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SYSTEM OPTIMIZATION OF PARAMETERS OF DIAMOND
GRINDING
Yenikieiev A.F.
doctor of technical sciences, reader, Donbas State Engineering Academy, professor
Yevsiukova F.M.
National Technical University "Kharkiv Polytechnic Institute", senior lecturer
Zykov I.S.
candidate of technical sciences, docent, NTU "KhPI", professor
Prihodko O.Y.
candidate of technical sciences, docent, NTU "KhPI", docent
Abramska I.B.
Ukrainian State University of Railway Transport, senior lecturer
ABSTRACT
It is offered the indirect methods for measuring of the roughness of the machined surface of the workpiece and cutting characteristics of the grinding wheel. It is built a three-level multiprocessor system for optimizing the parameters of the diamond grinding process based on these indirect methods as well as the method of coordinate-wise control and the principle of deviation control. It is developed mathematical models of hardware of the system taking into account the uncertainty factors which are caused by interference and inaccuracies in the measurement of input signals. It is built the schemes of computer modeling of processes of transformation of input information by hardware.
Keywords: multiprocessor system, hardware, indirect measurements, computer simulation.
Introduction. The use of well-known means of automation of the diamond grinding process (DG) has a goal to reduce the processing time of the surface of the workpiece and get a predetermined roughness. Traditional technologies are focused on the maintenance of
grinding modes by the one-dimensional computer system, which stores its data bank in the form of the processing program of the party details. The software doesnt use the information about the production quality of the machined surface of the workpiece, as to re-
ceive the signal through the direct measurement is impossible because of the lack of the necessary primary converter. This makes inefficient use of known computer systems.
The aim of the article is the development of the multiprocessor system (MS) optimization of parameters of the technological process DG in conditions of incomplete information. His goal is achieved by the solution of such tasks:
- development of the indirect methods for measuring of the roughness of the machined surface of the workpiece and the cutting characteristics of the grinding wheel (GW);
- development of the combined principle of the structural construction of three-dimensional MS;
- development of mathematical models of hardware of the system taking into account the uncertainty factors which are caused by interference and inaccuracies in the measurement of input signals;
- synthesis of hardware for processing of the input signals.
The combined principle of structural construction of MS. The basis of its development is based on: method of coordinate-wise control; hierarchical principle; indirect measurements of the expected quality of the machined surface of the workpiece and cutting characteristics of the grinding wheel; principle of deviation control. The structural features of the machine and the technological possibilities of GD process allow the use of such sensors: the rotational speed (o), transversal (Sj ) and longitudinal (S2 ) feeds of grinding wheel. Accordingly, the vector of parameters of the DG process in the combined principle of the MS construction has coordinates
X = (Sl, S2, of . (1)
The DG feeds, together with the rotational speed, determine the dynamics of the cutting process of the workpiece material. A shaft with DG is presented in the form of a beam with the clamped end. It is attached the cutting force of the workpiece material to its free end. Under the action of this force, the end of the beam makes forced oscillations, which are mathematically described by the differential equation
Jlgr(f)+Plg/(f)+ylg() = M(t), (2)
where Ji - moment of inertia, Pi - damping coefficient, Yi - torsional stiffness of the shaft, y(t) - swirl angle, M(t)= RF(t) - disturbing moment, R - radius of GW, F (t) - cutting force.
Deviations of speed are derived from the swirl angle of the shaft, accordingly to this, there is such an integral-differential equation
JjAffl' (t) + P1Arn(t) + — \Arn(t)dt = R1F(t). (3)
0
The standard levels of amplitude deviations are set as a result of the statistical processing of experimental data and they are entered in the MS database. According to the results of measurements of the deviation signal, MS evaluates the predicted amplitude of the micro-roughnesses and performs a comparison with the standards of database. As a result of comparison, the signals of coordinate setting of parameters of DG process are formed. For realization of these signals the MS database stores the vector of setpoints, which is represented by such mathematical model
Q=(a et, S1opt, S2opt, ®opt}. (4) Vectors of output signals of channels, technological parameters of DG process and microroughnesses are presented by such model
F(U, Q,£, A®, P,a>)= Y. (5)
The processes of transformation of input information channels are presented by the next generalized mathematical model
Fk = (uk, Qk ,lk )= 0. (6) Interference and errors of treatment if input signals form a vector with such coordinates
(7)
The new technology combined the processes of DG and DSG in the continuous cycle of treatment of the workpiece surface. To estimate the cutting ability of the grinding wheel, MS software uses such indirect methods:
• to increase by 5% of the power that is consumed by the drive of the main motion of the machine tool;
• to reduce by 7 % the average speed of the rotation wheel.
The structural construction of three-dimensional MS is shown in Fig. 1. The figure adopted symbols: MC1...MC5 - microcontrollers, GM - grinding machine, GW - grinding wheel, p - power main drive
of GM, Aet - admission to the amplitude of micro-
roughnesses, DAC - digital-to-analog converter, qt - input converter into a digital code, ED - electric drive, Ao - deviation of the rotation speed of GW, PMT - power measuring transducer, S - sensor, SA -servo actuator, I, U - signals of current and tension of the drive of the GM main motion, C1 - commutator of analog signals, T1.T5 - transducers, GIS - generator of impulsive signals.
GW correcting signal ►
from GM
from GM
Figure 1 - Structural construction of three-dimensional MS
The information link between MS and DG process is provided by channels of the first level which are built on the basis of MC1.. .3. Their hardware set the modes of cutting of the workpiece surface in the form of a vector (5). The output signals are formed by channels during the direct or reverse motion of the grinding wheel.
MC4 on the basis of direct measurements of the instantaneous speed and the information technology of the processing of this signal evaluates the predicted amplitude of microroughnesses. The speed signal processing algorithm consists of the following procedures: the deviation signal extraction and its presentation by the limited number of Fourier; the calculation of the active value and its comparison with databank standards. If the amplitude of microroughnesses exceeds the threshold, the size of which is defined by microrough-nesses, then MC4 analyzes the databank and forms the signals for the feed adjustment. The testing of these signals is performed by MC1 and MC2 during the pause between the direct and reverse motion of the grinding wheel. The drive of the transverse and longitudinal feeds set the new parameters of the grinding process.
The current and voltage signals of the drive of the main motion and also the speed signal of the wheel rotation provide the information link between DG process and hardware of the third level of MS. The information communication between this channel and DSG process is provided by the correcting signal of the grinding wheel which parameters are measured by the T5 block.
On the basis of the direct digital control method the MS functioning is organized by the real time operating system. Their software task [1] is performed as a result of the analysis of the databank information of the optimal parameters of the grinding process. The input information process is performed in the conditions of the limited time. It is developed the application software and the information database.
The modeling of the elements of the first level.
ED3 and SA3 modules are set by the following transfer function [2]
bo P
W (P ) =
T p+Уг* p+1)
(8)
where b0 ,Ti - transfer coefficient and constant
time. Z-transformations of the expression (3) is received in this way
bo(ci+c2 z)z ^_
W* (z ) = -
1 -
+ e
z-1 + e Tl e T2 z
c1 = 1 +
T2 e T* - T1e Tl
TTT
Ct = e
T „ T
0 T*e T* - Tie
To T
Tl - T*
where To - input sampling period. MC3 compares the acting speed of the grinding wheel rotation with the optimum value of the databank. The general case of the representation of its transfer function has the form [2]
W3 (p) =
1 - e -
(9)
where Z - constant time.
The specific of operation of S3 and T4 allows to present them in the form of the aperiodic circuit with a delay which has the transfer function [2]
W4 (P) = ^ - PT3
T3 P +1
(10)
T
T
T2
e
T
T
o
T
T
o
pz
T
r
where Ai - confidential interval of speed measurements of the wheel rotation made by the primary converter which is received on the basis of the information approach as a result of the statistical processing of experimental data.
The specific operation of the DAC3 block allows presenting it in the form
/ X 1 - e ~PT3
W5 (p) =-. (11)
p
The transfer function of the velocity command channel is received in this way
a 4 T1T2T3T3
a
a,
= T1T2 (T3 +t3 ) + (ti + t2 )T3t3 , = tt2 + T3T3 + Tr + (t + t2 )(t3 + ^ )
= t + T + T +h .
The transfer function of the device 3 is received by the minimization of the quadratic criterion of quality using the channel reference model [2]
W3 (z ) = 1.206
z2 - 0.1706z
z2 - 0.995z - 5.019-103
(13)
W( (p ) = ^Irrp[y3r3 p 2 + (T3 + ^3 ) P +1 ap + a3p3 + a2p2 + ap +1
(12)
The scheme of computer simulation of the processing of the velocity signal of the grinding wheel rotation is based on the expressions (12) and (13) (Fig. 2).
Figure 2 - Scheme of the computer simulation of the velocity signal processing of the grinding wheel rotation
T1 and SA1 models make being late for operation of the velocity processing channel of the GW transverse feed. Z-transformation of expressions for transfer functions of this channel is as follows [3]
¡L 0 1 1 | ——2 , _3
^ (z-1)= " " "
= z
-2 b0 + bI z + b2 z + b3 z
10 —1 o —2 o —3
+ a0 z + a0 z + a ° z
W (z 1 ) =
kM TM (1 -14 )z
1 -(1 +14 )z
-1
+14 z "
(14)
In these expressions the dead zone of the primary converter which is defined on the basis of the information approach as a result of the statistical processing of experimental data.
In determining the input signal of hardware for the setting of the transverse feed of the grinding wheel the following is considered:
• during the direct and reverse motion of the grinding wheel the phase of the output signal changes for 1800;
n
S (t ) = £-
4S1
cos
(2k - 1)Q
• the direct and reverse motion of the wheel forms the period of the main harmonica of the frequency representation of the input action.
On the basis of this verbal description such signal most fully fits this task [1]
i0 if -n<Q. t < -n+a,-a<Q. t <a, n-a<Q. t <n, S(t) = k if a<Qt <^-a, (15)
[-S if -tf+a<Q t < -a, a> 0,0 <a< 0.5^, where a = 0.5Q(t: — t2 ), t2 - interaction time of the grinding wheel with the detail surface.
The Fourier transform for the signal (15) has the following form
(2k — 1)Q t 2
After replacing of the harmonic components with the sum of exponential signals, we have this form
s1 (t)=4s1 £_l .
n t\2k -1
-cos
:(t1 -12 )
sin (2k - 1)Q t. (16)
— n(2k - 1)""L 2
The solution of the Cauchy problem for differential equations describing the transformation processes of information by the channel is performed using the convolution theorem. After applying the inverse Laplace Transform in the transfer function of the signal processing channel of the GW transverse feed and mathematical transformations we have received such form
(t1 - t2 )
|e;[(2k-1)Qt-0.5n] _e-j[(2k-1)Qt-0.5n]| ^
4
W9 (t) = £ Be pkt
k=1
Bk =
bpTr p + 1)
4a4p3 + 3a3p2 + 2a2p + ax
. (18)
p=pk
2
The input signal of the channel, which is applied using the convolution theorem on the basis of expresas projected in the determination of the effectiveness of sions (17) and (18) its hardware, has been received after the transformation
4 n OAT
2 A2i-lTk
yKi (t) = ZBke pk< £
k=1
Synthesis of MC1 in the conditions of interference acting is carried out by the minimization method of the quadratic criterion of quality using the channel reference model of the signal processing of the GW transverse feed. On the basis of the expression (14) its transfer function is received in this form [3]
Constant
< 1 V'1 + T2 (2i -1)2 - 2
Wmi (z - ) =
e
f[0.5tf-arctgTk (2i-1)Q]
767.113-
2.3 - 3.453 z+1.33 z ~2
.(19)
0.262 + 0.008z - + 0.2182 z ~2 The computer simulation scheme of the signal processing of the GW transverse feed is made on the basis of expressions (14) and (19) (Fig. 3).
Figure 3 - Scheme of the computer simulation of the signal processing of the GW transverse feed
The following expressions for transfer functions [4] are obtained from the analysis of the structural scheme of the signal processing channel of the GW longitudinal feed with the use of computer simulation and the mathematical apparatus of z-transformations.
_2 b; + b; z-1 + b; z -z + b; z
W1o(z-1 )= z
1 + a[ z + a 2 z
+ a?,z
p^-1 y kMz-11 - TM +(TM -14 )z
1 -(l +14 )z- +14 z "2 .
The delay introduced by T2 devices, the data hold device and ED2 is considered in these expressions. It is also taken into account the dead zone of the primary converter defined on the basis of the information approach as a result of the statistical processing of experimental data.
It is taken into account that the GW longitudinal feed using the input signal channel is essentially the drive speed
yK 2 (t ):
8 n
-- Z Bke-pkt
k=5
Z
B
s 2 (t)= Z B2k-1
k=1
2S
|eX2k-1)Qt + e-j(2k-1)qt ]
2k-1
np
^ (t1 - t2 )
. (20)
After applying the inverse Laplace Transform in the transfer function of the signal processing channel of the GW longitudinal feed and mathematical transformations, we have received such form
8
Mt)= Z Bk
k=5
Bk =
b0)Trp(T4 p + 1)
4a4 p 3 + 3a3 p 2 + 2a2 p + aj
- pkt
. (21)
B2i-1Tk
^+T2 (2i-1)2 Q2
The output signal of the channel applied as projected in the determination of the effectiveness of its hardware has been received using the convolution theorem on the basis of expressions (20) and (21)
, jarctgTk(2i-l)Q _ e -jarctgTk(2i-l)Q
The method of synthesis of MC2 using the channel reference model is the basis of its development in interference conditions. Its transfer function [4] is obtained by the minimization of the quadratic criterion of quality with the use of the device z-transformations
Wm 2
(z1 )■
4 )_ 0.04752 + 0.862z- 0.823z~2
, . (22) 0.374 - 0.374z-1
The scheme of the computer simulation of the signal processing of the GW longitudinal feed is shown in Figure 4.
n
e
p= pk
Figure 4 - Scheme of the computer simulation of the signal processing of the GW longitudinal feed.
Modeling of elements of the second level. The
following expression for the transfer function [4] is obtained from the analysis of the structural scheme of the signal processing channel of deviations with the use of computer simulation and the mathematical apparatus of z-transformations
i * —i b* z 1
10
A®(t )=I A
k=1
ej(kQt+Wk-0.5n) - e-j(kOt+Wk+0.5n)
. (24)
W0(z -1 )=
After applying the inverse Laplace Transform in the transfer function of the signal processing channel of deviations and mathematical transformations we have received such form
1 + axz
(23)
+ a o z
1 1
—t --1
T« , d „ T„
The channel delay and the error in measurement deviations which is determined on the basis of the information approach as a result of statistical processing of experimental data are taken into account in this expression.
The input signal channel is presented in the form of the limited number of Fourier. After replacement of harmonic components with a sum of exponential signals, we have
wh(t) = b9e t16 + b10e t17
(25)
where B _ A5kd3kn4 g _ A5kd3kn4
9 ^16 (T17 - T16 ) , 10 T17 T17 - T16 ) ,
5 - uncertainty of measure-nA - delay of the microcontrol-
A -
T16 = Td3 , T17 = Tn4 ,
ments of deviations, ler 4.
The output signal of the channel, that is applied as projected in the determination of the hardware effectiveness, has been received using the convolution theorem on the basis of expressions (24) and (25)
—110 ^4 (t) = B9e T16 I
AT16
k=1^1 + T 2 k 2Q2
ej(Wk+0.5n-arctgT6kQ) - e-j(Wk+0.5n+arctgT16kQ)
+
+ B^e
AkT17
I- _
k=^1 + T 2 k2 Q2
(Wk+0.5n-arctgT17kQ) - j(wk+0.5n+arctgT7kQ)
The method of synthesis of MC4 using the channel reference model is the basis of its development in interference conditions. Its transfer function [4] is obtained by the minimization of the quadratic criterion of quality with the use of the mathematical device of z-transfor-mations
-e
Wm 4
(z 1 )■
1.54 - 2.096z+ 0.616z~2 2.574 + 0.083 z - - 2.156 z ~2
(26)
The computer simulation scheme of the signal processing of the instantaneous speed by means of hardware of the second level of MS is made on the basis of
expressions (23) and (26) (Fig. 5).
T
7
e
Figure 5 - Scheme of the computer simulation of the signal processing of the instantaneous speed of GW
rotation.
Modeling of the elements of the third level. The
following expressions for the discrete transfer function
[5] is obtained from the analysis of the scheme of structural construction of the third level of MS with the use of the mathematical apparatus of z-transformations
W15(z ) = k1T20(T20 + k 2T19 )
T18z
T18z
(Ti7 -Tl6)
z - e
(Tl7 - Tl6)
z - e
(27)
After applying the inverse Laplace Transform in the transfer function of hardware of the third level and transformations, we have received such form
Wi6(i) = Bne Tl6 + 5i2e Tl7
46V
R _ k1T20(T20 + k2T19 ) 11 ^16(^17 - T16 ) '
output signal of the channel, that is applied as projected in the determination of the hardware effectiveness, has been received by means of the convolution theorem on the basis of the expression (19)
T17 , „ ™ T16 »
B12 =
k1T20(T20 + k2YT19 ) T17(T16 - T17)
The input signal of hardware of the third level is presented in the form of the Heaviside function. The
^ (t) = 1(t)+Mü e +1
511116 e T16+1 .(28)
T17 +1 T16 +1
The transfer function of MC5 is obtained by the minimization of the quadratic criterion of quality
k1T20(T20 + k 2У T19 )
WM 5 (z ) = -
T17 T16
( T
z 2 +
T0 Л
+ e
T17 0 T16 ■
( T
e T17 e T16 z2 -
(29)
e T17 + e T16
z +1
v у
The computer simulation scheme of the signal pro- is made on the basis of expressions (27) and (29) (Fig. cessing by means of hardware of the third level of MS 6).
Figure 6 - Scheme of the computer simulation
T
T
17
16
1
T
r
6
e
z
Conclusions. The combined principle and three-level structural construction of MS for improving the efficiency of GW process in conditions of incomplete information on the basis of indirect measurements of microroughnesses with the predicted amplitude and the GW cutting characteristics has been developed. Mathematical models of hardware system, taking into account their delay, have been built using the discrete Laplace Transform in conditions of interference. The transfer functions have been obtained from the analysis of schemes of the structural construction of MS. Microcontrollers for digital processing of the input signals have been synthesized by minimizing of the quadratic criterion of quality with the use of standard models of channels. The computer simulation schemes of the processing of the input signals have been made and it is determined that the hardware meet the requirements of MS for this processing speed.
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