- Email: [email protected]
The nonlinear dynamics of metal cutting
hr. 1. Engng Sci. Vol. 30, No. 10, Pp. 1433-1440, Printed in Great Britain. All rights reserved
THE NONLINEAR
ML?O-7225192 $5.00 + 0.00 Copyright @ 1992 Pergamon Press Ltd
1992
DYNAMICS
B. S. BERGER’,
OF METAL CUTTING
M. ROKNI’
and I. MINIS2
‘Department of Mechanical Engineering, University of Maryland, College Park, MD 20742 and *Department of Mechanical Engineering, University of Maryland, Baltimore County, Baltimore, MD 21228, U.S.A. Abstrret-The information dimension, D(O), is computed as a function of the dimension of the reconstruction space, E for the logistic equation, data sets of tool work-piece post transient acceleration and for the attractor associated with a cutting model. For the logistic equations, D(0) : E converges to an approximation of the known value of the information dimension. Measurements of tool work-piece post transient accelerations for pre- and mild-chatter yield plateaus for which D(0) = 2.38 and 2.51, respectively.
INTRODUCTION Recently [l], an orthogonal metal cutting process was modeled by a system of coupled discontinuous ordinary differential equations. Omitting regenerative effects, the cutting force was assumed to be a discontinuous function of the depth of cut and the relative velocity between the tool and cutting surface. The motion was shown to be chaotic for certain parametric ranges. In the following, the information dimension, D(O), is computed for data sets of tool work-piece post transient acceleration and for the attractor associated with a cutting model. Experimental data for cases of pre- and mild-chatter are studied for which it is shown that D(0) = 2.38 f 0.10 and 2.51 f 0.08, respectively. A model of the dynamics of cutting based on the Wu-Liu theory [2], is studied in which regenerative effects are included. The resulting system of nonlinear differential delay equations (DDE) are shown to have limit cycle behavior, D(0) = 1, when modelling heavy chatter and more complex behavior when modelling mild chatter, D(0) = 2.65 f 0.30. Since the DDE are infinite dimensional the existence of a low order attractor is of significance. Although the near equality of D(0) for the experimental data and model is suggestive, it should be noted that equality of attractor dimension does not imply identical asymptotic behavior. The non-integer values of D(0) found for the experimental data indicate that nonlinearities are of importance in the physical processes involved in metal cutting.
DIMENSION
The qualitative features of the time evolution of many systems are identical to those of the solution of a typical system of differential equations of the form 40
=f(x(t),
P)
(I)
where ~1=a vector of experimental parameters. The n components of x are taken as coordinates of points in an n-dimensional Euclidean phase space R".For dissipative systems volume elements of R" contract under the action of (1) and there exists a set of points, U c R", initial conditions, which asymptotically evolve to a compact set of points, A, called an attractor. The ergodic theorem asserts that for every continuous function #(x(t)) where x(t) = F(4O)P P> 4 lim 1
T--r-T ~‘44+(0),
CL,01dt = I, 1433
~44~)h
(2)
B. S. BERGER et
1434
al.
where, for almost all initial conditions x(O), p = ergodic invariant probability measure [3], Equation (2) relates asymptotic time averages to spatial integration over the attractor. Since asymptotic time behavior is of importance, in defining Lyapunov exponents for example, (2) indicates that properties of the attractor and probability measure p would be of interest [3]. Quantities characterizing properties of A are dimension and metric entropy [4]. The dimension of a point set, dim E, E c R”, has been defined in a variety of not always equivalent ways [5,6]. In [6], dimension is categorized as either fractal dimension or dimension of the natural measure. The latter involves a probability measure for their definition while the former do not. Most definitions of dimensions share the following properties where E and F c R”: E c F + dim E % dim F, dim(E U F) = max(dim E, dim F), F, open and F c R” =$ dim F = n, F finite or countable 3 dim F = 0, f a translation, rotation, similarity or affinity of R” j dim f (F) = dim F and M a smooth ~-dimensional manifold + dim tM = m [5]. In [4], the dimension function, D(y), is defined as:
D(Y) = - lim loh(n/k),/[ n-=
2 ST(~,E, n)]
(l/y)logb(l/m)
(3)
j=l
where ~tl= number of reference points yj, i = 1, . . . , m on the attractor, E = dimension of the reconstruction space, RE, y = (1 - q)D, where Dq = Renyi dimension [4], 6/(k, E, n) = distance between the reference points yi and their k-th nearest neighbors for an E dimensional reconstruction space and n = number of generic points of the attractor. It has been shown that Do>“DwzD,, DtsDtwhere DH = Hausdorff dimension, D(0) = D, , D1 = information dimension and DL = Lyapunov dime~ion [7-g]. Let the pointwise dimension ‘yi = lim In pjlln E e--r0
and the mass pj = kj(n)/n where E is the radius of a sphere centered at the reference point yj and kj(n) is the sphere’s ~pulation. In (3), the pointwise dimension, aj, is found by fixing the kj, the number of nearest neighbors to yj, and determining the sphere’s population, corresponding radius. These are called constant mass methods and have been shown to
AA=?
D(O1
=A.=2 0
0 60
0 0 55
Q
i
0 13 ~~-.-.-_-_-_____-_-‘---.-
._._._.-___-.-_~__“_.-~._~_~_ 0
cl
A
q
e
A=5
4
0
m ,_-_-.__I
A
A 0.50
‘F
045
-
0.40
~. ‘I
et
A
A
A
2
3
4
s
6
Fig. 1. D(0) : E for the logistic equation.
7
6
E
The nonlinear dynamics of metal cutting
1435
converge independently of the magnitude of D(y). The alternative fixed size method, sphere radius is fixed and population determined, does not converge for D(0) > 3 [lo]. As an example of the computation of D(O), consider the attractor associated with the logistic mapf(x) = r2x(l -x)_ The Hausdorff dimension for )L= 3.57 is known to be 0.538 151. Figure 1 shows D(0) as a function of E, D(0) :E, determined by (3) with m = 800, n = ~,~, y = 0, k = 200, 1 zzzE 59. In all cases, the determination of D(0) from (3) was based on reconstructions of the attractor formed by delaying x by i = 1, . . . , E multiples of A, Xi(~) = x(r + (i - 1)A) where the iteration index r and A are positive integers [3]. From Fig. 1, it appears that small values of A result in convergence to the exact solution for large values of E while large values of A result in an earlier attainment of the exact value with a slight overshoot for larger values of E. Similar behavior was noted in [7]. In the following, (3) is used in the study of a cutting model described by a system of differential delay equations and experimentally generated orthogonal cutting data.
MATHEMATICAL
MODEL
In [2], the cutting tool, supporting structure and cutting force are modeled by
& = -&$i x
f,
=
[a,@4
-a4x~
x
-
[%(X4
-
07x3
+
-X4@
a&x;?
-X40
(4)
&=x4
(5)
a&(t
-
-
i, =.X3
-
d)
-
agx4
-
a
-
-
r)
+
a2(X3
-
xl(t
-X3@
a,x,(t
%(X3
(x,
-
-
-
-
t))
r))]
z)
+
(x*
X3@
-
$)I
-
-
&I$
x,(t
-
62d
(6)
-
z))
(7)
where x1 and x2 = tool deflections normal and parallel to the cutting surface, respectively. The following physical properties are assumed: feed rate so = 0.13 mm/rev., cutting speed, u,= 3 m/s, the delay, r, takes on various values, shear modulus of cut material, CY= 6.0 e 10aN/m2, equivalent mass, stiffness and linear damping coefficients for the normal and parallel directions are 83.1 kg, 48.86 kg, 41.9 - lo6 N/m, 111 - lo6 N/m and 15 - ld N s/m, 13.3 - ld N s/m, respectively, depth of cut, w = 0.0052 m and tool nose damping coefficient, k = 0, cubic damping coefficients, a,, and al2 = 1.0 s2/m2. The coefficients, ui, in (6,7) follow from table 1, part 2 of [2], a, = 0.05212~, a2 = 0.01545~, a3 = O.O0426arw, a4 = 0.10557~, u, = 0.1805 - 103, a8 = 0+05212m + 0.5042 * 106, a9 = us = 0.01547&w, U6= 0.~17~, 0.2722 - ld and al0 = 2.2738. 106; initial conditions were x = 0 for -r r: t < 0 and x = (0.1 - 10m3}. I at t = 0. a,, and al2 were chosen to insure bounded solutions [ll].
EVALUATION
OF D(O)
All numerical solutions of (4)-(7) were calculated using an Adams-Bashforth predictor and an Adams-Moulton corrector both of 0(h4) with h = (0.2) 0 10e3 s. A plot of x2 vs x1 is shown in Fig. 2 for a delay of t = 0.1 s. From the figure, the attractor appears to be a limit cycle. This is corroborated by the plot of D(O):E for n = 75,000, m = 3000 with k = 100, 200 and 300 corresponding to the upper, middle and lower points in the triad of values of D(0) shown for each value of E. It is seen that for k = 300, D(0) has converged to 1.0 for E 2.2. D(0) for k = 100 and 200 initially overshoots 1.0, but the asymptotic value of D(0) = 1.0 is reached for E 2 9, Fig. 3.
1436
B. S. BERGER
et al.
-0.0014
0.0014
x2 o.OQ70
0.0042
0.0014
-0.0014
-0.0042
-0M70 -0.0070
-0.0042
Fig. 2. xzvsx,,
0.0042
0.0070
z=O.ls.
Increasing the delay to z = 0.4 s results in the complex behavior shown in Figs 4-6. These are 120~t~135and14O~tz~155 projections of the attractor onto the x2, x,planefor6Oz%tzz75, respectively. In this case a dimension of D(0) = 2.65 f 0.30 was found [ll]. D(0) may be found for experimentally determined point sets provided that the power levels associated with noise are not excessive and that data acquisition times are short enough to prevent the occurrence of experimental drift ]8]. The question as to whether or not D(0) may be approximated for chatter data can be answered in the affirmative. D(O) was computed for
D(O)
.600
.400
,200
0.
i 0.
I
I
I
I
I
3.50
7.00
10.5
14.0
17.5
Fig. 3. D(O):&
t=O.is.
,
21 .o
E
The nonlinear dynamics of metal cutting
1437
x4 5
-
3
-
1
-
-1
-
-3
-
-5
8
’
-0.005
-0.003
-0.001 Fig. 4. xqvsx2,
0.001 t=0.4s,
0.003
x2
0.005
6Ost~75.
two sets of measurements of f, the relative tool-workpiece acceleration normal to the cutting surface. Geometric and material parameters were held constant except for the depth of cut w. Data sets were ident~ed by visual and auditory observations as pre-chatter and mild chatter cutting states corresponding to depths of cut of w = 2.2 mm and 2.3 mm, respectively. Figures 7 and 8 give D(O):E with m=3000, n=60,000, w=2.2mm, and m=3000, n=70,000, w = 2.3 mm. A plateau or region of reduced slope is evident in D(0) : E for 3 s E I 7. This is x4 5
-
3
-
1
-
-1
.
-3
.
-5
’
-0.005
1 x2 -0.003
-0.001 Fig. 5. x.,vsxz,
0.001 r=O.4s,
14O~t~155.
0.003
0.005
B. S. BERGER
et al.
x4 5
3
1
-1
-3
4 x2
-5 -0.005
-0.001
-0.003
Fig. 6. x4vsx2,
0.001 r=O.4s,
0.003
0.005
14Ost=155.
6.0
3.5
7.0
10.5
14.0
E
Fig. 7. D(O):&
w = 2.2 mm.
17.5
21.0
1439
The nonlinear dynamics of metal cutting 6.0
-
1.0
)
t
0
I
I
I
I
I
I
35
70
10 5
14.0
17.5
21.0
E
Fig. 8. D(O):& w = 2.3 mm.
indicative of the presence of for w = 2.2 mm, and 2.3 mm, D(0): E for E = 4,5,6 and demonstrating a known affect
an attractor for which D(0) = 2.38 f 0.10 and D(0) = 2.51 f 0.08 respectively. This estimate is the average and mean deviation of k = 100, 200 and 300. For E > 8, D(0): E increases with E of noise on D(0) : E for large values of E.
CONCLUSIONS
It has been shown that despite the presence of noise estimates of the dimension, D(O), of attractors associated with orthogonal cutting data can be found. For the pre and mild chatter cases D(0) = 2.38 f 0.10 and D(0) = 2.51 f 0.08 for w = 2.2 mm and 2.3 mm respectively. A model of the cutting dynamics based on the Wu-Liu theory [2], was expressed as a system of non-linear differential delay equations. Although the system is infinite dimensional it was shown that the associated attractor is of low dimension. For r = 0.1 s, heavy chatter, D(0) = 1.0 indicating limit cycle behavior while D(0) = 2.63 f 0.3 for t = 0.4 s. Non-integer values of D(0) for the experimental data indicate that nonlinearities play an important role in cutting dynamics. Acknowledgements-The authors wish to take this opportunity to gratefully acknowledge the continuing stimulus and inspiration which Professor A. C. Eringen has provided the mechanics community. The assistance of Dr G. Broggi and J. Young, the support of the Computer Science Center and the computer facility of the Department of Mechanical Engineering, University of Maryland, is acknowledged.
REFERENCES [l] [2] [3] [S]
I. GRABEC, Phys. L&t. 117A(8), 384-386 (1986). D. W. WU and C. R. LIU, ASME J. Engng Ind. 105, 107-111; 112-118 (1985). J. P. ECKMANN and S. RUELLE, Rev. Mod. Phys. 57(3), 617-656 (1985). P. BAD11 and A. PGLJTI, 1. Stat. Phys. 40(516), 725-750 (1985).
1440
B. S. BERGER
et al.
[5] K. FALCONER, Fractal Geometry. Wiley, New York (1990). [6] J. D. FARMER, E. OTT and J. A. YORKE, Physica 7D, 153-180 (1983). (71 G. BROGGI, Numerical characterization of experimental chaotic signals. Dissertation (1988). (81 G. BROGGI, 1. Opt. Sot. Am. B S(5), 1020-1028 (1988). [9] J. L. FREDRICKSON et al., /. Di#. Engs 49, 185 (1983). (lo] E. J. KOSTELICH and H. L. SWINNEY, Phys. Scripta 40, 436-441 (1989). [ll] B. S. BERGER, M. ROKNI and I. MINIS, Q. Appl. Math. To appear.
der Universitit
Ziirich
THE NONLINEAR
ML?O-7225192 $5.00 + 0.00 Copyright @ 1992 Pergamon Press Ltd
1992
DYNAMICS
B. S. BERGER’,
OF METAL CUTTING
M. ROKNI’
and I. MINIS2
‘Department of Mechanical Engineering, University of Maryland, College Park, MD 20742 and *Department of Mechanical Engineering, University of Maryland, Baltimore County, Baltimore, MD 21228, U.S.A. Abstrret-The information dimension, D(O), is computed as a function of the dimension of the reconstruction space, E for the logistic equation, data sets of tool work-piece post transient acceleration and for the attractor associated with a cutting model. For the logistic equations, D(0) : E converges to an approximation of the known value of the information dimension. Measurements of tool work-piece post transient accelerations for pre- and mild-chatter yield plateaus for which D(0) = 2.38 and 2.51, respectively.
INTRODUCTION Recently [l], an orthogonal metal cutting process was modeled by a system of coupled discontinuous ordinary differential equations. Omitting regenerative effects, the cutting force was assumed to be a discontinuous function of the depth of cut and the relative velocity between the tool and cutting surface. The motion was shown to be chaotic for certain parametric ranges. In the following, the information dimension, D(O), is computed for data sets of tool work-piece post transient acceleration and for the attractor associated with a cutting model. Experimental data for cases of pre- and mild-chatter are studied for which it is shown that D(0) = 2.38 f 0.10 and 2.51 f 0.08, respectively. A model of the dynamics of cutting based on the Wu-Liu theory [2], is studied in which regenerative effects are included. The resulting system of nonlinear differential delay equations (DDE) are shown to have limit cycle behavior, D(0) = 1, when modelling heavy chatter and more complex behavior when modelling mild chatter, D(0) = 2.65 f 0.30. Since the DDE are infinite dimensional the existence of a low order attractor is of significance. Although the near equality of D(0) for the experimental data and model is suggestive, it should be noted that equality of attractor dimension does not imply identical asymptotic behavior. The non-integer values of D(0) found for the experimental data indicate that nonlinearities are of importance in the physical processes involved in metal cutting.
DIMENSION
The qualitative features of the time evolution of many systems are identical to those of the solution of a typical system of differential equations of the form 40
=f(x(t),
P)
(I)
where ~1=a vector of experimental parameters. The n components of x are taken as coordinates of points in an n-dimensional Euclidean phase space R".For dissipative systems volume elements of R" contract under the action of (1) and there exists a set of points, U c R", initial conditions, which asymptotically evolve to a compact set of points, A, called an attractor. The ergodic theorem asserts that for every continuous function #(x(t)) where x(t) = F(4O)P P> 4 lim 1
T--r-T ~‘44+(0),
CL,01dt = I, 1433
~44~)h
(2)
B. S. BERGER et
1434
al.
where, for almost all initial conditions x(O), p = ergodic invariant probability measure [3], Equation (2) relates asymptotic time averages to spatial integration over the attractor. Since asymptotic time behavior is of importance, in defining Lyapunov exponents for example, (2) indicates that properties of the attractor and probability measure p would be of interest [3]. Quantities characterizing properties of A are dimension and metric entropy [4]. The dimension of a point set, dim E, E c R”, has been defined in a variety of not always equivalent ways [5,6]. In [6], dimension is categorized as either fractal dimension or dimension of the natural measure. The latter involves a probability measure for their definition while the former do not. Most definitions of dimensions share the following properties where E and F c R”: E c F + dim E % dim F, dim(E U F) = max(dim E, dim F), F, open and F c R” =$ dim F = n, F finite or countable 3 dim F = 0, f a translation, rotation, similarity or affinity of R” j dim f (F) = dim F and M a smooth ~-dimensional manifold + dim tM = m [5]. In [4], the dimension function, D(y), is defined as:
D(Y) = - lim loh(n/k),/[ n-=
2 ST(~,E, n)]
(l/y)logb(l/m)
(3)
j=l
where ~tl= number of reference points yj, i = 1, . . . , m on the attractor, E = dimension of the reconstruction space, RE, y = (1 - q)D, where Dq = Renyi dimension [4], 6/(k, E, n) = distance between the reference points yi and their k-th nearest neighbors for an E dimensional reconstruction space and n = number of generic points of the attractor. It has been shown that Do>“DwzD,, DtsDtwhere DH = Hausdorff dimension, D(0) = D, , D1 = information dimension and DL = Lyapunov dime~ion [7-g]. Let the pointwise dimension ‘yi = lim In pjlln E e--r0
and the mass pj = kj(n)/n where E is the radius of a sphere centered at the reference point yj and kj(n) is the sphere’s ~pulation. In (3), the pointwise dimension, aj, is found by fixing the kj, the number of nearest neighbors to yj, and determining the sphere’s population, corresponding radius. These are called constant mass methods and have been shown to
AA=?
D(O1
=A.=2 0
0 60
0 0 55
Q
i
0 13 ~~-.-.-_-_-_____-_-‘---.-
._._._.-___-.-_~__“_.-~._~_~_ 0
cl
A
q
e
A=5
4
0
m ,_-_-.__I
A
A 0.50
‘F
045
-
0.40
~. ‘I
et
A
A
A
2
3
4
s
6
Fig. 1. D(0) : E for the logistic equation.
7
6
E
The nonlinear dynamics of metal cutting
1435
converge independently of the magnitude of D(y). The alternative fixed size method, sphere radius is fixed and population determined, does not converge for D(0) > 3 [lo]. As an example of the computation of D(O), consider the attractor associated with the logistic mapf(x) = r2x(l -x)_ The Hausdorff dimension for )L= 3.57 is known to be 0.538 151. Figure 1 shows D(0) as a function of E, D(0) :E, determined by (3) with m = 800, n = ~,~, y = 0, k = 200, 1 zzzE 59. In all cases, the determination of D(0) from (3) was based on reconstructions of the attractor formed by delaying x by i = 1, . . . , E multiples of A, Xi(~) = x(r + (i - 1)A) where the iteration index r and A are positive integers [3]. From Fig. 1, it appears that small values of A result in convergence to the exact solution for large values of E while large values of A result in an earlier attainment of the exact value with a slight overshoot for larger values of E. Similar behavior was noted in [7]. In the following, (3) is used in the study of a cutting model described by a system of differential delay equations and experimentally generated orthogonal cutting data.
MATHEMATICAL
MODEL
In [2], the cutting tool, supporting structure and cutting force are modeled by
& = -&$i x
f,
=
[a,@4
-a4x~
x
-
[%(X4
-
07x3
+
-X4@
a&x;?
-X40
(4)
&=x4
(5)
a&(t
-
-
i, =.X3
-
d)
-
agx4
-
a
-
-
r)
+
a2(X3
-
xl(t
-X3@
a,x,(t
%(X3
(x,
-
-
-
-
t))
r))]
z)
+
(x*
X3@
-
$)I
-
-
&I$
x,(t
-
62d
(6)
-
z))
(7)
where x1 and x2 = tool deflections normal and parallel to the cutting surface, respectively. The following physical properties are assumed: feed rate so = 0.13 mm/rev., cutting speed, u,= 3 m/s, the delay, r, takes on various values, shear modulus of cut material, CY= 6.0 e 10aN/m2, equivalent mass, stiffness and linear damping coefficients for the normal and parallel directions are 83.1 kg, 48.86 kg, 41.9 - lo6 N/m, 111 - lo6 N/m and 15 - ld N s/m, 13.3 - ld N s/m, respectively, depth of cut, w = 0.0052 m and tool nose damping coefficient, k = 0, cubic damping coefficients, a,, and al2 = 1.0 s2/m2. The coefficients, ui, in (6,7) follow from table 1, part 2 of [2], a, = 0.05212~, a2 = 0.01545~, a3 = O.O0426arw, a4 = 0.10557~, u, = 0.1805 - 103, a8 = 0+05212m + 0.5042 * 106, a9 = us = 0.01547&w, U6= 0.~17~, 0.2722 - ld and al0 = 2.2738. 106; initial conditions were x = 0 for -r r: t < 0 and x = (0.1 - 10m3}. I at t = 0. a,, and al2 were chosen to insure bounded solutions [ll].
EVALUATION
OF D(O)
All numerical solutions of (4)-(7) were calculated using an Adams-Bashforth predictor and an Adams-Moulton corrector both of 0(h4) with h = (0.2) 0 10e3 s. A plot of x2 vs x1 is shown in Fig. 2 for a delay of t = 0.1 s. From the figure, the attractor appears to be a limit cycle. This is corroborated by the plot of D(O):E for n = 75,000, m = 3000 with k = 100, 200 and 300 corresponding to the upper, middle and lower points in the triad of values of D(0) shown for each value of E. It is seen that for k = 300, D(0) has converged to 1.0 for E 2.2. D(0) for k = 100 and 200 initially overshoots 1.0, but the asymptotic value of D(0) = 1.0 is reached for E 2 9, Fig. 3.
1436
B. S. BERGER
et al.
-0.0014
0.0014
x2 o.OQ70
0.0042
0.0014
-0.0014
-0.0042
-0M70 -0.0070
-0.0042
Fig. 2. xzvsx,,
0.0042
0.0070
z=O.ls.
Increasing the delay to z = 0.4 s results in the complex behavior shown in Figs 4-6. These are 120~t~135and14O~tz~155 projections of the attractor onto the x2, x,planefor6Oz%tzz75, respectively. In this case a dimension of D(0) = 2.65 f 0.30 was found [ll]. D(0) may be found for experimentally determined point sets provided that the power levels associated with noise are not excessive and that data acquisition times are short enough to prevent the occurrence of experimental drift ]8]. The question as to whether or not D(0) may be approximated for chatter data can be answered in the affirmative. D(O) was computed for
D(O)
.600
.400
,200
0.
i 0.
I
I
I
I
I
3.50
7.00
10.5
14.0
17.5
Fig. 3. D(O):&
t=O.is.
,
21 .o
E
The nonlinear dynamics of metal cutting
1437
x4 5
-
3
-
1
-
-1
-
-3
-
-5
8
’
-0.005
-0.003
-0.001 Fig. 4. xqvsx2,
0.001 t=0.4s,
0.003
x2
0.005
6Ost~75.
two sets of measurements of f, the relative tool-workpiece acceleration normal to the cutting surface. Geometric and material parameters were held constant except for the depth of cut w. Data sets were ident~ed by visual and auditory observations as pre-chatter and mild chatter cutting states corresponding to depths of cut of w = 2.2 mm and 2.3 mm, respectively. Figures 7 and 8 give D(O):E with m=3000, n=60,000, w=2.2mm, and m=3000, n=70,000, w = 2.3 mm. A plateau or region of reduced slope is evident in D(0) : E for 3 s E I 7. This is x4 5
-
3
-
1
-
-1
.
-3
.
-5
’
-0.005
1 x2 -0.003
-0.001 Fig. 5. x.,vsxz,
0.001 r=O.4s,
14O~t~155.
0.003
0.005
B. S. BERGER
et al.
x4 5
3
1
-1
-3
4 x2
-5 -0.005
-0.001
-0.003
Fig. 6. x4vsx2,
0.001 r=O.4s,
0.003
0.005
14Ost=155.
6.0
3.5
7.0
10.5
14.0
E
Fig. 7. D(O):&
w = 2.2 mm.
17.5
21.0
1439
The nonlinear dynamics of metal cutting 6.0
-
1.0
)
t
0
I
I
I
I
I
I
35
70
10 5
14.0
17.5
21.0
E
Fig. 8. D(O):& w = 2.3 mm.
indicative of the presence of for w = 2.2 mm, and 2.3 mm, D(0): E for E = 4,5,6 and demonstrating a known affect
an attractor for which D(0) = 2.38 f 0.10 and D(0) = 2.51 f 0.08 respectively. This estimate is the average and mean deviation of k = 100, 200 and 300. For E > 8, D(0): E increases with E of noise on D(0) : E for large values of E.
CONCLUSIONS
It has been shown that despite the presence of noise estimates of the dimension, D(O), of attractors associated with orthogonal cutting data can be found. For the pre and mild chatter cases D(0) = 2.38 f 0.10 and D(0) = 2.51 f 0.08 for w = 2.2 mm and 2.3 mm respectively. A model of the cutting dynamics based on the Wu-Liu theory [2], was expressed as a system of non-linear differential delay equations. Although the system is infinite dimensional it was shown that the associated attractor is of low dimension. For r = 0.1 s, heavy chatter, D(0) = 1.0 indicating limit cycle behavior while D(0) = 2.63 f 0.3 for t = 0.4 s. Non-integer values of D(0) for the experimental data indicate that nonlinearities play an important role in cutting dynamics. Acknowledgements-The authors wish to take this opportunity to gratefully acknowledge the continuing stimulus and inspiration which Professor A. C. Eringen has provided the mechanics community. The assistance of Dr G. Broggi and J. Young, the support of the Computer Science Center and the computer facility of the Department of Mechanical Engineering, University of Maryland, is acknowledged.
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