Generation of engineered surfaces by the surface-shaping system

Int, J. Mach. Tools Manufact. Vol. 35, No. 9. pp. 1269-129(I. 1995 Copyright~) 1995 ElsevierScienceLtd Printed in Great Britain. All rights reserved (1890-6955/9559.50 + .0(1

Pergamon

0890-6955(94)00114-6 GENERATION

OF ENGINEERED SURFACE-SHAPING

SURFACES SYSTEM

BY THE

MIN S. H O N G t and KORNEL F. EHMANNt (Received 15 March 1994; in final form 22 August 1994) Abstract--Engineering surfaces are generated by a variety of manufacturing processes, each of which produces a surface with its own characteristic topography. A method for the prediction of the topography of engineering surfaces has been developed based on models of machine tool kinematics and cutting tool geometry. The model termed the surface-shaping system accounts for not only the nominal or global cutting motions but also takes into account errors during machining such as tool runout, machine deformation and vibration, as well as higher order motions.

INTRODUCTION

The geometric characteristics of engineering surfaces, generated by a variety of manufacturing processes, play a fundamental role in the prediction of the functional performance of machine components. In addition, it has been found that the properties of engineering surfaces, and in particular their roughness, have great effects on lubrication, friction, etc. [1, 2]. Many attempts have been made in the past to determine the topography of engineering surfaces through the characterization of their profiles measured by the stylus method. Stylus profilometer measurement methods are typically used to evaluate a number of parameters of measured profiles, and have become so widespread that they are embodied in many international surface roughness standards. Recently, many attempts have been made to develop numerical simulation methods for manufacturing processes. A model to predict the topography of end milled surfaces has been developed accounting for the presence of cutter runout, end mill deflection, and spindle speed variation [3, 4]. In addition, computer models have been introduced, which take into consideration the geometry and the kinematics of the motions of the milling cutter including a superimposed tertiary spindle motion onto the conventional milling process [5]. The agreement between the experiments and simulations has indicated that tertiary motion cutting and the computer simulation model predicting its outcome are practical and accurate [6]. In addition, numerous computer simulations have been performed to simulate the topography of the ground surface [7], of tooling systems for electrochemical machining (ECM) [8], and of the electric discharge machining (EDM) process for die sinking [9]. Nevertheless, a more powerful and general machining simulator is required as a tool for the evaluation of the accuracy and the verification of the machine's performance for integrating design and manufacturing processes. The objective of this paper is, therefore, to define a generalized analytical model and procedure for the simulation of the surface generation process. The basic idea of the surface generation process is introduced first. Then, the cutting tool geometry is defined. General concepts of cutting tool geometry are presented, which can be applied to the majority of metal removal based surface generating processes. Specifically, the major cutting edge region is defined by a surface-surface intersection algorithm and the cutting angles are obtained along the major cutting edges. Thereafter, by improving the existing method for modeling machine tool kinematics, embodied in the model of the "form-shaping system" (FS) [10], and combining it with the clear understanding

tDepartment of Mechanical Engineering, Northwestern University, Evanston, IL 60208, U.S.A. 1269

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Min S. Hong and K. F. Ehmann

of the cutting tool geometry, a new method to represent surface generating processes, termed the "surface-shaping system" (SS) [11], is introduced. The newly developed method incorporates not only the nominal motion of the machine tool but also the various errors caused by the machine's operation and also utilizes a precise description of the tool geometry. Finally, the feasibility of generating precisely controlled engineering surfaces in single-point cutting is examined by building upon the principles of the proposed surface-shaping system. For simplicity only the nominal motion of a turning operation with superimposed higher order motions is considered in this paper. THE SURFACE GENERATION PROCESS Engineering surfaces are generated by the combination of several motions that not only facilitate the chip-forming processes but also move the point of engagement along the surface. The workpiece may rotate around its axis; the tool is set to cut at a certain depth and receives a continuous, longitudinal feed motion. The geometrical shape of the machined surface depends on the tool geometry and its path during the machining operation. The simulation of the machine tool's kinematics is vitally important in the design of automated control systems, computer-aided design systems, etc. The specific features of the machine's kinematics are dictated by stringent requirements for accuracy, by the complexity of the workpiece and operations performed, by a variety of error sources during machining, as well as by the diversity of the types of machine tools, machine setups, etc. A precise description of the cutting tool geometry is needed for the simulation of the engineeering surface generation processes. The cutting angles at the cutting edge influence not only the cutting conditions, the characteristics of chip formation, and tool wear but also the resultant surface texture. Moreover, the cutting angles influence the static and dynamic cutting forces. Thus, for a precise modeling and simulation of the surface generating process, it is necessary to develop a new method which can accommodate any kind of machining operation performed by any general cutter geometry. In other words, once the user selects a specific machine and the geometry of the cutter and workpiece, the new model of the surface generation process should provide not only a clear relationship between the cutter and workpiece but also define the tool path equations and provide the necessary information about the resultant surface texture. The new system should be suitable for single-point tool machining as well as multi-axis machining with complex cutter geometries. The term "surface-shaping system" will be used in this paper to designate a system model which includes the machine tool's kinematics and the geometry of the cutting tools and workpieces and derives the relationship between the surface-shaping points on the cutting tool and workpieces with the aim to predict the texture of the resulting machined surface. The model of the surface-shaping system will include the following features: (a) nominal motion of the tool with respect to the workpiece; (b) tool runout, spindle error and uneven spacing of inserts in multi-tooth machining; (c) errors caused by the elastic deformation of the mechanical elements of the machine, fixture, tool and workpiece; (d) errors caused by machine tool vibrations; (e) higher order motions used to control the desired surface texture; and (f) clear representation of the complex geometry of the cutting edges including the cutting angles. In order to generate the general tool path equation which defines the relationship between the cutting tool and the workpiece, a suitable set of coordinate systems should be established. Figure 1 schematically shows the standard right-hand Cartesian coordinate systems used to describe the kinematics of the surface generation process:

Generation of Engineered Surfaces by the Surface-shaping System

1271

Zi÷3

{Si~.~.

Zi

/~'0i÷3 Xi+3J'/ /

-

l

---

/

Y~÷6

/

{S,) ,"

Zi4-6

Yi

Actual axis ~ Za, Z ~ Xi÷6~

/ •

or# I~a

Fig. 1. Schematics of the surface-shaping system.

{So} is a reference flame placed on the workpiece; (St}, {$2} . . . . . and {Si} are machine flames with i transformation matrices with respect to flame {So}; {Si+~}, {Si÷2} and {Si+3} represent flames in which all elastic deformations of the machine, tool and workpiece with respect to frame {Si} are defined; {Si+4}, {Si+5} and {Si+6} are flames with errors caused by machine vibrations with respect to frame {S~+3}; ...; {Sn.p.s}, {Sn.p4} . . . . . and {S,.o} tool rotation and translational flames with x, y and z directional tool runout and spindle offsets with respect to frame {Sn.p-6), respectively; {Sn_p} a flame at the center of the tool in its nominal position; and, finally, {S,} the flame at the center of the tool in its actual position with higher order motions (pth order motions in this case). The frame Sk (k = 1, 2 . . . . . i, i+1 . . . . . n-p, ..., n) designates the motion of the kth link of the surface-shaping system relative to the (k1)th link. Usually the zeroth link (k = 0) is the workpiece [10]; n is the total number of transformation matrices. Higher order motions are defined as motions which are arbitrary, realizable functions of time and control the desired surface texture by superimposing additional intended motions of the machine tool. In the absence of higher order motions, flames {S,_p} and {S,} coincide; {STa} is the tool base frame which coincides with flame {S,} and defines the tool surfaces and the cutting edge; {ST} is the tool frame which is set on the cutting edge and defines the cutting angles. The tool base frame {STB} and the tool frame {ST} are used to define the cutting tool's geometry. In the following sections, in order to develop an analytical representation of the surface generation process, general concepts of the cutting tool geometry are introduced and the model of the surface-shaping system is addressed.

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Min S. Hong and K. F. Ehmann GENERAL DEFINITION OF CUTTING TOOL GEOMETRY

In order for a tool to generate an engineering surface, it must possess certain geometric characteristics, which differ for different machining operations. Since the characteristics of a single-point cutting tool are recognizable on every tool, its characteristics will be examined and then generalized to encompass multi-point tools as well.

Cutting edges The most important features of a single-point tool are its cutting edges and the angles between the tool's faces and flanks. The major cutting edge is intended to be responsible for the transient surface on the workpiece. As shown schematically in Fig. 2, the major cutting edge is defined by the intersection of the tool face f a y ) and the tool flank (A~,). Let the tool face and flank surfaces be parametric surfaces l}l(U,W) and PE(S,t) which are defined parametrically in the tool base frame {STB} as PI(u,w) = {

(1)

P2(s,t) =

(2)

x(u,w), y(u,w), z(u,w) } { x(s,t), y(s,t), Z(S,t) }

where u, w, s and t are E [0,1], respectively. The tool base frame {STB} is situated at the point which connects the machine and the cutting tool and coincides with the machine frame {S,}. In the tool base frame {STB), the intersection curve between two surfaces is defined in a general parametric form as g(v) =

(x(v),y(v),z(v) }

(3)

where v is the parameter, v E [0,1] if normalized. Except for some surface types such as quadrics, the surface intersection problem cannot be attacked analytically [12], i.e. there is no closed-form representation for g(v). No matter what kind of approximation is to be used, at first it is always necessary to calculate a string of intersection points that can be defined as Assumed direction of

ZT primary motion: V

v)

X

direction )tion: F {ST

Fig. 2. Definition of angles and surfaces in the tool-in-hand system.

Generation of Engineered Surfaces by the Surface-shaping System gj = g(vj)

1273 (4)

where vj is a fixed value of v in equation (3); gj can be determined by calculating the intersection between a surface-curve on P1 (primary surface) and P2. The solution of the surface-intersection between P1 (u,w) and P2(s,t) can be defined by the vector equation PI(u,w) - Pz(s,t) = 0.

(5)

Mathematically, the solution of the surface-intersection problem involves solving three simultaneous equations, one for each of the Cartesian coordinates or vector components. The application of the Newton-Raphson method [13] to equation (5) leads to [-PY - P~ ~2 Pt~]i {Au, Aw, As, At}iT = Pl(Ui,Wi) - P2(si,ti)

(6)

{ u, w,s, t}i+l = {u, w,s, t}i + {Au, Aw, As, At}i

(7)

and

where subscript i stands for the ith iteration and superscripts u, w, s and t represent the partial derivatives of P1 and Pz with respect to them. The initial values for {u, w, s, t} are decided and the termination condition for the intersection curve is obtained based on Chen and Ozsoy's method [13].

Cutting angles Once the tool-in-hand system [14] is set at a point gj on the major cutting edge g(v), the normal rake angle (~,) and the normal clearance angle (a.) are obtained by defining three planes: (i) the tool reference plane (Pr), (ii) the tool cutting-edge plane (Ps) and (iii) the cutting-edge normal plane (P,). The analytical expressions for these planes and for the normal rake and clearance angles are obtained in the tool frame {ST} as follows. Tool referenceplane (Pr)- The tool reference plane, at point gj on the major cutting edge, is parallel to the tool base or normal to the direction of the primary motion V. The equation of the tool reference plane is given by the scalar product ql" V = 0

(8)

where ql is any point on the tool reference plane. Tool cutting.edge plane (Ps)- The tool cutting-edge plane, at point gj on the major cutting edge, is tangential to the major cutting edge and perpendicular to the tool reference plane or the principal unit normal vector nj. The equation of the tool cuttingedge plane is given by q2 • nj = 0

(9)

where q2 is a generic point on the tool cutting-edge plane. The principal unit normal vector nj at a point gj on the major cutting edge is normal to the major cutting edge and as such must lie in the cutting-edge normal plane. The principal unit normal vector nj can be written as [15] ki

where

1274

Min S. Hong and K. F. Ehmann - - g VVOv

kj =

gj~

j oj2 gy, Ig~'l

(11)

and g~' and g}'V are the tangent vector and second derivative vector of the position vector gj, respectively, i.e. gY _ d g j ( j and g},V_ d2gj(v) J dv dv 2

(12)

Cutting-edge normal plane (Pn). The cutting-edge normal plane, at a point gj on the major cutting edge, is a plane through gj perpendicular to the unit tangent vector tj. The unit tangent vector at point gj on the cutting edge is denoted by

tj = g}' . Ig l

(13)

The equation of the cutting-edge normal plane is given by the scalar product q3-tj = 0

(14)

where q3 is any point on the cutting-edge normal plane. Normal rake angle (3'n). The normal rake angle is the angle between the tool face (Av) and the tool reference plane (Pr) in the cutting-edge normal plane (Pn). The intersection of the tool face and the cutting-edge normal plane at a point gj can be defined as ( a l -- k . n ) .

k.n = 0

(15)

where a~ is a position vector indicating any point on the dual image curve of the intersection in the parameter plane u-w and knn is the normal vector to the cuttingedge normal plane from the reference system {STB) of the tool base. Therefore, the normal rake angle ('Yn) at a point gj can be obtained as ~n = c o s - l ( % " nj)

(16)

where nj is the principal unit normal and "ru is the unit tangent vector at point a I = g j . Normal clearance angle (eto). The normal clearance angle is the angle between the major flank (A,,) and the tool cutting-edge plane (Ps) in the cutting-edge normal plane (Pn). The intersection of the major flank and the cutting-edge normal plane at a point gj can be defined as (a2 - knn) • k.n = 0

(17)

where a 2 is a position vector indicating any point on the dual image curve of the intersection in the parameter plane s-t. Therefore, the normal clearance angle (~tn) at a point gj can be obtained as an = cos-'(r2j • (nj x tj))

(18)

where T2j is the unit tangent vector at point a 2 = g j . In addition, the unit tangent vectors ,r~ and T2 in equations (16) and (18) are obtained as follows

Generation of Engineered Surfaces by the Surface-shaping System

a'l=

dat(v') dr'

da2(v")

and'r2= '-----7dv

1275

(19)

where v' and v" are the new parameters of the intersections.

Cutting tool model If the tool surfaces are considered as continuous functions then based on the above derivations the essential geometric forms of the edge can be defined. They consist of the angles and the coordinates of the cutting edge. From equations (10), (13), (16) and (18) the cutting angles ~g at the given point gi on the cutting edge can be expressed as

= ~g(nj,tj, rlj,r2j)

(20)

where "y. and a . are the normal rake and clearance angles at the given point gj, respectively. Vectors nj,tj, a'lj and T2j are principal unit normal, unit tangent and unit tangent vectors of the intersections at the given point gj between the tool face or the flank and the cutting normal plane, respectively. Let the frame {S,) and the radius vector rn coincide with the tool base frame {STB} and the position vector gj, respectively. Then, the cutting edge can be represented by the position vector r, as m

r. = g(vj) =

.. .e 4 @AILI,,

(21)

where vi is a fixed value of v. The expression for the radius vector r. can be always represented as a product of a number of homogeneous transformation matrices AJi 1.i between the ith and i - l t h links. Vector e 4 represents the origin of the tool frame {ST} and the superscript @ represents the characteristics of the transformation matrices AJ. Equations (20) and (21) jointly represent the model of the cutting tool. For a single-point tool, the position vector r. can be written as r. = gj = [gxj gyj gzj 1 ]T

(22)

where gxj, gyj and gzj are the components of the position vector gj, respectively. For a single- and multi-blade tool, then the position vector r. is a vector function of one independent variable v and the parameter defining one of the blades ci as r. =g(vj,ci)

i = 0, 1 .... , N

(23)

where N is the number of cutting blades. For an abrasive, the position vector r. can be written by a vector parametric equation as

r. = g(vj, Ck)

k = 0, 1 ..... M

(24)

where Ck are the parameters defining the coordinates of a grit of the abrasive and M is the number of parameters.

Ideal discrete form of the cutting tool model For simulation purposes, it is necessary to discretize the cutting edge and to evaluate the cutting angles at the discrete points. A string of intersection points gi between the

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Min S. Hong and K. F. Ehmann

tool face and flank is obtained numerically as shown in Fig. 3. The intersection points gj are determined by the value of the propagation step which depends on a user specified tolerance [13]. Since the intersection points gj obtained by the surface-surface intersection algorithm are not uniformly distributed, it is necessary to reorganize them into a form more suitable for a computer simulation model. For a more efficient and easier implementation, an evenly spaced piecewise continuous string Pk is defined as follows. If it is assumed that the discrete interval of the string gj is infinitesimal, an even or uniform length A can be defined as n--I

h = 1 /~c [gj+l n

gj[

(25)

If go = Po and g, = Pm, then the discrete interval between Pk and Pk-* can be written as [Pk -- Pk--l[ ---- A,

(26)

k = 1, 2, .., m .

Major cutting edge: g(v)

tc

+

P~+2

/P+" Y'&+2 g n - i / g ,

g2 go

ro

~-'---I~

{STB}

X,ra

Fig. 3. Discrete form of cutting edge and cutting angles.

Generation of Engineered Surfaces by the Surface-shaping System

1277

Starting from j = 0 to ] = n and k = 0 to k = m, the point Pk is determined by Pk = g j + l - I

+ Cl(gj+l

-

g j + H ) when A ---Igi+l - gjl,

Pk = gj+l "]- C2(gj+l+l - gj+l) w h e n A > Igj+, -- gjl,

0<_C,<1 0 -< C2 < 1

(27)

where l = 1, 2, .., n-j. With known vectors Pk-1, gj+l-,, gj+l and gj+~+l, one can obtain the constants C~ or (72 by substituting equations (27) into equation (26), respectively, and solving the resulting equation. Then, the new Pk can be calculated from equations (27). After the intersection points Pk are obtained by utilizing equation (20), the discrete expression for the new cutting angles ~p = {~/.k, Otnk}p is obtained in one of two possible ways as follows: (i) by linear interpolation IP__kk_--g_j+l__[ . 'Ynk = ~/nj+l "1- Igj+l+l --

IPk- gj+d

Otnk= Otnj+l Jr Igj+l+l -

-

.

gi+~l~'~/"j+~+l - ~"J+~) ,

-

-

(28)

where { % j + H , '~nj+l} and {etnj+t+l, o%+1} are the normal rake and clearance angles at gj+~+l and gj+l, respectively, or: (ii) by recomputation ~nk = COS--t('r]k " nk)

et.k = COS-l(a'2k • (nk X tk)) (29) where the vectors nk, tk, Xlk and "r2k are the principal unit normal, unit tangent and unit tangent vectors at the new point Pk as defined by equations (10), (13), (16) and (18), respectively.

Actual discrete model of the cutting tool Using the ideal discrete form of the cutting tool, defined by the new points Pk, the radius vector r. of the nominal cutter in equations (22), (23) and (24) can be rewritten in a modified general form. By considering the radius vector r. in frame {S. } and the discrete form of the cutting edge Pk in frame {S-rB}, and letting the tool base frame {S-ra} coincide with the reference frame {Sn} (see Fig. 1), the position of the given points Pk on the cutting edges of the tool can be written as r. = p(Vk, Cm)

(30)

where Vk is a fixed value of v and Cm are the parameters defining the coordinates of Ok, i.e., ci and Ck in equations (23) and (24), respectively. In real machining, geometric inaccuracies such as tool eccentricity, tool inclination, spindle runout, etc. make the tooth trajectories more complex. Further complications are due to the imperfect geometry of the tool and of the insert, often collectively described in terms of insert throw or runout [16]. When insert type cutters are used, runout is more likely due to irregularities in the cutter pockets, insert size, regrinding operations, or the setting of the inserts. If these geometric errors of the tool are included, then the model of the actual cutting tool can be expressed by using the ideal discrete form of the cutting tool Pk as m

rn = PO(Vk, Cl) Jr p.(Vk, Cl) ----~

@Aii_l.i e4

(31)

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Min S. Hong and K. F. Ehmann

where Po and p, indicate the ideal cutting model and geometric errors of the cutting tool, respectively. Equation (31) applies to not only deterministic but also to nondeterministic tools. Similarly, the actual discrete cutting angles ~p can be expressed by ¢~p = (I~p0 "~ (I~l~ = ('~nk, 0tnk}pO 3¢ ( ' ~ k , 0[~k)p¢

(32)

where again ~po and ~m indicate the ideal cutting angles and angular errors of the cutting tool, respectively. The general expression of the surface-shaping points Pk of the tool can now be also written as r, = B-re4

(33)

where B-r indicates the matrix of the homogeneous transformations between the tool base frame {S-rB} and the tool frame {ST}. The matrix B-r can always be represented by a product of m matrices A j as m

ji l , i BT = [1 Ai-

(34)

"

i=l

Thus, the discrete expressions of the cutting edge, equation (33), in combination with the corresponding expression for the cutting angles, equation (20), provide a clear description of the tool geometry and are essential not only for the simulation of the engineering surfaces but also for modeling of the cutting dynamics. In this paper only the information on the cutting edge geometry will be utilized. The connection between the cutting tool geometry and the machine tool kinematics will be discussed next. THE SURFACE-SHAPINGSYSTEM (SS) The analytical equation of the ideal or nominal motion of the surface-shaping system can be written in general form as n

ro(t,Np) = Bo.,(t,Np) rn = ,=IS[ Aiil.i(t,Np)rn

(35)

where Bo., is the transformation matrix between the frames {So} and {S,}; t is time and N o = {Fx(t), Fy(t), Fz(t), cos} defines the nominal machining parameters, i.e. feedrate and spindle speed. The analytical equations of the machine's operation including various errors caused by the kinematic tool runout, elastic deformation and machine vibration, etc. as well as the higher order motions can, in turn, be written, in general form, as

ro(t,Np,Er,Ed,E ...... Bin) = Bo.n(t,Np,Er,Ed,E ...... Hm)r, n

= where

A,_ l.i(t,Np,Er,Ed,E ...... Hm)r.

(36)

: error caused by radial, axial, and angular tool runout : error caused by elastic deformation E~ = {ed, edy, edz} : error caused by machine vibrations E~ = {ew, e~y, e~} Hm = {h~x(t), hmy(t), hmz(t)} : higher order motions.

Er = { r, 13, p,}

Equation (36) can be further expanded to describe more complex situations by accounting for the desired factors. The expansion can be readily achieved by defining

Generation of EngineeredSurfacesby the Surface-shapingSystem

1279

the appropriate new transformation matrices. Thus, the general model of the surfaceshaping system represented by equation (36) can be used to analyze not only simpler models but also to account for more realistic machining situations. In order to explicitly define the surface-shaping system, it is necessary to introduce a specific symbolic expression for the surface-shaping system. The matrices AJ in equations (35) and (36), which represent the specific transformation between two consecutive SS links, need to be identified. The following symbols for transformation matrices are defined:

NAJ transformation matrices of the nominal motion EAJ transformation matrices of the errors caused by the tool runout, the elastic deformation, machine vibration, etc. HAJ transformation matrices of the higher order motions Thus, the relation between the coordinates of the points of the cutting edge in system {S,} and the coordinates of the same points in reference system {So} of the workpiece can be obtained as ro = Bo.. r. = ~

@A[kl,kr.

(37)

where the superscript @ represents the characteristics of the transformation matrices AJ. Since the equations of the tool face and flank and the cutting edge are known, all required properties such as the principal unit normal vector and the unit tangent vector on the cutting edge can be calculated. Furthermore, the normal rake and clearance angles at the given point on the cutting edge are obtained by equation (20) with known quantities. Substituting rn from equation (33) into (37), the model of the surface-shaping system is obtained as ro -----Bo.r BTe4 •

(38)

Thus, equation (38), which is the model of the surface-shaping system, links the coordinates of the surface-shaping points of the tool with the coordinates of the workpiece. Since the surface-shaping system establishes the relationship between the cutting edge and the workpiece, it provides not only the geometric properties but can also be used to estimate the dynamic properties of machining processes. The instantaneous chip cross section can be obtained by the precise description of the cutting tool geometry at different time intervals and used to calculate the applied forces. Once the applied forces at a given time interval are known by the utilization of the machine's structural dynamic characteristics, the instantaneous relative vibrations between the tool and workpiece can be defined. These vibrations can be superimposed on the ideal kinematic motion in order to obtain a more accurate surface model. In the following section, general tool path equations of the turning operation will be derived from equation (38) of the surface-shaping system. Then, practical examples of the turning operation will be addressed. SURFACE GENERATING PROCESSES IN TURNING The surface-shaping system provides a general mathematical model for the prediction of the surface characteristics of metal-removal manufacturing processes. The following examples of the turning process illustrate the use of the surface-shaping system model to simulate different implementations of a turning operation. HTM35:9-F

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Min S. Hong and K. F. Ehmann

Turning is one of the most widely used single-point cutting operations. In the simulation of the turning process, the following parameters, referring to Fig. 4, will be used: Fz tos 0

: component of the feedrate in the Z direction : angular velocity of the spindle in the counterclockwise direction : angle of the rotating spindle, i.e. 0(t) = tost.

Figure 4 shows the traditional turning process with a single-point tool. The radius vector ro in the reference frame {So} is derived from equation (38) as follows

ro

=

Bo,4r4

= NA6(0)NAI(x)NAE(y)NA3(z)r4

(39)

where NA6(0) represents the nominal rotation 0 of the frame {S~} about the Z-axis and NAI(x), NA2(y) and NA3(z) are the nominal translations x, y and z of the frames {$2}, {$3} and {$4} in the X, Y and Z directions with respect to frame {So), respectively. Note that z = Fz't, and that r4 is the radius vector of the surface-shaping points of the tool in frame {$4}. Figure 5(a) shows in detail the cutting tool geometry in a turning process. In the simplest case with a single-point tool, as shown in the figure, the cutting edge along the cutter nose becomes the main engaged cutting edge during surface generation. When it is assumed that the tool face (A~) is a plane and a tool flank (A~,) is a cylindrical surface, then the major cutting edge Pk is defined by a plane-cylinder intersection. Let the cutting edge on the cutter nose be an arc with a radius p. Then, the radius vector r4 of the surface-shaping points Pk on the cutting edge can be written as r4 = NA5(t~)nAl(p)NA2(Pyk)e4

(40)

where NAS(0) is the nominal rotation 0 of the radius vector r4 about the Y-axis and NAI(p) a n d NA2(Pyk) are the nominal translations p and Pyk of the radius vector r 4 in

X1,X2 ~Zo,Z1 X3 COs

Y4 Fig. 4. Surface-shaping system of the turning process.

Generation of EngineeredSurfacesby the Surface-shapingSystem

1281

,YT

XT

X4,

Y 4,YTB

{ST}

p

Cutting t ° ~ section -"-] i

~ ~

~ i

! i

~ /

v'-~ °tnk

/PYk

|

"l li!iiiiiiii!!i!ii7 Y 4 ,YTB Fig. 5. (a) Cutter nose of the single-point tool. (b) Sectional view of the cutter nose.

the direction of the X and Y axes with respect to frame ($4 }, respectively. After some manipulation, the matrix form of the tool path equation of the surface-shaping points l~, in frame (So} of the SS can be expressed as

Yo

xcos0 - y sin0 - Pyk sin0 + p Cos0 cos0 ] l X sin0 + y COS0 + Pyk COS0 + O sin0 cos~ ] F~ • t - p sin0 | 1 J

(41)

In comparison with the general tool geometry, the single-point tool is a special case. Therefore, the discrete form of the cutting edge can be obtained readily with an equivalent increment A:

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Min S. Hong and K. F. Ehmann

A = IPk - Pk--l[

p(At~)

=

(42)

where

¢.

-

,o

n

and ~o and 0n are the entry and exit of the engaged cutting edge and n is the number of cutting edge segments, respectively. Figure 5(b) shows the sectional view of the cutter and its rake and clearance angles at Pk. Once the surface-shaping points Pk are known, then the normal rake and clearance angles are obtained in the Z.r--Y-r plane, i.e. the cutting normal plane P,, as ~/nk = a r c t a n hOykj - Ybk and ank = arctan p - Xbk

p

(43)

Loykl

where Xbk and Ybk are defined in Fig. 5(b). If there exists an error caused by machine vibrations in the Z direction of the frame {$4} such as, for example, a random vibrational cutting motion, equations (39) can be rewritten as r o = Bo,5r 5 = NA6(O)NAI(x)NA2(y)NA3(z)EA3(evz)r

(44)

5

where evz represents the random vibrational motion, and r 4 coincides with r5 in equation (40). Then, equation (41) can be rewritten as

Xo Yo Zo

x cos0 - y sin0 - Pyk sin0 + p cos0 cosO

1

1

x sin0 + y cos0 + Pyk COS0 + p sin0 cosO

(45)

Fz • t + ew - p sinO

For the given working parameters in Table 1, a synthesized surface resulting from the conventional turning process, equation (41), is graphically shown in Fig. 6(a). In Fig. 6(b), a random vibrational motion is superimposed onto the nominal turning process, equation (45), and results in a synthesized surface with a random magnitude of 0.04 mm. Apparently, Figs 6(a) and (b) show totally different surface textures. The Ra and R q (r.m.s.) values of the former are 2.19 Ixm and 2.55 txm and those of the latter are 3.83 ~m and 4.71 ixm, respectively. As a further example a controlled depth cutting process with miniature single point tools can be considered. The simulation of this process can be accomplished by the introduction of higher order motions in the form of, for example, controlled sinusoidal oscillations of the tool. Let the sinusoidal depth cutting motion in the X direction represent the higher order motion, then equations (37) and (38) can be rewritten as

(46)

ro = Bo.srs = NA6(O)NAI(x)NA2(y)NA3(z)HAI(hmx(t))rs

(47)

Table 1. Working parameters of the turning process Feedrate Workpiece rad. Grid matrix

2.0 mm/s 10 mm 60 × 60

Cutting speed Cutter nose rad. Grid dimension

1570 mm/s 0.1 mm 0.01 mm

Generation of Engineered Surfaces by the Surface-shaping System

1283

(a)

"•

0.050 0.025 0

0.050

.~ o.o2s •~

0

0.2 0.2

•

(b)

0.050

0.6

-'~eCtioo ..

0.4"~

%

*'¢"

02

Fig. 6. (a) Conventional simulated turned surface. (b) Presence of random vibrations.

where hmx(t) = G sintot and G and to are the magnitude and the frequency of the sinusoidal motion, respectively, and r5 coincides with r4 in equation (39). Then, equation (40) can be rewritten as

Xo

leo Zo 1

{x + G .sin(tot)} cos0 - y sin0 - Pyk sin0 + p cos0 cos0

I

{x + G.sin(tot)} sin0 + y cos0 + Pyk COS0 + p sin0 cosd/

(48)

Fz" t 1 p sinO

For the working parameters given in Table 2, the simulated surface resulting from Table 2. Working parameters for simulation Feedrate Cutting speed Workpiece radius Grid size Cutting depth Cutter nose radius

5 mm/s 800 mm/s 10 mm 0.01 mm 0.01 mm 0.10 mm

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Min S. Hong and K. F. Ehmann

,/

0.050 0.025 0

0.050 ~ee0.025 0.6

r'~.~;

~

q/v

0.2 0.2 Fig. 7. Simulated surface with sinusoidal motion. the superposition of a sinusoidal motion onto the conventional turning process is graphically shown in Fig. 7. The sinusoidal depth motion results in a synthesized surface with a magnitude of 0.01 ram.

PHYSICALREALIZATIONOF A SURFACETEXTURINGSYSTEM In order to investigate the feasibility of method for the dynamic generation of controlled surface topographies, based on the surface-shaping system, the experimental system shown in Fig. 8 was developed [17]. As mentioned earlier, only the nominal motion of the single-point cutting process with the higher order motions is investigated in this experiment. In spite of its simplicity the experimental system contains a number of key elements of a general surface-shaping system and as such is able to verify the key postulates of the earlier theoretical developments. The essential elements are:

WORKPIECE Higher Order

Motion

l

.v

Amplifier al

Feed Direction

Microcomputer

ControlSystem

~

t---IOscill°sc°p e]

PC

Fig. 8. Schematic diagram of the surface texturing system.

I

Generation of Engineered Surfaces by the Surface-shaping System

1285

(a) a micro-positioning stage (MPS), (b) a conventional turning system with spindle position feedback and (c) a computer control system. In the process, the controlled surface topography is imparted by a single-point cutting tool in conjunction with a suitable tertiary motion performed in the X direction, i.e. normal to the workpiece. In the following subsections, each of these elements will be described briefly and experimental results will be addressed.

Experimental apparatus Micro-positioning device (MPS). A compound micro-positioning device consisting of a parallel springlike stage, a piezoelectric actuator element and a single-point cutter was designed and built. The basic design requirements were dictated by the need for high frequency and high fidelity controlled motions in the 1-50 p,m range. The piezoelectric actuator used was a model P-243.20 made by Physik Instrumente capable of generating 20,000 N of force within a 0-30 I~m range at a maximal operating voltage of -1500 V. The tool used was a Kennametal cutting tool with a triangular insert No. TNMG 331K. The monolithic stage was made of spring steel by means of a wire cut electric discharge machine. The basic design was developed by using geometric construction through a CAD system (SDRC's IDEAS VI) as shown in Fig. 9(a). The MPS uses parallel

(a) MPS Structure from IDEAS VI - CAD system t d

F

~..

o~°~°°°•

........

•°°°.°.

•

...

. ! ! ..............................

(b) Parallel spring Fig. 9. The parallel spring stage.

1286

Min S. Hon8 and K. F. Ehmann

springs proposed by Katoh et a1.[18]. Figure 9(b) shows the essential dimensions of the parallel springs with semi-circular flexure hinges [19]. In the static analysis, a constant displacement 10 p,m was applied at point "a", see Fig. 9(a), resulting in a deformed displacement of 17.3 I~m at point "b" yielding a 1.73 displacement gain. For dynamic analysis the MPS was restrained at the base of the stage and a force was applied at point "a" in the X-axis direction, resulting in a natural frequency of ton = 484.59 Hz. For verifying the FEM and analytical results, both static and dynamic experiments were performed [17]. Turning process with feedback. A rotary incremental optical encoder was mounted to the end of the spindle of a conventional engine lathe. The principal function of the encoder is to provide interrupts to the control computer at equal angular intervals to facilitate a precise control of the surface features around the circumference. Computer control system. To generate the desired surface topography, suitable command sequences need to be generated and executed by the control computer. The hardware used in the developed system consisted of a Motorola 68020 based computer in conjunction with A / D and D / A converters and interrupt circuitry. In addition an IBM PC based system was used for off-line analysis and evaluations. Since the command sequence is a key factor defining the resulting surface, it will be treated in detail in the next section.

Command generation In conventional cutting processes, the surface topography is basically determined by the tool geometry and nominal tool motions [20]. The tertiary motion of the cutting tool [5, 6] introduces an important factor influencing surface topography and allows the generation of controlled patterns. The generation of the desired topography requires, however, the formulation of appropriate commands to the MPS. In this study, singlesinusoidal cutting motion commands by equation (48) will be considered: this command structure generates "valley" type patterns resembling the surface topography of a lasertextured surface. A continuous sinusoidal wave is employed to generate holes on the workpiece in the X direction as shown in Fig. 10. The command equation is [ /2~ri _ hmx(i)=Gi[sint-~ 2)+1]

(49)

where i is the encoder feedback signal increment and Ns is the number of angular increments, i.e. the number of encoder counts per wave. In this type of machining, Gi is given with a constant number G. Thus, the maximum cutting depth is 2G as shown in Fig. 10(a), while the cross section of the machined workpiece is shown in Fig. 10(b). The maximum transverse length of the resultant holes, Pw, is obtained as Pw -~ 2x/P2-(p-2G)2 (P is the radius of the cutter) when the cutting depth is maximum. If pw > fd, the resulting surface patterns will be formed by the intersection of two overlapping sinusoidal cutting paths. If, however, Pw is kept in the range 0 < pw < fo, no overlap between adjacent cutting paths occurs. Figure 10(c) shows the phase shift of the generated patterns on the workpiece surface which depends on the ratio L/p~, i.e.

_/~ sr "q Pl - N~ "

(50)

Alternatively, the relationship between N~ and N~ can be simply expressed in terms of the integer and the remainder portions as

Ur = tCiU~+/Or

(51)

Generation of Engineered Surfaces by the Surface-shaping System

1287

X (b) O',osmcl)n oOheWol'k[~ with the maximum cutting depth

(a) Cuidq motions

Y A

T

Wo~q~em

L(=xD)

+II

O000L O0 00-

Y

oooo -'-I ~1"-

Lz

(c) L/P~ = Integer (( = 0 )

1

Workmce

A

L

FFS+ ] A

(d) L/P~ ,~ Integer ( ~ = co )

Fig. 10. Schematics of surface generation through a simple sinusoidal motion.

where Ki is an integer which is equal to the total number of sinusoidal waves in one revolution, and Kr is the remainder. If ~ is an integer, there will be no phase shift on the surface of the workpiece between texture features (Fig. 10(c)), while if lq is not an integer, the surface topography is generated with a natural phase shift between two adjacent cutting paths in the feed direction. The phase shift, e, can be obtained as

yielding a shift in the circumferential direction, Fig. 10(d), equal to 8, = ~PtE .

(53)

E X P E R I M E N T A L RESULTS

The experiments were performed by the use of the developed MPS in conjunction with the single-sinusoidal cutting motion based on the algorithms in the preceding section. The cutting conditions for these experiments were identical to those of the simulations (Table 2). The workpiece material was brass (73.8 RB). The encoder resolution, Nr, was 4096 pulses per revolution. A surface profilometer was used to collect surface profiles and plane views of the surface were photographed by a SEM (Scanning Electron Microscope).

1288

Min S. Hong and K. F. Ehmann

A

B. "B

A

I

4

(a) Plane view photograph

-'

:

B-B

I

Hole

i

ii

I

i

i

• i

I

' i

I

!

I

I

lOOgm (b) Surface profiles Fig. 11. Surface with sinusoidal holes. By using a continuous sinusoidal motion, a surface topography consisting of valleys can be generated. Figure 11(a) shows a plane view photograph of t. e machined surface topography while Fig. 11(b) shows the profiles in the X - Y and X - Z planes. The surface pattern is generated continuously without phase shift between adjacent patterns. In order to compare the profiles to the theoretically predicted value, six parallel traces in the Z direction were measured. Figure 12 shows that the experimental profiles

Generation of Engineered Surfaces by the Surface-shaping System 0.01 r ~a)

1289

I

;b), '~0

0

DOC

!

~mrnl

(c),

-0.01

~v/: -0.02

0.2

-0.2

0.6

Feed direction{ mm}

(a) Simulated profiles

. ":

i :~i[: ii:,.:: 1:4:i !

-

~...

['.; ~1::.~ .

,

. 'i

!

,

.-:i: !:.:,ii;

i

j

~:!

:::C~, !

'

;

i:;::~;"~!:l"-,-qr-"

(a) .. ,

(b;

!.,: ,_

i¸¸¸:i::1 •

i

i ¸

J

(c)

(d) i r-

::

I i

~

'

I ~ I:;i

"

I

i

I

i

!,, (e)

lO0~m (b)

(D

Experimental profiles

Fig. 12. Profiles of simulated and experimental surfaces.

closely trace those of the simulation results along the Y direction. T h e even spacing b e t w e e n adjacent profiles is p = L/n = 0.05 m m ( L , the longitudinal length of the hole in the Y direction is 0.3 m m and n = 6). T h e a b o v e e x a m p l e relates only to a simple type of surface t o p o g r a p h y . It is possible to obtain m a n y o t h e r types of surfaces m a c h i n e d by different cutting tool g e o m e t r i e s a c c o m p a n i e d by suitable tertiary motions. CONCLUSIONS

B a s e d on the theoretical and e x p e r i m e n t a l results, the following conclusions can be drawn.

1290

Min S. Hong and K. F. Ehmann

(1) T h e p r o p o s e d surface-shaping system model facilitates the evaluation and simulation o f the surface texture o f m a c h i n e d c o m p o n e n t s by accounting for realistic tool g e o m e t r y and machine kinematics conditions. (2) G e n e r a l c o n c e p t s for representing deterministic and non-deterministic cutting tools are presented. Specifically, precise expressions for the m a j o r cutting edge and angles are defined by a s u r f a c e - s u r f a c e intersection algorithm. (3) Since the surface-shaping system establishes actual tool path equations and describes the cutting tool g e o m e t r y precisely, it establishes the foundation for the estimation o f the static and d y n a m i c cutting forces. (4) A series o f single-point cutting tests was c o n d u c t e d to confirm the simulation results and investigate the characteristics of m a c h i n e d surfaces. The experimental results were f o u n d to be consistent with the predictions. REFERENCES [1] N. Patir and H. S. Cheng, An application of average flow model to lubrication between rough sliding surfaces, ASME J. Lubrication Technol. 101,220--230 (1979). [2] W. T. Lai and H. S. Cheng, Computer simulation of elastic rough contacts, ALSE Trans. 22, 184-189 (1985). [3] T. S. Babin, J. M. Lee, J. W. Sutherland and S. G. Kapoor, A model for end milled surface topography, Proc. 13th NAMRC, pp. 362-368 (1985). [4] T. S. Babin, J. W. Sutherland and S. G. Kapoor, On the geometry of end milled surfaces, Proc. 14th NAMRC, pp. 168-176 (1986). [5] S. J. You and K. F. Ehmann, Synthesis and generation of surfaces milled by ball nose end mills under tertiary cutter motion, ASME J. Engng Ind. 113, 17-24 (1991). [6] M. S. Hong and K. J. Ehmann, Practical implementation of tertiary cutter motions for the improvement of 3-D sculptured surface characteristics in milling, Trans. NAMRI of SME 18, 222-229 (1990). [7] K. J. Stout and P. J. Sullivan, The analysis of the three dimensional topography of the grinding process, Ann. CIRP 38(1), 545-548 (1989). [8] G. B. Gorjala, K. P. Rajurkar and V. K. Jain, Computer modeling and graphical simulation of ECM tool design, Trans. NAMRI of SME 19, 121-127 (1991). [9] C. Tricaico, R. Delpretti and D. F. Dauw, Geometrical simulation of the EDM die-sinking process, Ann. CIRP 37(1), 191-196 (1988). [10] D. N. Reshetov and V. T. Portman, Accuracy of Machine Tools. ASME Press, U.S.A. [11] M. S, Hong, Generation, characterization and synthesis of engineering surfaces, Ph.D. thesis, Northwestern University, Evanston, Illinois, U.S.A. (1994). [12] I. D. Faux and M. J. Pratt, Computational Geometry for Design and Manufacture, Chapter 9. Ellis Horwood Ltd, Chichester, U.K. (1979). [13] J. J. Chen and T. M. Ozsoy, Predictor-corrector type of intersection algorithm for C2 parametric surfaces, Comput.-Aided Des. 20(6), 347-352 (1988). [14] G. Boothroyd and W. Knight, Fundamentals of Machining and Machine Tools. Marcel Dekker, U.S.A. (1989). [15] M. Mortenson, Geometric Modeling. John Wiley, New York (1985). [16] E. J. A. Armarego and N. P. Deshpande, Computerized predictive cutting models for forces in endmilling including eccentricity effects, Ann. CIRP 38(1), 45-49 (1989). [17] C. J. Wang, Surface topography control in single-point cutting, M.S. thesis, Northwestern University, Evanston, Illinois, U.S.A. (1993). [18] T. Katoh, N. Tsuda and M. Sawabe, One piece compound parallel spring with reduction flexure levers, Bull. Jpn Soc. Precision Engng 18, 329-334 (1984). [19] Y. Hara, S. Montishi and K. Yashida, A new micro-cutting device with high stiffness and resolution, Ann. CIRP 39(1), 375-378 (1990). [20] G. M. Zhang and S. G. Kapoor, Dynamic generation of machined surfaces II. Construction of surface topography, Trans. ASME J. Engng. Ind. 113, 145-153 (1991).