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Explanation of random vibrations in cutting on grounds of deterministic chaos
Robotics & Computer-Integrated Manufacturing, Vol. 4, No. 1/2, pp. 129-134, 1988
0736-5845/88 $3.00 + 0.00 Pergamon Press plc
Printed in Great Britain.
• Paper
E X P L A N A T I O N O F R A N D O M V I B R A T I O N S IN C U T T I N G ON G R O U N D S O F DETERMINISTIC CHAOS I. G R A B E C Fakulteta za Strojni~tvo, 61000 Ljubljana, Yugoslavia This article presents a model of orthogonai cutting on an elastic system. The dynamics of the system is described by the differential equations in which the forces are expressed by empirical nonlinear relations. The oscillations of the system determined by numerical solutions of nonlinear differential equations exhibit chaotic characters in certain domains of typical parameters and thus represent a new example of deterministic chaos. This indicates that, besides the irregular properties of cut material, nonlinear dynamics must also be taken into account in order to explain the random character of vibrations observed in cutting systems.
INTRODUCTION Cutting processes can excite vibrations in a manufacturing system which diminish the surface quality of products. The explanation of the generation mechanism of vibrations therefore represents an essential step in the effort to improve the operation of manufacturing systems.1 The material and surface properties of the workpiece are usually more or less irregular so that the forces generated in cutting fluctuate randomly and can excite vibrations in a nonrigid cutting system. 2 It seems therefore acceptable to describe the character of the forces separately from the response characteristics of the system. However, random vibrations can be excited also by the cutting of very homogeneous workpieces with polished surfaces, such as plastic rods or plates. 3 In such cases random character and vibrations themselves are inherently determined by the dynamics of the complete cutting system. This indicates that the dynamical effects can be as important for the explanation of random vibrations in cutting as the irregularity of the applied raw material. The purpose of this paper is to confirm this supposition by explaining the dynamical behavior of a simple model of an orthogonal cutting system. It is well known from dynamics that random vibrations can be inherently generated only in unstable nonlinear systems.4 The main nonlinearity in a cutting system is related to the deformation zone, while the rest of the workpiece, the tool and the frame of the system can be treated as elastic parts. 1 The properties of the deformation zone have been
studied extensively, so that we can describe the dependence of cutting forces on system parameters by the empirical relations of other authors. 5 The inclusion of these relations into the dynamic description of the cutting system model leads to a system of nonlinear differential equations. Numerically determined solutions of the equations indeed exhibit random oscillations in certain domain of cutting parameters. As they describe the movement of a deterministic system stemming from fixed initial conditions, they represent an example of deterministic chaos. DESCRIPTION OF THE CUTTING SYSTEM The basic material removal process can be represented by a wedge shaped cutting edge. We limit our attention to the most simple case of orthogonal machining in which the straight cutting edge is set parallel to the workpiece surface and normal to the cutting direction. The cross section of the cutting zone can then be represented by the scheme in Fig. 1. In order to formulate the system properties we make the following assumptions. The cut width is small compared with its depth so that the cutting process can be described in two dimensions only. The workpiece of homogeneous material and straight surface is pushed against the knife with constant velocity vi. The cut depth at the begining is set to hi. The knife, the frame and the workpiece altogether can be represented as a two-dimensional oscillator (Fig. 2) which is coupled only by the cutting zone. The flow of the material against the knife 129
130
Robotics & Computer-lntegrated Manufacturing • Volume 4, Number 1/2, 1988
Both values as well as the constants c1, c 2 and c3 are specific for a given material. 5 It is well known that the friction coefficient depends on the tangential velocity Vd between surfaces in contact. This velocity is reduced with respect to flow velocity for a certain factor R:
V
¥
I
V (6)
Vd - R "
Fig. 1. Scheme of cutting process. m
Fj
With this velocity the friction coefficient can be expressed as
k
I
c
K=Ko(ce(Vv'Ro
generates the cutting force F = (Fx, Fy) which causes displacement ( x , y ) of the knife edge. Due to the elastic properties of the cutting system, the knife displacement can be determined from differential equations mJ~ + rxk + k x x = F~
(1)
m y + ryy + k y y = Fy.
(2)
In these equations m denotes the inertial mass of the system, r and k are coefficients of damping and rigidity and k = d t / d t . The position of the unloaded knife edge corresponds to x = 0 and y = 0, and the right angle 0t = 0 is assumed. The flow of the material generates the main cutting force F x and the friction force Fy which can be expressed by the friction coefficient K: (3)
Fy = K.Fx.
The main cutting force and friction coefficient depend on the material and the tool properties as well as on the cut depth and velocity. For a wide class of technical materials cut under steady conditions, the dependence on cut depth h and flow velocity v can be approximately described by the nonlinear relations: 5
K=Ko(c2(~o-1)2+
) )
- 1
+ 1
(7)
(8)
R = ctg rk.
Fig. 2. Model of elastic cutting system.
1
+ 11(c3(~-~-1f).
By the reduction factor a characteristic shear angle ~b is determined through the relation
\\\
F~ =F~0h--~0
-l)Z
(4)
1)(c3(ff--S - 1)2+ 1). (5)
Shear angle depends on the flow velocity and the cut depth but the latter dependence is weak so that R can be approximately expressed by
Here again R0 and c4 are specific parameters of cut material. Under nonsteady cutting the cut depth, the flow and friction velocity are changing due to the moving knife tip. The following expressions: h =hi-y;v
=vi-k;
must be inserted into equations describing cutting forces. Beside this it must be taken into account that F ~ = 0 for h < 0 or v < 0, K = 0 for F ~ < 0 and K < 0 for r e < 0. All these conditions can be simply expressed by the help of unit step function U and signum function Sgn. For further treatment, it is also convenient to introduce the nondimensional variables. X = x/ho; Y = y/ho; T = t'vo/ho = rwo; X' = dX/dT
= k/Vo; H = Hi - Y = (h, - y ) / h o ;
V = (v, - k)/Vo = Vi - X ' .
(11)
The dynamical equations of the cutting system can then be represented in the form: X" + D~X' + AX
= F
Y" + DyY' + BY = KF
The parameters v0 and ho represent the values at which the cutting force and friction coefficient are minima.
(10)
(12) (13)
with: F = F o H ( C ~ ( V - 1) / + 1)U(H) U ( V )
(14)
Random vibrations in cutting • I. GRABEC
131
choose H i = hi/ho = 0.5 and Vi = vi/vo = 0.5. F
/
0
j
'/
Fig. 3. Dependence of main cutting force on velocity.
1) 2 + 1) (C3(H - 1) z + 1) • U ( F ) . Sgn ( V / )
K : Ko(Q(V/-
(15)
R = R o ( C 4 ( V - 1) 2 + 1)
(16)
V t = V - RY'
(17)
A : k x / m W ~ = ( W J W o ) z, B = ky/mW:o = ( W ~ / W o ) z, Dx = r d m Wo2
(18)
Fo = Fxo/homW~.
(19)
The dependence of the main cutting force on V is shown in Fig. 3. Similar to this is also the dependence of the friction coefficient on friction velocity. Due to the negative slope of both dependencies in velocity interval from 0 to 1, an instability similar to the stick-slip p h e n o m e n o n can develop in the cutting system. The corresponding oscillations in both degrees of freedom are coupled through the cutting force represented by nonlinear terms. It is well known that coupled nonlinear oscillators can exhibit chaotic movement without random excitation from outside, which can be expected also in the cutting system. Unfortunately, the complicated, nonlinearity of the derived system of differential equations ~3'14 prevents us from finding analytically its solution which would confirm this expectation. Therefore, we proceed to find its solution numerically for the case of natural initial conditions X(0) = Y(0) -- 0 and X ' (0) = Y' (0) - 0.
SPECIFICATIONS OF THE PARAMETERS AND NUMERICAL SOLUTION For a wide class of materials such as low carbon content steals, the specific parameters of the cutting process are approximately given by the following set of values: s c 1 = 0 . 3 ; c 2 = 0 . 7 ; c 3 = 1 . 5 ; Ca=1.2; R 0 = 2 . 2 , h0 = 0 . 2 5 m m ; v 0 = 6.6ms-1; k 0 = 0.36, W 0 = 2 . 7 x 104s -1. In order to obtain unstable system it is reasonable to
The resonant frequency of a free knife clamped into a frame on a lathe corresponds to the order of m a g n i t u d e 103 Hz. For the sake of simplicity we therefore choose w~0 = w0 to obtain A = 1. The rigidity of the cutting system in y direction is usually less than that in the x direction, so that the selection B = 1 is reasonable. The damping of the cutting system is normally weak. Here we assume Dx = D1 = D < 1. The only unspecified parameter is then the force F0 which is approximately proportional to the arbitrary chip width. A strong dynamical effect could be expected if the steady state friction force corresponds to the displacement of the knife for hi in the y direction. This corresponds to Y = y/ho ~ hi/ho ~ 0.5 or K F / B = Y -~ 0.5. From Eqs (14) to (16) and steady state conditions: H = 1/2, V = 1/2 it follows then F0 = 1.6 B. The dependence of the cutting force on y stabilizes, so that under intensive cutting by F0 = 1.6 B, the knife is indeed not pushed out of the material. From the meaning of the parameter F 0 one can expect that the amplitude of oscillations in the cutting system mainly depends on it. It is therefore instructive to find solutions for various values of F0. For the purpose of a numerical treatment the differentials in Eqs (12) and (13) are represented by differences of sample values. It is reasonable to take the width of the sampling interval T1 as much smaller than the typical period To = 2n/x/A of the free oscillator. In our calculations we use T1 = 0.025 and apply Euler's method of integration of ordinary differential equations. Figures 4 - 8 show grpahs of calculated functions X(T) and Y(T) for various combinations of parameters F0, B and D. Figure4 shows a typical example of random oscillations obtained with parameters F0 = 0.45, B = 0.3 and D = 0 which correspond to intensive cutting (F0 = 1.5 B). In the x and y directions the characteristic periods are approximately 7 and 9 which closely correspond to periods of free knife oscillations. Nonlinear coupling between both degrees of freedom mainly causes random variation of amplitude and phase of each mode, while the frequences are not influenced appreciably. Both records represent an example of deterministic chaos. Figures 5 and 6 demonstrate the influence of diminished value of F0 at B = 0.3 and D = 0. With decreasing F0 vibrations exhibit even longer transient intervals at the beginning of cutting after which stationary and nearly periodic oscillations are established. The oscillations in the x direction are not
132
Robotics & Computer-Integrated
M a n u f a c t u r i n g • V o l u m e 4, N u m b e r 1 / 2 , 1 9 8 8
IvV~~V"~V'~Vv vlvvlvIvvv~Ivl~vv'qvVV~I~~Iv~ vl~I~vvv~
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Fo
,'v
=
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;
B
vv~
=
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;
I3 =
v~v~/'~
,~y~/~IT=5OO
0
Fig. 4. O s c i l l a t i o n s o f a n u n d a m p e d s y s t e m w i t h F = 0 . 4 5 , B = 0 . 3 ; D = 0.
IIAIIIIIIIIII/III AiL IIAII IIIII/IIIIII/II I//II/ IIIAII II vv'vvvvvvvvvvvvlvvlvvlvlvvvVVVIvvVvIWlWVWVVVVVVVlVVlVVivvvvlvvvVwwlwvwl
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oX~^~^~^^^AA ~AAAAIAAAAIAAAAAAIAAIAIAIIIAIIIAIAAIAAIAIAIIAIIAIAAAAIAAIIIIIIII i '"'~""~'vv'VvvvwvvvvvvvvwlvvvvvvvVlVVVVVlIIWVVWlVVlWVV'VlVVlIVIVVWVVVl 1 v
'o~AAAAAAAAA .vvv.~vv .,.
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;
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Fig. 6. S o l u t i o n s o b t a i n e d f o r F 0 = 0 . 1 5 ; B = 0 . 3 ; D = 0.
,
IT=500
133
Random vibrations in cutting • I. GRABEC
IvllVIIV1V IIllVI/llll/lllltlll l/lII/ltllll/tltl/ll//lllllllt
vVVVVVVVVVVVVVVVVVVVVI?VIVV.V2 I' - VVV VV Fig. 7. Solutions obtained for F0 = 0.45; B = 1; D = 0.
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Fo
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I
;
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.3
;
I'
/]
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~
'
IT=500
. I
Fig. 8. Oscillations of a damped system with F0 = 0.45; B = 0.3; D = 0.1. influenced appreciably while the oscillations in the y direction are drastically decreased with decreasing F 0. The reasons for this effect are still not explained. Figure 7 corresponds to F0 = 0.45, B = 1 and D = 0. In this case, the frequencies of a free knife in both directions are the same. This results in strong resonant coupling between both modes and amplified oscillations. As would be expected the phase shift between both oscillations is 180 ° . In spite of greater rigidity in the y direction with that corresponding to Fig. 4, the amplitude of oscillations is not decreased. It appears that the equivalence of both free knife frequencies is not favorable for the cutting process. Figure 8 c o r r e s p o n d s to a d a m p e d knife (F0 = 0.45; B = 0.3; D = 0.1). Increased damping
prevents development of proper oscillations in the y direction, so that mainly the oscillations in x direction are left. Their influence on the y component is weak. CONCLUSIONS Our treatment has shown that the most simple model of orthogonal cutting on an elastic system exhibits nonlinear oscillations which exhibit a random character at least in certain domains of characteristic parameters. It seems that the random character is a consequence of nonlinear coupling of strong oscillations with different frequencies. At the amplitudes of oscillations determined by our calculations, nonlinear effects do not prevail. This might be
134
Robotics & Computer-Integrated Manufacturing • Volume 4, Number 1/2, 1988
also a consequence of the fact that the empirical nonlinear relations are always obtained by some averaging process in which essential nonlinear properties can be lost. Detailed properties of deterministic chaos in cutting process are now still under investigation.
REFERENCES 1. Tobias, S.A.: Schwingungen an Werkzeugmaschinen. Miinchen, Carl Hauser, 1961.
2. Peklenik, J., Mosedale, T., Kwiatkowski, A.W.: Proceedings of the International Machine Tool Design and Research Conference. Oxford, Pergamon Press, pp. 209 683, 1967. 3. Thomson, P.J., Sarwar, M.: Proceedings of the 15th International Machine Tool Design and Research Conference. London, MacMillan, p. 218, 1974. 4. Lichtenberg, A.J., Lieberman, M.A.: Regular and Stochastic Motion. Berlin, Springer, 1983. 5. Hastings, W.F., Oxley, P.L.B., Stevenson, M.G.: Proceedings of the 12th International Machine Tool Design and Research Conference. London p. 507, 1971.
0736-5845/88 $3.00 + 0.00 Pergamon Press plc
Printed in Great Britain.
• Paper
E X P L A N A T I O N O F R A N D O M V I B R A T I O N S IN C U T T I N G ON G R O U N D S O F DETERMINISTIC CHAOS I. G R A B E C Fakulteta za Strojni~tvo, 61000 Ljubljana, Yugoslavia This article presents a model of orthogonai cutting on an elastic system. The dynamics of the system is described by the differential equations in which the forces are expressed by empirical nonlinear relations. The oscillations of the system determined by numerical solutions of nonlinear differential equations exhibit chaotic characters in certain domains of typical parameters and thus represent a new example of deterministic chaos. This indicates that, besides the irregular properties of cut material, nonlinear dynamics must also be taken into account in order to explain the random character of vibrations observed in cutting systems.
INTRODUCTION Cutting processes can excite vibrations in a manufacturing system which diminish the surface quality of products. The explanation of the generation mechanism of vibrations therefore represents an essential step in the effort to improve the operation of manufacturing systems.1 The material and surface properties of the workpiece are usually more or less irregular so that the forces generated in cutting fluctuate randomly and can excite vibrations in a nonrigid cutting system. 2 It seems therefore acceptable to describe the character of the forces separately from the response characteristics of the system. However, random vibrations can be excited also by the cutting of very homogeneous workpieces with polished surfaces, such as plastic rods or plates. 3 In such cases random character and vibrations themselves are inherently determined by the dynamics of the complete cutting system. This indicates that the dynamical effects can be as important for the explanation of random vibrations in cutting as the irregularity of the applied raw material. The purpose of this paper is to confirm this supposition by explaining the dynamical behavior of a simple model of an orthogonal cutting system. It is well known from dynamics that random vibrations can be inherently generated only in unstable nonlinear systems.4 The main nonlinearity in a cutting system is related to the deformation zone, while the rest of the workpiece, the tool and the frame of the system can be treated as elastic parts. 1 The properties of the deformation zone have been
studied extensively, so that we can describe the dependence of cutting forces on system parameters by the empirical relations of other authors. 5 The inclusion of these relations into the dynamic description of the cutting system model leads to a system of nonlinear differential equations. Numerically determined solutions of the equations indeed exhibit random oscillations in certain domain of cutting parameters. As they describe the movement of a deterministic system stemming from fixed initial conditions, they represent an example of deterministic chaos. DESCRIPTION OF THE CUTTING SYSTEM The basic material removal process can be represented by a wedge shaped cutting edge. We limit our attention to the most simple case of orthogonal machining in which the straight cutting edge is set parallel to the workpiece surface and normal to the cutting direction. The cross section of the cutting zone can then be represented by the scheme in Fig. 1. In order to formulate the system properties we make the following assumptions. The cut width is small compared with its depth so that the cutting process can be described in two dimensions only. The workpiece of homogeneous material and straight surface is pushed against the knife with constant velocity vi. The cut depth at the begining is set to hi. The knife, the frame and the workpiece altogether can be represented as a two-dimensional oscillator (Fig. 2) which is coupled only by the cutting zone. The flow of the material against the knife 129
130
Robotics & Computer-lntegrated Manufacturing • Volume 4, Number 1/2, 1988
Both values as well as the constants c1, c 2 and c3 are specific for a given material. 5 It is well known that the friction coefficient depends on the tangential velocity Vd between surfaces in contact. This velocity is reduced with respect to flow velocity for a certain factor R:
V
¥
I
V (6)
Vd - R "
Fig. 1. Scheme of cutting process. m
Fj
With this velocity the friction coefficient can be expressed as
k
I
c
K=Ko(ce(Vv'Ro
generates the cutting force F = (Fx, Fy) which causes displacement ( x , y ) of the knife edge. Due to the elastic properties of the cutting system, the knife displacement can be determined from differential equations mJ~ + rxk + k x x = F~
(1)
m y + ryy + k y y = Fy.
(2)
In these equations m denotes the inertial mass of the system, r and k are coefficients of damping and rigidity and k = d t / d t . The position of the unloaded knife edge corresponds to x = 0 and y = 0, and the right angle 0t = 0 is assumed. The flow of the material generates the main cutting force F x and the friction force Fy which can be expressed by the friction coefficient K: (3)
Fy = K.Fx.
The main cutting force and friction coefficient depend on the material and the tool properties as well as on the cut depth and velocity. For a wide class of technical materials cut under steady conditions, the dependence on cut depth h and flow velocity v can be approximately described by the nonlinear relations: 5
K=Ko(c2(~o-1)2+
) )
- 1
+ 1
(7)
(8)
R = ctg rk.
Fig. 2. Model of elastic cutting system.
1
+ 11(c3(~-~-1f).
By the reduction factor a characteristic shear angle ~b is determined through the relation
\\\
F~ =F~0h--~0
-l)Z
(4)
1)(c3(ff--S - 1)2+ 1). (5)
Shear angle depends on the flow velocity and the cut depth but the latter dependence is weak so that R can be approximately expressed by
Here again R0 and c4 are specific parameters of cut material. Under nonsteady cutting the cut depth, the flow and friction velocity are changing due to the moving knife tip. The following expressions: h =hi-y;v
=vi-k;
must be inserted into equations describing cutting forces. Beside this it must be taken into account that F ~ = 0 for h < 0 or v < 0, K = 0 for F ~ < 0 and K < 0 for r e < 0. All these conditions can be simply expressed by the help of unit step function U and signum function Sgn. For further treatment, it is also convenient to introduce the nondimensional variables. X = x/ho; Y = y/ho; T = t'vo/ho = rwo; X' = dX/dT
= k/Vo; H = Hi - Y = (h, - y ) / h o ;
V = (v, - k)/Vo = Vi - X ' .
(11)
The dynamical equations of the cutting system can then be represented in the form: X" + D~X' + AX
= F
Y" + DyY' + BY = KF
The parameters v0 and ho represent the values at which the cutting force and friction coefficient are minima.
(10)
(12) (13)
with: F = F o H ( C ~ ( V - 1) / + 1)U(H) U ( V )
(14)
Random vibrations in cutting • I. GRABEC
131
choose H i = hi/ho = 0.5 and Vi = vi/vo = 0.5. F
/
0
j
'/
Fig. 3. Dependence of main cutting force on velocity.
1) 2 + 1) (C3(H - 1) z + 1) • U ( F ) . Sgn ( V / )
K : Ko(Q(V/-
(15)
R = R o ( C 4 ( V - 1) 2 + 1)
(16)
V t = V - RY'
(17)
A : k x / m W ~ = ( W J W o ) z, B = ky/mW:o = ( W ~ / W o ) z, Dx = r d m Wo2
(18)
Fo = Fxo/homW~.
(19)
The dependence of the main cutting force on V is shown in Fig. 3. Similar to this is also the dependence of the friction coefficient on friction velocity. Due to the negative slope of both dependencies in velocity interval from 0 to 1, an instability similar to the stick-slip p h e n o m e n o n can develop in the cutting system. The corresponding oscillations in both degrees of freedom are coupled through the cutting force represented by nonlinear terms. It is well known that coupled nonlinear oscillators can exhibit chaotic movement without random excitation from outside, which can be expected also in the cutting system. Unfortunately, the complicated, nonlinearity of the derived system of differential equations ~3'14 prevents us from finding analytically its solution which would confirm this expectation. Therefore, we proceed to find its solution numerically for the case of natural initial conditions X(0) = Y(0) -- 0 and X ' (0) = Y' (0) - 0.
SPECIFICATIONS OF THE PARAMETERS AND NUMERICAL SOLUTION For a wide class of materials such as low carbon content steals, the specific parameters of the cutting process are approximately given by the following set of values: s c 1 = 0 . 3 ; c 2 = 0 . 7 ; c 3 = 1 . 5 ; Ca=1.2; R 0 = 2 . 2 , h0 = 0 . 2 5 m m ; v 0 = 6.6ms-1; k 0 = 0.36, W 0 = 2 . 7 x 104s -1. In order to obtain unstable system it is reasonable to
The resonant frequency of a free knife clamped into a frame on a lathe corresponds to the order of m a g n i t u d e 103 Hz. For the sake of simplicity we therefore choose w~0 = w0 to obtain A = 1. The rigidity of the cutting system in y direction is usually less than that in the x direction, so that the selection B = 1 is reasonable. The damping of the cutting system is normally weak. Here we assume Dx = D1 = D < 1. The only unspecified parameter is then the force F0 which is approximately proportional to the arbitrary chip width. A strong dynamical effect could be expected if the steady state friction force corresponds to the displacement of the knife for hi in the y direction. This corresponds to Y = y/ho ~ hi/ho ~ 0.5 or K F / B = Y -~ 0.5. From Eqs (14) to (16) and steady state conditions: H = 1/2, V = 1/2 it follows then F0 = 1.6 B. The dependence of the cutting force on y stabilizes, so that under intensive cutting by F0 = 1.6 B, the knife is indeed not pushed out of the material. From the meaning of the parameter F 0 one can expect that the amplitude of oscillations in the cutting system mainly depends on it. It is therefore instructive to find solutions for various values of F0. For the purpose of a numerical treatment the differentials in Eqs (12) and (13) are represented by differences of sample values. It is reasonable to take the width of the sampling interval T1 as much smaller than the typical period To = 2n/x/A of the free oscillator. In our calculations we use T1 = 0.025 and apply Euler's method of integration of ordinary differential equations. Figures 4 - 8 show grpahs of calculated functions X(T) and Y(T) for various combinations of parameters F0, B and D. Figure4 shows a typical example of random oscillations obtained with parameters F0 = 0.45, B = 0.3 and D = 0 which correspond to intensive cutting (F0 = 1.5 B). In the x and y directions the characteristic periods are approximately 7 and 9 which closely correspond to periods of free knife oscillations. Nonlinear coupling between both degrees of freedom mainly causes random variation of amplitude and phase of each mode, while the frequences are not influenced appreciably. Both records represent an example of deterministic chaos. Figures 5 and 6 demonstrate the influence of diminished value of F0 at B = 0.3 and D = 0. With decreasing F0 vibrations exhibit even longer transient intervals at the beginning of cutting after which stationary and nearly periodic oscillations are established. The oscillations in the x direction are not
132
Robotics & Computer-Integrated
M a n u f a c t u r i n g • V o l u m e 4, N u m b e r 1 / 2 , 1 9 8 8
IvV~~V"~V'~Vv vlvvlvIvvv~Ivl~vv'qvVV~I~~Iv~ vl~I~vvv~
AAAAAAAMAAAAAAAAAAAAAMAAAAAAAA VVY
......
Fo
,'v
=
VV,VVvY,~
.45
;
B
vv~
=
.3
vvvv" ~
;
I3 =
v~v~/'~
,~y~/~IT=5OO
0
Fig. 4. O s c i l l a t i o n s o f a n u n d a m p e d s y s t e m w i t h F = 0 . 4 5 , B = 0 . 3 ; D = 0.
IIAIIIIIIIIII/III AiL IIAII IIIII/IIIIII/II I//II/ IIIAII II vv'vvvvvvvvvvvvlvvlvvlvlvvvVVVIvvVvIWlWVWVVVVVVVlVVlVVivvvvlvvvVwwlwvwl
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.
.
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.
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.
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.
.
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;
B =
'T=5 O0
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Fig. 5. S o l u t i o n s o b t a i n e d f o r F 0 = 0 . 3 ; B = 0.3; D = 0.
oX~^~^~^^^AA ~AAAAIAAAAIAAAAAAIAAIAIAIIIAIIIAIAAIAAIAIAIIAIIAIAAAAIAAIIIIIIII i '"'~""~'vv'VvvvwvvvvvvvvwlvvvvvvvVlVVVVVlIIWVVWlVVlWVV'VlVVlIVIVVWVVVl 1 v
'o~AAAAAAAAA .vvv.~vv .,.
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,
,
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,
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=
.15
'
;
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.3
Fig. 6. S o l u t i o n s o b t a i n e d f o r F 0 = 0 . 1 5 ; B = 0 . 3 ; D = 0.
,
IT=500
133
Random vibrations in cutting • I. GRABEC
IvllVIIV1V IIllVI/llll/lllltlll l/lII/ltllll/tltl/ll//lllllllt
vVVVVVVVVVVVVVVVVVVVVI?VIVV.V2 I' - VVV VV Fig. 7. Solutions obtained for F0 = 0.45; B = 1; D = 0.
o AIIIIII AAIIII A III IIIII IIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIII vVv'vvvIvvvff VVVVV VV IVfflV Vt IVV IIVffYffI II VY VVI VlVVI Vff IVIIIVffVV/
l
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;
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IT=500
. I
Fig. 8. Oscillations of a damped system with F0 = 0.45; B = 0.3; D = 0.1. influenced appreciably while the oscillations in the y direction are drastically decreased with decreasing F 0. The reasons for this effect are still not explained. Figure 7 corresponds to F0 = 0.45, B = 1 and D = 0. In this case, the frequencies of a free knife in both directions are the same. This results in strong resonant coupling between both modes and amplified oscillations. As would be expected the phase shift between both oscillations is 180 ° . In spite of greater rigidity in the y direction with that corresponding to Fig. 4, the amplitude of oscillations is not decreased. It appears that the equivalence of both free knife frequencies is not favorable for the cutting process. Figure 8 c o r r e s p o n d s to a d a m p e d knife (F0 = 0.45; B = 0.3; D = 0.1). Increased damping
prevents development of proper oscillations in the y direction, so that mainly the oscillations in x direction are left. Their influence on the y component is weak. CONCLUSIONS Our treatment has shown that the most simple model of orthogonal cutting on an elastic system exhibits nonlinear oscillations which exhibit a random character at least in certain domains of characteristic parameters. It seems that the random character is a consequence of nonlinear coupling of strong oscillations with different frequencies. At the amplitudes of oscillations determined by our calculations, nonlinear effects do not prevail. This might be
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also a consequence of the fact that the empirical nonlinear relations are always obtained by some averaging process in which essential nonlinear properties can be lost. Detailed properties of deterministic chaos in cutting process are now still under investigation.
REFERENCES 1. Tobias, S.A.: Schwingungen an Werkzeugmaschinen. Miinchen, Carl Hauser, 1961.
2. Peklenik, J., Mosedale, T., Kwiatkowski, A.W.: Proceedings of the International Machine Tool Design and Research Conference. Oxford, Pergamon Press, pp. 209 683, 1967. 3. Thomson, P.J., Sarwar, M.: Proceedings of the 15th International Machine Tool Design and Research Conference. London, MacMillan, p. 218, 1974. 4. Lichtenberg, A.J., Lieberman, M.A.: Regular and Stochastic Motion. Berlin, Springer, 1983. 5. Hastings, W.F., Oxley, P.L.B., Stevenson, M.G.: Proceedings of the 12th International Machine Tool Design and Research Conference. London p. 507, 1971.