Dynamic Modeling for Control of the Milling Process

L. K. Lauderbaugh Assistant Professor. Mem. ASME Department of Mechanical Engineering, Aeronautical Engineering and Mechanics, Rensselaer Polytechnic Institute, Troy, New York 12180-3590

A. G. Ulsoy Associate Professor. Mem. ASME Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, Michigan 48109-2125

Dynamic Modeling for Control of the Milling Process In the interest of maximizing the metal removal rate and preventing tool breakage in the milling process, it has been proposed that fixed gain feedback controllers, which manipulate the feed rate to maintain a constant cutting force, be implemented. These process controllers have resulted in substantial improvements in the metal removal rate; however, they may have very poor performance when the process parameters deviate from the design conditions. To address these performance problems, an empirical second order model of the force response for a milling system to feedrate changes is presented along with experimental results which show that the parameters of this model vary significantly with cutting conditions. These variations are shown to have significant effects on the performance of fixed-gain proportional plus integral action and linear model following controllers. This is demonstrated using machining tests as well as through digital simulations.

Introduction Numerically Controlled (NC) and Computer Numerically Controlled (CNC) machine tools have greatly reduced operator input, resulting in significant improvements in productivity. However, further improvements can be made by online manipulation of the feeds and speeds. These improvements result from the fact that in NC and CNC machining, the feeds and speeds are fixed throughout the operation. These feeds and speeds are selected based on the most severe conditions expected. Thus, by on-line manipulation of the feeds and speeds, the metal removal rate (volume of metal removed per unit time), and hence the productivity, could be increased. This observation has precipitated the application of process control to manipulate the feedrates in an attempt to optimize the metal removal rate (MRR). When these process controllers are used, improvements in the MRR of 20 percent to 80 percent have been reported [1]. Along with these improvements, the cutting process controllers have introduced some new problems. They may not perform as designed and may become unstable or cause tool breakage [2]. These performance problems are believed to be the result of the large variations in the static and dynamic behavior of the cutting process. To address these problems, this paper presents an empirical second order dynamic model of the milling system and also examines how the parameters of this model vary with cutting conditions. The milling system model was based on slot milling 1020 cold-rolled steel with a 12.7 mm. diameter high speed steel (HSS) milling cutter with the depth of cut varied over the range of 1.91 mm to 3.81 mm. This modeling work was directed towards obtaining a model structure and determining if the parameters of the model change with cutting conditions. This limited scope permitted us to perform a limited number

Contributed by the Production Engineering Division for publication in the JOURNAL OF ENGINEERING FOR INDUSTRY. Manuscript received October 1985, revised April 1988.

of modeling experiments and to use simple modeling techniques. Once developed, the model was also used to design two types of fixed gain controllers and to demonstrate the effects of the parameter variations on the controller performance. The following sections discuss the experimental setup and conditions, the experimental results, and the implications for controller design. Experimental Set-up and Conditions This section describes the components of the experimental setup shown in Fig. 1. Also included in this section is a discussion of the experimental conditions. Milling Machine. The CNC milling machine was a standard commercial unit, equipped with a two horsepower spindle drive and an open-loop controlled, stepping motor driven table. This mill was modified so that the programmed feedrate could be overriden with a voltage from the control computer's digital-to-analog converter (DAC). This feedrate override circuit allowed the computer to adjust the feedrate from 0 to 125 percent of the programmed value. This feedrate override took place in the analog circuitry that generates the pulse train to the stepping motors. Therefore, the override was not effected by the acceleration and deceleration ramping that occurs during programmed feedrate changes. Dynamometer. A three-component, strain gage based, dynamometer was used to measure the cutting forces. This dynamometer was mounted on the table of the milling machine and the work piece was mounted on the dynamometer. The dynamometer we used was built in-house, and had natural frequencies of 300 Hz or higher in each direction (with the workpiece mounted). While higher natural frequencies can be found in commercial units, the 300 Hz unit worked well in our studies since the important frequencies in our experiments were all below 50 Hz.

Journal of Engineering for Industry

NOVEMBER 1988, Vol. 1 1 0 / 3 6 7

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" D/A Converters

Feedrate Override Circuit

CNC

Laboratory Microcomputer

Milling

System

Machine

3 Component Force Dynamometer A/D Converters

and Analog Signal Processing

\r Fig. 1 Schematic of the laboratory data acquisition and control system for milling

with a four-tooth cutter. Unfortunately, the effects of the runout are still present. The signal processing would be further complicated if we had used a cutter with a number of teeth that was not a multiple of four, or a width of cut less than the diameter of the cutter, because there would have been periodic fluctuations in the resultant force signal due to the mechanics of the process as well as the tool runout. Then, the problem is: how do we determine if the force level has changed. Some researchers have used peak forces [8], while others have used a scheme that integrates the forces over one revolution [9]. The force signal used should be determined by the application: If the primary constraint on the force is due to tool breakage, peak force is probably the most important; if the primary concern is tool deflection, the average force is probably the most useful. Experimental Conditions. The system described above was used to perform the modeling experiments. We conducted a series of milling tests o n 1020 cold rolled steel with a fourtooth 12.7 m m HSS cutter. A step input in voltage was applied to the feedrate override circuit and the forces were sampled. The step took the feed from 25.4 m m / m i n t o 50.8 m m / m i n . This experiment was repeated at three different depths (1.91 mm, 2.54 m m , and 3.81 mm) and at two different spindle speeds (550 rpm and 780 rpm).

Filtering and Signal Processing. Because of the mechanics of milling and the fact that the dynamometer resolves the forces into three components, some signal processing is required. First, the signals from t h e dynamometer were amplified, then passed through a n analog circuit which reconstructed the resultant force from the force components by implementing:

Experimental Results Now that the experimental setup has been explained, we can proceed t o the development a n d evaluation of a dynamic model. As mentioned above, we are not concerned with finding an exact, highly detailed model of the cutting process. Some researchers are working on developing such models [8, 14, 15, 16], These models, due to our lack of understanding of (Fx •+F CO the cutting process, require that empirical constants b e evaluated. Also, much of the modeling work done in the past where Fx and Fy are the x a n d y force components, which lie in has not been intended for control purposes and cannot be the plane of the table. Since the z component (Fz) is small, it readily applied to controller design. Fortunately, for our conwas omitted t o reduce the computation time. T h e circuit troller design, we only need to determine the structure of the diagram is available in [17]. process model for the milling system, a n d whether t h e This analog resultant force signal was then filtered to pre- parameters in this model vary significantly with cutting condivent aliasing problems due to sampling, as well as to eliminate tions. This reduced demand on the modeling allowed us to run noise resulting from the spindle and the stepping motor drives. a limited number of experiments a n d use simple modeling The filtered signal was then sampled with the 12 bit analog-to- techniques. digital converter (ADC) using a sampling period of r = 0 . 0 5 The section on modeling for the entire system analyzes the sec., thus the analog filter cutoff frequency was chosen to be experimental results, and presents a model for the entire con10 H z (i.e., half the sampling frequency). trolled system. Then in the section on milling system modeling This digitized signal (data sequence) may still n o t be in a the results are further analzyed by considering the effect of the form that can be used for control because the interrupted cut, machine and measurement system dynamics resulting in a the variation in chip thickness, and the tool runout (caused by model of the dynamics of the milling system. Finally, these eccentric mounting of the tool in the spindle) that occur in results are compared with other work reported in the milling, can cause periodic fluctuations in the force signal [6, literature. 7]. However, the periodic fluctuations in force due to the interrupted cut and the variation in chip thickness are not Results and Modeling for the Entire Controlled present in the case of slot-milling with a four-tooth cutter [6]. System. Some of the experimental results are shown in Figs. Therefore, in this study all of the results were for slot-milling 2 to 4. Two interesting general observations can be made from 2

Nomenclature a = axial depth of cut (mm) a = feed exponent i6 = depth exponent FR = resultant force (N) FB* model output (N) Fx = ^-component of the force (N) FY = /-component of the force (N) Fz = z-component of the force (N) 368/Vol. 110, NOVEMBER 1988

/ = feed per tooth (mm/tooth) fp = programmed feedrate (mm/min) = transfer function for the o„ dynamometer OR = transfer function for the feedrate override h = average chip thickness K = gain of the controlled system &, = integral control gain

K „ N N, r T V "n

r

proportional control gain spindle speed (rpm) number of teeth reference input (N) sampling period (sec) feedrate override voltage (volts) natural frequency (rad/sec) damping ratio Transactions of the ASME

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H E = d •=r-T:; 6

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Reduced block diagram lor the mill

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6.35

Block diagram of mill

%

Fig. 8(b)

-:±s\

7.62

G

ll^ltii =0=1

4

6

8

io

12

14

Time, t, sec.

Spindle

Fig. 8(d) Table position vs time for step change in teed

Position Transducer

Mill Table

Fig. 8(c)

V(k)

Table position measurement system

g-'(1.594+1.537 q~l) 1-1.879 q~l + .9 - 2

A{q-1)

(7)

with, A(q-l)

=

l+alq-l+a2q~

system. Their work resulted in a fourth order model of the milling system with all of the eigenvalues having imaginary parts indicating an oscillatory system. Their work also shows how the eigenvalues change for 3 sets of cutting conditions. Their results indicate the same general trend as our data: the system order is higher than one and the response can become oscillatory under cutting conditions. These modeling results, although somewhat diverse, are not inconsistent. Such models reflect the milling system, including drive, sensor, structural, and cutting effects. It is of interest to isolate such effects for a better understanding of the underlying process. However, our goal here is to obtain a model of our milling system relating changes in the control voltage (proportional to the feedrate command) to changes in the resultant cutting force. The model in equation (4), and its discrete time counterpart in equation (7), is then suitable for our goal of discrete controller design. Implications for Controller Design

and B(q-*) = b0 + blqwith the a, and the b, as defined above. Equation (7) is the model required for digital controller design. Comparison With Previous Models. The model given in equation (4) is somewhat different from the models researchers usually use for control of milling machines. While the process has generally been treated as a first-order system [9, 9a], our data indicate that a second-order model may be more appropriate. Furthermore, the overall system gain has been modeled somewhat differently than in previous studies. A static model, developed for use in turning [10], and also used to describe milling operations, is given by, F=Cah"

(8)

where Cis a constant, h is the average chip thickness (which in milling, is proportional to the feed per tooth) and n is a constant, usually less than one. The gain of equation (8) is proportional to the axial depth of cut while our experimental results indicate that the gain is proportional to the depth raised to some exponent, as shown in equation (|4) (to compare equation (4) to (8) consider steady operation with a constant feedrate where the derivatives in equation (4) go to zero). An interesting similarity between the two models is that research in turning shows the value of n to be 0.7 [2]; we found a value of 0.73 for the corresponding exponent, a, from our milling experiments. Garcis-Gardea at al. [16] have used the Dynamic Data Systems methodology to evaluate the dynamics of the milling Journal of Engineering for Industry

Now that a suitable model has been developed, candidate controller designs can be evaluated. We have evaluated a proportional plus integral action (PI) [12] and linear model reference controller (LMRC) [11] using simulations, and (in the case of the PI controller) machining tests. Both controllers were designed for a depth of cut of 2.5 mm. Then, when a suitable design was completed, the controller performance was simulated with the depth of cut varying from 2 mm to 4 mm in 1 mm steps, as shown in Fig. 9. The controller performance was evaluated by using digital simulations as well as actual machining tests. These simulations were performed using the SIMULA Digital Control System Simulation Package [3] developed by the authors for this project. The SIMULA package simulates the discrete time digital controller, the continuous time plant as well as the analog-to-digital converter and the digital-to-analog converter. In these digital simulations, equations (4) and (5) were used as the model of the plant with the values of Ks, /3, and a found in our experiments. Linear Model Reference Controller. In the case of the LMRC, a reference model specifies the desired performance of the closed loop system. In this study, the reference model was selected to be: FR*(z) .0104Z+.0097 (9) r(z) ~ z2-l.767z+JS7 Where FR * (z) is the output of the reference model, and r(z) is the reference input (the reference force). This model has the NOVEMBER 1988, Vol. 110/371

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.0077

Table 1 Milling system model parameters calculated from experiments (low and high refer to feeds of 25.4 mm/min. and 50.8 mm/min, respectively) Test Niumber

Depth

Spindle Speed

Damping Ratio

a (mm)

N (rev/min)

Y

Natural Frequency 0)

«

.0163

/

Process Gain

Resultant Force

200 -

(N)

K (N/volt)

low high

low high

FR

(rad/sec)

3 0 0 -j

1

1.91

550

.1

2.3

375 500

191 128

2

2.54

550

7

2.97

336 550

171 140

3

2.54

780

3

3.24

270 490

138 125

4

2.54

550

6

2.89

370 556

189 142

5

3.81

550

9

2.97

600 960

306 245

6

3.81

780

8

2.53

570 900

291 230

/ /

100-

o

i

0

1

1.27

2.54

Depth-of-cut, •»• .011 mm/tooth 300

•

Fig. 6

1

3.81

a(mm)

Gain K versus depth-of-cut with N = 780 rpm

.022 mm/tooth

» 78 0 rpm e

550 rpm

1.0

/ • /

100 -

/

0 0

I 2.54

T 1.27

' /

"I 3.81

/

/

0 0

1.27

2.54

3.81

a (mm) Fig. 5

Depth-of-cut,

Gain versus depth of cut with N = 550 rpm Fig. 7

input to the milling system described by the transfer function Gcp. Then, the dynamics of the dynamometer and amplifier are lumped together in GD. As mentioned in the section on the experimental system, the dynamometer (with workpiece mounted) has natural frequencies of 300 Hz or higher in each direction (measured with a spectrum analyzer). Thus, for signals with frequencies below 300 Hz, the dynamometer could be treated as a simple gain, and its dynamics did not contribute to the dynamics shown in Figs. 2 to 4. Therefore, over the frequency range of interest, GD was treated as a simple gain that is removed in the computer (i.e., the gains of the dynamometer, amplifier, and the analog-to-digital converter were cancelled). Thus, we could treat the system as if the resultant force were sampled directly. Then the final block contains the dynamics of the signal processing and reconstruction of the resultant force. Combining the first three blocks in Fig. 8(a) and removing the measurement dynamics, GD and Gcp, results in the block diagram of Fig. 8(b). Since the first block contains only a gain, the dynamics observed in Figs. 2-4 cannot be attributed to the drive or measurement system. The dynamics in the data are apparently due to the dynamics of the milling system. The dynamics of the milling system can be modeled by fitting a second-order nonlinear differential equation to the data (see Appendix). Such a model represents the milling system including the cutting process and structural characteristics of the tool and spindle. However, as noted above, the drive and measurement system do not contribute to the dynamics in this model of the milling system. The following model was used, FR + 2fanFR + oon2FR =KsaVfa-x(2fco„/+ oi„2f) 370/ Vol. 110, NOVEMBER 1988

a(mm)

Damping ratio versus depth of cut

From Fig. 7, the variation in the parameter f can be modeled as, f=0.4a-.65

(5)

where a is the depth of cut in mm and is in the range of 1.91 to 3.81 mm. The specific cutting force coefficient, Ks, includes the effects of tool and workpiece material, tool sharpness, and cutting speed. In this case Ks was found to be 2500 N/mm/tooth. The constant (3 was found to be 1.4 and a was found to be 0.73. Equations (2) and (4) are continuous time representations of the plant and milling system dynamics. To use these relationships for digital controller design we must express them in discrete time. Equation (2) can be transformed to discrete time by using 7=0.05 sec, a zero-order hold type of digital to analog converter, and a damping ratio less than 1.0. This transformation results in a pulse transfer function of the form, FR(z) V(z)

b0z + bx z + axz + a2 2

(6)

where b0, bx, ax and a2 can be found from theK, f, and cu„ of equation (2) as shown in [17, 18]. For a depth of cut of 2.5 mm on 1020 CR steel and a feedrate of 50.8 mm/min and a spindle speed of 550 rpm, a, = - 1.879, a2 = .9, b0 = 1.594, and bx = 1.537 and defining (4) ? ^ ' a s a one step delay operator, Transactions of the AS ME

Downloaded From: http://manufacturingscience.asmedigitalcollection.asme.org/ on 11/17/2014 Terms of Use: http://asme.org/terms

H E = d •=r-T:; 6

1

CS

1

-

Fig. 8(a)

>

6CP

*-

So

5P

'

p

5.08

c

3.81

ISO

&.

1£

Reduced block diagram lor the mill

•

:

•

—^~—i}:W^e^Yy S E ^ ; --:.:-. T-j/?~. r ::L r-- z ^ i r : = ~ ^ £ n-i -zr.: -

—

-.:-;:rr

:::-.. —-.TT:;:-: •iri-::--:— ..::= -:---.-- ^Tii: ; =

• r!i ? s ^

: : ; ^ r i\:\rz

--:5f:=>, H-1

1.27

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=

•!-:L-3~

•

!

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n 0

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HW'f: [ l i f T T ^

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7 : i - ^ = ~1H L3 .::-::_::;•:::: -::tr^r ^r:^:::j

:-^i ? ; : : ; : -

6.35

Block diagram of mill

%

Fig. 8(b)

-:±s\

7.62

G

ll^ltii =0=1

4

6

8

io

12

14

Time, t, sec.

Spindle

Fig. 8(d) Table position vs time for step change in teed

Position Transducer

Mill Table

Fig. 8(c)

V(k)

Table position measurement system

g-'(1.594+1.537 q~l) 1-1.879 q~l + .9 - 2

A{q-1)

(7)

with, A(q-l)

=

l+alq-l+a2q~

system. Their work resulted in a fourth order model of the milling system with all of the eigenvalues having imaginary parts indicating an oscillatory system. Their work also shows how the eigenvalues change for 3 sets of cutting conditions. Their results indicate the same general trend as our data: the system order is higher than one and the response can become oscillatory under cutting conditions. These modeling results, although somewhat diverse, are not inconsistent. Such models reflect the milling system, including drive, sensor, structural, and cutting effects. It is of interest to isolate such effects for a better understanding of the underlying process. However, our goal here is to obtain a model of our milling system relating changes in the control voltage (proportional to the feedrate command) to changes in the resultant cutting force. The model in equation (4), and its discrete time counterpart in equation (7), is then suitable for our goal of discrete controller design. Implications for Controller Design

and B(q-*) = b0 + blqwith the a, and the b, as defined above. Equation (7) is the model required for digital controller design. Comparison With Previous Models. The model given in equation (4) is somewhat different from the models researchers usually use for control of milling machines. While the process has generally been treated as a first-order system [9, 9a], our data indicate that a second-order model may be more appropriate. Furthermore, the overall system gain has been modeled somewhat differently than in previous studies. A static model, developed for use in turning [10], and also used to describe milling operations, is given by, F=Cah"

(8)

where Cis a constant, h is the average chip thickness (which in milling, is proportional to the feed per tooth) and n is a constant, usually less than one. The gain of equation (8) is proportional to the axial depth of cut while our experimental results indicate that the gain is proportional to the depth raised to some exponent, as shown in equation (|4) (to compare equation (4) to (8) consider steady operation with a constant feedrate where the derivatives in equation (4) go to zero). An interesting similarity between the two models is that research in turning shows the value of n to be 0.7 [2]; we found a value of 0.73 for the corresponding exponent, a, from our milling experiments. Garcis-Gardea at al. [16] have used the Dynamic Data Systems methodology to evaluate the dynamics of the milling Journal of Engineering for Industry

Now that a suitable model has been developed, candidate controller designs can be evaluated. We have evaluated a proportional plus integral action (PI) [12] and linear model reference controller (LMRC) [11] using simulations, and (in the case of the PI controller) machining tests. Both controllers were designed for a depth of cut of 2.5 mm. Then, when a suitable design was completed, the controller performance was simulated with the depth of cut varying from 2 mm to 4 mm in 1 mm steps, as shown in Fig. 9. The controller performance was evaluated by using digital simulations as well as actual machining tests. These simulations were performed using the SIMULA Digital Control System Simulation Package [3] developed by the authors for this project. The SIMULA package simulates the discrete time digital controller, the continuous time plant as well as the analog-to-digital converter and the digital-to-analog converter. In these digital simulations, equations (4) and (5) were used as the model of the plant with the values of Ks, /3, and a found in our experiments. Linear Model Reference Controller. In the case of the LMRC, a reference model specifies the desired performance of the closed loop system. In this study, the reference model was selected to be: FR*(z) .0104Z+.0097 (9) r(z) ~ z2-l.767z+JS7 Where FR * (z) is the output of the reference model, and r(z) is the reference input (the reference force). This model has the NOVEMBER 1988, Vol. 110/371

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Time,

t

(sec) Time,

Fig. 9

t (SGC)

Step changes in depth-of-cut Fig. 10

Simulated LMRC, at the designed conditions

same structure as the actual plant [see equation (6)] but has an overall gain of 1 (because the input is now force and in equation (6) the input was a voltage), a damping ratio of 0.8 and a natural frequency of 3 rad/sec. This model can be expressed algebraically as, FR*{k)

q'Hiq^) (10) E(q^) r(k) where£'( 1 5'~ 1 )=l+e 1 q~l + e2q1 and H(q~i) = h0 + h{ q~l. Then to design the controller we express equation (7) in predictor form [11], = y(q~l)FR(k)+d(q-[)V(k)

E(q-')FR(k+l)

(11)

where y(q-i)

G(q-i)

=

and F(qi)B(q~l)

&{q->) =

Fig. 11

Simulated LMRC, depth of cut varying

l

F(q "') and G(q~ ) are the unique polynomials satisfying E(q-i)

= F(q~i)A(q-i)

+ q->G(q->)

(12)

since the control objective is to have the output of the plant, FR, follow the output of the model, FR*, the control law can then be derived by equating E(q~[) FR(k+l) (equation 11) with E(q~i)FR*(k+ 1) (equation 10) which yields, y{q-i)FR(k)+8(q-i)V(k)=H(q~1)r(k)

(13)

Proportional Plus Integral Controller. Due to the poor performance of the LMFC, a proportional plus integral PI controller was designed. The PI controller was tuned so that at a depth of cut of 2.5 mm. the closed loop poles were at .944 + 0.137i, .944-0.1371, and 0.99, in the z domain. These values were selected to yield stable performance over the range of depths from 2 mm to 4 mm.The control law is given by, V(k+ 1)= V(k) + (Ki + Kp)E(k)

Then solving for the control output, V{k), gives, V(k)=-}-[h0r(k)+filr(k

-KpE(k-

1)

(16)

with

l)-(ei-ai)FR(k)

E(k)=r(k)-FR{k) (e2-a2)FR(k-l)

+

blV(k-l)]

(14)

Then, substituting the values from equation (6) and equation (9) into equation (14) V(k) = .0065r(£) + .006lr(k-1)-

.0703F* (k)

+ .0109FR(k-l)-.964V(k-l)

(15)

For the design conditions, this control law produces the performance shown in Fig. 10, where the reference model output and the controlled system output are plotted. This figure shows that, despite the process nonlinearities, there is only a small overshsoot and the system response has been improved. However, Fig. 11 shows the resultant force and the model output when the depth of cut is allowed to vary as shown in Fig. 9. This performance is unacceptable due to the steady state tracking errors, resulting from the sensitivity of the controller to parameter variation [11]. 372/ Vol. 110, NOVEMBER 1988

Kp =.0006 Ki = .0003 The performance of this controller was simulated, and the results are shown in Fig. 12. In this simulation the variation in depth of cut is the same as was used for the LMFC, and the performance is somewhat better. The controller is doing a good job of maintaining the reference force, but there are rather large "overshoots" at the step changes. These "overshoots" can cause excessive tool deflection and may even break the tool. Machining Tests. To further evaluate the performance of the PI controller, machining tests were conducted. The machining test was run on 1020 CR steel, with a four-tooth HSS cutter, a spindle speed of 550 rpm, and a programmed feedrate of 50.8 mm/min. The results are shown in Fig. 13, Transactions of the ASME

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where the resultant force is plotted, and Fig. 14, where the feedrate override is plotted. Notice, in Fig. 14, that in the first section of the cut, at the 2 mm. depth, the controller output saturated before the reference value was reached, at the 3 mm depth, the feedrate was reduced, and at 4 mm depth, further

reduced. Finally, the feedrate was increased to the maximum value as the tool exited the cut. The overshoot shown in Fig. 13 is not as large as predicted by the simulations. In the simulations, the changes in depth

15

30 Time, t

Fig. 15

45

60

(sec)

Simulated PI controller with ramp changes in depth-of-cut

Time, t (sec)

Fig. 12

Simulated P.I. controller with depth-of-cut variation

Reference

y

15

30 Time, t

Fig. 16 Time, t

Fig. 13

Ramp changes in depth-of-cut

(sec)

Actual PI controller cutting stepped 1020 CR steel

Time, t

Fig. 14

45 (sec)

(sec)

Feedrate override voltage for the PI controller

Journal of Engineering for Industry

Time, t

Fig. 17

(sec)

Actual PI controller cutting 2024 Al at 4 mm depth

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occur instantaneously across the entire tool. In the actual operation, the tool gradually advanced into the new depth allowing the controller to reduce the feed before the new depth affected the entire tool. The simulation results in Fig. 15 show the controller performance when the depth of cut changes as a ramp as shown in Fig. 16. The length of the ramp in Fig. 16 was calculated based on a 12.7 mm diameter cutter arid a feedrate of 50.8 mm/mm. Comparing Fig. 15 with the machining results in Fig. 13 shows good agreement. However, at higher feedrates we expect this effect to be small, and the behavior of the controller to be more like that shown in Fig. 12. While we now have an understanding of how a fixed gain controller responds to changes in depth of cut, we do not know what will happen when the work-piece material properties vary. To address this, a machining test was conducted on 2024 aluminum. The results, when aluminum is machined with the PI controller designed for machining steel, are shown in Fig. 17. Here the depth of cut was constant at 4 mm., and the spindle speed was increased to 1200 rpm. In this, case, the performance was much worse than that for the steel. When the tool begins to enter the work, there is a 50 percent overshoot which takes almost 30 seconds to decay. These large errors represent serious performance problems.

Summary and Conclusions The purpose of the work described in this paper was to develop a model for the milling system that we could use for controller design and to examine the effects of changes in the process on the parameters of the model. We were also interested in how these changes affect the performance of fixed gain controllers.

0

Fig. A1 Lumped parameter model of the milling system

A P P E N D I X A Simple Model for the Milling System Dynamics A model for the dynamics of the milling system can be developed from the system shown in Fig. A l . The parameters M, k, and c are a lumped representation of all mass, stiffness, and damping effects in the milling system which lead to discrepancies between the actual velocity of the work piece relative to the end of the tool, v0 (related to chip thickness) and the input table velocity, vr These effects may arise, for example, from the cutting process as well as the spindle structural characterstics. The input to the system is the table velocity, Vj. The output, as shown in Fig. Al is the actual velocity of the tool relative to the work piece, v0. The differential equation for the system shown is: c(vi-vo)

+ k(xi-x0)

= mv0

Conclusions. The primary conclusion is that normal variations in the process can cause degradation in the performance of fixed gain controllers. Furthermore, these problems may occur unpredictably. Because the changes in damping and gain have opposite effects on the stability of the system, there may be local regions where the performance is acceptable, others where the performance is sluggish but stable, and still others where the controller is unstable. This work has also pointed out two areas where further research is needed: (i) Further work is required to accurately characterize and explain the dynamics of the milling system. (2) Further research is needed in applying adaptive controllers or robust controllers to the milling system. 374/Vol. 110, NOVEMBER 1988

(Al)

The transfer function for this system can be obtained by taking the laplace transform, noting that x=v/s. F

Summary. First, note that the milling system mechanics may produce a periodic fluctuation in forces. This is due to the interrupted cut, the varying chip thickness, and the runout problems. The choice of signal processing to be applied is dependent on the cutting conditions and the nature of the force constraint [6, 7]. The main results of this work can be summarized as follows: (J) The data presented in this paper indicate that the milling system dynamics are approximately represented by a nonlinear and nonstationary second-order differential equation. (2) The parameters of this second-order model vary significantly with cutting conditions; the gain of the system increases nonlinearly with the depth of cut; the damping of the system increases linearly with depth. (5) Both the PI controller and the LMRC have performance problems when the process parameters (i.e., depth of cut, or material properties) vary.

1

cs + k o(s) = Vj(s) ins2 + cs + k Then applying Taylor's equation [6], FR=Krv0 equation (3), v0 = GOR V:

(A2)

and from

FR __ K(cs + k) (A3) V ms2 + cs + k where K=KTGOR, and dividing through by m, defining o>„2 = k/m and 2fw„ = c/m gives the form in equation (2), FR(s) V(s)

K(2^ns + un2) s2 + 2$uns + wn2

(A4)

Acknowledgments We are pleased to acknowledge the financial support of the National Science Foundation under grant, MEAM-8112629, and the equipment and materials donated by TRW Corp. and Kennametal Inc. We would also like to acknowledge the assistance of Lynn Buege, Fred Rasmussen, Steve Culp, and Steve Erskin with the experimental studies. References 1 Ulsoy, A. G., Koren, Y., and Rasmussen, F., "Principal Developments in the Adpative Control of Machine Tools," ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 105, No. 2, June 1985, pp. 107-112. 2 Masory, O., and Koren, Y., "Adpative Control System for Turning," Annals of the CJRP, Vol. 29, 1980, pp. 281-284. 3 Lauderbaugh, L. K., and Ulsoy, A. G., SIMULA: Digital Control System Simulation Package, Technical Report No. NSF/MEA-84008, National Technical Information Service, Springfield, VA, 1984. 4 Nakazawa, K., "Improvement of Adaptive Control of Milling Machines by Non-Contact Cutting Force Detector," Proceedings of the 16th International Machined Tool Design and Research Conference, pp. 109-116.

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5 Bedini, R., Lisini, G., and Pinotti, P., "Experiments on Adaptive Control of a Milling Machine," ASME JOURNAL OF ENGINEERING FOR INDUSTRY, February 1976, pp. 239-245. 6 Tlusty, T., and MacNeil, P., "Dynamics of Cutting Forces in End Miling," Annals ofCIRP, Vol. 24, 1975, pp. 21-25. 7 Fu, H., DeVor, R., and Kapoor, S., "A Mechanistic Model for the Prediction of the Force System in Face Milling Operations," ASME JOURNAL OF ENGINEERING FOR INDUSTRY, Feb.

1984,

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106, pp. 81-88.

8 Tlusty, J., and Elbestawi, M., "Analysis of Transients in an Adaptive Control Servo Mechanism for Milling with Constant Force," ASME JOURNAL OF ENGINEERING FOR INDUSTRY, Vol. 99, No. 3, August, 1977, pp. 766-772. 9 Tomizuka, M., Oh, J. H., and Dornfeld, D. A., "Model Reference Adaptive Control of the Milling Process," in Control of Manufacturing Processes arid Robotic Systems, Edited by D. E. Hardt and W. J. Book, ASME, New York, 1983. 9a Attanis, Y., Yellowley, I., and Tlusty, J., "The Detection of Tool Breakage in Milling," in Sensors and Controls for Manufacturing, Edited by E. Kannatey-Asibu, A. G. Ulsoy, and R. Komanduri, ASME, New York, 1985, pp. 41-48. 10 Koren, Y., and Masory, O., "Adaptive Control and Process Estimation," Annals ofCIRP, Vol. 30, No. 1, 1981, pp. 373-376.

Journal of Engineering for Industry

11 Goodwin, G. C , and Sin, K. S., Adaptive Filtering Prediction and Control, Prentice-Hall Inc., New Jersey, 1984. 12 Takahashi, Y., Rabins, M., and Auslander, D. M., Control and Dynamic Systems, Addison-Wesley Publishing Co., California, 1972. 13 Beachley, N. H., and Harrison, H. L., Introduction to Dynamic System Analysis, Harper and Row Publishers, New York, 1978. 14 Watanabe, T., and Iwai, S., "A Geometric Adaptive Control System to Improve the Accuracy of Finished Surfaces Generated by Milling Operations," in Measurement and Control for Batch Manufacturing, Edited by D. E. Hardt, ASME, New York, 1982. 15 Watanabe, T., and Iwai, S., "Design of an Adaptive Control Constraint System of a Milling Machine Tool," Proceedings of the IFAC Symposium on Computer Aided Design of Multivariate Technological Systems, Sept. 15-17, 1982 West Lafayette, Ind. 16 Garcia-Gardea, E., Barney, F. A., and Wu, S. M., "Determination of True Cutting Signal by Separation of Instrumentation Dynamics from Measured Response," ASME JOURNAL OF ENGINEERING FOR INDUSTRY, August 1979, Vol. 101, pp. 264-268. 17 Lauderbaugh, L. K., Implementation of Model Reference Adaptive Force Control in Milling, Ph.D Thesis The University of Michigan, Ann Arbor, MI., 1986. 18 Astrom, K. J., and Wittenmark, B., Computer Controlled Systems Theory and Design, Prentice-Hall Inc., New Jersey, 1984.

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