- Email: [email protected]
Optimal passive vibration control of cutting process stability in milling
Journal of Materials Processing Technology, 28 ( 1991 ) 285-294
Elsevier
Optimal passive vibration control of cutting process stability in
milling
K. J. Liu and K. E. Rouch Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506, U.S.A.
Abstract The paper proposes a design concept for an optimal passive dynamic absorber in the milling process. An investigation and analysis into the characteristics of, and interaction between, forced vibration and linear chatter in milling has been carried out by use of a two-degree--of-freedom structural model simulation. A dynamic absorber mass is then connected to the main system through passive elements (spring and damper). Optimization procedures to determine the optimum values of spring and damping coefficients for a specific mass ratio have been developed by employing the ADS optimization package. A theoretical approach for the prediction of machine tool chatter in milling including dynamic cutting forces permits calculation of borderlines of stability for a structure model with an optimal absorber. Compared with the limit of stability under no control, the stable range under optimal passive control has been improved over one hundred percent. Transient responses can also be obtained from numerical integration of the system's equation of motion in the state variable representation based on the fourth-order Runge-Kutta method. The theoretical predictions can be verified by the results of numerical simulation of transient response.
1. INTRODUCTION In the machining operation, a violent relative vibration between the workpiece and the cutting tool, which is called chatter, is developed under some operating conditions. Chatter is not desirable because it has an adverse effect on the machining accuracy, the surface finish, and the tool life, and also because it results in reduced output by requiring a lower metal removal rate to obtain vibration-free performance. Therefore, the prediction of machine chatter and the vibration control in the machining operation becomes increasingly significant. The major step towards understanding the regenerative machine tool chatter was presented by Merritt (1965). This study, which focused on a stability theory for regenerative chatter in an orthogonal turning process with a feedback loop, was based on the theory of steady orthogonal cutting process given by Merchant (1944). It has been found that the onset of machine chatter is a function of the spindle speed. Nigm et al. (1973, 1977a,b) provided a recent contribution to the study of 0924-0136/91/$03.50 © 1991 ElsevierSciencePublishers B.V. All fightsreserved.
286 the dynamics of oscillatory metal cutting and the prediction of the dynamic cutting coefficients from steady--state cutting data. Wu and Liu (1985a,b) also developed a mathematical model of machine chatter, using an analytical approach to predict dynamic cutting forces from the steady--state cutting, tests. Although machine tool chatter occurs in many different machining operations, most of the research work has traditionally focused on the turning operation, which is a simple and basic metal cutting process. In turning, the cutting forces are fixed in direction relative to the machine tool structure, while they rotate with respect to the structure in milling. The extension from turning to milling results in a time-varying dynnmlc system. The earlier detailed mathematical models of the milling system have been proposed by Sridhar et al. (1968a,b) and Hohn et al. (1968). The general formulation of the milling process equation and a stability algorithm have been developed. Recently Minis et al. (1990b) proposed a general mathematical model to describe all aspects of milling dynamics and to predict the linear and nonlinear machining chatter in milling. Because the regenerative chatter limits the rate and quality of material removal during machining, a considerable amount of research has been done to reduce the machine chatter and to increase the borderline of stability of the machine tool. Some vibration control strategies have been successfully applied to the turning and boring operations (Glaser and Nachtigal, 1979; Tewani et al. 1990), but limited research has been reported using optimal vibration control in the milling process. In this paper, the theory of optimal control with a damped vibration absorber against machine tool chatter in the milling process under actual cutting conditions is developed and the stability analysis is done. The model for cutting dynamics and the theory for stability analysis given by Minis et al. (1990b) is used in this study. The threshold of stability is obtained for the milling process under no control and then is compared with the limit of cut for the milling process with the optimal passive control. Optimization procedures for the damped dynamic absorber employ the numerical ADS optimization design package (Vanderplaats, 1984). Computer simulations of transient responses can be obtained from numerical integration of the system's dynamic equations of motion based on the fourth-order Runge--Kutta method. The theoretical prediction of the machine tool stability for the milling process can be verified by the transient responses of the system in the time domain under varying cutting conditions.
2. THE DYNAMICS OF MILLING 2.1. Modeling of the Milling The milling system is considered to be a two-degree-of-freedom lumped mass system, therefore the dynamics of the machine tool's structure are described by two orthogonal modes (Minis, 1990b). The X-Y coordinate system is fixed with respect to the machine tool structure, and its axes are aligned with the principal modes of oscillation, as shown in Fig. 1. A commonly used arrangement of springs and dampers called the damped dynamic absorber (Ball, Sheth, and Rouch, 1986) is attached to the system, as shown in Fig. 2. The absorber system, consisting of a comparatively small vibratory system m3, b3, and k3, can reduce the motion of the main system (Den Hartog, 1947). Newton's law applied to the system gives n-I
mx~+bl~+klX=
~, Fxi i=0
(2.1)
287 n-I
m2 ~' 4- b2 jr + k2 y + bs (~ - ~) + ks (y - z)= i~0=Fyi,
(2.2)
ms ~. + bs ( ~ - ~ ) + ks ( z - y ) = 0
(2.3)
where mr, bl, kt, m2, b2, and k2 are the modal parameters of the machine tool structure, ms, bs, and ks are the modal parameters of the vibratory absorber, Fxi and Fyi are the components of the cutting force that is applied on the i - t h tooth, n is the number of the teeth of the cutter.
///////
~
V
b2
I o FIB:. 1. Two--degree--of-freedom milling system.
I
Fig. 2. Milling system with vibration absorber.
2.2. Mode.ling of the Cutting Process For an orthogonal cutting process with continuous chip formation without built-up edge, the cutting forces applied on the cutting tool depends on the instantaneous uncut chip thickness. For a single-point cutting tool, the cutting force equation has been investigated by Merritt (1965). According to the theory of orthogonal cutting, the cutting force applied on the i-th tooth can be viewed as single-point cutting process. Therefore the equations for the components of the cutting force are derived (see Minis, 1990b for more details of the derivation). The components of the dynamic cutting force, Fui and Fvi, can be represented in terms of the instantaneous depth of cut as follows,
where
Fui = - h g(0i) [kd ( u i - u i ° + ft sin 0i) + bd ui] Fvi = - h g(0i) [ kc (Ui -- Ui 0 "{" ft sin 0i) + bc ui]
(2.4) (2.5)
i is the i - t h tooth of the cutter; h is the axial depth of cut; 0i is the angle of the i - t h tooth between the global coordinate system X-Y and the local coordinate s y s t e m U i - V i ; kd and bd are the stiffness and the damping of the cutting process, respectively, in the direction of the thrust force Furl kc and bc are the stiffness and the damping of the cutting process, respectively, in the direction of the cutting force Fvi; ui is the modulation of the surface being cut by_ the i - t h tooth;, ui 0 is the modulation of the top surface of the chip; ft is the feed of milling table per tooth, and the function g(0i) is used to indicate the force components are nominally non---zero between the entry angle and exit angle.
288 Therefore the cutting force components in X - Y coordinate system can be obtained Fxi = - h g(Oi) p(x,y,0i) [ ai]'x+~i~'$+Ti~'x~ i)
y+'Yl2
+6t ]
(2.6)
Fyi = - h g(0i) p(x,y,0i) [ ail)x+~i~'~+Ttl)x~ i,
where a ~ ) , Z~), 7~1, . . . , 5~i) are all periodic time functions as given in the reference (Minis, 1990b), and p(x,y,0i) is used to indicate the cutting forces are non---zero under linear chatter condition. By summing the forces for all teeth from Equations (2.6) and (2.7), we can obtain the dynamic equations of the milling process with damped vibration absorber as in Equations (2.1)-(2.3).
3. STABILITY ANALYSIS Because we are interested in the conditions which result in the onset of chatter only, we can simplify the model to be a linear dynamic system. The critical axial depth of cut can be calculated in terms of the spindle speed and the stability diagram can be constructed. Under linear chatter conditions, we have
(3.1) (3.2) (3.3)
x~ il (t) = x(t-T), y~i) (t) = y(t-T), p(x,y,0i) = 1. Therefore the dynamic equations of the system become ml ~ + bl i: + kl x = -h[anx+~u±+Tllx(t-T)+a12y+~12j'+712y(t-T)]
(3.4)
m2~+b2~'+k2y+b3
(3.5)
(~-
z)+
ka(y-
z)
= -h[a2 lx+f~21k + 72,x(t-T)+ a2uv+~2j'+ 72~y(t-T)] m3 ~ + ba ( ~ - j') + ka ( z - y ) = 0
(3.6)
where all,/~11, ~11," • ", 52 represent the sums of the corresponding terms. Rewriting Equations (3.4)-(3.6) in terms of the differential operator D and rearranging them in the operational form as follow
289
[G~,(D) G,,(D)] [A,(D,t) A,2(D,t)]
[G,t(D) G,~(D)] [~,]= 0 + h LG2t(D) G~(D)J 52
(3.7)
or
]~ + h G(D) A(D,t) ]~ + h G(D) ~ = 0
(3.8)
where G(D) is the flexibility matrix of the structure, and its components are
1 G11(D) = mxD_~+blD+kl GI2(D) = 0
(3.9) (3.10)
G2,(D) = 0 G2 (D) =
(3.11) I
m2D2+(b~+bs)D+(k2+k3)_ and
(3.12)
(b 3+k3) 2 (m3D2+b~D+ka)
Akm(D,t)= a'km(t) + ~km(t) D + %re(t) e-TD k,m = 1,2
(3.13)
Equation (3.8) represents a system of differential--
1 + h (Wn + W22) + h 2 (Wll-W22 -W12"W2t) = 1 + ~(h,A,w) = 0
(3.14)
A is the eigenvalue, w is the rotational frequency (w=nfl), and
w2~ w22 J- G22(~).(~2,+~2,~+~2,e-~T) g22(~).(~22+~22~+~2e-~T J (3.15) a n , " ' , ~2 obtained by using the numerical Gauss integration represent the mean values of the corresponding functions. For the system to be stable, the eigenvalue A must be located in the negative half of the complex plane. A numerical technique has been developed, along with a computer program, to find the critical axial depth of cut and the chatter frequency under varying cutting conditions.
4. NUMERICAL OPTIMIZATION TECHNIQUE In order to improve the threshold of stability of the milling process by passive control technique, a damped vibration absorber has been designed. Optimization procedures to determine the optimum values of spring and damping coefficient under a specific mass ratio for the vibration absorber have been applied by employing the ADS optimization package (Vanderplaats, 1984). The objective function for this purpose is chosen as the minimum value of critical axial depth of cut over a wide range of spindle speed. The system will be always stable if the milling machine is operating under or below this minimum axial
290 depth of cut. The design variables are the spring constant and the damping coefficient of the vibration absorber. The physical constraints were the spring constant ratio and damping coefficient ratio for the vibration absorber to the main system. Those two ratios were assumed to be less than one for general design purposes. As a matter of fact, we should try to choose those two ratios as small as possible to achieve our design. Based on the objective function, the design variables, and the constraints we set, the optimization design problem can be implemented by numerical techniques. ADS, a general purpose numerical optimization program, contains a wide variety of algorithms to solve the problem as Minimize Subject to :
F(x)
(4.1)
Gj(x) .LE. 0 Hk(x) .EQ. 0 XLi .LE. Xi .LE. XUi
j = 1, m k = 1, L i = 1, n
(4.2) (43) (414)
where F(x) is the objective function, Gj(x) is the inequality constraint, Hk(x) is the equality constraint, Eq.(4.4) is the side constraint,and m, L, n are the numbers of the corresponding constraints. In the application, half-immersion up-mining with an 8-tooth cutter is considered. We use modal parameters of the milling structure obtained from the reference (Minis, 1990a), m1=0.55 Kg, b1=330 N--see/m, k1=2 x 107 N/m , m2=1.2 Kg, b2=498 N--sec/m, and k~=2 x 107 N/m, and the cutting process parameters are kd=1000 Kg/mm 2, bd=--0.75 Kg---sec/mm2, kc=1900 Kg]mm~, and bc=--0.45 Kg-sec/mm2, to design the optimal vibration absorber for a specific mass ratio (taken as 0.1). Two local optimum values for the objective function have been obtained by different initial guess to check the global nature of the optimization process. They are x1=0.1627, x2=0.0946, F(x)=0.6598 and x1=0.4659, x~=0.0735, F(x)=0.6567. In an alternative approach, those optimum values are observed from the contour of two design variables, as shown in Fig. 3. By using the optimum values of the spring and damping coefficient for the damped vibration absorber, we construct the new stability chart under optimal passive control. Those results and comparisons are shown in Fig. 4.
,-xc~ v 1.0 --
1.0
~'~"
0.8
!~ o.6 E
~0.5
v 0.4 0.2 0.0 2OO .£I
• 0.0
0.5
1.0
Design Variable X(1) Fig. 3. Contour of two design variables.
I
I
400
600
800
Spindle Speed (RPM) Fig. 4. Stability Diagram in Milling.
~
°
~1~ o
~
~I~ ° ~I~ ° ~
o
o
II
i
I~
°
o
i
II
~
~
II
i
'~'
II
~
II
, +
II
.~
0~ P,rl('D
0
~'~ 0 ~.~
°~u ~2e
~'~
cn ~
OP~l.
0
E
0
292
0 0 --hTll -h712 mt ml B(t) =
D(t) =
0 0 -h721 -h722 m2 m2 0
0
0
0
(5.12) 6x2
- h 0~2
(5.13)
0 0
~xi
The simulation scheme is based on the Runge-Kutta method. We find the transient responses for X and Y directions by giving an axial depth of cut h and a spindle speed ft. Therefore, we can determine if the system is stable under those cutting conditions from the system's responses. If the amplitudes of the responses are decreasing or in steady state, the system is stable. Otherwise, the system is unstable. From the responses, we also can determine the critical depth of cut under a specific rotational speed fl to construct the stability diagram. We examined the system responses under the rotational speed of cutter fl = 570 RPM and the feed ft = 0.127 mm/tooth. The axial depth of cut h = 0.300 mm and h = 0.350 m m were used for the milling system without control, and h = 0.700 m m and h = 0.800 mm for the milling system with optimal passive control. The Y---direction responses are shown in Fig. 5-Fig. 8. The system is stable for h = 0.300 m m and h = 0.700 mm and unstable for h = 0.350 m m and h = 0.800 mm. From the stability diagram (Fig. 4), the critical axial depth of cut is 0.305 mm for milling without control and is 0.725 mm with optimal control. Therefore, the stability analysis and transient response simulation are consistent. 0 -1 ,",--2
6000-
4000 -
E 2ooo o:
E-3
"~,-5 v--
2ooo -4000" -6000:
6
-
-7 -B 0.0 0.1
0.2 0.3 0.4 0.5 t (see)
Fig. 5. Response for fl=570RPM ft=0.127mm/tooth, h=0.300mm.
:.
0.0 0.1 0.2 0..'=1 0.4 0.5
t (sec) Fig. 6. Response for fl=570RPM
ft=O.127mm/tooth, h=0.350mm.
293
-
N
5
200
~-,
"
100
E v
0
-I0
%=f
-15
"~ -~oo -200
-20 '
I
i
0.0 0.1
I
I
0.2 0.3 0.4 0.5
t (sec) Fig. 7. Response for ~=570RPM
ft=O.127mm/tooth, h=0.700mm.
0.0 0.i
0.2
0.3 0.4 0.5
t (-eo) Fig. 8. Response for ~=570RPM
ft=O.127mm/tooth, h=0.800mm.
6. CONCLUSIONS This paper presented a vibration control technique in milling with an optimal damped vibration absorber. A dynamic model of the whole system was derived. The regeneration phenomenon, which is due to the previous cutting experience in the workpieee, was included in the dynamic equations as a time-delay term. The dynamics of the system became a set of differential-difference equations with periodic coefficients. The theory of the stability analysis of the milling process, with a two degrees of freedom structural model, given by Minis (1990b) was considered to be accurate and was therefore used in this study. A numerical technique has been developed to find the threshold of stabihty of a milling process with the vibration absorber added. This method has been verified to be a more efficient approach for constructing the stability chart rather than any other graphic methods. The state variable representation was obtained from the system dynamics, and was used to find the transient responses by the fourth-order Runge--Kutta numerical integration method. An optimal vibration control algorithm was developed to fulfill our goal set, which is to reduce the vibration and to increase the stability of the cutting process. It is seen that there is one hundred percent* improvement by the comparison between a plain milling process and a milling with an optimal passive vibration absorber. The theoretical prediction of the machine tool stability was verified by the transient response of the system in the time domain under varying cutting conditions.
ACKNOWLEDGEMENT This research was supported in part by the Center for Robotics and Manufacturing Systems at the University of Kentucky.
294 REFERENCES Ball, J. H., Sheth, P. N., and Rouch, K. E., 1986, United States Patent Number 4,583,912, "Damped Dynamic Vibration Absorber," April 22. Den Hartog, J. P., 1947, Mechanical Vibrations, Third Edition, McGraw-Hill Book Company, New York, pp. 87-102. Glaser, D. J. and Nachtigal, C. L., 1979, "Development of a Hydraulic Chambered, Actively Controlled Boring Bar," ASME Journal of Engineering for Industry, Vol. 101, pp. 362-368. Hohn, R. E., Sridhar, R., and Long, G. W., 1986, "A Stability Algorithm for a Special Case of the Milling Process---Contribution to Machine--Tool Chatter Research---6," ASME Journal of Engineering for Industry, Vol. 90, No. 2, pp. 325-329. Merchant, M. E., 1944, "Basic Mechanics of the Metal Cutting Process," ASME Journal of applied mechanics, Vol. 66, pp. A168-A175. Merritt, H. E., 1965, "Theory of Self-Excited Machine-Tool Chatter----Contribution to Machine-Tool Chatter Research---l," ASME Journal of Engineering for Industry, Vol. 87, No. 4, pp. 447-454. Minis, I. E., Magrab, E. B., and Pandelidis, I. O., 1990a, "Improved Method for the Prediction of Chatter in Turning, Part 2 : Determination of Cutting Process Parameters," ASME Journal of Engineering for Industry, Vol. 112, pp. 21-27. Minis, I., Yanushevsky, A., Tembo, A., and I-Iocken, R., 1990b, "Analysis of Linear and Nonlinear Chatter in Milling," Annals of the CIRP, Vol. 39, No. 1, pp. 459-462. Nigm, M. M. I., 1973, The Dynamic Characteristics of the Metal Cutting Process, Ph.D. Thesis, University of Birmingham. Nigm, M. M., Sadek, M. M., Tobias, S. A., 1977a, "Dimensional Analysis of the Steady State Orthogonal Cutting Process," Int. J. Mach. Tool. Des. Res., Vol. 17, pp. 1-18. Nigm, M. M., Sadek, M. M., Tobias, S. A., 1977b, "Determination of Dynamic Cutting Coefficients from Steady State Cutting Data," Int. J. Mach. Tool. Des. Res., Vol. 17, pp. 19-37. Sridhar, R., Hohn, R. E., and Long, G. W., 1968a, "A General Formulation of the Milling Process Equation--Contribution to Machine-Tool Chatter Research------5," ASME Journal of Engineering for Industry, Vol. 90, No. 2, pp. 317-324. Sridhar, R., Hohn, R. E., and Long, G. W., 1968b, "A Stability Algorithm for the General Milling Process--Contribution to Machine---Tool Chatter Research--7," ASME Journal of Engineering for Industry, Vol. 90, No. 2, pp. 330-334. Tewani, S. G., Rouch, K. E., and Walcott, B. L., 1990, "Cutting Process Stability of a Boring Bar with Active Dynamic Absorber," (To appear). Vanderplaats, G. N., 1984, Numerical Optimization Techniques for Engineering Design : With Application, McGraw-Hill Book Company, New York. Wu, D. W. and Liu, C. R., 1985a, "An Analytical Model of Cutting Dynamics. Part 1 : Model Building," ASME Journal of Engineering for Industry, Vol. 107, pp. 107-111. Wu, D. W. and Liu, C. R., 1985b, "An Analytical Model of Cutting Dynamics. Part 2 : Verification," ASME Journal of Engineering for Industry, Vol. 107, pp. 112-118.
Elsevier
Optimal passive vibration control of cutting process stability in
milling
K. J. Liu and K. E. Rouch Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506, U.S.A.
Abstract The paper proposes a design concept for an optimal passive dynamic absorber in the milling process. An investigation and analysis into the characteristics of, and interaction between, forced vibration and linear chatter in milling has been carried out by use of a two-degree--of-freedom structural model simulation. A dynamic absorber mass is then connected to the main system through passive elements (spring and damper). Optimization procedures to determine the optimum values of spring and damping coefficients for a specific mass ratio have been developed by employing the ADS optimization package. A theoretical approach for the prediction of machine tool chatter in milling including dynamic cutting forces permits calculation of borderlines of stability for a structure model with an optimal absorber. Compared with the limit of stability under no control, the stable range under optimal passive control has been improved over one hundred percent. Transient responses can also be obtained from numerical integration of the system's equation of motion in the state variable representation based on the fourth-order Runge-Kutta method. The theoretical predictions can be verified by the results of numerical simulation of transient response.
1. INTRODUCTION In the machining operation, a violent relative vibration between the workpiece and the cutting tool, which is called chatter, is developed under some operating conditions. Chatter is not desirable because it has an adverse effect on the machining accuracy, the surface finish, and the tool life, and also because it results in reduced output by requiring a lower metal removal rate to obtain vibration-free performance. Therefore, the prediction of machine chatter and the vibration control in the machining operation becomes increasingly significant. The major step towards understanding the regenerative machine tool chatter was presented by Merritt (1965). This study, which focused on a stability theory for regenerative chatter in an orthogonal turning process with a feedback loop, was based on the theory of steady orthogonal cutting process given by Merchant (1944). It has been found that the onset of machine chatter is a function of the spindle speed. Nigm et al. (1973, 1977a,b) provided a recent contribution to the study of 0924-0136/91/$03.50 © 1991 ElsevierSciencePublishers B.V. All fightsreserved.
286 the dynamics of oscillatory metal cutting and the prediction of the dynamic cutting coefficients from steady--state cutting data. Wu and Liu (1985a,b) also developed a mathematical model of machine chatter, using an analytical approach to predict dynamic cutting forces from the steady--state cutting, tests. Although machine tool chatter occurs in many different machining operations, most of the research work has traditionally focused on the turning operation, which is a simple and basic metal cutting process. In turning, the cutting forces are fixed in direction relative to the machine tool structure, while they rotate with respect to the structure in milling. The extension from turning to milling results in a time-varying dynnmlc system. The earlier detailed mathematical models of the milling system have been proposed by Sridhar et al. (1968a,b) and Hohn et al. (1968). The general formulation of the milling process equation and a stability algorithm have been developed. Recently Minis et al. (1990b) proposed a general mathematical model to describe all aspects of milling dynamics and to predict the linear and nonlinear machining chatter in milling. Because the regenerative chatter limits the rate and quality of material removal during machining, a considerable amount of research has been done to reduce the machine chatter and to increase the borderline of stability of the machine tool. Some vibration control strategies have been successfully applied to the turning and boring operations (Glaser and Nachtigal, 1979; Tewani et al. 1990), but limited research has been reported using optimal vibration control in the milling process. In this paper, the theory of optimal control with a damped vibration absorber against machine tool chatter in the milling process under actual cutting conditions is developed and the stability analysis is done. The model for cutting dynamics and the theory for stability analysis given by Minis et al. (1990b) is used in this study. The threshold of stability is obtained for the milling process under no control and then is compared with the limit of cut for the milling process with the optimal passive control. Optimization procedures for the damped dynamic absorber employ the numerical ADS optimization design package (Vanderplaats, 1984). Computer simulations of transient responses can be obtained from numerical integration of the system's dynamic equations of motion based on the fourth-order Runge--Kutta method. The theoretical prediction of the machine tool stability for the milling process can be verified by the transient responses of the system in the time domain under varying cutting conditions.
2. THE DYNAMICS OF MILLING 2.1. Modeling of the Milling The milling system is considered to be a two-degree-of-freedom lumped mass system, therefore the dynamics of the machine tool's structure are described by two orthogonal modes (Minis, 1990b). The X-Y coordinate system is fixed with respect to the machine tool structure, and its axes are aligned with the principal modes of oscillation, as shown in Fig. 1. A commonly used arrangement of springs and dampers called the damped dynamic absorber (Ball, Sheth, and Rouch, 1986) is attached to the system, as shown in Fig. 2. The absorber system, consisting of a comparatively small vibratory system m3, b3, and k3, can reduce the motion of the main system (Den Hartog, 1947). Newton's law applied to the system gives n-I
mx~+bl~+klX=
~, Fxi i=0
(2.1)
287 n-I
m2 ~' 4- b2 jr + k2 y + bs (~ - ~) + ks (y - z)= i~0=Fyi,
(2.2)
ms ~. + bs ( ~ - ~ ) + ks ( z - y ) = 0
(2.3)
where mr, bl, kt, m2, b2, and k2 are the modal parameters of the machine tool structure, ms, bs, and ks are the modal parameters of the vibratory absorber, Fxi and Fyi are the components of the cutting force that is applied on the i - t h tooth, n is the number of the teeth of the cutter.
///////
~
V
b2
I o FIB:. 1. Two--degree--of-freedom milling system.
I
Fig. 2. Milling system with vibration absorber.
2.2. Mode.ling of the Cutting Process For an orthogonal cutting process with continuous chip formation without built-up edge, the cutting forces applied on the cutting tool depends on the instantaneous uncut chip thickness. For a single-point cutting tool, the cutting force equation has been investigated by Merritt (1965). According to the theory of orthogonal cutting, the cutting force applied on the i-th tooth can be viewed as single-point cutting process. Therefore the equations for the components of the cutting force are derived (see Minis, 1990b for more details of the derivation). The components of the dynamic cutting force, Fui and Fvi, can be represented in terms of the instantaneous depth of cut as follows,
where
Fui = - h g(0i) [kd ( u i - u i ° + ft sin 0i) + bd ui] Fvi = - h g(0i) [ kc (Ui -- Ui 0 "{" ft sin 0i) + bc ui]
(2.4) (2.5)
i is the i - t h tooth of the cutter; h is the axial depth of cut; 0i is the angle of the i - t h tooth between the global coordinate system X-Y and the local coordinate s y s t e m U i - V i ; kd and bd are the stiffness and the damping of the cutting process, respectively, in the direction of the thrust force Furl kc and bc are the stiffness and the damping of the cutting process, respectively, in the direction of the cutting force Fvi; ui is the modulation of the surface being cut by_ the i - t h tooth;, ui 0 is the modulation of the top surface of the chip; ft is the feed of milling table per tooth, and the function g(0i) is used to indicate the force components are nominally non---zero between the entry angle and exit angle.
288 Therefore the cutting force components in X - Y coordinate system can be obtained Fxi = - h g(Oi) p(x,y,0i) [ ai]'x+~i~'$+Ti~'x~ i)
y+'Yl2
+6t ]
(2.6)
Fyi = - h g(0i) p(x,y,0i) [ ail)x+~i~'~+Ttl)x~ i,
where a ~ ) , Z~), 7~1, . . . , 5~i) are all periodic time functions as given in the reference (Minis, 1990b), and p(x,y,0i) is used to indicate the cutting forces are non---zero under linear chatter condition. By summing the forces for all teeth from Equations (2.6) and (2.7), we can obtain the dynamic equations of the milling process with damped vibration absorber as in Equations (2.1)-(2.3).
3. STABILITY ANALYSIS Because we are interested in the conditions which result in the onset of chatter only, we can simplify the model to be a linear dynamic system. The critical axial depth of cut can be calculated in terms of the spindle speed and the stability diagram can be constructed. Under linear chatter conditions, we have
(3.1) (3.2) (3.3)
x~ il (t) = x(t-T), y~i) (t) = y(t-T), p(x,y,0i) = 1. Therefore the dynamic equations of the system become ml ~ + bl i: + kl x = -h[anx+~u±+Tllx(t-T)+a12y+~12j'+712y(t-T)]
(3.4)
m2~+b2~'+k2y+b3
(3.5)
(~-
z)+
ka(y-
z)
= -h[a2 lx+f~21k + 72,x(t-T)+ a2uv+~2j'+ 72~y(t-T)] m3 ~ + ba ( ~ - j') + ka ( z - y ) = 0
(3.6)
where all,/~11, ~11," • ", 52 represent the sums of the corresponding terms. Rewriting Equations (3.4)-(3.6) in terms of the differential operator D and rearranging them in the operational form as follow
289
[G~,(D) G,,(D)] [A,(D,t) A,2(D,t)]
[G,t(D) G,~(D)] [~,]= 0 + h LG2t(D) G~(D)J 52
(3.7)
or
]~ + h G(D) A(D,t) ]~ + h G(D) ~ = 0
(3.8)
where G(D) is the flexibility matrix of the structure, and its components are
1 G11(D) = mxD_~+blD+kl GI2(D) = 0
(3.9) (3.10)
G2,(D) = 0 G2 (D) =
(3.11) I
m2D2+(b~+bs)D+(k2+k3)_ and
(3.12)
(b 3+k3) 2 (m3D2+b~D+ka)
Akm(D,t)= a'km(t) + ~km(t) D + %re(t) e-TD k,m = 1,2
(3.13)
Equation (3.8) represents a system of differential--
1 + h (Wn + W22) + h 2 (Wll-W22 -W12"W2t) = 1 + ~(h,A,w) = 0
(3.14)
A is the eigenvalue, w is the rotational frequency (w=nfl), and
w2~ w22 J- G22(~).(~2,+~2,~+~2,e-~T) g22(~).(~22+~22~+~2e-~T J (3.15) a n , " ' , ~2 obtained by using the numerical Gauss integration represent the mean values of the corresponding functions. For the system to be stable, the eigenvalue A must be located in the negative half of the complex plane. A numerical technique has been developed, along with a computer program, to find the critical axial depth of cut and the chatter frequency under varying cutting conditions.
4. NUMERICAL OPTIMIZATION TECHNIQUE In order to improve the threshold of stability of the milling process by passive control technique, a damped vibration absorber has been designed. Optimization procedures to determine the optimum values of spring and damping coefficient under a specific mass ratio for the vibration absorber have been applied by employing the ADS optimization package (Vanderplaats, 1984). The objective function for this purpose is chosen as the minimum value of critical axial depth of cut over a wide range of spindle speed. The system will be always stable if the milling machine is operating under or below this minimum axial
290 depth of cut. The design variables are the spring constant and the damping coefficient of the vibration absorber. The physical constraints were the spring constant ratio and damping coefficient ratio for the vibration absorber to the main system. Those two ratios were assumed to be less than one for general design purposes. As a matter of fact, we should try to choose those two ratios as small as possible to achieve our design. Based on the objective function, the design variables, and the constraints we set, the optimization design problem can be implemented by numerical techniques. ADS, a general purpose numerical optimization program, contains a wide variety of algorithms to solve the problem as Minimize Subject to :
F(x)
(4.1)
Gj(x) .LE. 0 Hk(x) .EQ. 0 XLi .LE. Xi .LE. XUi
j = 1, m k = 1, L i = 1, n
(4.2) (43) (414)
where F(x) is the objective function, Gj(x) is the inequality constraint, Hk(x) is the equality constraint, Eq.(4.4) is the side constraint,and m, L, n are the numbers of the corresponding constraints. In the application, half-immersion up-mining with an 8-tooth cutter is considered. We use modal parameters of the milling structure obtained from the reference (Minis, 1990a), m1=0.55 Kg, b1=330 N--see/m, k1=2 x 107 N/m , m2=1.2 Kg, b2=498 N--sec/m, and k~=2 x 107 N/m, and the cutting process parameters are kd=1000 Kg/mm 2, bd=--0.75 Kg---sec/mm2, kc=1900 Kg]mm~, and bc=--0.45 Kg-sec/mm2, to design the optimal vibration absorber for a specific mass ratio (taken as 0.1). Two local optimum values for the objective function have been obtained by different initial guess to check the global nature of the optimization process. They are x1=0.1627, x2=0.0946, F(x)=0.6598 and x1=0.4659, x~=0.0735, F(x)=0.6567. In an alternative approach, those optimum values are observed from the contour of two design variables, as shown in Fig. 3. By using the optimum values of the spring and damping coefficient for the damped vibration absorber, we construct the new stability chart under optimal passive control. Those results and comparisons are shown in Fig. 4.
,-xc~ v 1.0 --
1.0
~'~"
0.8
!~ o.6 E
~0.5
v 0.4 0.2 0.0 2OO .£I
• 0.0
0.5
1.0
Design Variable X(1) Fig. 3. Contour of two design variables.
I
I
400
600
800
Spindle Speed (RPM) Fig. 4. Stability Diagram in Milling.
~
°
~1~ o
~
~I~ ° ~I~ ° ~
o
o
II
i
I~
°
o
i
II
~
~
II
i
'~'
II
~
II
, +
II
.~
0~ P,rl('D
0
~'~ 0 ~.~
°~u ~2e
~'~
cn ~
OP~l.
0
E
0
292
0 0 --hTll -h712 mt ml B(t) =
D(t) =
0 0 -h721 -h722 m2 m2 0
0
0
0
(5.12) 6x2
- h 0~2
(5.13)
0 0
~xi
The simulation scheme is based on the Runge-Kutta method. We find the transient responses for X and Y directions by giving an axial depth of cut h and a spindle speed ft. Therefore, we can determine if the system is stable under those cutting conditions from the system's responses. If the amplitudes of the responses are decreasing or in steady state, the system is stable. Otherwise, the system is unstable. From the responses, we also can determine the critical depth of cut under a specific rotational speed fl to construct the stability diagram. We examined the system responses under the rotational speed of cutter fl = 570 RPM and the feed ft = 0.127 mm/tooth. The axial depth of cut h = 0.300 mm and h = 0.350 m m were used for the milling system without control, and h = 0.700 m m and h = 0.800 mm for the milling system with optimal passive control. The Y---direction responses are shown in Fig. 5-Fig. 8. The system is stable for h = 0.300 m m and h = 0.700 mm and unstable for h = 0.350 m m and h = 0.800 mm. From the stability diagram (Fig. 4), the critical axial depth of cut is 0.305 mm for milling without control and is 0.725 mm with optimal control. Therefore, the stability analysis and transient response simulation are consistent. 0 -1 ,",--2
6000-
4000 -
E 2ooo o:
E-3
"~,-5 v--
2ooo -4000" -6000:
6
-
-7 -B 0.0 0.1
0.2 0.3 0.4 0.5 t (see)
Fig. 5. Response for fl=570RPM ft=0.127mm/tooth, h=0.300mm.
:.
0.0 0.1 0.2 0..'=1 0.4 0.5
t (sec) Fig. 6. Response for fl=570RPM
ft=O.127mm/tooth, h=0.350mm.
293
-
N
5
200
~-,
"
100
E v
0
-I0
%=f
-15
"~ -~oo -200
-20 '
I
i
0.0 0.1
I
I
0.2 0.3 0.4 0.5
t (sec) Fig. 7. Response for ~=570RPM
ft=O.127mm/tooth, h=0.700mm.
0.0 0.i
0.2
0.3 0.4 0.5
t (-eo) Fig. 8. Response for ~=570RPM
ft=O.127mm/tooth, h=0.800mm.
6. CONCLUSIONS This paper presented a vibration control technique in milling with an optimal damped vibration absorber. A dynamic model of the whole system was derived. The regeneration phenomenon, which is due to the previous cutting experience in the workpieee, was included in the dynamic equations as a time-delay term. The dynamics of the system became a set of differential-difference equations with periodic coefficients. The theory of the stability analysis of the milling process, with a two degrees of freedom structural model, given by Minis (1990b) was considered to be accurate and was therefore used in this study. A numerical technique has been developed to find the threshold of stabihty of a milling process with the vibration absorber added. This method has been verified to be a more efficient approach for constructing the stability chart rather than any other graphic methods. The state variable representation was obtained from the system dynamics, and was used to find the transient responses by the fourth-order Runge--Kutta numerical integration method. An optimal vibration control algorithm was developed to fulfill our goal set, which is to reduce the vibration and to increase the stability of the cutting process. It is seen that there is one hundred percent* improvement by the comparison between a plain milling process and a milling with an optimal passive vibration absorber. The theoretical prediction of the machine tool stability was verified by the transient response of the system in the time domain under varying cutting conditions.
ACKNOWLEDGEMENT This research was supported in part by the Center for Robotics and Manufacturing Systems at the University of Kentucky.
294 REFERENCES Ball, J. H., Sheth, P. N., and Rouch, K. E., 1986, United States Patent Number 4,583,912, "Damped Dynamic Vibration Absorber," April 22. Den Hartog, J. P., 1947, Mechanical Vibrations, Third Edition, McGraw-Hill Book Company, New York, pp. 87-102. Glaser, D. J. and Nachtigal, C. L., 1979, "Development of a Hydraulic Chambered, Actively Controlled Boring Bar," ASME Journal of Engineering for Industry, Vol. 101, pp. 362-368. Hohn, R. E., Sridhar, R., and Long, G. W., 1986, "A Stability Algorithm for a Special Case of the Milling Process---Contribution to Machine--Tool Chatter Research---6," ASME Journal of Engineering for Industry, Vol. 90, No. 2, pp. 325-329. Merchant, M. E., 1944, "Basic Mechanics of the Metal Cutting Process," ASME Journal of applied mechanics, Vol. 66, pp. A168-A175. Merritt, H. E., 1965, "Theory of Self-Excited Machine-Tool Chatter----Contribution to Machine-Tool Chatter Research---l," ASME Journal of Engineering for Industry, Vol. 87, No. 4, pp. 447-454. Minis, I. E., Magrab, E. B., and Pandelidis, I. O., 1990a, "Improved Method for the Prediction of Chatter in Turning, Part 2 : Determination of Cutting Process Parameters," ASME Journal of Engineering for Industry, Vol. 112, pp. 21-27. Minis, I., Yanushevsky, A., Tembo, A., and I-Iocken, R., 1990b, "Analysis of Linear and Nonlinear Chatter in Milling," Annals of the CIRP, Vol. 39, No. 1, pp. 459-462. Nigm, M. M. I., 1973, The Dynamic Characteristics of the Metal Cutting Process, Ph.D. Thesis, University of Birmingham. Nigm, M. M., Sadek, M. M., Tobias, S. A., 1977a, "Dimensional Analysis of the Steady State Orthogonal Cutting Process," Int. J. Mach. Tool. Des. Res., Vol. 17, pp. 1-18. Nigm, M. M., Sadek, M. M., Tobias, S. A., 1977b, "Determination of Dynamic Cutting Coefficients from Steady State Cutting Data," Int. J. Mach. Tool. Des. Res., Vol. 17, pp. 19-37. Sridhar, R., Hohn, R. E., and Long, G. W., 1968a, "A General Formulation of the Milling Process Equation--Contribution to Machine-Tool Chatter Research------5," ASME Journal of Engineering for Industry, Vol. 90, No. 2, pp. 317-324. Sridhar, R., Hohn, R. E., and Long, G. W., 1968b, "A Stability Algorithm for the General Milling Process--Contribution to Machine---Tool Chatter Research--7," ASME Journal of Engineering for Industry, Vol. 90, No. 2, pp. 330-334. Tewani, S. G., Rouch, K. E., and Walcott, B. L., 1990, "Cutting Process Stability of a Boring Bar with Active Dynamic Absorber," (To appear). Vanderplaats, G. N., 1984, Numerical Optimization Techniques for Engineering Design : With Application, McGraw-Hill Book Company, New York. Wu, D. W. and Liu, C. R., 1985a, "An Analytical Model of Cutting Dynamics. Part 1 : Model Building," ASME Journal of Engineering for Industry, Vol. 107, pp. 107-111. Wu, D. W. and Liu, C. R., 1985b, "An Analytical Model of Cutting Dynamics. Part 2 : Verification," ASME Journal of Engineering for Industry, Vol. 107, pp. 112-118.