Mechanistic Modelling of the Milling Process using Complex Tool Geometry D. ROTH( ) 1 , F. ISMAIL1, S. BEDI1

1Department of Mechanical Engineering, University of Waterloo, CAD CAM Research Group, Waterloo, Ont., Canada N2L 3G1

djroth@engmail.uwaterloo.ca 1 (519) 888-4567 x 7089 1 (519) 888-6197

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

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.

Milling forces can be described by a cutting force and a plowing (or edge) force

component. That is:

bKbhKF ec += (1)

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:

bKvQKF ec += / (2)

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

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

. 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.

Figure 2

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 = pmax-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:

)()( 0max0min ppfppfb −−−′= (3)

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

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 RA215.44-25MN25-09C flat bottom 3-

tooth endmill with an outer diameter of 25.4mm. The cutter runout, was measured to be

0.013335mm. The inserts used actually have a nose radius measured at 0.79375 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

It can be seen that the radiused tool stops cutting 0.160 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.

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, 1982. The Prediction of Cutting Forces in End Milling with

Application to Cornering Cuts. Int. J. Mach. Tool Des. Res. 22(1), pp. 7-22.

[2] B. M. Imani, M. H. Sadeghi, and M. A. Elbestawi, 1998. An Improved Process Simulation System for Ball-end

Milling of Sculptured Surfaces. Int. J. of Mach. Tools and Manu., 38(9), pp 1089-1107.

[3] R.B. Jerard, B.K. Fussell, and M.T. Ercan, 2001. On-line Optimization of Cutting Conditions for NC

Machining. In 2001 NSF Design, Manufacturing and Industrial Innovation Research Conference. Jan 7-10.

[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, 1996. OBBTree: A Hierarchical Structure for Rapid Interference

Detection. Computer Graphics (SIGGRAPH 96 Proc.) pp 171-180.

[6] Kral, 1986. Numerical Control Programming in APT. Prentice-Hall, Englewood Cliffs, NJ.

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

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