Mechanistic Modelling of the Milling Process using Complex Tool Geometry

Transcription

1 Mechanistic Modelling of the Milling Process using Complex Tool Geometry D. ROTH( ) 1, F. ISMAIL 1, S. BEDI 1 1 Department of Mechanical Engineering, University of Waterloo, CAD CAM Research Group, Waterloo, Ont., Canada N2L 3G1 djroth@engmail.uwaterloo.ca 1 (519) x (519) fismail@mecheng1.uwaterloo.ca sbedi@mecheng1.uwaterloo.ca Abstract Mechanistic models of the milling process must calculate the chip geometry and the cutter edge contact length in order to predict milling forces accurately. This task becomes increasingly difficult for the machining of three dimensional parts using complex tool geometry, such as bull nose cutters. In this paper, a mechanistic model of the milling process based on an adaptive and local depth buffer of the computer graphics card is compared to a traditional simulation method. Results are compared using a 3-axis wedge shaped cut a tool path with a known chip geometry in order to accommodate the traditional method. Effects of cutter nose radius on the cutting and edge forces are considered. It is verified that there is little difference (1.4% at most) in the predicted force values of the two methods, thereby validating the adaptive depth buffer approach. The numerical simulations are also verified using experimental cutting tests of aluminum, and found to agree closely (within 12%). Keywords: Mechanistic Modelling 3-Axis Cutting Force Complex Tool Geometry Introduction The vector based mechanistic model of Kline et al. [1] is a well known and accepted force model of the milling process. However, this model is only effective for modelling cuts with known chip geometry. For more complex tool paths, a more comprehensive method

2 of predicting the chip geometry must be employed. Currently the most common methods of determining chip geometry are constructive solid geometry (CSG) (for example, see Imani et al. [2]) and discrete models using extended Z-buffers (see Jerard et al. [3] for a good review). Mechanistic models allow the planning of tool paths from the point of view of efficient tool pass selection, feed rate scheduling, deflection compensation and vibration control. An accurate mechanistic model will then increase the efficiency of the tool path and the quality of the resulting product. All the methods developed so far, to the authors knowledge, rely on generic software implementations of their respective algorithms. However, many of the elements necessary for these simulations have been implemented cheaply and efficiently in computer graphics hardware such as homogeneous matrix transformations, and depth buffer functionality. Roth et al., [4], describe how readily available and affordable computer graphics hardware may be used to implement a mechanistic model of the milling process. Limitations of the implementation identified by the authors include the use of a brute force rendering algorithm, sizing the depth buffer to the tool and only considering the case of flat end mills. In this paper the method is improved upon in three areas. First, the algorithm is updated to handle more complex tool shapes. Secondly, the depth buffer is sized to the cutter teeth, thereby improving both the memory requirements and efficient usage of the depth buffer. Finally, as mentioned by Gottschalk et al. [5], the algorithm takes advantage of view frustum culling, to improve the rendering efficiency. Mechanistic model A typical rendering engine is used to compose a rasterized (a pixelized representation) view of a scene. The rendering engine makes use of a viewing direction and a viewing volume, which determines what portion of the scene is visible. A rendering engine also uses a depth buffer to determine the distance from the current view datum (the eye location) to each rasterized object in the scene. As each pixel has a given area and an associated depth, rendering a scene results in a volumetric model.

3 Milling forces can be described by a cutting force and a plowing (or edge) force component. That is: F = K bh K b (1) c + e where K is an experimentally determined proportionality constant, h the instantaneous chip thickness, b the edge contact length and the subscripts c and e refer to cutting and edge force coefficients respectively. This model is based on the instantaneous chip area bh and can be converted to a volumetric force model by: F = K Q / v K b (2) c + e where Q is the instantaneous metal removal rate and v is the tangential cutting speed of the tool. The mechanistic model in this paper uses an adaptive and local depth buffer to calculate Q and b. By adaptive, it is meant that the orientation of the depth buffer changes to be aligned constantly with the tool. By local it is meant that the depth buffer is sized to the tool as opposed to the stock material. The reader should note that a depth buffer differs from an extended Z-buffer in two important ways: first the extended Z-buffer has a fixed orientation while the depth buffer is aligned to the view direction. Secondly, as a consequence of the first limitation, an extended Z-buffer must store multiple intersection points, whereas a depth buffer need only store a single value. With the aid of the rendering engine, the depth buffer is implemented directly in graphics hardware. While the implementation of the mechanistic model using the depth buffer is discussed extensively in [4], a brief summary is provided here for the sake of completeness. For every position in which the cutting forces are to be determined, a scene is composed of the current tool position, every previous tool position and the stock material. This is shown in Figure 1a. The reader should note that a rendered tool position consists of the swept sector of each tooth for a given time step. For each scene in the simulation, the depth buffer direction is adaptively set, aligned with the current tool axis. In graphics parlance, this is accomplished by viewing the scene along the tool axis. When a scene is rendered, the graphics engine determines the state of the depth buffer by determining how those objects would be positioned in the scene, relative to

4 the current eye location (the viewing datum). The current state of the stock material is determined by rendering every previous tool position to the scene and saving the state of the depth buffer. The current tool position is then rendered to the scene and the state of the depth buffer is determined again. Essentially, the difference between the two states determines the in-process chip geometry. Improvements in the Determination of the Metal Removal Rate, Q The determination of Q is discussed extensively in [4]. However, in that paper, the depth buffer was sized to the tool diameter. While this is an improvement over extended Z-buffer methods, where the Z-buffer is sized according to the stock material, most of the depth buffer is not used for modelling the in process chip geometry. As can be seen in Figure 1a, most of the buffer holds information about the previous tool positions, leading to an inefficient usage of the depth buffer. An efficient usage of the depth buffer would concentrate a greater majority of the depth buffer elements (sometimes, referred to as dexels) to determining in process chip geometry. In this case, instead of allocating one depth buffer sized to the tool diameter, one could allocate a number of depth buffers equal to the number of tool teeth (Figure 1b). In this manner, the simulation could effectively zoom in on the cutting teeth, allowing the simulation to more accurately determine the in process chip geometry (Figure 1c). Notice the depth buffer sized to the tool diameter and the depth buffers zoomed to the tool teeth are approximately the same size, yet the latter is a more efficient usage. Figure 1 Furthermore, the tool shape used in [4] was a simple flat end mill. This is but one type of tool used in milling. The theoretical 7-parameter (d,e,f,h,r,α,β) APT tool is shown in Figure 2. Through a judicious choice of parameters, this tool may mathematically model flat end, ball nose, bull nose and even more complicated tool shapes. This exact shape tool will rarely, if ever, be used to machine a part. However, if the simulation is able to handle its complicated geometry, then it is capable of handling the wide assortment of tool shapes used in manufacturing.

5 Figure 2 Improvements in the Determination of the Edge Contact Length, b The simulation described herein is ideally suited for determining the edge contact length, b. The depth buffer measures the depth of each element from the current viewing datum, and is denoted by the variable p. As the simulation is concerned with two states of the depth buffer, p refers to the state of the depth buffer before the rendering of the current tool position and p the state after. For a simple flat end mill, the maximum depth of cut can be calculated as p = p max -p min, and the edge contact length is simply b = p/cos ψ, where ψ is the cutter helix angle. However, for the APT tool of Figure 2, the edge contact length may only be calculated by projecting the segment p onto the tool shape. Referring to Figure 2, the tool shape consists of three distinct piecewise sections a linear section OA, described by the angle α, a circular arc AB described by centre point (e,f) and radius r, and a linear section BC., described by the angle β. The linear section projections are determined by similar triangles and the circular projection are determined by the subtended angle of the depth intersection point. If f(z) is a function that computes the edge contact length of a height z in the shown tool coordinate system, then the overall edge contact length, b, can be calculated from: b = f p p ) f ( p p ) (3) ( min 0 max 0 Analysis of the Method Comparison to Traditional Simulation Two simulation experiments, that included the effect of cutter runout, were carried out in which a wedge shaped cut was taken from rectangular stock. The wedge measured 6mm deep and 12.7mm wide at the start of the cut and linearly tapered to 0mm in both the depth and width over a length of 90mm. While the experiments considered both a climb cut and a

6 conventional cut case, in the interest of brevity, only results of the climb cut case will be presented here. Results of the traditional simulation method of Kline et al. [1] vs. the graphical method are given in Figure 3. The results show an excellent agreement between the two methods. At most, the X-direction forces differ by 12.7N (1.4% relative to the traditional peak-to-peak). The minimum y-direction forces differ by 7.2N (0.8%) and the maximum y-direction forces differ by 1.0N (0.1%). This indicates the rendering engine approach is valid to the same extent as the traditional method. Figure 3 One can notice a stepping in the maximum negative y-direction forces of traditional method (Figure 3b), due to discretization effects in the calculation of the entry angle. Since the simulation was carried out with a discretized angle of θ = 2, whenever the radial immersion changes by an amount greater than the arc length subtended by θ, a step will occur. One may estimate the step period from 1/ θt, with θ measured in radians and t the total time of the cut. This gives 4.77 steps/sec, which is seen to be the case in Figure 3b. The graphical method does not suffer from this stepping as its results are effectively averaged over the rendered sweep sector. Effect of Nose Radius The cutter used in this experiment was a Sandvik RA MN25-09C flat bottom 3- tooth endmill with an outer diameter of 25.4mm. The cutter runout, was measured to be mm. The inserts used actually have a nose radius measured at mm using a Mitutoyo PH 350 shadow graph projector. To investigate the effect of the nose radius, an additional simulation was carried out using a tool with no nose radius. This simulation reveals the effect of the nose radius is significant at shallow depths of cut. The results of this simulation as compared to the original simulation are shown in Figure 4. Figure 4

7 It can be seen that the radiused tool stops cutting seconds ahead of the zero radiused tool. This means, wedge length is 2.41mm shorter than the expected path. The force profiles of Figure 4 also reveal that the forces near the ends differ by up to 97.9% in the x- direction and 98.1% in the y-direction because of the effect of the nose radius. Comparison with experimental results Figure 5 Shown in Figure 5 are experimental results conducted on an OKK MCV 410 vertical machining centre. As both the traditional and graphical simulation methods do not account for vibrational effects, in order to allow for experimental to simulation comparisons, any frequencies above the tooth passing frequency were filtered out from the experimental results. While the agreement between the simulated and experimental results is good, the simulation results of Figure 3 differ from the experimental results of Figure 5 in some areas. For example, the minimum X-direction forces differ by at most 13.8N (1.5% relative to simulation peak-to-peak). The negative y-direction forces differ by 101.8N (11.7%) while the positive y-direction forces differ by 6.8 N (0.8%). In each case, the simulated forces over predict the actual cutting forces. Conclusions In this paper, a graphical mechanistic model is presented and compared with both a traditional simulation method and experimental results. Both simulation methods compute the in process chip geometry at discretized tool positions to predict forces. The simulations predict almost identical values of force components. The simulation results also compare well to filtered experimental results, with differences attributed to unmodelled effects, such as cutter deflection. This paper builds on the accomplishments of a previous paper. Improvements to the graphical method include the modelling of more complex tool shapes, view frustum culling for increased simulation efficiency and zoomed in depth buffers for more efficient use of the depth buffer.

8 Acknowledgments The authors appreciate the funding provided by the Natural Sciences and Engineering Research Council of Canada as well as the research infrastructure provided by the Ontario Innovation Trust and the Canada Foundation for Innovation. References [1] W.A. Kline, R.E. DeVor, and J.R. Lindberg, The Prediction of Cutting Forces in End Milling with Application to Cornering Cuts. Int. J. Mach. Tool Des. Res. 22(1), pp [2] B. M. Imani, M. H. Sadeghi, and M. A. Elbestawi, An Improved Process Simulation System for Ball-end Milling of Sculptured Surfaces. Int. J. of Mach. Tools and Manu., 38(9), pp [3] R.B. Jerard, B.K. Fussell, and M.T. Ercan, On-line Optimization of Cutting Conditions for NC Machining. In 2001 NSF Design, Manufacturing and Industrial Innovation Research Conference. Jan [4] D. Roth, F. Ismail, and S. Bedi. Mechanistic Modelling of the Milling Process Using an Adaptive Depth Buffer. To appear in CAD. [5] S. Gottschalk, M.C. Lin, and D. Manocha, OBBTree: A Hierarchical Structure for Rapid Interference Detection. Computer Graphics (SIGGRAPH 96 Proc.) pp [6] Kral, Numerical Control Programming in APT. Prentice-Hall, Englewood Cliffs, NJ.

9 Figure 1 Sizing the depth buffer; (a) to the tool diameter, (b) to the tool teeth, (c) zoomed in view of the teeth depth buffers Figure 2 Calculating the edge contact length on the 7-parameter APT tool [6]. Figure 3 (a), (b), traditional simulation results vs. (c), (d) graphical simulation results Figure 4 comparing (a) x-direction, (b) y-direction simulated force profiles Figure 5 experimental (a) x-direction, (b) y-direction force profiles

10 Current tool position Bounds of the viewing volume sized to the tool diameter Previous tool positions (a) (i) (ii) Bounds of the viewing volume sized to the tool teeth (iii) (b) Zoomed in view of the tool teeth (i) (ii) (iii) (c) Figure 1 Sizing the depth buffer; (a) to the tool diameter, (b) to the tool teeth, (c) zoomed in view of the teeth depth buffers

11 view datum z C h p min p ma x p 0 f r e O A B b p x d Figure 2 Calculating the edge contact length on the 7-parameter APT tool [6].

12 0 200 F X [N] Time [s] 400 (a) F Y [N] Time [s] 0 (b) 200 F X [N] Time [s] 400 (c) N]

13 Figure 3 (a), (b), traditional simulation results vs. (c), (d) graphical simulation results F X [N] 30 With nose radius Without nose radius Time [s] 25 (a) 20 F Y [N] Time [s] (b) Figure 4 comparing (a) x-direction, (b) y-direction simulated force profiles

14 F X [N] Filtered Unfiltered Filtered Time [s] 400 (a) Unfiltered F Y [N] Figure 5 experimental (a) x-direction, (b) y-direction force profiles