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Characteristic Shapes and Wavelength Decomposition of Surfaces in Machining
Characteristic Shapes and Wavelength Decomposition of Surfaces in Machining S. M. Pandit, Michigan Technological University/USA
- Submitted by M. C. Shaw (3)
Sumnary: In the past, characterization of machined surfaces by a few parameters, even at the risk of a less comprehensive description, was essential for the common manual shop floor use. In this era of explosive growth of electronic instrumentation and computers, restriction to few numbers is no longer required. Therefore this paper presents a comprehensive characterization of surfaces including detailed quantitative estimate of their wavelength content, random or uncorrelated component and a graphical representation of characteristic shapes. It includes the autocorrelation and autospectrum in a natural way, but its main advantage is that each wavelength component can be related and analyzed with respect to cutting conditions and the physical mechanisms or processes generating the surfaces. Examples of surface profiles generated in Electro Discharge Machining, turning and grinding are used to illustrate the effects of machining parameters on the characteristic shapes. The characterization can be readily derived by the Data Dependent Systens methodology and is particularly relevant and useful in computer aided manufacturing and design.
INTRODUCTION The geometrical characteristics of the topography of machined surfaces form the key link in solving the fundamental problem in manufacturing: Specify and control manufacturing conditions such as speed, feed, depth of cut, grit size and hardness,to ensure the functional performance of the machined surface in friction and wear, electrical resistance, thermal conductance, etc. A crucial requirement in the attack on this problem is a method which will provide a comprehensive mathematical description of more or less random surface topography. Such a description or characterization must readily relate to the physics, mechanics or chemistry of the manufacturing process and the functional behavior of the surface in use. Beginning in the pre-computer era with CLA or RHS values, attempts at characterization of surface profiles as well as cutting spaces of grinding wheels include the height density function [ll, the autocorrelation [21 or its Fourier transform, the Spectrum [,I, and a suitable combination of these 141. At the same time, a comprehensive methodology of modeling and characterization called Data Dependent Systems (DDS) [5,61 based on observed digital data was being developed to include autocorrelation and spectra in a natural way. The DDS approach was applied to obtain a succinct but physically meaningful characterization of the grinding wheel profiles 171, and later on to nodel the grinding process itself as a feedback systen [a]. Such applications of DDS to manufacturing processes and machined surface topography have been summarized in 191, whereas the computation of functionally important parameters has been illustrated in [10,111 and a general methodology for such investigations has been outlined in 1121. These applications have established the feasibility of using DDS to provide a succinct characterization of surfaces which can readily relate to the manufacturing process generating the surface and its functional behavior. Attempts to circumvent the standard DDS methodology by mechanically following the so called AR-modeling have led to discouraging results and unnecessarily pessimistic conclusions [13,141. In particular, low order AR models are claimed to give a poor description of profiles which are unsymmetric or contain high deterministic (periodic) component. In contrast, the models obtained by the standard DDS procedure are fully capable of capturing unsymnetric and/or highly periodic nature of profiles and faithfully reproducing their behavior, as illustrated in [15,161. A survey and assessment of surface typology analysis techniques, primarily based o n empirical ordinate distributions, autocorrelation function and amplitude versus frequency graphs, has been recently presented in a key-note paper [171. Conclusions from this survey have expressed misgivings about the interpretation using a purely statistical approach, based on only a few independent characteristics obtained from an empirical distribution, and stressed the need for a more detailed characterization, including the wavelength content, made possible by digital conputers 1181. Such a detailed characterization with a comprehensive analysis of the wavelength decomposition by DDS has indeed been demonstrated for surface profiles generated in turning [19,201. Thus, in this era of explosive growth of electronic instrumentation and computers, restriction to a few parameters in characterizing machined surfaces is no longer necessary as in the past. This paper shows how the DDS methodology provides a comprehensive char-
Annals of the ClRP Vol. 30/1/1981
acterization of surfaces including detailed quantitative estimates of their wavelength content, random or uncorrelated component and a graphical representation of characteristic shapes. The overall DDS approach to surface topography is outlined in section 1 to show how it naturally includes the autocorrelation and spectrum with much improved estimates. Section 2 briefly explains the DDS modeling procedure as that of extracting the random or white noise component. Section 3 shows that the impulse response or the Green's function of this model yields the characteristic shapes of the surfaces. Wavelength decomposition of the profiles into component mode shapes is discussed in Section 4. Examples of characteristic shapes, wavelength decomposition and their relations to the machining processes are given in Section 5 from Electro Discharge Machining (EDM), turning and grinding. 1.
DATA DEPENDENT SYSTEMS (DDS) APPROACH TO TOPOGRAPHY
Characterization of a surface profile requires two different kinds of information: The first dealing with the overall variation of heights or height distribution, and the second describing the average trend along the line of measurement,which is the statistical dependence or correlation between successive observations. Empirically, the first is represented bv a height frequency histogram and the second by the sample autocorrelation or its Fourier transform-spectrum. However, mere plots of histograms, autocorrelation or spectra, although instructive as a visual aid in distinguishing different profiles, are not very useful in applications. Hence attempts have been made to fit these with mathematical functions, specified by a few parameters which can be estimated by standard statistical methods. The most c o m o n distributions used in statistics are Gbssian or normal for symmetric histograms and Beta for unsymmetric ones, both of which have been proposed for profile characterization. These distributions are specified by only two parameters, which can be optimally estimated directly fron the observed digital data, rather than using the histograms. If no distribution is assumed, conventional empirical Barameters such as Ra, Rp, Rq, Rt, Sk and k can be used. Although computation of these parameters is rather straightforward, due to their essential empirical nature they cannot reflect peculiarities of the processes generating them, and therefore considerably overlap for different machining processes 117,181. This is also true, to a lesser degree, of empirical autocorrelation or spectra. The problem is further conplicated by the fact that estinated autocorrelation or spectra have undesirable statistical properties such as bias, variance and dependence 1211 and hence present a distorted picture of the "true" autocorrelation and spectrum. A better alternative, just like the theoretical dis-
tribution for the empirical histogram, is to assume the autocorrelation function to be of sone reasonable foru, such as exponential or sinusoidal, and estimate the parmeters determining these forms fron the profile data. Such a form of the autocorrelation is equivalent to a model for the profile in the form of differential/difference equation. Conjecturing or guessing the correct shape of the autocorrelation function, or equivalently the form of the nodel, from the profile data or the empirical autocorrelation/ spectra,is virtually impossible in all but very simple first or second order models as seen from 1221.
487
However, the recorded data fully incorporates the information about the model. Therefore a systematic quantitative analysis of the recorded data should directly lead to the correct form of the model, without having to go through empirical autocorrelation or spectra. Such a quantitative analysis is indeed possible by treating the observed data as the response of a system governed linear differential/difference equations with white noise (uncorrelated process with flat spectrum) input. The system, as represented by these equations derived from and dependent upon the data alone, may be called as the "Data Dependent System (DDS). This concept of obtaining system equations from the recorded or observed data alone was advanced in [ 5 1 and later developed with the requisite mathematical foundation in 161. Although the DDS nethodology primarily uses differential equations, characteristic shapes and wavelength decomposition can also be readily obtained from difference equations. This paper is therefore restricted to difference equations, which are easier to model by the computer routines than differential equations.
Once the DDS models are obtained directly fron observed data by computer routines, their theoretical autocorrelation and spectra can be easily plotted with almost trivial computation compared to Fourier transformation. Such plots are much smoother and easily interpretable than those of empirical autocorrelation and spectra [e.g. 19.201. However, these plots are no longer required in the DDS methodology. Characteristic roots and parameters of the m d e l together with its impulse function or the Green's function provide a far more fundamental, useful and readily interpretable characterization. In particular,the Green's function or the impulse response function,in the form of a linear combination of ex-nentials and damped sinusoidals,is intimately related to the solution of differential equations. Hence it can be easily related to the physics, mechanics or chemistry of the manufacturing process generating the surface as well as its functional behavior, as illustrated in [7-11,15,16,19,201. This provides a powerful analytical methodology for control of topography by the manufacturer and for control of functional performance by the user, conceptually outlined in the block diagram shown in Figure 1.
I
DDS MODELING BY RESEARCH ENGINEER OR COMPUTER
I I
,,TOPOGRAPHY
d2X:
+
. . . . + dnik
for large enough n. A complex conjugate pair of Ails represent a damped sinusoidal, whereas a real $i represents an exponential. However, as explained in the last section, estimating n and parameter di, Xi of equation (1) from empirical or sampled autocorrelation is impossible for large n. It is much easier to estimate the parameters of a difference equation model for xt equivalent to (1):
... (l-inB)xt =
(l-).lB) (1-X2B)
.
(1-B1B-j2B2-. .-9,-1B
n-1 )at
(2)
where B is backshift operator defined by Bxt = x ~ - ~ , xt is the profile height and at is the independent series with variance cra2 i.e. discrete white noise with flat spectrum of height oa2/2n. Equation ( 2 ) is called Autoregressive Moving Average model of order n,n-1, denoted by AFWA (n,n-1), when the underlying continuous time system is ignored. If one wants to consider the underlying continuous system governed by a differential equation with characteristic roots iJi, then they are related to discrete i.i by
FUNCTIONAL PER-
CHARACTERISTIC FAAPE (IMPULSE
4 c
U.
at
DDS Modeling as Reduction of Profile to White Noise: Equation ( 4 )
LJiA (3)
where A is the sampling interval, and then model (2) is called USA'l(n,m) [15,231. Although in this paperthe underlying continuous time system is not considered, relation (3) is still useful for interpretation of models . The DDS modeling procedure consists of fitting AFT& (n,n-1) models by standard nonlinear least squares routines until the residual sum of squares of at's cannot be significantly reduced as judged by F-test [6,15,23,24]. In other words,n is increased until the residual at's cannot be further "whitened". DDS modeling is thus the reduction of correlated surface profile to an uncorrelated or white noise process which is the random component. This is schematically represented in Figure 2. The function which accomplishes this whitening or reduction to white noise is the Green's function discussed in the next section. 3.
GREEN'S FUNCTION-CHARACTERISTIC SHAPES
Assuming that hi's are distant and moving the left hand side operator in euuation (2) to the right, the m d e l becomes ,n-1 (1 - 91B - S2S2en-lu ) x+ = at (1 ilB) ( 1 X2B) (1 - inB)
-
-
......
(4)
'
where the Green's function G. can be shown by applying partial fractions to be 16, 20,231 G 1. = glAi
+ g2hi +...+
(5)
g n A n1
i#k k = 1,2,
488
-
Fig. 1: Schematic of DDS Approach to Control of Topography
>.. = e
Although the DDS modeling procedure is based on two theorems called Fundamental Theoren and Uniform Sanpling Theorem i6.231, its rationale can also be interpreted in terms of the autocorrelation or autocovariance function, commonly used in surface characterization. If obvious nonstationary trends such as form error are removed, the autocorrelation/autocovariance between two points on a surface profile sufficiently far apart tends to zero. Therefore the (theoretical, not sampled) autocovariance can always be represented by a sum of large enough numbers of exponentials and decaying sine waves. Thus the covariance y k between - ~be expressed by two points xt and x ~ may
+
*
I-----J
DDS MODELING-REDUCTION TO !mITE NOISE
yk = dlA!
BY I
!V+NUFACl'URER
Figure 2:
2.
-
4 - CONTROTOF;-
...n
Equation (4) shows that the AIUlA (n,n-l) model (2) essentially represents the profile xt as convolution of white noise at with the kernel G. The only parameter needed to specify the zero "mean white noise process is its variance aa2, which, together with the Green's function G . completely characterizes the surface profile x Although autocorrelation and spectrum can be rehdily obtained from equation (2), they contain no more information than G. and oa2. In fact comparison of equations (1) and (5? shows that their graphical forms are similar, so that even visually yk does not add anything t o G.. 1 On the other hand, whereas the autocorrelation function is merely a statistical concept devoid of any physical interpretation, G. as an impulse response function is amenable to a far'more fundamental interpretation and analysis. A s Figure 2 shows, a 's are a series of uncorrelated random shocks or imphses. To each such impulse, the system responds following Gi, in the form of Gjatej, and the profile xt is a sum of all these responses, as equation (4) states.
.
In processes such as ED!! and grinding, where the random a 's can be clearly identified as random discharges or rhndom grain positions, the Green's function (or parts of it) can even be interpreted as a typical or characteristic crater and grain. In processes such as turning and milling the components of the Green's function are the results of inherent periodicities interacting with various process elements. Therefore the Green's function may be truly said to represent a "characteristic shape" of the random profile, and together with a 's specified by a single parameter ua2, provides an e&haustive and physically meaningful characterization. 4.
MODAL ANALYSIS-IIAVELENGTH DECONPOSITION
Equations (5-6) provide a component by component wavelength decomposition of the profile. The characteristic roots hi, via relation (3),provide the natural frequency or the central wavelength of each component and the damping ratio indicating the extent to which neighboring wavelengths are present 119,201. The coefficients g . in equation (5) give the amplitude and phase of eack component. This is really the "modal analysis" familiar in structural vibrations,except that in the present case it applies to random vibration; indeed the surface profile can be considered as a random vibration signal. It is also similar to Fourier analysis. But, instead of decomposing into sinusoidals like Fourier analysis, it decomposes the Green's function into exponentials and damped sinusoidals, with possibly different danping for each component. It may thus be thought of as a generalized Laplane transform of the random profile 161. Rather than using the amplitude and phase given by gi's in equation (5-6), one may want to estimate "modal power", in analogy with modal analysis and Fourier transform, to get an idea of the relative contribution of each component of the characteristic shape, which can then be related easily to cutting conditions and other related information. This is readily obtained from equation (1) since the total power is the variance of the profile [6,19,20,23,241 yo = dl +d2
L
i,j=1
EXAWLES
5.1
Electro-Discharge Machining (EDM)
The first order model given by equation (2) with n=l, A1 = +1, has been found to be adequate for many E m profiles, and even when not strictly adequate, has been found to provide good enough approximation 125. 261. The Green's function of this model is an exponential : G. = 3
3
1
1
(9)
which represents the characteristic shape. However, since this would be strongly dependent on the sampling interval :, it is easier to deal with continuous time Green's function G(t) = e-PLot
(9a)
where so is obtained by equation (3): (10)
and the intensity of continuous time white noise:
Since e-4'5 = 0.0111, the exponential Green's function G . or G(t) is practically zero at t = jA = 4.5. Revalving this exponential around a vertical axis at gives a "characteristic cra er" I261 of Cit = 4.5/a aneter toodepth ratio 9/cio an6 depthuz2, fron which the volume and erosion rate can be easily found. Dimensions of characteristic craters obtained from EDMed AISID2 tool steel surface profiles by DDS in [271 are reproduced in Table 1 and the correspondinc: plots are shown in Figure 3. Technological paraneters such as erosion rates and RMS obtained from these are quite close to the experimental values I271 and the agreement can be further inproved by using higher order models. As expected, the craters become wider and deeper and their volume increases with increase in current and/or pulse duration. CHAXACTERISTIC CIiATER
+...+dn n
where d2. =
5.
The DDS modeling procedure provides a large number of models starting from the simple first order to a usually high order statistically adequate model and beyond. Although the lowest order statistically adequate model is the optimal choice based on data alone, many of the lower order models can be chosen as reasonable approximations for specific analysis purposes. Even with high order adequate models, one can choose only one or two modes (real roots or pairs of complex complex conjugate roots) contributing most to the power, as given by equation (7). The first example of Em1 surfaces illustrates the use of a simple first order approximate model, the second example o f turned surfaces illustrates the analysis o f a high order adequate model and the third example of the grinding wheel and ground surface illustrates the use of only two main modes from a high order adequate model in studying the wheel workpiece interaction.
(7)
gigj l-A.A,
, a = 1,2,
...,n
A j
WORKPIECE (STEEL) Current A
Note that di 1s the power due to a real root b i , whereas d i + di+l 1s the power due to a complex conjugate pair of roots A i , Xi+l. Thus the plot of components and the sum G. given by equation (5) yields a visual picture of tie characteristic shapes and their component, providing a qualitative characterization. Tabulated values of characteristic roots Xi, natural frequencies (wavelengths) and damping ratios found from roots pi of equation (31, and the relative contribution of each conponent as a % of the total power y , provide a complete quantitative characterization 09 the wavelength decomposition. These are illustrated in Figures 3-6 and Tables 1 - 3 discussed in connection with specific examples in the next section.
10
TOOL (COPPER)
Pulse Duration
us 40 *0° 400
1000
3.29 2.59 2.32 2.25
17.22 29.60 32.45 37.11
0.0195 0.1597 0.2640 0.4200
40
25
2oo 400
1000
Table 1.
EDM CHARACTERISTIC CRATER DIME~!SIOXS
489
50um
I
CURRENT: 25A WORKPIECE (STEEL)
CURRENT: 10A WORKPIECE,(STEEL)
v
40us
K--; J
1
I
I
-I--
--
--
200lls
L2
4001.1s
looops I I
I TOOL (COPPER) I
w
40us
I
x-:--J I
zoous
I
I
I
c.7
400ps
I
F i g u r e 3:
C h a r a c t e r i s t i c Craters i n EDM
Table 2 WAVELENGTH DECOXPOSITION OF ARMA ( 1 4 , 1 2 1 HODEL FOR FEED 0.122iTm/REV
#
NATURAL FREQUENCY RCOTS( X.) REAL I h G . (CYC/INII) (HZ)
WAVE DAMPLENGTH I N G POWER hm) RATIO ( 8 )
1
.925
.OOO
1.40
.42
.7146
2
.878
-.434 - 4 34
8.20
2.44
.1219
3
.454
.OOO
14.10
4.19
.0709
-
7.49
4
.571
.787 -.787
16.85
5.01
.0593
.0295
5
.132
-.727
.727
25.42
7.55
.0393
.2121
...................................................... -
1
-980
-..196 196
3.52
1.05
-2843
.0046
17.47
2
.527
-.1 7 5
.175
11.97
3.56
.0835
.8785
12.41
5.53
3
.914
75 -.. 3375
6.95
2.07
.1438
.0301
10.10
2.27
4 -.412
-.7 7 1
.771
36.88
10.96
.0271
.0650
-.07
5 -.433
.837 -.a37
36.58
10.87
.0273
.0290
.06
6
.625
-..750 750
15.65
4.65
.0639
.0271
-06
7
.336
-.894
.894
21.64
6.43
.0462
.0381
-.02
8 -.006
.962 -.962
28.17
8.37
-0355
.0245
-.01
9 -.a09
32 -.. 5532
45.70
13.58
.0219
.0124
-.oo
.170
52.89
15.72
.0189
.0171
42.56
.0447 4 0 . 8 4
6 -.295
.OOO
21.77
6.47
.0459
-
1.54
7 -.336
.388 -.388
42.48
12.62
.0235
.2835
-.50
8 -.465
.a85 885
36.69
10.90
0.273
-0003
.OOO
2.00
.59
.5304
-
9 -.894
-.
.28
-.01
What i s more i m s o r t a n t i s t h a t a t h e r m a l n o d e l for t h e ED11 process h a s b e e n d e v e l o p e d o n t h e basis o f c h a r a c t e r i s t i c crater. T h i s m o d e l i s m a t h e m a t i c a l l y simpler a n d y e t p r o v i d e s better p r e d i c t i o n s t h a n t h o s e a v a i l able i n t h e l i t e r a t u r e , e v e n i n t h e case of c e m e n t e d c a r b i d e s w h i c h are d i f f i c u l t to nodel a n d p r e d i c t 1281. 5.2
Turning
S t a t i s t i c a l l y adequate AE4A models of order ( 1 2 , l l ) to ( 2 0 , 1 9 ) h a v e been used t o e l u c i d a t e e f f e c t s o f c u t t i n g c o n d i t i o n s on wavelength decomposition o f s u r f a c e s g e n e r a t e d i n t u r n i n g [201. For i l l u s t r a t i v e purposes, w a v e l e n g t h d e c o m p o s i t i o n of two s u r f a c e s g e n e r a t e d a t f e e d s 0 . 1 2 2 m/rev. a n d 0 . 2 8 8 mm/rev., o t h e r c o n d i t i o n s b e i n g p r a c t i c a l l y t h e same, are g i v e n i n T a b l e s 2 a n d 3
10 - . 9 3 5
-.1 7 0
.oo
and t h e corresponding c h a r a c t e r i s t i c shapes w i t h t h e i r component6 are p l o t t e d i n F i g u r e s 4 a n d 5 , f o l l o w i n g e q u a t i o n s (4-7). For t h e s a k e o f c l a r i t y o n l y f i v e h i g h e s t c o n t r i b u t i n g components are plotted, a l t h o u g h Since t h e r e is t h e tables l i s t a l l the c o m p o n e n t s . too much i n f o r m a t i o n ' i n t h e t a b l e s , o n e can make c h a r t s of o n l y f r e q u e n c i e s o r d a m p i n g r a t i o s etc. t o see t h e v a r i a t i o n w i t h c u t t i n g c o n d i t i o n s a t a glance [ c . f . 201.
C o n s i d e r i n g t h e w a v e l e n g t h d e c o m p o s i t i o n T a b l e s 2 and 3, i t i s s e e n t h a t t h e f e e d w a v e l e n g t h s o f 0.122 and 0.288 a r e q u i t e p r e c i s e l y r e p r o d u c e d i n the root g2 and 1 r e s p e c t i v e l y w i t h c o n t r i b u t i o n s 40.8% and 77.5%. Near h a r m o n i c s o f t h e f e e d f r e q u e n c y a r e p r e s e n t i n root l ' s 4 and 5 i n T a b l e 1 and rwt 1 3 i n T a b l e 2 . 1.5
Note t h a t a l t h o u g h t h e tool n o s e g e n e r a t e s a p e r i o d i c s u r f a c e , it i s n o t s i n u s o i d a l and t h e r e f o r e h a r n o n i c s o f the f u n d a m e n t a l f e e d f r e q u e n c y s h o u l d b e p r e s e n t , a s seen from F o u r i e r series o f p e r i o d i c s i g n a l s . The f e e d and i t s h a r m o n i c s a c c o u n t f o r a b o u t 49% o f t h e power i n T a b l e 2 a t f e e d 0.122 m / r e v . a s a g a i n s t 88% i n T a b l e 3 a t f e e d 0.288 m / r e v . Increasing contrib u t i o n of f e e d wavelength w i t h i n c r e a s i n g feed is s e l f e v i d e n t from t h e dynamics o f t u r n i n g . T h i s i s a l s o q u a l i t a t i v e l y r e f l e c t e d i n t h e c h a r a c t e r i s t i c shape p l o t s i n F i g u r e s 4 and 5.
1.0
Host of t h e r e m a i n i n g c o n t r i b u t i o n comes from t h e real root # 1 i n T a b l e 2 w i t h e x p o n e n t i a l r e s p o n s e and t h e complex root 32 i n T a b l e 3 w i t h s u c h h i g h damping rat i o t h a t t h e response is p r a c t i c a l l y exponential, a s a l s o s e e n from t h e p l o t s i n F i g u r e s 4 and 5. These roots c a n b e r e l a t e d t o t o o l w o r k - p i e c e v i b r a t i o n s , w h e r e a s some o f t h e micrometer w a v e l e n g t h s w i t h n e g l i g i b l e c o n t r i b u t i o n s i n T a b l e s 2 and 3 seem t o p r o v i d e a n i n d e p e n d e n t estimate of c h i p s u r f a c e l a m e l l a r w i d t h s i n micro-morphology [ 2 0 ] .
0.5
5.3
0.0
Surface Grindinq
A s i m i l a r i n v e s t i g a t i o n h a s been c a r r i e d o u t f o r s u r -
0
0.05
0.15
0.10
0.20
0.25
j (m) F i g u r e 4:
C h a r a c t e r i s t i c Shape ( G r e e n ' s F u n c t i o n ) a n d its Components a t Feed 0.122 mm/rev. i n T u r n i n g
f a c e g r i n d i n g u s i n g w h e e l and g r o u n d s u r f a c e p r o f i l e s a s t h e c u t p r o g r e s s e s [291. Two p r i m a r y c o n t r i b u t o r s w i t h t h e l a r g e s t % c o n t r i b u t i o n were i s o l a t e d from For an a b r a s i v e t a b l e s s i m i l a r t o T a b l e s 2 and 3. wheel AA46HSV40, t h e s e w a v e l e n g t h s r a n g e d from 0.54mm t o 0 . 6 5 m and 0.053mm t o 0.14mm, w i t h t h e c o n t r i b u t i o n from t h e s m a l l e r w a v e l e n g t h o f t h e o r d e r of 1%. F o r 46 g r a i n s i z e , d i a m e t e r o f a r o u g h l y s p h e r i c a l g r a i n i s a b o u t 0.38 nun, t h e r e f o r e t h e larger w a v e l e n g t h can b e r e l a t e d t o t h e g r a i n s randomly s p a c e d w i t h bond material. Then t h e s m a l l e r w a v e l e n g t h c a n b e t a k e n a s a r i s i n g from t h e c u t t i n g e d g e s . R a t h e r h i g h damping r a t i o s o f a b o u t 0.25 t o 0.46 f o r t h e l a r g e r and 0.40 t o 0.78 f o r t h e s m a l l e r i n d i c a t e t h a t t h e p e r i o d i c i t y is v e r y weak and t h e n a t u r e o f t h e c h a r a c t e r i s t i c T h i s is a l s o c o n f i r m e d shapes is s t r o n g l y exponential. by t h e p l o t of a t y p i c a l " c h a r a c t e r i s t i c g r a i n " shown i n F i g u r e 6. S i m i l a r a n a l y s i s of t h e g r o u n d s u r f a c e p r o f i l e s i n d i c a t e s t h a t t h e p r i m a r y w a v e l e n g t h , w i t h a l m o s t 100% c o n t r i b u t i o n , i s of t h e same o r d e r o f m a g n i t u d e a s t h e s m a l l e r wavelength i n t h e wheel. These wavelengths a r e reproduced i n T a b l e 4 f o r an experiment. S i nce t h e c u t t i n g edges of t h e g r a i n s w i l l imprint t h e i r marks on the s u r f a c e , t h e s e w a v e l e n g t h s are u n d e r s t a n d a b l e . However, t h e w a v e l e n g t h s on t h e w o r k p i e c e a r e c o n s i s t e n t l y s m a l l e r t h a n t h o s e on t h e w h e e l . Assumi n g t h i s d i f f e r e n c e t o be due t o t h e e l a s t i c d e f l e c t i o n , t h e l a s t column o f T a b l e 4 g i v e s t h e e l a s t i c d e f l e c t i o n c o e f f i c i e n t computed from t h e s e w a v e l e n g t h s . V a l u e s of these c o e f f i c i e n t s agree w e l l with t h e experimental and t h e o r e t i c a l i n v e s t i g a t i o n s a v a i l a b l e i n t h e l i t e r a t u r e , as d o the e s t i m a t e s o f p r o g r e s s o f wheel wear made from t h e l a r g e r w a v e l e n g t h on t h e wheel 1291.
1.
1.
0.
WHEEL-WORK
TABLE 4 DEFLECTION FROM WAVELENGTHS
0. BEFORE AND AFTER CUT
WAVELENGTH SURFACE PROFILE I?s
(mm)
WAVELENGTH WHEEL
PROFILE "Iw
ELASTIC DEFLECTION COEF. PIw- WS
(mm)
% wW
-0.
--I 0
, , ,, 0.05
, ,, ,
, , , ,
0.10
0.15
, , ,
,
0.20
.,
,
,
1.
0.0474
0.0525
9.97
2.
0.0836
0.0938
10.8
3.
0.0935
0.1097
14.73
4.
0.1203
0.1417
15.10
0.25
j (m)
F i g u r e 5:
C h a r a c t e r i s t i c Shape ( G r e e n ' s F u n c t i o n ) and i t s Components a t Feed 0.288mm/rev. i n Turning
49 1
I111
P a n d i t , S. !l.,N a s s i r p o u r F . , and Uu, S. !I., " s t o c h a s t i c Geometry o f A n i s o t r o p i c Randon S u r f a c e s w i t h A p p l i c a t i o n t o Coated A b r a s i v e s , " T r a n s . ASHE, J o u r . o f E n g i n e e r i n g f o r I n d u s t r y , voi. W B , ~ 9 7 3 p?. , 218-224.
2
[121
P a n d i t , S . :!., " Q u a n t i t a t i v e E v a l u a t i o n of A b r a s i v e T o o l and 'Iachined S u r f a c e Topography," PTOC. 1977 Symposium o n New Developments i n T o o l I ' l a t e r i a l s and A p p l i c a t i o n s , IIT, C h i c a a o , 1 9 7 7 , pp. 107-113.
1.5
[131
S t a u f e r t , G . , " C h a r a c t e r i z a t i o n of Random Roughn e s s P r o f i l e s - A Comparison o f AR-Ilodeling Techn i q u e s and P r o f i l e D e s c r i p t i o n by :leans o f C o m o n l y Used P a r a m e t e r s , " A n n a l s o f C I R P , 28/1, ( 1 9 7 9 ) , pp. 431-435.
1141
D e V r i e s , V . R.,
[151
P a n d i t , S . .pi., " S t o c h a s t i c L i n e a r i z a t i o n by Data Dependent S y s t e m s , " T r a n s . ASIlE, J. D n. S st. :leas. and C o n t r-o l , V
1
0.51
"Autoregressive Tine S e r i e s !lodels f o r S u r f a c e P r o f i l e C h a r a c t e r i z a t i o n , " A n n a l s o f CIRP, 28/1, (19791. pp. 437-440.
2 2 6 .
[16]
P a n d i t , S . I!., and K. P. R a j u r k a r , "Data Depend e n t S y s t e m s Approach t o S o l a r Energy Simulat i o n InDUts." PrOC. o f AS!€E/SSEA ence , ~ c .o . -n f e r ~ Reno, Nbvada, A p r i l 1 9 8 1 , T r a n s . ASHE, J o u r n a l of S o l a r Energy E n g i n e e r i n g , 1981. ~
-0.5
i;/' 0.2
0
0.4
P e r e r s , J . , Vanherck, P . , S a s t r o d i n o t o , M., Assessment of S u r f a c e Typology A n a l y s i s T e c h n i q u e s , : A n n a l s o f CIRP, 28/2, ( 1 9 7 9 ) , pp. 539-554.
[18]
P e t e r s , J . , Vanherck, P., S a s t r o d i n o t o , Evaluation of Surface Describing Parameters," :lorth Amer. ! l a n u f a c t u r i n g R e s e a r c h C o n f e r e n c e (1980) , pp. 352-357.
[191
C h a r a c t e r i s t i c Shape ( G r e e n ' s F u n c t i o n ) and i t s Components f o r AA46H8V40 Wheel i n S u r f a c e G r i n d i n g
11
P e s a n t c , H., " D e t e r r i n a t i o n o f S u r f a c e Zoughness Typoloay by !leano o f m p l i t u d e D e n s i t y C u r v e s , " A n n a l s of CIRP, 1 2 , ( 1 9 6 4 ) , pp. 61-68.
[ 21
P e k l e n i k , J . , " I n v e s t i g a t i o n of. S u r f a c e Typolo g y , A n n a l s o f C I W , 1 5 , ( 1 9 6 7 1 , pp. 381-385.
[ 31
B e r , A. and S. Braun, " S p e c t r a l A n a l y s i s o f S u r f a c e F i n i s h , " Annals o f C I R P , 1 6 , ( 1 9 6 8 ) , pp. 53-59.
I 41
W h i t e h o u s e , D. J . , "Typology o f l l a n u f a c t u r e d S u r f a c e s , " A n n a l s o f CIRP, 1 9 , ( 1 9 7 1 ) . pp. 417431.
51
P a n d i t , S. ! l . , "A B a y e s i a n Approach t o Time S e r ies A n a l y s i s i n A d a p t i v e C o n t r o l , " !l.S. T h e s i s , The P e n n s y l v a n i a S t a t e U n i v e r s i t y , U n i v e r s i t y Park, (1970).
[ 61
P a n d i t , S . M., "Data Dependent Systems: !lodeli n g A n a ly sis and Optimal C o n tr o l v i a Ti ne S e r i e s , " Ph.D. T h e s i s , U n i v e r s i t y o f WisconsinMadison, ( 1 9 7 3 ) .
I 71
P a n d i t , S. 3. and S . !I. Wu, " C h a r a c t e r i z a t i o n o f A b r a s i v e Tools by C o n t i n u o u s Time S e r i e s , " T r a n s . ASIE, S . of Eng. f o r I n d u s t r y , 95B, ( 1 9 7 3 1 , pp. 821-826.
[ 81
P a n d i t , S. 11. and S. Revach, "A Data Dependent S y s t e m s Approach t o Dynamics of S u r f a c e Generat i o n i n T u r n i n g , " ASilE P a p e r No. 80-V!A/PROD-25, T r a n s . ASME, J o u r n a l o f E n g i n e e r i n g f o r I n d u s t r y , V o l 103B. 1981.
211
J e n k i n s , G. ?1. and D. G. W a t t s , S p e c t r a l A n a l y s i s and I t s A p p l i c a t i o n s , Kolden-Day, San F r a n c i s c o , 1968.
221
Box, G. E . P . , and G. 11. J e n k i n s , T i m e S e r i e s Analysis: F o r e c a s t i n g and C o n t r o l , HoldenDay, S a n f r a n s i s c o , 1970.
[231
P a n d i t , S. tl., " D a t a Dependent S y s t e m s and Exp o n e n t i a l Smoothing," i n A n a l y s i n g T i m e S e r i e s , Ed. 0. D. Anderson, N o r t h H o l l a n d , 1 9 8 0 , pp. 217-238.
[24]
P a n d i t , S. 11.. a n d 9. H. r . 7 ~ . Time S e r i e s and System A n a l y s i s , F o r c h c o m i n g book, t o b e published.
[25]
P a n d i t , S. M . , and K. P. R a j u r k a r , " i i a t h e n a t i c a l k!odel f o r E l e c t r o - D i s c h a r g e :lachined S u r f a c e Roughness," P r o c . o f t h e N o r t h Aner. l l e t a l Working R e s e a r c h C o n f e r e n c e , V I , 1078, pp. 339-34 5.
[261
P a n d i t , S. tl., and K . P. R a j u r k a r , " C r a t e r Geom e t r y a n d V o l u m e from E l e c t r o - D i s c h a r g e Mac h i n e d S u r f a c e P r o f i l e s by Data Dependent S y s t e m s , " T r a n s . ASME, J o u r n a l o f E n g i n e e r i n q f o r I n d u s t r y , V o l . 1028, 1980 I PP. 289-295.
P a n d i t , S . ! 1 . , S u r a t k a r P. T . , and Wu, S. I;., " l l a t h e m a t i c a l :lode1 o f a Ground S u r f a c e Prof i l e w i t h t h e G r i n d i n g P r o c e s s a s a Feedback 39, (19761, pp. 205-217. System,"
1271
P a n d i t , S . 51., and K. P. R a j u r k a r , " D a t a Depend e n t S y s t e m s Approach to EDI.1 P r o c e s s Modeling from S u r f a c e Roughness P r o f i l e s , " A n n a l s o f C I R p , V o l . 29/1, 1 9 6 0 , pp. 107-112.
P a n d i t , S. i-I. and S. M. Wu, "Data Dependent to Xanufacturina S y s t e m s : E New AuDroach -System A n a l y s i s , " P r o c . I n t ' l Conf. i n PGod. P a r t I , Tokyo, J a p a n , 119741, pp. 82-87.
[281
P a n d i t , S. M . , and K. P. R a j u r k a r , " A n a l y s i s of E l e c t r o - D i s c h a r g e r4achining of Cemented Carb i d e s , " A n n a l s of C I R P , Vol. 30/1, 1981.
[23]
P a n d i t , S. M . , and G. S a t h y a n a r a y a n a n , " A N e w Approach t o t h e A n a l y s i s o f V h e e l - l l o r k p i e c e I n t e r a c t i o n i n S u r f a c e G r i n d i n g , " Elorth Amer. M a n u f a c t u r i n g R e s e a r c h C o n f e r e n c e , I X , 1981.
s,
I 91
a.,
P a n d i t , S . i d . , and W , S. : I . , A S t o c h a s t i c Approach t o Node o f D e f o r m a t i o n
[lo] S u z a t h a r , P. T . ,
a n d C o n t a c t Betvieen Rough S u r f a c e s , " ( 1 9 7 6 ) , pp. 239-250.
492
P a n d i t , S . Y . , and S. Revach, "T'avelenqth Decomp o s i t i o n of S u r f a c e Roughness i n T u r n i n g ," P r o c . N o r t h Amer. M a n u f a c t u r i n g R e s e a r c h Conf e r e n c e V I I I , ( 1 9 8 0 ) , pp. 358-365.
201 EFERENCES [
!!.,
=,
0.6
j (m) F i g u r e 6:
[171
T
s
,
39,
- Submitted by M. C. Shaw (3)
Sumnary: In the past, characterization of machined surfaces by a few parameters, even at the risk of a less comprehensive description, was essential for the common manual shop floor use. In this era of explosive growth of electronic instrumentation and computers, restriction to few numbers is no longer required. Therefore this paper presents a comprehensive characterization of surfaces including detailed quantitative estimate of their wavelength content, random or uncorrelated component and a graphical representation of characteristic shapes. It includes the autocorrelation and autospectrum in a natural way, but its main advantage is that each wavelength component can be related and analyzed with respect to cutting conditions and the physical mechanisms or processes generating the surfaces. Examples of surface profiles generated in Electro Discharge Machining, turning and grinding are used to illustrate the effects of machining parameters on the characteristic shapes. The characterization can be readily derived by the Data Dependent Systens methodology and is particularly relevant and useful in computer aided manufacturing and design.
INTRODUCTION The geometrical characteristics of the topography of machined surfaces form the key link in solving the fundamental problem in manufacturing: Specify and control manufacturing conditions such as speed, feed, depth of cut, grit size and hardness,to ensure the functional performance of the machined surface in friction and wear, electrical resistance, thermal conductance, etc. A crucial requirement in the attack on this problem is a method which will provide a comprehensive mathematical description of more or less random surface topography. Such a description or characterization must readily relate to the physics, mechanics or chemistry of the manufacturing process and the functional behavior of the surface in use. Beginning in the pre-computer era with CLA or RHS values, attempts at characterization of surface profiles as well as cutting spaces of grinding wheels include the height density function [ll, the autocorrelation [21 or its Fourier transform, the Spectrum [,I, and a suitable combination of these 141. At the same time, a comprehensive methodology of modeling and characterization called Data Dependent Systems (DDS) [5,61 based on observed digital data was being developed to include autocorrelation and spectra in a natural way. The DDS approach was applied to obtain a succinct but physically meaningful characterization of the grinding wheel profiles 171, and later on to nodel the grinding process itself as a feedback systen [a]. Such applications of DDS to manufacturing processes and machined surface topography have been summarized in 191, whereas the computation of functionally important parameters has been illustrated in [10,111 and a general methodology for such investigations has been outlined in 1121. These applications have established the feasibility of using DDS to provide a succinct characterization of surfaces which can readily relate to the manufacturing process generating the surface and its functional behavior. Attempts to circumvent the standard DDS methodology by mechanically following the so called AR-modeling have led to discouraging results and unnecessarily pessimistic conclusions [13,141. In particular, low order AR models are claimed to give a poor description of profiles which are unsymmetric or contain high deterministic (periodic) component. In contrast, the models obtained by the standard DDS procedure are fully capable of capturing unsymnetric and/or highly periodic nature of profiles and faithfully reproducing their behavior, as illustrated in [15,161. A survey and assessment of surface typology analysis techniques, primarily based o n empirical ordinate distributions, autocorrelation function and amplitude versus frequency graphs, has been recently presented in a key-note paper [171. Conclusions from this survey have expressed misgivings about the interpretation using a purely statistical approach, based on only a few independent characteristics obtained from an empirical distribution, and stressed the need for a more detailed characterization, including the wavelength content, made possible by digital conputers 1181. Such a detailed characterization with a comprehensive analysis of the wavelength decomposition by DDS has indeed been demonstrated for surface profiles generated in turning [19,201. Thus, in this era of explosive growth of electronic instrumentation and computers, restriction to a few parameters in characterizing machined surfaces is no longer necessary as in the past. This paper shows how the DDS methodology provides a comprehensive char-
Annals of the ClRP Vol. 30/1/1981
acterization of surfaces including detailed quantitative estimates of their wavelength content, random or uncorrelated component and a graphical representation of characteristic shapes. The overall DDS approach to surface topography is outlined in section 1 to show how it naturally includes the autocorrelation and spectrum with much improved estimates. Section 2 briefly explains the DDS modeling procedure as that of extracting the random or white noise component. Section 3 shows that the impulse response or the Green's function of this model yields the characteristic shapes of the surfaces. Wavelength decomposition of the profiles into component mode shapes is discussed in Section 4. Examples of characteristic shapes, wavelength decomposition and their relations to the machining processes are given in Section 5 from Electro Discharge Machining (EDM), turning and grinding. 1.
DATA DEPENDENT SYSTEMS (DDS) APPROACH TO TOPOGRAPHY
Characterization of a surface profile requires two different kinds of information: The first dealing with the overall variation of heights or height distribution, and the second describing the average trend along the line of measurement,which is the statistical dependence or correlation between successive observations. Empirically, the first is represented bv a height frequency histogram and the second by the sample autocorrelation or its Fourier transform-spectrum. However, mere plots of histograms, autocorrelation or spectra, although instructive as a visual aid in distinguishing different profiles, are not very useful in applications. Hence attempts have been made to fit these with mathematical functions, specified by a few parameters which can be estimated by standard statistical methods. The most c o m o n distributions used in statistics are Gbssian or normal for symmetric histograms and Beta for unsymmetric ones, both of which have been proposed for profile characterization. These distributions are specified by only two parameters, which can be optimally estimated directly fron the observed digital data, rather than using the histograms. If no distribution is assumed, conventional empirical Barameters such as Ra, Rp, Rq, Rt, Sk and k can be used. Although computation of these parameters is rather straightforward, due to their essential empirical nature they cannot reflect peculiarities of the processes generating them, and therefore considerably overlap for different machining processes 117,181. This is also true, to a lesser degree, of empirical autocorrelation or spectra. The problem is further conplicated by the fact that estinated autocorrelation or spectra have undesirable statistical properties such as bias, variance and dependence 1211 and hence present a distorted picture of the "true" autocorrelation and spectrum. A better alternative, just like the theoretical dis-
tribution for the empirical histogram, is to assume the autocorrelation function to be of sone reasonable foru, such as exponential or sinusoidal, and estimate the parmeters determining these forms fron the profile data. Such a form of the autocorrelation is equivalent to a model for the profile in the form of differential/difference equation. Conjecturing or guessing the correct shape of the autocorrelation function, or equivalently the form of the nodel, from the profile data or the empirical autocorrelation/ spectra,is virtually impossible in all but very simple first or second order models as seen from 1221.
487
However, the recorded data fully incorporates the information about the model. Therefore a systematic quantitative analysis of the recorded data should directly lead to the correct form of the model, without having to go through empirical autocorrelation or spectra. Such a quantitative analysis is indeed possible by treating the observed data as the response of a system governed linear differential/difference equations with white noise (uncorrelated process with flat spectrum) input. The system, as represented by these equations derived from and dependent upon the data alone, may be called as the "Data Dependent System (DDS). This concept of obtaining system equations from the recorded or observed data alone was advanced in [ 5 1 and later developed with the requisite mathematical foundation in 161. Although the DDS nethodology primarily uses differential equations, characteristic shapes and wavelength decomposition can also be readily obtained from difference equations. This paper is therefore restricted to difference equations, which are easier to model by the computer routines than differential equations.
Once the DDS models are obtained directly fron observed data by computer routines, their theoretical autocorrelation and spectra can be easily plotted with almost trivial computation compared to Fourier transformation. Such plots are much smoother and easily interpretable than those of empirical autocorrelation and spectra [e.g. 19.201. However, these plots are no longer required in the DDS methodology. Characteristic roots and parameters of the m d e l together with its impulse function or the Green's function provide a far more fundamental, useful and readily interpretable characterization. In particular,the Green's function or the impulse response function,in the form of a linear combination of ex-nentials and damped sinusoidals,is intimately related to the solution of differential equations. Hence it can be easily related to the physics, mechanics or chemistry of the manufacturing process generating the surface as well as its functional behavior, as illustrated in [7-11,15,16,19,201. This provides a powerful analytical methodology for control of topography by the manufacturer and for control of functional performance by the user, conceptually outlined in the block diagram shown in Figure 1.
I
DDS MODELING BY RESEARCH ENGINEER OR COMPUTER
I I
,,TOPOGRAPHY
d2X:
+
. . . . + dnik
for large enough n. A complex conjugate pair of Ails represent a damped sinusoidal, whereas a real $i represents an exponential. However, as explained in the last section, estimating n and parameter di, Xi of equation (1) from empirical or sampled autocorrelation is impossible for large n. It is much easier to estimate the parameters of a difference equation model for xt equivalent to (1):
... (l-inB)xt =
(l-).lB) (1-X2B)
.
(1-B1B-j2B2-. .-9,-1B
n-1 )at
(2)
where B is backshift operator defined by Bxt = x ~ - ~ , xt is the profile height and at is the independent series with variance cra2 i.e. discrete white noise with flat spectrum of height oa2/2n. Equation ( 2 ) is called Autoregressive Moving Average model of order n,n-1, denoted by AFWA (n,n-1), when the underlying continuous time system is ignored. If one wants to consider the underlying continuous system governed by a differential equation with characteristic roots iJi, then they are related to discrete i.i by
FUNCTIONAL PER-
CHARACTERISTIC FAAPE (IMPULSE
4 c
U.
at
DDS Modeling as Reduction of Profile to White Noise: Equation ( 4 )
LJiA (3)
where A is the sampling interval, and then model (2) is called USA'l(n,m) [15,231. Although in this paperthe underlying continuous time system is not considered, relation (3) is still useful for interpretation of models . The DDS modeling procedure consists of fitting AFT& (n,n-1) models by standard nonlinear least squares routines until the residual sum of squares of at's cannot be significantly reduced as judged by F-test [6,15,23,24]. In other words,n is increased until the residual at's cannot be further "whitened". DDS modeling is thus the reduction of correlated surface profile to an uncorrelated or white noise process which is the random component. This is schematically represented in Figure 2. The function which accomplishes this whitening or reduction to white noise is the Green's function discussed in the next section. 3.
GREEN'S FUNCTION-CHARACTERISTIC SHAPES
Assuming that hi's are distant and moving the left hand side operator in euuation (2) to the right, the m d e l becomes ,n-1 (1 - 91B - S2S2en-lu ) x+ = at (1 ilB) ( 1 X2B) (1 - inB)
-
-
......
(4)
'
where the Green's function G. can be shown by applying partial fractions to be 16, 20,231 G 1. = glAi
+ g2hi +...+
(5)
g n A n1
i#k k = 1,2,
488
-
Fig. 1: Schematic of DDS Approach to Control of Topography
>.. = e
Although the DDS modeling procedure is based on two theorems called Fundamental Theoren and Uniform Sanpling Theorem i6.231, its rationale can also be interpreted in terms of the autocorrelation or autocovariance function, commonly used in surface characterization. If obvious nonstationary trends such as form error are removed, the autocorrelation/autocovariance between two points on a surface profile sufficiently far apart tends to zero. Therefore the (theoretical, not sampled) autocovariance can always be represented by a sum of large enough numbers of exponentials and decaying sine waves. Thus the covariance y k between - ~be expressed by two points xt and x ~ may
+
*
I-----J
DDS MODELING-REDUCTION TO !mITE NOISE
yk = dlA!
BY I
!V+NUFACl'URER
Figure 2:
2.
-
4 - CONTROTOF;-
...n
Equation (4) shows that the AIUlA (n,n-l) model (2) essentially represents the profile xt as convolution of white noise at with the kernel G. The only parameter needed to specify the zero "mean white noise process is its variance aa2, which, together with the Green's function G . completely characterizes the surface profile x Although autocorrelation and spectrum can be rehdily obtained from equation (2), they contain no more information than G. and oa2. In fact comparison of equations (1) and (5? shows that their graphical forms are similar, so that even visually yk does not add anything t o G.. 1 On the other hand, whereas the autocorrelation function is merely a statistical concept devoid of any physical interpretation, G. as an impulse response function is amenable to a far'more fundamental interpretation and analysis. A s Figure 2 shows, a 's are a series of uncorrelated random shocks or imphses. To each such impulse, the system responds following Gi, in the form of Gjatej, and the profile xt is a sum of all these responses, as equation (4) states.
.
In processes such as ED!! and grinding, where the random a 's can be clearly identified as random discharges or rhndom grain positions, the Green's function (or parts of it) can even be interpreted as a typical or characteristic crater and grain. In processes such as turning and milling the components of the Green's function are the results of inherent periodicities interacting with various process elements. Therefore the Green's function may be truly said to represent a "characteristic shape" of the random profile, and together with a 's specified by a single parameter ua2, provides an e&haustive and physically meaningful characterization. 4.
MODAL ANALYSIS-IIAVELENGTH DECONPOSITION
Equations (5-6) provide a component by component wavelength decomposition of the profile. The characteristic roots hi, via relation (3),provide the natural frequency or the central wavelength of each component and the damping ratio indicating the extent to which neighboring wavelengths are present 119,201. The coefficients g . in equation (5) give the amplitude and phase of eack component. This is really the "modal analysis" familiar in structural vibrations,except that in the present case it applies to random vibration; indeed the surface profile can be considered as a random vibration signal. It is also similar to Fourier analysis. But, instead of decomposing into sinusoidals like Fourier analysis, it decomposes the Green's function into exponentials and damped sinusoidals, with possibly different danping for each component. It may thus be thought of as a generalized Laplane transform of the random profile 161. Rather than using the amplitude and phase given by gi's in equation (5-6), one may want to estimate "modal power", in analogy with modal analysis and Fourier transform, to get an idea of the relative contribution of each component of the characteristic shape, which can then be related easily to cutting conditions and other related information. This is readily obtained from equation (1) since the total power is the variance of the profile [6,19,20,23,241 yo = dl +d2
L
i,j=1
EXAWLES
5.1
Electro-Discharge Machining (EDM)
The first order model given by equation (2) with n=l, A1 = +1, has been found to be adequate for many E m profiles, and even when not strictly adequate, has been found to provide good enough approximation 125. 261. The Green's function of this model is an exponential : G. = 3
3
1
1
(9)
which represents the characteristic shape. However, since this would be strongly dependent on the sampling interval :, it is easier to deal with continuous time Green's function G(t) = e-PLot
(9a)
where so is obtained by equation (3): (10)
and the intensity of continuous time white noise:
Since e-4'5 = 0.0111, the exponential Green's function G . or G(t) is practically zero at t = jA = 4.5. Revalving this exponential around a vertical axis at gives a "characteristic cra er" I261 of Cit = 4.5/a aneter toodepth ratio 9/cio an6 depthuz2, fron which the volume and erosion rate can be easily found. Dimensions of characteristic craters obtained from EDMed AISID2 tool steel surface profiles by DDS in [271 are reproduced in Table 1 and the correspondinc: plots are shown in Figure 3. Technological paraneters such as erosion rates and RMS obtained from these are quite close to the experimental values I271 and the agreement can be further inproved by using higher order models. As expected, the craters become wider and deeper and their volume increases with increase in current and/or pulse duration. CHAXACTERISTIC CIiATER
+...+dn n
where d2. =
5.
The DDS modeling procedure provides a large number of models starting from the simple first order to a usually high order statistically adequate model and beyond. Although the lowest order statistically adequate model is the optimal choice based on data alone, many of the lower order models can be chosen as reasonable approximations for specific analysis purposes. Even with high order adequate models, one can choose only one or two modes (real roots or pairs of complex complex conjugate roots) contributing most to the power, as given by equation (7). The first example of Em1 surfaces illustrates the use of a simple first order approximate model, the second example o f turned surfaces illustrates the analysis o f a high order adequate model and the third example of the grinding wheel and ground surface illustrates the use of only two main modes from a high order adequate model in studying the wheel workpiece interaction.
(7)
gigj l-A.A,
, a = 1,2,
...,n
A j
WORKPIECE (STEEL) Current A
Note that di 1s the power due to a real root b i , whereas d i + di+l 1s the power due to a complex conjugate pair of roots A i , Xi+l. Thus the plot of components and the sum G. given by equation (5) yields a visual picture of tie characteristic shapes and their component, providing a qualitative characterization. Tabulated values of characteristic roots Xi, natural frequencies (wavelengths) and damping ratios found from roots pi of equation (31, and the relative contribution of each conponent as a % of the total power y , provide a complete quantitative characterization 09 the wavelength decomposition. These are illustrated in Figures 3-6 and Tables 1 - 3 discussed in connection with specific examples in the next section.
10
TOOL (COPPER)
Pulse Duration
us 40 *0° 400
1000
3.29 2.59 2.32 2.25
17.22 29.60 32.45 37.11
0.0195 0.1597 0.2640 0.4200
40
25
2oo 400
1000
Table 1.
EDM CHARACTERISTIC CRATER DIME~!SIOXS
489
50um
I
CURRENT: 25A WORKPIECE (STEEL)
CURRENT: 10A WORKPIECE,(STEEL)
v
40us
K--; J
1
I
I
-I--
--
--
200lls
L2
4001.1s
looops I I
I TOOL (COPPER) I
w
40us
I
x-:--J I
zoous
I
I
I
c.7
400ps
I
F i g u r e 3:
C h a r a c t e r i s t i c Craters i n EDM
Table 2 WAVELENGTH DECOXPOSITION OF ARMA ( 1 4 , 1 2 1 HODEL FOR FEED 0.122iTm/REV
#
NATURAL FREQUENCY RCOTS( X.) REAL I h G . (CYC/INII) (HZ)
WAVE DAMPLENGTH I N G POWER hm) RATIO ( 8 )
1
.925
.OOO
1.40
.42
.7146
2
.878
-.434 - 4 34
8.20
2.44
.1219
3
.454
.OOO
14.10
4.19
.0709
-
7.49
4
.571
.787 -.787
16.85
5.01
.0593
.0295
5
.132
-.727
.727
25.42
7.55
.0393
.2121
...................................................... -
1
-980
-..196 196
3.52
1.05
-2843
.0046
17.47
2
.527
-.1 7 5
.175
11.97
3.56
.0835
.8785
12.41
5.53
3
.914
75 -.. 3375
6.95
2.07
.1438
.0301
10.10
2.27
4 -.412
-.7 7 1
.771
36.88
10.96
.0271
.0650
-.07
5 -.433
.837 -.a37
36.58
10.87
.0273
.0290
.06
6
.625
-..750 750
15.65
4.65
.0639
.0271
-06
7
.336
-.894
.894
21.64
6.43
.0462
.0381
-.02
8 -.006
.962 -.962
28.17
8.37
-0355
.0245
-.01
9 -.a09
32 -.. 5532
45.70
13.58
.0219
.0124
-.oo
.170
52.89
15.72
.0189
.0171
42.56
.0447 4 0 . 8 4
6 -.295
.OOO
21.77
6.47
.0459
-
1.54
7 -.336
.388 -.388
42.48
12.62
.0235
.2835
-.50
8 -.465
.a85 885
36.69
10.90
0.273
-0003
.OOO
2.00
.59
.5304
-
9 -.894
-.
.28
-.01
What i s more i m s o r t a n t i s t h a t a t h e r m a l n o d e l for t h e ED11 process h a s b e e n d e v e l o p e d o n t h e basis o f c h a r a c t e r i s t i c crater. T h i s m o d e l i s m a t h e m a t i c a l l y simpler a n d y e t p r o v i d e s better p r e d i c t i o n s t h a n t h o s e a v a i l able i n t h e l i t e r a t u r e , e v e n i n t h e case of c e m e n t e d c a r b i d e s w h i c h are d i f f i c u l t to nodel a n d p r e d i c t 1281. 5.2
Turning
S t a t i s t i c a l l y adequate AE4A models of order ( 1 2 , l l ) to ( 2 0 , 1 9 ) h a v e been used t o e l u c i d a t e e f f e c t s o f c u t t i n g c o n d i t i o n s on wavelength decomposition o f s u r f a c e s g e n e r a t e d i n t u r n i n g [201. For i l l u s t r a t i v e purposes, w a v e l e n g t h d e c o m p o s i t i o n of two s u r f a c e s g e n e r a t e d a t f e e d s 0 . 1 2 2 m/rev. a n d 0 . 2 8 8 mm/rev., o t h e r c o n d i t i o n s b e i n g p r a c t i c a l l y t h e same, are g i v e n i n T a b l e s 2 a n d 3
10 - . 9 3 5
-.1 7 0
.oo
and t h e corresponding c h a r a c t e r i s t i c shapes w i t h t h e i r component6 are p l o t t e d i n F i g u r e s 4 a n d 5 , f o l l o w i n g e q u a t i o n s (4-7). For t h e s a k e o f c l a r i t y o n l y f i v e h i g h e s t c o n t r i b u t i n g components are plotted, a l t h o u g h Since t h e r e is t h e tables l i s t a l l the c o m p o n e n t s . too much i n f o r m a t i o n ' i n t h e t a b l e s , o n e can make c h a r t s of o n l y f r e q u e n c i e s o r d a m p i n g r a t i o s etc. t o see t h e v a r i a t i o n w i t h c u t t i n g c o n d i t i o n s a t a glance [ c . f . 201.
C o n s i d e r i n g t h e w a v e l e n g t h d e c o m p o s i t i o n T a b l e s 2 and 3, i t i s s e e n t h a t t h e f e e d w a v e l e n g t h s o f 0.122 and 0.288 a r e q u i t e p r e c i s e l y r e p r o d u c e d i n the root g2 and 1 r e s p e c t i v e l y w i t h c o n t r i b u t i o n s 40.8% and 77.5%. Near h a r m o n i c s o f t h e f e e d f r e q u e n c y a r e p r e s e n t i n root l ' s 4 and 5 i n T a b l e 1 and rwt 1 3 i n T a b l e 2 . 1.5
Note t h a t a l t h o u g h t h e tool n o s e g e n e r a t e s a p e r i o d i c s u r f a c e , it i s n o t s i n u s o i d a l and t h e r e f o r e h a r n o n i c s o f the f u n d a m e n t a l f e e d f r e q u e n c y s h o u l d b e p r e s e n t , a s seen from F o u r i e r series o f p e r i o d i c s i g n a l s . The f e e d and i t s h a r m o n i c s a c c o u n t f o r a b o u t 49% o f t h e power i n T a b l e 2 a t f e e d 0.122 m / r e v . a s a g a i n s t 88% i n T a b l e 3 a t f e e d 0.288 m / r e v . Increasing contrib u t i o n of f e e d wavelength w i t h i n c r e a s i n g feed is s e l f e v i d e n t from t h e dynamics o f t u r n i n g . T h i s i s a l s o q u a l i t a t i v e l y r e f l e c t e d i n t h e c h a r a c t e r i s t i c shape p l o t s i n F i g u r e s 4 and 5.
1.0
Host of t h e r e m a i n i n g c o n t r i b u t i o n comes from t h e real root # 1 i n T a b l e 2 w i t h e x p o n e n t i a l r e s p o n s e and t h e complex root 32 i n T a b l e 3 w i t h s u c h h i g h damping rat i o t h a t t h e response is p r a c t i c a l l y exponential, a s a l s o s e e n from t h e p l o t s i n F i g u r e s 4 and 5. These roots c a n b e r e l a t e d t o t o o l w o r k - p i e c e v i b r a t i o n s , w h e r e a s some o f t h e micrometer w a v e l e n g t h s w i t h n e g l i g i b l e c o n t r i b u t i o n s i n T a b l e s 2 and 3 seem t o p r o v i d e a n i n d e p e n d e n t estimate of c h i p s u r f a c e l a m e l l a r w i d t h s i n micro-morphology [ 2 0 ] .
0.5
5.3
0.0
Surface Grindinq
A s i m i l a r i n v e s t i g a t i o n h a s been c a r r i e d o u t f o r s u r -
0
0.05
0.15
0.10
0.20
0.25
j (m) F i g u r e 4:
C h a r a c t e r i s t i c Shape ( G r e e n ' s F u n c t i o n ) a n d its Components a t Feed 0.122 mm/rev. i n T u r n i n g
f a c e g r i n d i n g u s i n g w h e e l and g r o u n d s u r f a c e p r o f i l e s a s t h e c u t p r o g r e s s e s [291. Two p r i m a r y c o n t r i b u t o r s w i t h t h e l a r g e s t % c o n t r i b u t i o n were i s o l a t e d from For an a b r a s i v e t a b l e s s i m i l a r t o T a b l e s 2 and 3. wheel AA46HSV40, t h e s e w a v e l e n g t h s r a n g e d from 0.54mm t o 0 . 6 5 m and 0.053mm t o 0.14mm, w i t h t h e c o n t r i b u t i o n from t h e s m a l l e r w a v e l e n g t h o f t h e o r d e r of 1%. F o r 46 g r a i n s i z e , d i a m e t e r o f a r o u g h l y s p h e r i c a l g r a i n i s a b o u t 0.38 nun, t h e r e f o r e t h e larger w a v e l e n g t h can b e r e l a t e d t o t h e g r a i n s randomly s p a c e d w i t h bond material. Then t h e s m a l l e r w a v e l e n g t h c a n b e t a k e n a s a r i s i n g from t h e c u t t i n g e d g e s . R a t h e r h i g h damping r a t i o s o f a b o u t 0.25 t o 0.46 f o r t h e l a r g e r and 0.40 t o 0.78 f o r t h e s m a l l e r i n d i c a t e t h a t t h e p e r i o d i c i t y is v e r y weak and t h e n a t u r e o f t h e c h a r a c t e r i s t i c T h i s is a l s o c o n f i r m e d shapes is s t r o n g l y exponential. by t h e p l o t of a t y p i c a l " c h a r a c t e r i s t i c g r a i n " shown i n F i g u r e 6. S i m i l a r a n a l y s i s of t h e g r o u n d s u r f a c e p r o f i l e s i n d i c a t e s t h a t t h e p r i m a r y w a v e l e n g t h , w i t h a l m o s t 100% c o n t r i b u t i o n , i s of t h e same o r d e r o f m a g n i t u d e a s t h e s m a l l e r wavelength i n t h e wheel. These wavelengths a r e reproduced i n T a b l e 4 f o r an experiment. S i nce t h e c u t t i n g edges of t h e g r a i n s w i l l imprint t h e i r marks on the s u r f a c e , t h e s e w a v e l e n g t h s are u n d e r s t a n d a b l e . However, t h e w a v e l e n g t h s on t h e w o r k p i e c e a r e c o n s i s t e n t l y s m a l l e r t h a n t h o s e on t h e w h e e l . Assumi n g t h i s d i f f e r e n c e t o be due t o t h e e l a s t i c d e f l e c t i o n , t h e l a s t column o f T a b l e 4 g i v e s t h e e l a s t i c d e f l e c t i o n c o e f f i c i e n t computed from t h e s e w a v e l e n g t h s . V a l u e s of these c o e f f i c i e n t s agree w e l l with t h e experimental and t h e o r e t i c a l i n v e s t i g a t i o n s a v a i l a b l e i n t h e l i t e r a t u r e , as d o the e s t i m a t e s o f p r o g r e s s o f wheel wear made from t h e l a r g e r w a v e l e n g t h on t h e wheel 1291.
1.
1.
0.
WHEEL-WORK
TABLE 4 DEFLECTION FROM WAVELENGTHS
0. BEFORE AND AFTER CUT
WAVELENGTH SURFACE PROFILE I?s
(mm)
WAVELENGTH WHEEL
PROFILE "Iw
ELASTIC DEFLECTION COEF. PIw- WS
(mm)
% wW
-0.
--I 0
, , ,, 0.05
, ,, ,
, , , ,
0.10
0.15
, , ,
,
0.20
.,
,
,
1.
0.0474
0.0525
9.97
2.
0.0836
0.0938
10.8
3.
0.0935
0.1097
14.73
4.
0.1203
0.1417
15.10
0.25
j (m)
F i g u r e 5:
C h a r a c t e r i s t i c Shape ( G r e e n ' s F u n c t i o n ) and i t s Components a t Feed 0.288mm/rev. i n Turning
49 1
I111
P a n d i t , S. !l.,N a s s i r p o u r F . , and Uu, S. !I., " s t o c h a s t i c Geometry o f A n i s o t r o p i c Randon S u r f a c e s w i t h A p p l i c a t i o n t o Coated A b r a s i v e s , " T r a n s . ASHE, J o u r . o f E n g i n e e r i n g f o r I n d u s t r y , voi. W B , ~ 9 7 3 p?. , 218-224.
2
[121
P a n d i t , S . :!., " Q u a n t i t a t i v e E v a l u a t i o n of A b r a s i v e T o o l and 'Iachined S u r f a c e Topography," PTOC. 1977 Symposium o n New Developments i n T o o l I ' l a t e r i a l s and A p p l i c a t i o n s , IIT, C h i c a a o , 1 9 7 7 , pp. 107-113.
1.5
[131
S t a u f e r t , G . , " C h a r a c t e r i z a t i o n of Random Roughn e s s P r o f i l e s - A Comparison o f AR-Ilodeling Techn i q u e s and P r o f i l e D e s c r i p t i o n by :leans o f C o m o n l y Used P a r a m e t e r s , " A n n a l s o f C I R P , 28/1, ( 1 9 7 9 ) , pp. 431-435.
1141
D e V r i e s , V . R.,
[151
P a n d i t , S . .pi., " S t o c h a s t i c L i n e a r i z a t i o n by Data Dependent S y s t e m s , " T r a n s . ASIlE, J. D n. S st. :leas. and C o n t r-o l , V
1
0.51
"Autoregressive Tine S e r i e s !lodels f o r S u r f a c e P r o f i l e C h a r a c t e r i z a t i o n , " A n n a l s o f CIRP, 28/1, (19791. pp. 437-440.
2 2 6 .
[16]
P a n d i t , S . I!., and K. P. R a j u r k a r , "Data Depend e n t S y s t e m s Approach t o S o l a r Energy Simulat i o n InDUts." PrOC. o f AS!€E/SSEA ence , ~ c .o . -n f e r ~ Reno, Nbvada, A p r i l 1 9 8 1 , T r a n s . ASHE, J o u r n a l of S o l a r Energy E n g i n e e r i n g , 1981. ~
-0.5
i;/' 0.2
0
0.4
P e r e r s , J . , Vanherck, P . , S a s t r o d i n o t o , M., Assessment of S u r f a c e Typology A n a l y s i s T e c h n i q u e s , : A n n a l s o f CIRP, 28/2, ( 1 9 7 9 ) , pp. 539-554.
[18]
P e t e r s , J . , Vanherck, P., S a s t r o d i n o t o , Evaluation of Surface Describing Parameters," :lorth Amer. ! l a n u f a c t u r i n g R e s e a r c h C o n f e r e n c e (1980) , pp. 352-357.
[191
C h a r a c t e r i s t i c Shape ( G r e e n ' s F u n c t i o n ) and i t s Components f o r AA46H8V40 Wheel i n S u r f a c e G r i n d i n g
11
P e s a n t c , H., " D e t e r r i n a t i o n o f S u r f a c e Zoughness Typoloay by !leano o f m p l i t u d e D e n s i t y C u r v e s , " A n n a l s of CIRP, 1 2 , ( 1 9 6 4 ) , pp. 61-68.
[ 21
P e k l e n i k , J . , " I n v e s t i g a t i o n of. S u r f a c e Typolo g y , A n n a l s o f C I W , 1 5 , ( 1 9 6 7 1 , pp. 381-385.
[ 31
B e r , A. and S. Braun, " S p e c t r a l A n a l y s i s o f S u r f a c e F i n i s h , " Annals o f C I R P , 1 6 , ( 1 9 6 8 ) , pp. 53-59.
I 41
W h i t e h o u s e , D. J . , "Typology o f l l a n u f a c t u r e d S u r f a c e s , " A n n a l s o f CIRP, 1 9 , ( 1 9 7 1 ) . pp. 417431.
51
P a n d i t , S. ! l . , "A B a y e s i a n Approach t o Time S e r ies A n a l y s i s i n A d a p t i v e C o n t r o l , " !l.S. T h e s i s , The P e n n s y l v a n i a S t a t e U n i v e r s i t y , U n i v e r s i t y Park, (1970).
[ 61
P a n d i t , S . M., "Data Dependent Systems: !lodeli n g A n a ly sis and Optimal C o n tr o l v i a Ti ne S e r i e s , " Ph.D. T h e s i s , U n i v e r s i t y o f WisconsinMadison, ( 1 9 7 3 ) .
I 71
P a n d i t , S. 3. and S . !I. Wu, " C h a r a c t e r i z a t i o n o f A b r a s i v e Tools by C o n t i n u o u s Time S e r i e s , " T r a n s . ASIE, S . of Eng. f o r I n d u s t r y , 95B, ( 1 9 7 3 1 , pp. 821-826.
[ 81
P a n d i t , S. 11. and S. Revach, "A Data Dependent S y s t e m s Approach t o Dynamics of S u r f a c e Generat i o n i n T u r n i n g , " ASilE P a p e r No. 80-V!A/PROD-25, T r a n s . ASME, J o u r n a l o f E n g i n e e r i n g f o r I n d u s t r y , V o l 103B. 1981.
211
J e n k i n s , G. ?1. and D. G. W a t t s , S p e c t r a l A n a l y s i s and I t s A p p l i c a t i o n s , Kolden-Day, San F r a n c i s c o , 1968.
221
Box, G. E . P . , and G. 11. J e n k i n s , T i m e S e r i e s Analysis: F o r e c a s t i n g and C o n t r o l , HoldenDay, S a n f r a n s i s c o , 1970.
[231
P a n d i t , S. tl., " D a t a Dependent S y s t e m s and Exp o n e n t i a l Smoothing," i n A n a l y s i n g T i m e S e r i e s , Ed. 0. D. Anderson, N o r t h H o l l a n d , 1 9 8 0 , pp. 217-238.
[24]
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