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An adaptive-control system for the grinding process
Journal of Materials Processing Technology, 40 (1994) 433-442 Elsevier
433
An adaptive-control system for the grinding process D a o y u an Yu, Zhengcheng Duan, J i a n c h u n Den, Hongwen Ha Department # 1 of Mechanical Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China
(Received October 15, 1992; accepted March 29, 1993)
Industrial Summary A new adaptive-control system has been successful in the external, cylindrical crossgrinding process in terms of raising the material removal rate and improving the geometrical accuracy of slender shafts. In comparison with the conventional method in the crossgrinding process, this method of adaptive-control grinding is superior in respect of the material removal rate and the roundness of the slender shafts. The preceding beneficial effects are independent of the grinding parameters chosen. It is practicable, therefore, for the adaptive control system to be applied in practical production.
1. Introduction In the external cylindrical cross-grinding process, the geometrical error of slender shafts being machined is caused by the combined influence of many factors. However, the most important of these factors under constant circumstances or conditions is the grinding force. It is known t hat slender shafts are weak in their radial rigidity and t hat the radial rigidity is sometimes unequal rotationally, for example, crank shafts, during the course of one rotation; thus, the machining remainder is not uniform and the grinding force is not constant in the grinding process. Because of the variation of the grinding forces, the elastic deformation of slender shafts is changing constantly during the grinding process, which not only results in the material removal rate at different machining portions of the slender shafts being different, but also produces geometric errors in the shafts. In addition, a correlative analysis [1] has shown t h a t the correlation coefficient between the grinding force and the geometrical error of slender shafts is 0.73-0.81, which means t hat the grinding force is the
Correspondence to: Daoyuan Yu, Department # 1 of Mechanical Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China.
0924-0136194]$07.00 O 1994 Elsevier Science B.V. All rights reserved. SSDI 0924-0136(93)E0046-J
434
D. Yu et al./An adaptive-control system
Fig. 1. Forces in the cross-grinding process.
most important factor directly affecting the geometrical accuracy of the shafts. It is important to control the grinding force in the grinding process in order to raise the geometrical accuracy and machining efficiency of the slender shafts. The grinding force in the cross-grinding process consists of a tangential and a radial grinding force, as illustrated in Fig. 1. Since the geometrical accuracy of slender shafts is sensitive to variation of the radial grinding force, how to control the radial grinding force in the external cylindrical cross-grinding process is the subject of the present discussion. In the past, a constant force grinding was applied in the grinding process in production by keeping the cross-feed velocity of the wheel constant. From the early 1970s, a new technology - Adaptive Control - has been developed abroad and applied to manufacture. However, up to now, the adaptive control of grinding force is still retained at an experimental stage owing to its complicated construction: e.g. the adaptively-controlled rotational velocity of the workpieces in cylindrical grinding by Yamanashi University, Japan; the cylindrical grinder with vibration-excited grinding head adaptive control by Tokyo University; the grinding force adaptively-controlled cylindrical grinder by Rzeszow University, Poland and etc. Taking into account the peculiarity of cross grinding, a new and simpler adaptive-control system used for controlling the radial grinding force has been designed and applied to the cross-grinding process, the results proving to be satisfactory.
2. Principle As shown in Fig. 2, the adaptive-control system is composed of two strain gauges attached to a tailstock center, a half-bridge circuit, a pre-amplifier, a post-amplifier, an electro-hydraulic servo valve and a hydraulic cylinder. F 8 is the radial grinding force and F¢ is the compensatory force that the adaptivecontrol system produces, acting in opposition to Fg and in the negative direction.
D. Yu et al./ An adaptive-control system
435
When the two forces are equal, the half-bridge circuit keeps a balance and there is no output signal, but when the forces are not equal there is an output signal t h a t is in proportion to a difference between the two forces. Once the output signal is returned to the servo valve through amplification, the hydraulic cylinder adds AFt to Fc or subtracts AF~ from F~, dependent on whether the output signal is positive or negative. This causes the F¢ to follow the Fg until the half-bridge circuit is restored to balance once more, and so on. Thus, the difference between the two forces acting on the slender shafts tends to zero. As the result, there is no elastic deformation of the slender shafts in the crossgrinding process. The adaptive-control system, therefore, can avoid producing geometrical error due to the elastic deformation of the slender shafts being machined.
Fig. 2. The adaptive-control system.
_ I frequency I instrument
I
Fig. 3. The frequency-response test set-up.
436
D. Yu et al./An adaptive-control system
3. Simulation experiment I n o r d e r t h a t t h e a d a p t i v e - c o n t r o l s y s t e m c a n be s u c c e s s f u l i n t h e crossg r i n d i n g p r o c e s s , a f r e q u e n c y - r e s p o n s e e x p e r i m e n t h a s to b e m a d e so as to v e r i f y w h e t h e r o r n o t its s t a b i l i t y , a c c u r a c y a n d r e s p o n s e v e l o c i t y s a t i s f y t h e r e q u i r e m e n t s . As s h o w n i n Fig. 3, t h e f r e q u e n c y t e s t i n s t r u m e n t e x p o r t s a s i n e s i g n a l , t h e a m p l i t u d e of w h i c h is c o n s t a n t a t u n i t y a n d t h e p h a s e of w h i c h is e q u a l to 0.1, 0.3, 0.5, 1, 2, ..., 27 Hz. D a t a of t h e f r e q u e n c y - r e s p o n s e e x p e r i m e n t of t h e a d a p t i v e - c o n t r o l s y s t e m a r e l i s t e d i n T a b l e 1.
Table 1 Experimental data of the system's frequency response m
Amplitude ratio
Phase difference (°)
Frequency (Hz)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.8932 0.9009 0.9126 0.9204 0.9243 0.9670 0.9709 0.9845 1.0214 1.0369 1.0485 1.0777 1.1126 1.1390 1.1728 1.2019 1.2560 1.2854 1.7864 1.5922 1.4369 1.2291 1.2214 1.1243 1.0447 0.9553 0.8796 0.8117 0.7456 0.6680
0 1.1 1.5 2.5 4.1 5.5 8.1 10.0 12.2 13.4 16.0 17.4 19.4 21.2 23.3 26.4 28.4 30.1 163.5 170.5 178.0 184.2 187.2 189.5 192.2 194.1 197.3 199.0 201.0 205.5
0.1 0.3 0.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
D. Yu et al./An adaptive-controlsystem
437
4. Frequency response The transfer function of a linear dynamic system is usually represented as the ratio of two frequency-dependent polynomials and it can be evaluated from the data of a frequency-response experiment [2]. According to Ref. [3], the transfer function of an adaptive-control system is expressed as the ratio of two dependent polynomials, namely:
GR(jog) ao N(j~) G(j(o) l +Gk(j(9) bs(j(o)s +...+bl (flo)+ l=D(jo~)
(1)
where Ao
Gk(rio) = B5 (jco)s + " "B1 (rio) + 1
(2)
Gk(flo) is the open-loop transfer function and G(jcg) is the closed-loop transfer function of the system. The integral squared error between the system's impulse response and the approximate impulse response, which is a representation of the transfer function to be found in the time domain, is defined as: Et= i {ga(t)-g(t)}z dt --00
where
ga(t)=L -1G~(s) g(t)=L -1 G(s) According to Parseval's theorem, the error function in the frequency domain is derived as follows:
E~,= f IGa(flo)-G(jco)l 2 dco -oo
where G.(jo)) is the frequency response and G(flo) is the approximate transfer function of the system to be found. The sum of the approximate error of all the experimental points is defined by
E= ~ [G.(jcoi)-G(jcoi)[ 2 A(~i i~1
Substituting ¢i
1
[D(flo;)12
weighted factor - in the above equation
E= ~ ¢ilD(jcoi)Va(Jcoi)-N(jcoi)[ 2 Acoi i=1
(3)
D. Yu et al./ An adaptive-control system
438 where G~(j0)i) a(.o I
= Ai cos Oi+ j Ai sin Oi= Ri + jIi A(.om~_0)m__0)ra_l
0)1+('02
A0)i =(19i+1-0)i-1
2
2
and m is the ruth g r o u p of the e x p e r i m e n t a l data. F o l l o w i n g accepted s t a n d a r d m a t h e m a t i c a l procedures, eqn. (3) is now diff e r e n t i a t e d with respect to e a c h of the u n k n o w n coefficients ao, bl, b2 ..... b5 and t h e n set equal to zero. The p o l y n o m i a l coefficients are d e t e r m i n e d so as to minimize the w e i g h t e d sum of squares of the errors of the experimental d a t a in the iterative c o m p u t a t i o n . The problem can be f o r m u l a t e d by a set of the linear s i m u l t a n e o u s algebraic e q u a t i o n s of the form:
Qx =p
(4)
where
ao
Q=
2o.
""
TUI°
x= bl
P=
.
T5
whence
i=1
i=1
Uh= ~ ~,co~(Ri+I,)2A0), Th = ~ c~,0)~I,Aoo, i=l
i=1
F r o m the d a t a in Table 2, the u n k n o w n coefficients of eqn. (1) h a v e been solved from eqn. (4). The closed-loop t r a n s f e r f u n c t i o n of the a d a p t i v e - c o n t r o l system is 0.6199
G(S) = 0.53 x 10- 7 s 5 + 0.41 x 10- 5 s 4 + 0.98 x 10 -4 s 3 + 0.55 x 10- 2 s 2 + 0.29 x 10- ' s + 1
5. System analysis The c h a r a c t e r i s t i c e q u a t i o n of the adaptive c o n t r o l system is 0.53 x 10-7s5+0.41 x 10 -5s4+0.98 x 10-4s3+0.55 x 10-2s2+0.29 x 10 - i s + l = 0
~ ~ ~ ~ 1~ 1~ 1~ 1~
I d m m = O . l mm.
1 2 3 4 5 6 7 8
workpiece rotationrate (rpm)
0.9 0.9 1.2 1.2 0.9 0.9 1.2 1.2
infeed velocity of w h e e l (ram/rain)
Grinding parameters
Table 2
10 15 15 10 15 10 10 15
(dram)
(s)
0.9 1.8 0.9 1.8 0.9 1.8 0.9 1.8
wheel depth of c u t
sparkout period
5.0 3.9 4.8 4.3 2.0 5.0 3.8 3.4
ellipticity (dmm)
3.1 3.0 2.8 2.9 1.9 3.6 3.3 2.1
radial error (dmm)
radial error (dmm)
0.2 0.2 0.5 0.3 0.1 0.4 0.2 0.2
ellipticity (dmm)
grinding
0.7 0.8 0.9 0.9 0.8 0.9 0.9 0.7
2.8 1.1 1.2 1.5 1.1 1.7 2.7 1.0
ellipticity (dmm)
conventional grinding
conventional grinding ACC
m a t e r i a l 45 #
m a t e r i a l HT15-33
Measurement data after grinding
1.9 1.2 1.3 1.2 1.4 1.2 2.3 1.3
radial error (dmm)
0.2 0.2 0.5 0.2 0.2 0.2 0.5 0.4
ellipticity (dram)
ACC grinding
0.8 0.9 0.9 0.7 0.7 0.8 0.6 0.8
radial error (dmm)
440
D. Yu et al./An adaptive-control system
2 0 x LOG/G k (W)/ 18-~ !
00V""'~
,
i
PLOT
A-P
,
i
1'5
1'8
2'1
24
27 Hz
2'1
2'4
2'7 Hz
-60-120-180-24C
P-P
PLOT
Fig. 4. The Bode diagram.
In terms of the Aorwitz-Routh Stability Criterion, the adaptive-control system is stable. Knowing the closed-loop frequency responsed of the system, its open-loop frequency response is determined easily Gk(S)=
0.6199 0.53 × 10- vs s + 0.41 x 10 s s4+ 0.98 x 10- ¢s 3+ 0.55 x 10- 2s 2+ 0.29 × 10 1s + 0.3801
The Bode diagram of G k ( j o ) is shown in Fig. 4, the stability margin of the system being evaluated from it. The gain margin of the system is equal to 4.8 db and the phase margin of the system is equal to 14 degrees. As shown in Table 1, the amplitude versus frequency (0
6. Grinding experiment The adaptive-control grinding has been carried out on an NC external grinder (M120N-W). Under the same set of grinding parameters and the same original errors of the slender shafts being machined, conventional cross-grinding is carried out first and then adaptive-control grinding. The results of the two different grinding methods are presented in Table 2 and illustrated in Fig. 5, showing that adaptive-control grinding is superior to conventional
441
D. Yu et al./ An adaptive-control system Al1211314M
A~12/1314M
(a) Al1211314F.f
A~12/1314~1
HOSSONLEICE
Fig. 5. Comparison between the two grinding methods: (a) conventional grinding roundness before grinding (urn) 64, roundness after grinding (um) 26; (b) adaptive-control grinding - roundness before grinding (um) 62, roundness after grinding (um) 10.
g r i n d i n g in respect of the g e o m e t r i c a l a c c u r a c y and the m a t e r i a l r e m o v a l r a t e of the slender shafts. 7. C o n c l u s i o n s T h e c o m p e n s a t o r y force t h a t the a d a p t i v e - c o n t r o l system produces can fl~l. low rapidly a n y v a r i a t i o n of the radial grinding force in the external ¢,xlir~ drical cross-grinding process. A d a p t i v e - c o n t r o l grinding increases gr¢,.t I~ t!,
442
D. Yu et al./ An adaptive-control system
grinding efficiency (i.e. the m a t e r i a l removal rate) and the geometrical accur a c y of slender shafts, especially with long and thin shafts. The geometrical a c c u r a c y of the slender shafts being m a c h i n e d is i n d e p e n d e n t of the grinding p a r a m e t e r s chosen.
References [1] Yu Gao, A correlative analysis of grinding force and geometrical error, Master's thesis, Hauzhong University, China, 1982. [2] K. Yamashita, Y. Suzuki and K. Fujii, A method to find transfer function from frequency response data, J. Jpn. Assoc. Aurora. Contr. Eng., 14(11) (1970) 667-674. [3] Hongren Li, Hydraulic Control System, Press of National Defence Industry, China, 1978.
433
An adaptive-control system for the grinding process D a o y u an Yu, Zhengcheng Duan, J i a n c h u n Den, Hongwen Ha Department # 1 of Mechanical Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China
(Received October 15, 1992; accepted March 29, 1993)
Industrial Summary A new adaptive-control system has been successful in the external, cylindrical crossgrinding process in terms of raising the material removal rate and improving the geometrical accuracy of slender shafts. In comparison with the conventional method in the crossgrinding process, this method of adaptive-control grinding is superior in respect of the material removal rate and the roundness of the slender shafts. The preceding beneficial effects are independent of the grinding parameters chosen. It is practicable, therefore, for the adaptive control system to be applied in practical production.
1. Introduction In the external cylindrical cross-grinding process, the geometrical error of slender shafts being machined is caused by the combined influence of many factors. However, the most important of these factors under constant circumstances or conditions is the grinding force. It is known t hat slender shafts are weak in their radial rigidity and t hat the radial rigidity is sometimes unequal rotationally, for example, crank shafts, during the course of one rotation; thus, the machining remainder is not uniform and the grinding force is not constant in the grinding process. Because of the variation of the grinding forces, the elastic deformation of slender shafts is changing constantly during the grinding process, which not only results in the material removal rate at different machining portions of the slender shafts being different, but also produces geometric errors in the shafts. In addition, a correlative analysis [1] has shown t h a t the correlation coefficient between the grinding force and the geometrical error of slender shafts is 0.73-0.81, which means t hat the grinding force is the
Correspondence to: Daoyuan Yu, Department # 1 of Mechanical Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China.
0924-0136194]$07.00 O 1994 Elsevier Science B.V. All rights reserved. SSDI 0924-0136(93)E0046-J
434
D. Yu et al./An adaptive-control system
Fig. 1. Forces in the cross-grinding process.
most important factor directly affecting the geometrical accuracy of the shafts. It is important to control the grinding force in the grinding process in order to raise the geometrical accuracy and machining efficiency of the slender shafts. The grinding force in the cross-grinding process consists of a tangential and a radial grinding force, as illustrated in Fig. 1. Since the geometrical accuracy of slender shafts is sensitive to variation of the radial grinding force, how to control the radial grinding force in the external cylindrical cross-grinding process is the subject of the present discussion. In the past, a constant force grinding was applied in the grinding process in production by keeping the cross-feed velocity of the wheel constant. From the early 1970s, a new technology - Adaptive Control - has been developed abroad and applied to manufacture. However, up to now, the adaptive control of grinding force is still retained at an experimental stage owing to its complicated construction: e.g. the adaptively-controlled rotational velocity of the workpieces in cylindrical grinding by Yamanashi University, Japan; the cylindrical grinder with vibration-excited grinding head adaptive control by Tokyo University; the grinding force adaptively-controlled cylindrical grinder by Rzeszow University, Poland and etc. Taking into account the peculiarity of cross grinding, a new and simpler adaptive-control system used for controlling the radial grinding force has been designed and applied to the cross-grinding process, the results proving to be satisfactory.
2. Principle As shown in Fig. 2, the adaptive-control system is composed of two strain gauges attached to a tailstock center, a half-bridge circuit, a pre-amplifier, a post-amplifier, an electro-hydraulic servo valve and a hydraulic cylinder. F 8 is the radial grinding force and F¢ is the compensatory force that the adaptivecontrol system produces, acting in opposition to Fg and in the negative direction.
D. Yu et al./ An adaptive-control system
435
When the two forces are equal, the half-bridge circuit keeps a balance and there is no output signal, but when the forces are not equal there is an output signal t h a t is in proportion to a difference between the two forces. Once the output signal is returned to the servo valve through amplification, the hydraulic cylinder adds AFt to Fc or subtracts AF~ from F~, dependent on whether the output signal is positive or negative. This causes the F¢ to follow the Fg until the half-bridge circuit is restored to balance once more, and so on. Thus, the difference between the two forces acting on the slender shafts tends to zero. As the result, there is no elastic deformation of the slender shafts in the crossgrinding process. The adaptive-control system, therefore, can avoid producing geometrical error due to the elastic deformation of the slender shafts being machined.
Fig. 2. The adaptive-control system.
_ I frequency I instrument
I
Fig. 3. The frequency-response test set-up.
436
D. Yu et al./An adaptive-control system
3. Simulation experiment I n o r d e r t h a t t h e a d a p t i v e - c o n t r o l s y s t e m c a n be s u c c e s s f u l i n t h e crossg r i n d i n g p r o c e s s , a f r e q u e n c y - r e s p o n s e e x p e r i m e n t h a s to b e m a d e so as to v e r i f y w h e t h e r o r n o t its s t a b i l i t y , a c c u r a c y a n d r e s p o n s e v e l o c i t y s a t i s f y t h e r e q u i r e m e n t s . As s h o w n i n Fig. 3, t h e f r e q u e n c y t e s t i n s t r u m e n t e x p o r t s a s i n e s i g n a l , t h e a m p l i t u d e of w h i c h is c o n s t a n t a t u n i t y a n d t h e p h a s e of w h i c h is e q u a l to 0.1, 0.3, 0.5, 1, 2, ..., 27 Hz. D a t a of t h e f r e q u e n c y - r e s p o n s e e x p e r i m e n t of t h e a d a p t i v e - c o n t r o l s y s t e m a r e l i s t e d i n T a b l e 1.
Table 1 Experimental data of the system's frequency response m
Amplitude ratio
Phase difference (°)
Frequency (Hz)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.8932 0.9009 0.9126 0.9204 0.9243 0.9670 0.9709 0.9845 1.0214 1.0369 1.0485 1.0777 1.1126 1.1390 1.1728 1.2019 1.2560 1.2854 1.7864 1.5922 1.4369 1.2291 1.2214 1.1243 1.0447 0.9553 0.8796 0.8117 0.7456 0.6680
0 1.1 1.5 2.5 4.1 5.5 8.1 10.0 12.2 13.4 16.0 17.4 19.4 21.2 23.3 26.4 28.4 30.1 163.5 170.5 178.0 184.2 187.2 189.5 192.2 194.1 197.3 199.0 201.0 205.5
0.1 0.3 0.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
D. Yu et al./An adaptive-controlsystem
437
4. Frequency response The transfer function of a linear dynamic system is usually represented as the ratio of two frequency-dependent polynomials and it can be evaluated from the data of a frequency-response experiment [2]. According to Ref. [3], the transfer function of an adaptive-control system is expressed as the ratio of two dependent polynomials, namely:
GR(jog) ao N(j~) G(j(o) l +Gk(j(9) bs(j(o)s +...+bl (flo)+ l=D(jo~)
(1)
where Ao
Gk(rio) = B5 (jco)s + " "B1 (rio) + 1
(2)
Gk(flo) is the open-loop transfer function and G(jcg) is the closed-loop transfer function of the system. The integral squared error between the system's impulse response and the approximate impulse response, which is a representation of the transfer function to be found in the time domain, is defined as: Et= i {ga(t)-g(t)}z dt --00
where
ga(t)=L -1G~(s) g(t)=L -1 G(s) According to Parseval's theorem, the error function in the frequency domain is derived as follows:
E~,= f IGa(flo)-G(jco)l 2 dco -oo
where G.(jo)) is the frequency response and G(flo) is the approximate transfer function of the system to be found. The sum of the approximate error of all the experimental points is defined by
E= ~ [G.(jcoi)-G(jcoi)[ 2 A(~i i~1
Substituting ¢i
1
[D(flo;)12
weighted factor - in the above equation
E= ~ ¢ilD(jcoi)Va(Jcoi)-N(jcoi)[ 2 Acoi i=1
(3)
D. Yu et al./ An adaptive-control system
438 where G~(j0)i) a(.o I
= Ai cos Oi+ j Ai sin Oi= Ri + jIi A(.om~_0)m__0)ra_l
0)1+('02
A0)i =(19i+1-0)i-1
2
2
and m is the ruth g r o u p of the e x p e r i m e n t a l data. F o l l o w i n g accepted s t a n d a r d m a t h e m a t i c a l procedures, eqn. (3) is now diff e r e n t i a t e d with respect to e a c h of the u n k n o w n coefficients ao, bl, b2 ..... b5 and t h e n set equal to zero. The p o l y n o m i a l coefficients are d e t e r m i n e d so as to minimize the w e i g h t e d sum of squares of the errors of the experimental d a t a in the iterative c o m p u t a t i o n . The problem can be f o r m u l a t e d by a set of the linear s i m u l t a n e o u s algebraic e q u a t i o n s of the form:
Qx =p
(4)
where
ao
Q=
2o.
""
TUI°
x= bl
P=
.
T5
whence
i=1
i=1
Uh= ~ ~,co~(Ri+I,)2A0), Th = ~ c~,0)~I,Aoo, i=l
i=1
F r o m the d a t a in Table 2, the u n k n o w n coefficients of eqn. (1) h a v e been solved from eqn. (4). The closed-loop t r a n s f e r f u n c t i o n of the a d a p t i v e - c o n t r o l system is 0.6199
G(S) = 0.53 x 10- 7 s 5 + 0.41 x 10- 5 s 4 + 0.98 x 10 -4 s 3 + 0.55 x 10- 2 s 2 + 0.29 x 10- ' s + 1
5. System analysis The c h a r a c t e r i s t i c e q u a t i o n of the adaptive c o n t r o l system is 0.53 x 10-7s5+0.41 x 10 -5s4+0.98 x 10-4s3+0.55 x 10-2s2+0.29 x 10 - i s + l = 0
~ ~ ~ ~ 1~ 1~ 1~ 1~
I d m m = O . l mm.
1 2 3 4 5 6 7 8
workpiece rotationrate (rpm)
0.9 0.9 1.2 1.2 0.9 0.9 1.2 1.2
infeed velocity of w h e e l (ram/rain)
Grinding parameters
Table 2
10 15 15 10 15 10 10 15
(dram)
(s)
0.9 1.8 0.9 1.8 0.9 1.8 0.9 1.8
wheel depth of c u t
sparkout period
5.0 3.9 4.8 4.3 2.0 5.0 3.8 3.4
ellipticity (dmm)
3.1 3.0 2.8 2.9 1.9 3.6 3.3 2.1
radial error (dmm)
radial error (dmm)
0.2 0.2 0.5 0.3 0.1 0.4 0.2 0.2
ellipticity (dmm)
grinding
0.7 0.8 0.9 0.9 0.8 0.9 0.9 0.7
2.8 1.1 1.2 1.5 1.1 1.7 2.7 1.0
ellipticity (dmm)
conventional grinding
conventional grinding ACC
m a t e r i a l 45 #
m a t e r i a l HT15-33
Measurement data after grinding
1.9 1.2 1.3 1.2 1.4 1.2 2.3 1.3
radial error (dmm)
0.2 0.2 0.5 0.2 0.2 0.2 0.5 0.4
ellipticity (dram)
ACC grinding
0.8 0.9 0.9 0.7 0.7 0.8 0.6 0.8
radial error (dmm)
440
D. Yu et al./An adaptive-control system
2 0 x LOG/G k (W)/ 18-~ !
00V""'~
,
i
PLOT
A-P
,
i
1'5
1'8
2'1
24
27 Hz
2'1
2'4
2'7 Hz
-60-120-180-24C
P-P
PLOT
Fig. 4. The Bode diagram.
In terms of the Aorwitz-Routh Stability Criterion, the adaptive-control system is stable. Knowing the closed-loop frequency responsed of the system, its open-loop frequency response is determined easily Gk(S)=
0.6199 0.53 × 10- vs s + 0.41 x 10 s s4+ 0.98 x 10- ¢s 3+ 0.55 x 10- 2s 2+ 0.29 × 10 1s + 0.3801
The Bode diagram of G k ( j o ) is shown in Fig. 4, the stability margin of the system being evaluated from it. The gain margin of the system is equal to 4.8 db and the phase margin of the system is equal to 14 degrees. As shown in Table 1, the amplitude versus frequency (0
6. Grinding experiment The adaptive-control grinding has been carried out on an NC external grinder (M120N-W). Under the same set of grinding parameters and the same original errors of the slender shafts being machined, conventional cross-grinding is carried out first and then adaptive-control grinding. The results of the two different grinding methods are presented in Table 2 and illustrated in Fig. 5, showing that adaptive-control grinding is superior to conventional
441
D. Yu et al./ An adaptive-control system Al1211314M
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HOSSONLEICE
Fig. 5. Comparison between the two grinding methods: (a) conventional grinding roundness before grinding (urn) 64, roundness after grinding (um) 26; (b) adaptive-control grinding - roundness before grinding (um) 62, roundness after grinding (um) 10.
g r i n d i n g in respect of the g e o m e t r i c a l a c c u r a c y and the m a t e r i a l r e m o v a l r a t e of the slender shafts. 7. C o n c l u s i o n s T h e c o m p e n s a t o r y force t h a t the a d a p t i v e - c o n t r o l system produces can fl~l. low rapidly a n y v a r i a t i o n of the radial grinding force in the external ¢,xlir~ drical cross-grinding process. A d a p t i v e - c o n t r o l grinding increases gr¢,.t I~ t!,
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D. Yu et al./ An adaptive-control system
grinding efficiency (i.e. the m a t e r i a l removal rate) and the geometrical accur a c y of slender shafts, especially with long and thin shafts. The geometrical a c c u r a c y of the slender shafts being m a c h i n e d is i n d e p e n d e n t of the grinding p a r a m e t e r s chosen.
References [1] Yu Gao, A correlative analysis of grinding force and geometrical error, Master's thesis, Hauzhong University, China, 1982. [2] K. Yamashita, Y. Suzuki and K. Fujii, A method to find transfer function from frequency response data, J. Jpn. Assoc. Aurora. Contr. Eng., 14(11) (1970) 667-674. [3] Hongren Li, Hydraulic Control System, Press of National Defence Industry, China, 1978.