Mechanistic modelling of 5-axis milling using an adaptive depth buffer D. Rotha
a
F. Ismaila,∗
S. Bedia
University of Waterloo, Department of Mechanical Engineering, Waterloo, Ontario, Canada N2L 3G1
Abstract Mechanistic modelling of surface machining with a toroidal end mill is presented in this work for 5-axis machining. A graphical representation of tool movements (using tooth swept sectors or the swept surface of the tool) is rendered using a widely available rendering engine, whereby the in-process chip geometry and tool edge contact length are determined by an adaptive and local depth buffer. Owing to the chip geometry being highly dependent on the feedrate, a detailed derivation of the relative tool-workpiece velocity is presented based on the kinematics of a tilt-rotary table configuration. The mechanistic modelling is verified with experimental results and found to agree to within 7% of the peak-to-peak forces. Keywords: Mechanistic 5-axis force modelling; Adaptive depth buffer; Swept surface ∗
Corresponding author. tel: +1-519-888-4567; fax: +1-519-888-6197; web: www.me.uwaterloo.ca/ ~machlab E-mail-addresses:
[email protected] (D. Roth),
[email protected] (S. Bedi), fi
[email protected] (F. Ismail)
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1
Introduction
Mechanistic modelling of the milling process entails relating the in-process chip geometry and tool contact length to forces generated in the cut. It is through a thorough understanding of the forces generated at the tool workpiece interface that the machining process can be analysed according to criteria such as tool deflection, machine utilization and tool path efficiency. Mechanistic modelling has traditionally been approached using either solid modelling or discrete methodologies. Using boolean difference operations of the tool and workpiece, Bertok et al. [1] were able to calculate average force and torque values for a given NC code block. However, they did not present a method to predict peak values. Takata [2] used a solid modelling approach for 2- 12 -axis machining and feedrate scheduling as did Spence and Altintas [3]. Mounayri et al. [4] used a solid modelling system to compute the volume removed for a given 3-axis tool path followed by a ray casting technique to compute the instantaneous chip thickness. Imani et al. [5] developed a simulation system for 3-axis ball end milling of sculptured surfaces using c the ACIS solid modelling kernel. The authors used a NURBS curve to model the cutting
edge of a tool. However, extracting the instantaneous chip thickness from the model was computationally inefficient, and therefore the authors relied on the well known geometry of the ball-end mill and simple inclined cuts. To overcome the efficiency concerns of solid modelling techniques, discrete implementations (well reviewed by Jerard et al. [6]), where the stock material is represented by a 2-dimensional array of z-direction vectors (ZDV), are typically used. Choi et al. [7] used a Z-map for tool path planning and verification of 3-axis machining. Fussell et al. [8] employed a Z-map for feed-scheduling of 3-axis tool paths. Later, Fussell et al. [9] presented feed scheduling results (simulation only) for 5-axis milling using an extended z-map approach (a z-map in which multiple heights per ZDV are stored). The authors simplified the 5-axis
2
case by presenting results for only a ball nose cutter and by representing each tool move as a simple linear 3-axis move that neglects tool rotation. Kim et al. [10] discussed the application of a Z-map to the prediction of 3-axis ball-end milling forces, but relied heavily on the simple geometry of the sphere and linear movements. Although Z-buffer algorithms have been implemented for ball, flat and round-end mill cutters [11], different procedures for calculating the chip geometry are necessary for each method.
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Graphical method
An adaptive depth buffer method for mechanistic modelling of 3-axis machining was presented by Roth, Ismail and Bedi [12]. This method has been subsequently improved to include the effects of complex tool geometry [13]. The purpose of the current paper is to expand the methodology to 5-axis machining and present experimental results that provide verification of the method. The graphical method described in [12] and expanded upon in this paper is based on an adaptive and local depth buffer, and has been implemented using OpenGL and a standard PC graphics card (a 64MB nVidia GeForce3). By adaptive, it is meant that the depth buffer is aligned constantly to the tool, and hence eliminates the need for an extended z-buffer. By local, it is meant that the depth buffer is sized to the tool, as opposed to the workpiece. As described in [13], not only is it possible to size the depth buffer to the tool, but also it is possible to size the depth buffer to individual teeth. This leads to a very efficient use of the depth buffer, since the majority of the dexels (pixels with an associated depth) are used to calculate in-process chip geometry.
2.1
Mechanistic model
Shown in figure 1 are the radial and tangential force components on a tooth of a milling cutter acting during a cutting operation. The tool rotation is given by the angle Θ (although some 3
Y
Fx Ft
Fy
f
Fr Q X
Figure 1: Force components on a tool researchers prefer the angle φ measured in the direction of tool rotation). Each of these milling force components are described by a combination of a plowing (i.e., edge) force proportional to the instantaneous edge contact length, b, and a cutting force proportional to the instantaneous undeformed chip area, bh,
F = Kc bh + Ke b
(1)
where K is an experimentally determined proportionality constant, h the instantaneous chip thickness and the subscripts c and e refer to cutting and edge force coefficients respectively. The associated volumetric force model is
F = Kc Q/v + Ke b
(2)
where Q is the instantaneous metal removal rate and v is the tangential cutting speed of the tool. The previous papers [12, 13] describe in detail how each of these parameters can be determined in 3-axis machining, but the methodology will be briefly reviewed here for the sake of completeness.
4
q
Q
Initial tool shape
tool rotation by q
q
resulting tooth sectors
Figure 2: Tooth swept sectors for a tool rotating by angle θ 2.1.1
Simulation methodology
The simulation method is based on rendering representations of tool positions and the stock material, and determining the in-process chip geometry graphically. In this context, rendering implies the use of a rendering engine and graphics hardware. While many rendering engines are available, the two most popular are OpenGL (which is the method of choice for this paper) and DirectX. The methodology is unaffected by the choice of rendering engine. Graphical engines are limited to drawing very basic shapes, such as points, lines, triangles and quadrilaterals [14] so both the stock and tool positions must be represented as a set of these rendering primitives. It is usually very easy to render the initial stock shape since it is normally rectangular or cylindrical. However, representing the tool movement is more difficult. Nonetheless, with reference to figure 2, in a finite time period ∆t, each tooth will sweep out a tooth sector, subtended by the angle θ. This tooth swept sector can be rendered using graphics primitives and any complex tool shape can be represented in this manner. For example, shown in figure 3a-b is a schematic of a 7-parameter APT tool [15] and its corresponding tooth swept sector. Note that the representation of the tooth swept sector is composed of quadrilaterals arranged into strips. OpenGL facilitates the rendering of these types of related quadrilaterals
5
camera (view position)
^
ty
near
top
left
q
right
^
tz
Q ^
r
tx bottom
far view target
(a)
(b)
(c)
(d)
Figure 3: (a) Tooth swept sector for an APT tool, (b) Parameters to locate the view, (c) View volume enclosing the tooth swept sector, (d) Resulting plan view through the GL_QUAD_STRIP specifier, while DirectX has the D3DPT_TRIANGLESTRIP specifier, where a quadrilateral is simply composed of 2 coplanar triangles. Because the simulation method makes use of a rendering engine, one must define a view direction, a view orientation and a view volume. The view direction is specified by defining a view target and a view position (often referred to as the eye or camera position). The view orientation determines which direction is considered up when one looks along the view direction. The view volume defines what scene information is visible, and its size is specified by defining a viewing frustum using 6 bounding planes - left and right, top and bottom, near and far. In order to accurately calculate the process parameters, the simulation is always viewed along the current tool axis, ˆ tz . In this manner, the current tool position is always seen as a plan view. The view orientation (i.e., the up vector) for a given tooth swept sector is defined using the current rotation of the tool and the pitch angle of the tool. The view volume is constrained to the size of a tooth sector. Consider figure 3c-d – for a constant sector size (given by the angle θ), the viewing volume size may be chosen to tightly enclose the sector (i.e., the viewing volume is a bounding box to the swept sector). To view the tooth swept
6
sector as a plan view, with reference to the tool coordinate system, the view target is set as
et =
r (cos(Θ + θ/2), sin(Θ + θ/2)) 2
(3)
and the eye position is given by tz ep = et + H ˆ
(4)
where H is some predefined height set according to the tool dimensions. Because the tooth swept sector rotates about the tool axis, ˆ tz , the up-vector, eu , is also eu = et A key aspect of the simulation is its use of the rendering engine depth buffer. A depth buffer is used to store information about the distance of visible primitives in the scene from the current camera location. One depth value is stored per pixel in the scene. The combination of pixel information (colour) and depth information is termed a “dexel.” The current state of the stock material is defined by rendering all previous tool positions (i.e., the tooth swept sectors representing the tool positions) to the scene and reading from the depth buffer. The effect of the current tool position is determined by rendering it to the screen and re-reading the depth buffer. The difference in the two states of the depth buffer give the volume of material removed, V , while the projection of the low and high depths onto the tool shape (easily facilitated due to the plan view) gives the edge contact length. The instantaneous metal removal rate may then be calculated from Q = V /∆t. The in-process chip parameters are then substituted into equation (2) to determine the forces. While the basic methodology is unchanged from 3 to 5-axis machining, additional challenges, both graphical and physical, do arise. First, when rendering close to an edge of the stock material, the rendered scene must be clipped to that edge to accurately determine the in-process chip geometry. This is accomplished using clipping planes. While the use of clipping planes for 3-axis machining is relatively straight forward, the problem is exacerbated in 5-axis machining due to the need for additional clipping planes and the changing orientation of the tool. This challenge is discussed in detail in Appendix A. Secondly, the instantaneous 7
Figure 4: Schematic of the Deckel Maho 5-axis tilt-rotary table milling machine feedrate of the tool is profoundly affected by the kinematics of the 5-axis machine. This topic is discussed next.
2.2
Influence of machine kinematics on feedrate
The 5-axis machine used in this paper is of a tilt-rotary type configuration, capable of simultaneous movement in 3 translational, x, y and z, and 2 rotational axes, a and c. As shown in figure 4, the table moves linearly in the y-direction while the head moves linearly in the x and z-directions. The table also tilts about the x-axis and rotates about the z-axis (known as angles a and c, respectively). No matter what the machine configuration is, one must differentiate between the actual motions of the machine and how the tool path is specified (either in ISO G-Code or in a conversational programming language). Convention dictates that the part is programmed as if the stock material is stationary and the tool undergoes all translations and rotations. In reality, all machine tools use a combination of both tool and stock movements to achieve the necessary relative tool-workpiece positioning.
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To help distinguish between these two cases, the following convention is adopted in this paper: G-code statements will be in all capital letters (X, Y, Z, A, C), while actual machine movements will be in all lower case letters (x, y, z, a, c). 2.2.1
Coordinate systems
Shown in figure 5 are the various coordinate systems used in 5-axis machining with a tiltrotary table. Also shown in this figure is a simpler schematic of the spatial relationship between these coordinate frames, drawn in the Y-Z plane, with the positive x-direction out of the page. The machine tool builder defines a machine coordinate system, Cm , which is also known as the global coordinate system. This coordinate system is affixed to the most positive location on the machine tool, so that every point in the machine work volume has negative coordinates with respect to Cm . All machine movement and interpolation are done in this coordinate system - any other coordinate system referenced is for personal convenience. The machine tool builder is also responsible for identifying the location of the pivot point of each of the axes of rotation. These positions are critical for determining the correct inverse kinematics of the machine, and for post processing cutter location data into correct machine tool moves. These distances are measured in the machine tool “home” position. That is, the machine is positioned in such a way that translational motion is only possible in the negative direction, the C axis is rotated to align the table grooves to the global x-direction and the A axis is rotated to make the table perfectly horizontal. The location of the pivot point in the home position from Cm is labelled as coordinate system Cp . The origin of Cm is affixed to the most positive location in the machine work volume while the origin of Cp is aligned to the centre of the table. So, Cm and Cp are offset from one another in the x-direction, by, nominally, half the x-work volume distance. All the origins of the remaining coordinate system, except for the workpiece coordinate system, Cwp and the related datum coordinate system, Cdatum , lie in the Y-Z plane of frame Cp . 9
z Cm y
Cm r
m®p
Ctr Cp
C
a
r
Cwp
p®t r
rc®wp
ra®c d
C
c
Cwp
Cp
CtrC
a
C
c
Coordinate Aligned to Description Frame machine frame Cm machine frame - fixed to machine zero point Cp yes pivot point frame - fixed to machine home Ctr yes tilt-rotary table frame - moves with table Ca no tilt frame - moves with A axis Cc no rotary frame - moves with C axis Cwp no work piece frame - moves with workpiece Cdatum yes datum frame - fixed Figure 5: Relationship of defined coordinate systems to the tilt-rotary 5-axis machine. Inlay - Coordinate frame measuring chain.
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The tilt-rotary table will move away from Cp during machining. Let the coordinate system attached to the tilt-rotary table be Ctr - this coordinate system describes the movement of the table away from the home position. This coordinate system is attached to the table but does not rotate with the table. When the table is in its home position, the coordinate systems Cp and Ctr are coincident. Let the coordinate system about which the actual table rotation a occurs be Ca - this is the tilt axis. The origin of Ca is coincident with the origin of Ctr . Similarly, the c rotation occurs about the Cc coordinate system - this is the rotary axis. It is entirely possible (indeed likely) that the axes of rotation of the tilt-rotary table will not intersect. Let the perpendicular offset between the two vectors be δ, as shown in figure 5. Note that this is a distance measured in the y-direction of frame Ca . Furthermore, note that Ca and Cc move with the tilt-rotary table and move away from the coordinate system Cp . While the translations of one axis do not affect the translations of another, the c rotation is affected by the a rotation, because the rotary table is mounted on top of the tilt table. When a part is designed in any CAD system, it is designed with reference to its own part coordinate system, designated the workpiece coordinate system, Cwp . This coordinate system is located in the machine tool work volume via a datum vector and is designated by the coordinate system Cdatum (not pictured in figure 5), measured from Cm . When the machine tool is commanded to move to the location (X, Y, Z, A, C) = (0, 0, 0, 0, 0), Cwp will be coincident with Cdatum . 2.2.2
Effective feedrate calculation
In 3-axis machining, the feedrate is specified via the F code in either the G-code or conversational program. For example, assuming metric units, both the following lines specify a linearly interpolated movement from the current tool position to the specified point at a feedrate of 600 [mm/min]. G-Code: 11
N2112 G01 X13.876 Y-24.294 Z-12.982 F600; Conversational: L X13.876 Y-24.294 Z-12.982 F600 In general, the manner in which the controller computes each axis speed to achieve the desired feedrate is determined as follows: if the tool is currently located at P1 = (X1 , Y1 , Z1 ) and is commanded to travel to P2 = (X2 , Y2 , Z2 ) at a given feedrate, f, then the interpolated speed of each axis, v, is calculated as ∆X = X2 − X1
vX =
∆X f d
∆Y = Y2 − Y1 ∆Z = Z2 − Z1 √ d = ∆X 2 + ∆Y 2 + ∆Z 2 vY =
∆Y d
f
vZ =
(5)
∆Z f d
Note that the quantity f /d is the time required to complete the motion. However, in 5-axis machining, two complications arise due to the rotational axes. The first is that a rotational angle, in [◦ ], must be synchronised with a linear distance, in [mm]. The second problem is that the distance from the pivot point affects the relative speed of the tool to the workpiece (i.e., for a constant angular speed, ω, the greater the pivot point distance, r, the greater the tangential speed v = ωr). Machine controller designers solve the first problem by actually ignoring the mismatched units. That is, in order to synchronise the full 5-axis motion, a 5-axis “pseudo-distance” is calculated as d=
√
∆X 2 + ∆Y 2 + ∆Z 2 + ∆A2 + ∆C 2
(6)
where A and C are representatives of two axes of rotation. Obviously, the units of the linear motions (X, Y, Z) are different from the units of the rotational axes (A, C), and accordingly, equation (6) is not dimensionally consistent. However, for the purposes of axes synchronisation, it is considered adequate. 12
c
e3
c
g3
e2
a
g1
g2
e1
a u
(a)
(b)
Figure 6: Cylinder rotating simultaneously about two axes. Rotating coordinate frames: (a) Rotating with a only (b) rotating with both a and c The actual driven speeds of each of the axes are then calculated analogously to equation (5). The solution to the second problem is accomplished in two parts, by first calculating the speed contribution of the linear axes and then that contributed by the rotational axes. Calculation of the translational speed component is unchanged from the 3-axis case. To determine the rotational component, one must know the coordinates of the point under consideration either specified with respect to Cwp or Cdatum and use rotating coordinate frame analysis. 2.2.3
Velocity calculation from known workpiece coordinates
It is necessary to find the velocity of a point specified with respect to the workpiece coordinate system, pwp , caused by the rotations of the tilt-rotary table. The tilt-rotary table may be thought of as a cylinder rotating simultaneously about its longitudinal and vertical axes, as shown in figure 6. Two rotating coordinate systems are shown, Ce and Cg . In the first, e2 , e3 are fixed to the cylinder while e1 remains fixed to the global x-direction. In the second, all axes are fixed to and rotate with the cylinder. In either case, one may imagine the point pwp as lying on the surface of the cylinder.
13
In coordinate system Ce , the angular velocity of the coordinate system is ωe = aˆ ˙ e1 and the angular velocity of the cylinder is ωcyl = aˆ ˙ e1 + cˆ ˙ e3 . In the second coordinate system, Cg , the angular velocity of the cylinder can be thought of as ωcyl = aˆ ˙ e1 + cˆ ˙e3 = au ˙ + cˆ ˙g3 , where u is a fixed horizontal vector and g ˆ3 = ˆ e3 , since both remain aligned to the cylinder axis. This vector is resolved in the second frame as g2 . Therefore, u = cos(c)ˆ g1 − sin(c)ˆ ωcyl = a˙ cos(c)ˆ g1 − a˙ sin(c)ˆ g2 + cˆ ˙g3
(7)
As seen in figure 5, the location of the point pwp in the rotating coordinate frame Cg is given by pg = pc = rc→wp + pwp , because in reality, the coordinate systems Ce and Cg are identical to the coordinate systems Ca and Cc except for the small displacement δ. Then the velocity of this point caused by the tilt-rotary table is vpc = ωcyl × pc
(8)
Note that vector is specified in the rotating coordinates of Cg and should be converted to a global aligned coordinate system by means of the machine inverse kinematics. If one wishes to include the velocity caused by δ and the tilt angle, a, the term a˙ × ra→c = a˙ × (0, δ, 0) may be added to the above calculation. However, its contribution will be extremely small and may safely be neglected. 2.2.4
Velocity calculation from known datum coordinates
It is necessary to find the velocity of a point specified with respect to the datum coordinate system, pdatum , caused by the rotations of the tilt-rotary table. The coordinates of this point in coordinate system Ctr is given by ptr = pdatum + rtr→datum 14
(9)
As before, the angular velocity of the cylinder, ωcyl , is calculated from equation (7). However, ptr is specified in a coordinate system with axes parallel to the global coordinate system while ωcyl is specified in the rotating coordinate system Cg . Therefore, one must tr first transform ωcyl to a global aligned coordinate system to become ωcyl . The velocity of
the point due to the rotating axes is tr vptr = ωcyl × ptr
2.2.5
(10)
Relative velocity between tool and workpiece
The effective feedrate of the tool with respect to the workpiece is then v = vhead − vtable − vrotary
(11)
where the first two terms are due to the linear motion of the head and table (equations (5)(6)), and the last term is due to the rotational motion of the tilt-rotary table (equation (8) converted to global aligned coordinates or (10)).
2.3
Using the tool swept surface to increase efficiency
The rendering process is the most computationally expensive portion of the simulation. As described in section 2.1.1, the simulation requires a number of renderings proportional to the number of tooth swept sectors, which is dependent on the spindle speed. In previous research by Roth, Bedi, Ismail and Mann [16], a fast method for generating the swept surface of a toroidal tool was presented, which was based on determining the trajectory of “pseudo-inserts” of the tool. A generalization of that method to surfaces of revolution was later reported by Mann and Bedi in [17]. When a toroidal cutter is sliced with a plane through the tool axis, the resulting intersection is two circles (figure 7). For each circle, it is possible that zero, one or many points on 15
tz
Grazing curves
P1
d
r n P0
Figure 7: Calculating grazing points, P0 , P1 , on a torus the circle will contact the stock material and machine it. The candidate points are termed “grazing points”, and are determined as follows. Form the vector r = n × d, where n is the pseudo-insert normal (i.e., the normal of the slicing plane) and d is the pseudo-insert direction vector. If this vector is not ill-defined, a line in the direction of r will intersect the circle at exactly two points. Each of these points is a grazing point. If the vector is ill-defined (i.e., n is parallel to d), every point on the circle is a grazing point. Whether or not a grazing point machines the part is not the focus of this paper, but in general, machining points will lie on the bottom half of the circle. By slicing the tool with many planes, a grazing curve can be constructed by connecting each grazing point deemed to be machining the part in a piecewise linear fashion. The accuracy of this curve is related to the number of slicing planes. The benefit of this method is two-fold. First, its simplicity, even for complex tool shapes. Secondly, the swept surface of a tool motion is easily constructed, again in a piecewise linear fashion, by joining the grazing points of the grazing curves at the start and end points of the tool motion (figure 8). This is exactly the construction type needed in a graphical environment.
16
tz Grazing curves
Pseudo inserts Swept surface
Figure 8: Swept surface of a toroidal tool computed from grazing points
Figure 9: Tool shape appended to the front of swept surface 2.3.1
Swept surface simulation
A 3-axis approximation to the 5-axis tool path must be used as described by Fussell et al. [9] (i.e., where the 5-axis tool path is modelled as a superposition of numerous 3-axis moves, with orientation changes occurring between tool moves). This is due to the simulation requirement of viewing the tool positions as a plan view. For two successive tool positions, the viewing direction and target is chosen as the average of the tool orientations and positions. The viewing volume is no longer constrained to the tool tooth size, but rather to the swept surface size. It is likely that the front of the tool will cut material as well, and for this reason, the tool profile shape must be appended to the swept surface, as seen in figure 9. Now the entire swept surface is rendered against the stock material and from the depth
17
buffer, a “height map” of the machined material is obtained for the tool motion. In order to predict forces, the tool is discretized axially into circular sections which are swept against the height map. Instantaneous chip thicknesses can be predicted where the circle is found to intersect a height map dexel. Instantaneous edge contact lengths are also found in a similar manner. For more details, consult Fussell et al. [18]. The graphical method has several advantages over that of Fussell et al. [9]. First, since the depth buffer is always aligned to the tool orientation, the intersection calculations are greatly simplified since the swept surface is always seen as a plan view. That is the tool workpiece intersection problem has been reduced from the 5-axis case to the 3-axis case. Secondly, the tool contact area is easily determined by the depth buffer in graphics hardware, alleviating the need for user programmed calculations. Third, the methodology remains unchanged regardless of the tool shape (Fussell et al. presented results for a ball nose tool only). A final advantage is that the depth buffer is sized to the swept volume as opposed to the workpiece. By using the swept surface of the tool, the number of renderings required is no longer dependent on the spindle speed, but rather on the number of programmed tool moves. This increases the efficiency of the method dramatically.
3 3.1 3.1.1
5-axis experiments Experimental set-up Apparatus used
An experiment was conducted in 6061-T6 aluminum in order to determine the predictive capabilities of the simulation in full 5-axis machining. Initially a rectangular block of aluminum was roughed into the shape shown in figure 10 using a 3/4” flat end mill. The roughed stock was then machined to a cylindrical shape. The cylindrical shape was chosen because
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10.27 8.31 6.43 4.62 2.88 1.22
50
(a)
(b)
Figure 10: (a) Front and (b) isometric views of the roughed stock material of the full range of 5-axis motion required and due to the long rotations (and great impact on effective feedrate) necessary at the mid-point of the tool path. An inclined tool path (that is, a tool path not following the maximum or minimum curvature of the cylinder) with a slope of 3/4 was chosen to complicate the chip geometry seen by the tool, and is depicted in figure 10b. The tool path cutter location (CLDATA) file was generated based on the principal-axis method (PAM) [19]. During the course of machining, the tilt angle ranged from 2.064◦ ≤ A ≤ 43◦ while the rotary ranged from 92.522◦ ≤ C ≤ 267.175◦ . The tool used in the machining operation was a Widia brand TR360 1/2” toroidal milling cutter (figure 11), at a spindle speed of 6000 [rpm] and nominal feedrate of 600 [mm/min] (0.1 [mm/rev] chip load). Due to the rotating cutting force dynamometer limitations described in section 3.1.4, the tool has only one tooth and so runout was not a factor (although the simulation is quite capable of handling runout, as detailed in [12]). The stock blank measured 100 × 100 × 50 [mm] to begin with. The cylinder radius was chosen to be 50 [mm]. The maximum depth of the cylindrical section was calculated to be 11.7 [mm], based on the limiting angle of the A axis. The forces were measured using a Kistler type 9123C rotating cutting force dynamometer(RCD), connected to a 5223B 5-channel charge amplifier – a table type dynamometer is not useful in tilt-rotary 5-axis
19
Figure 11: Schematic of the Widia brand TR360 milling cutter Ktc [M P a] Krc [M P a] Kte [N/mm] Kre [N/mm]
1531.228 325.927 13.863 5.778
Table 1: Force coefficients obtained for aluminum
machining due to gravitational effects. Data acquisition was accomplished using Labview 5.0 and a National Instruments PCI-MIO-16E-4 analog to digital converter as well as a CB68LP interface card. The data acquisition sampling frequency used was 7200 [Hz] (giving 72 samples per revolution). 3.1.2
Determining force coefficients
Force coefficients were determined using a methodology similar to that described by Altintas [20]. A linear regression of the average measured forces (figure 12) vs. the chip load is computed for several slotting experiments. The cutting constants for this experimental setup are listed in table 1. 3.1.3
Force model and simulation
The mechanistic force model used in this experiment was described in section 2.1. However, as the force transducer used in this experiment is a RCD, the measurement XY reference 20
120
Average force [N]
100 80 60 40 20 0 -20 0
0.02
0.04 0.06 0.08 Chip load [mm/tooth]
0.1
0.12
Figure 12: Averaged measured forces vs. chip load for various slotting tests. ◦ y direction
X direction,
Fy
CRCD y
Fx
Fr
Ft v
x Qfix QRCD
Q
Figure 13: Force components on the tool in rotating coordinate system, CRCD frame, CRCD , rotates with the tool. The predicted simulation forces are converted to the rotating frame using Fx = −Fr cos(Θf ix ) − Ft sin(Θf ix )
(12)
Fy = −Fr sin(Θf ix ) + Ft cos(Θf ix )
(13)
where Θf ix is the constant angle between the x-axis of the rotating frame CRCD and cutting tooth, as depicted in figure 13.
21
3.1.4
RCD limitations and frequency range
The RCD employed in this paper is capable of measuring three components of cutting forces (Fx , Fy , Fz ) as well as the moment about the Z axis (Mz ), measured relative to the CRCD coordinate system. Accurate measurements with the RCD are only possible when the excitation frequency (i.e., the tooth passing frequency) is limited to fn /5, where fn is the measured natural frequency of the RCD while mounted in the machine spindle. Using a single hammer blow, the natural frequency of the system was measured at fn = 622 [Hz] and the damping ratio was estimated at 0.033. Therefore, the chosen 1 tooth cutter, operating at 6000 [rpm] (100 [Hz]) meets this criterion. Furthermore, measurements obtained with the RCD were low-pass filtered, with a cutoff frequency of 200 [Hz], to remove the natural frequency vibration component from the force signals.
3.2
Comparing experimental and simulation results
The CLDATA was post-processed into G-Code commands using the kinematic model of the 5-axis machine and by limiting the maximum CLDATA change to the lesser of a linear distance of 1 [mm] or a rotation of 1◦ . Two simulations were conducted – the first to obtain the complete cutting force profile over a limited length, and the second to obtain the peak forces for an entire tool pass. 3.2.1
Close-up profiles
The pixel density used in the calculation of the close-up profiles was 150 [mm−1 ]. The depth buffer was sized to the tooth swept sector, rather than the tool or stock material, to maximize the number of dexels used in the calculation of the in-process chip geometry and edge contact length. As such, the window size was 34 × 778, using a tooth swept sector angle of θ = 2◦ . The close-up profiles correspond to a time of approximately 9.1 [s] when
22
referenced to the overall time of machining. Figure 14a-b display the raw measured, filtered and simulated x and y-direction force profiles. Figure 14c displays the raw measured, filtered and simulated torque profile. Unfortunately, the transient vibration of the tool dominates the x-direction cutting force signal, as seen in the unfiltered profile of figure 14a. The magnitude of the vibration is most likely due to the reduced cutter stiffness caused by the excessive overhang length of the assembled RCD/tool measurement system. As a result, the x-direction signal is unsuitable for a comparison between simulated and experimental data. However, the x-direction forces represent a small portion of the magnitude of the force on the tool since it was aligned mostly with the radial forces on the tool. While the magnitude of the transient vibration in the y-direction force profile (figure 14b) is about the same as that in the x-direction, its ratio to the cutting forces is much less in this case. The low-pass digital filtering effectively removes this unwanted signal component, allowing a good comparison with simulated results. It can be seen from this figure that the filtered and the simulated results are in very good agreement - within 10 [N ] (or approximately 7%). In this case, the y-direction force is aligned mostly with the tangential force component. Figure 14c displays the measured and simulated torque profiles. Clearly, the amount of transient vibration in the torsional mode is virtually non-existent, eliminating the need for further digital filtering. The agreement between the simulated and measured force profiles is excellent with virtually no distinguishable difference in the profiles. 3.2.2
Overall profiles
In calculating the overall force and torque profiles, the swept surface approach of section 2.3 was used. In this manner, the number of renderings is proportional to the number of tool positions, rather than the spindle speed, leading to a much faster simulation. The results are presented in figure 15. For clarity, only the appropriate maximum or minimum value (per 23
250
80 60
200 40
150 FY(RCD) [N]
FX(RCD) [N]
20 0
100
−20 −40
50
−60 −80
0
−100 −120 0
0.01
0.02
0.03
0.04
−50 0
0.05
0.01
0.02
0.03
0.04
0.05
Time [s]
Time [s]
(a)
(b) 0.7 0.6
MZ(RCD) [Nm]
0.5 0.4 0.3 0.2 0.1 0 −0.1 0
0.01
0.02
0.03
0.04
0.05
Time [s]
(c)
Figure 14: Simulated and experimental (filtered , unfiltered constant programmed feedrate path (a) X, (b) Y, (c) MZ
24
) force profiles for a
0
160
−5
140
−10
120
[N]
−25
60
−30
40
−35
20
−40 −45 0
80
F
F
100
Y(RCD)
−20
X(RCD)
[N]
−15
10
20
30
40 50 Time [s]
60
70
80
0 0
90
10
20
30
(a)
60
70
80
90
60
70
80
90
(b) 200
0.7
180
0.6
160 Feed Rate [mm/min]
0.8
MZ(RCD) [Nm]
0.5 0.4 0.3
140 120 100
0.2
80 60
0.1 0 0
40 50 Time [s]
40
10
20
30
40 50 Time [s]
60
70
80
20 0
90
(c)
10
20
30
40 50 Time [s]
(d)
Figure 15: Simulated and experimental (filtered , unfiltered ) results for constant programmed feedrate path (a) X, (b) Y, (c) MZ, (d) Actual feedrate tool move for the simulation and per tool revolution for the measured data) was reported. For this approach, the pixel density used was 8 [mm−1 ], which resulted in a depth buffer size of (W × H) = 102 × 110. Twenty pseudo-inserts were used with an additional allocation of 6 points per side for degenerate points. The simulation time on a Pentium IV 2.4 [GHz] computer was approximately 13.0 [s], as compared to an actual machining time of 90 [s]. The number of tool positions simulated was 131. In figures 15a-c, peaks resulting from the stepped shaped of the roughed stock material are clearly visible at times of approximately 0-15 [s] and 75-90 [s]. Machining of the bottom
25
of the cylinder, where long rotations of the machine tool are encountered, represent the balance of the time. As expected, the simulation and experimental x-direction forces (figure 15a) are incomparable due to the excessive transient vibration amplitude of the measured signal, as discussed in section 3.2.1. A much butter result is obtained for the overall y-direction and torque, where the agreement between experimental and simulated results is very good. One area of concern in the swept surface simulation results is the time period of 20-70 [s], corresponding to the long rotational moves for machining the cylinder bottom. This brings to light a short coming of the linear mechanistic model. Plotted in figure 16 are the cutting and edge force simulation components corresponding to the first and second terms of equation (1) respectively. It can be seen that the cutting force component in the region of concern is essentially negligible, roughly 1/10th the edge force component. By implication, so too is the chip thickness (which was found to be as low as h = 0.0004 [mm]). The linear mechanistic model predicts the simulation force will be composed almost entirely of plowing forces, even though, at such a small chip load, the tool can hardly be described as plowing into the stock material. In effect, if the chip thickness falls to 0, so too should the total force. Therefore, a corrected piecewise linear mechanistic model is proposed, as shown in figure 17, where, below some threshold chip thickness value, hthresh , the mechanistic model is composed of entirely cutting forces
F = Kcthresh hb;
h 0 nx < 0
Front ±ˆ t × ˆi Back
ny > 0 ny < 0
Side-Left ˆ ±ˆ t×k Side-Right
Criteria
nx ≥ 0 nx < 0
Point on Plane nz ≥ 0 (0, 0, −zs ) nz < 0 (0, 0, 0) nz ≥ 0 (xs , 0, −zs ) nz < 0 (xs , 0, 0) nz ≥ 0 (0, 0, −zs ) nz < 0 (0, 0, 0) nz ≥ 0 (0, ys , −zs ) nz < 0 (0, ys , 0) ny ≥ 0 (0, 0, 0) ny < 0 (0, ys , 0) ny ≥ 0 (xs , 0, 0) ny < 0 (xs , ys , 0)
Table 2: Summary of clipping plane equations and criteria
The other planes are handled analogously and are summarized in table 2.
A.2
Degenerate clip plane equations
In the case of 3-axis machining, calculation of the side-clipping planes will result in ill-defined ˆ That is, ˆ ˆ = ∅, and the resulting clip normals, as the tool normal, ˆ tz , is parallel to k. tz × k plane equation will be 0 = 0. However, because a clip plane will retain any point that results in a value greater than or equal to 0 upon substitution into the plane equation, every point will be retained. Therefore an ill defined plane is not a problem and need not be even checked for, as degenerate clip plane equations are still valid.
B
Applying the methodology to feed scheduling
From equation (1), the total force component, F is comprised of an edge force component, Fe and a cutting force component, Fc . The edge force component is unaffected by the feedrate and depends only on the depth of cut. However, the cutting force varies linearly with the chip thickness, which is directly proportional to the feedrate. From this observation, assuming that the user has specified an acceptable level of force (e.g., from a static deflection analysis) 37
r
) F ct, F cr
Fc = (
l
Fe
=
,F et F (
F
) er
t
FA
Figure 24: Vector diagram used to determine adaptive feedrate the feedrate for any given tool movement can be programmed to maintain this force level. Let the adaptive force level be given by FA . Referring to the figure 24, this problem is akin to determining the point at which the circle of radius FA crosses the vector Fc so that FA = FcA + Fe , where FcA = βFc and β is some positive scalar value. The cutting and edge force vectors are plotted on axes corresponding to the tangential and radial directions. The vector corresponding to the cutting force passes through the origin, and has the equation Frc = ΛFtc , where Λ =
Krc . Ktc
The force threshold circle has its origin at (−Fte , −Fre ). Note
that the ratio of Fre /Fte = Frc /Ftc in general. There only exists an adaptive feed solution if FA > Fe . If this condition is not met, then it is an indication that the tool path is too aggressive and should be recalculated. That is, the cut must be split into two or more passes in order to decrease the depth of cut and reduce the force to a desired level. Stated mathematically, this criteria states that the solution of the intersection problem must be positive. It can be shown that the intersection is solved from FtcA =
−(Fte + ΛFre ) +
FA2 (1 + Λ2 ) − (Fre − ΛFte )2 1 + Λ2
38
(16)
The adaptive feedrate is then set based on modifying the un-cut chip thickness to match the required tangential force value. Note that the slope of the cutting force line is given by Λ, and is typically quite small – on the order of 0.3. The angle of this slope is also quite small meaning the magnitude of the tangential component is the better portion of the full magnitude of the this force vector. Therefore, for a sufficiently small slope, one should be able to estimate the adaptive feedrate based solely on the tangential force, and not on the combined vector. For the tangential component to be 95% of the full magnitude, one can compute the maximum allowable slope Λ according to Ftc = cos(θ) F cos(θ) = 0.95 θ = 18.19◦ Λ = tan θ Λ = 0.32864 Λ ≈ 1/3
(17)
Then, if the adaptive feedrate is to be set solely on the tangential force F − Kte b = Ktc bh
(18)
and a simple linear relationship arises. It can be seen that F − Kte b ∝ h ∝ ft ∝ f , so the adaptive feedrate, fa , is set using F − Kte b fa = Ftc − Kte b f
(19)
To the author’s knowledge, there has not until now been a non-iterative technique for
39
setting the adaptive feedrate presented in the literature.
40
