regions correctly, but the surface location errors predicted using the new method are more accurate when the deflections of the tool are high. However, a disadvantage of the approach based on a triangle mesh is the higher simulation time, which was increased by 58.9 % for the analyzed process. Therefore, the applicability of both techniques depends on the required accuracy and the application of the generated simulation results. If it is sufficient to locate chatter-effected regions, the CSG model should be preferred due to its higher efficiency. If the topographies of the
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