- Email: [email protected]
Integrated End Milling Optimization Development
Integrated End Milling Optimization Development V. A. Ostafiev (2). A. V. Globa. L. S. Globa, Kiev Polytechnic Institute - Submitted by T.N. Loladze (1)
Summary: Inteqrated method of end milling has been developed to optimize both cutter path,type of tools,number of cuts and cutting conditions.Method of Data Handlinq by Croups(MDHG) was used to sot up a model for complex non-homoqeneoa svstems with minimum data applied.Machining analytical introduction carrying from manufacturing process data directly with reliable accuracy was got by MDHG For creating technolorical process variables relationships now used manufacturtlig statistics taking in account the shop production conditions.Two staqes iterative mode interacting was carried out for optimization process: 1- the process structure has been optimized by cutter path,type of tools and number of cuts;2- using nonlinear uroqramminq the generalized LagranKe Multiplier Method for cutting condition outimization.Increase shop production rate has been confirmed by the method proposed. A feature inherent in designing production processes consists in a great number of alternative methods of machinina: a workpiece.Since it seems impossible to evaluate all these methods,computer means are widely used for constructing ontimum production processes which in turn requires the development of a mathematic model to describe adequately the process being designed and efficient procedure of its optimization [I ,2,4.6). The authors offer the procedure of integrated end milling o3timization process to permit oqtimization both of machininq conditions including cutter path and cuttinq tool qeometry.The process efficiency on the expensive equipment as mschininq centres with such statement can be increased by several times without any additional production expence.The development of the procedure of inteqrated optimization permits to qet the optimum selection of cutting conditions, required diameter of a cutters and its psthes in the machining grooves and pockets of complex confi,qurations. At present these parameters are specified separately,at best by varying one or two parameters,which does not allow to reveal all the reserves intrinsic in the milling process. The value of productive time T is selected as the objective function according P12] :
+ Tir.rouf Tir.rough * min9where Tj,rough is the machining time for the j-th roughing pass;^,,^ is the time for tool edqe preparation and chanqing;Trough is the tool life of the r o u q h i n q b ter;Tiefin is the machining time for the i-th finishinq pass;Tfin is the tool life of the finishing cutter;Taux is the auxiliary time;Tirafin is the idle time of the finishing cutter;Tir.rough is the idle time of the roughing cutter. In accordance with the objective function T the problem of intenrated optimization of the p&d milling process should be solved in two staqes with the use of the search methods of the iteration natu& The following parameters are selected as initial data for the integrated optimization of the end milling process: part drawing,drawing of workpiece,processing data bank whose bases contain the information on the cutting tools,machine-tools.The first stage starts with the check of the rough milling necessity. The rough cutter diameter,Drough,is limited by the dimensions and the inner angles of milling contour. The maximum deoth of cut a in the machiniq of in< 0.25 Drough. ner angles should not e x c e w aThe selection of an optimum diameter and geometry of a rough cutter is carried out according to tool base data using the gradient method [3]. Determination of milling depth, a, and width,b, presents a significant difficulty since they are vadable along the complicated surface of the workpiece profile in the general case.Considering stated above, the method has been develoned for reduced depth of milline.According to the method the nonuniform djstribution of the stock in the depth direction is substituted for the uniform distribution thus the volume of metal removed remains constant.The defined values of b and a, are optimized during the process of solving the problem. bring the optimization process the necessety of +
+
Annals of the CIRP Vol. 33/1/1984
rough machining in qeneral becomes clear.The criterion of the machining model ishT which is defined 8s difference between T (withoutP utilization of P1 rough milling cutter)and T (with utilization of p2 rough milling cutter). The process of cutter path optimization within the contour is carried out in accordance with the optimization criterion maximum production rate. Usually the optimization criterion the length of the end milling cutter path is chosaHowever,according to the production time criterion it has been proved the cutter path length could not be objective fcrt ction.Selection of T as the end milling cutter oriP herion willallowes to consider not only length of path,but the condition of machining as wel1;i.e. the modes of milling.At the first stage cutter diameter, geometry,its path and depth of cut is selected. For the second staqe of inteqrated optimization the mathematic model of milling process is presented
x
I
x
(2.W
)
Y t Y (X,Z,W) F -F(Y,Z) 9 min < R* 2 ma=, where X=X(X, ,X2. ,Xn) is the vector of optimizi variables(cutting speed and feed),Z.Z(Z ,Z Z? are input parameters (mechanical and ~he?mal-an8-physicalproperties of work and tool materials,kind of workpiece maintenaxe, data of machine tool and tooling);Y-Y(y, ,y2,...,y , )is the intermediate vector of process parameters(cutting force,roup;hness,machining accuracy,temperature,tool life etc.);F-F(f f f is the objective function vector(produc4jog'cost %nd rate);W-W(wl ,w2,. ,w$ is the vector of random parameters(mechanica1 property work iece and tool material change8,shape nonuniformityp. pmax, ie a certain number gmind machininq parameters constraints(feeds and speeds for a given machine too1,end cutters available at the shop etc.).Consequently,it is necessary to determine the vector X of optimizinq variables within permissible region. In the general case the mathematic model of the end milling process is represented by a system of= linear equations.To obtain necessary relationships of the process parameters,long-term laboratory studies are qenerally carried out. In addition to considerable time expenditures, large number of hiqhly skilled experts and siqnificant expenses,such data are introduced into the manufacture with a big delay.Though the same materials have been machined at the factories for many years and numerous data are available,the needed relationships have not yet been obtained because of their technique processing absence.A certain acceleration of this process materials machining had been achieved by implementing the experiments planning.However,the experiments carried out in accordance to a given plan do not allow to embrace the whole cutting conditions applicating field.Their extreme values are not always practicable or even fuefilled. At that time an enormous experience accumulated in the industry the first received the material for machininq,can be up to now evaluated only from the qualitative standpoint,not allowing for strategies. For example,to obtain a machining process relationships,long-term tool life experiments have been conducted and required expenditures a lot of work materials especially in the case of machinin ferrous allows as alumi~um,brass,bronzeetc.ke"?;nd
,...,
...,
-
..
-
29
of planning experiments predetermines a type of relationships and degree of its nonlipearfty. Therefore,it-isnot quite efficient to carry out an active experiment,because the relationships obtained durirq the laboratory experiment conditions can sinnificantlv differ from its in particular manufactGring. At the same time rather sood informative base there are in manufacture.HOwever,the lack of sufficientlv effective for a lone time led to absence of relationships from manufacturing data. For this reason,there is pro osed a Method of Data Handlinq by Groupa(MDHC) [5,79, allowing to process numerous statistical manufacture data and to obtain the required relationship8 as tool life,surface quality,etc.It provides for the Dossibility of taking into account specific features of each branch of industry,factory,workshop,and rapidly apulied the most effective cutting tool and its machining conditions.The MDHG is based on the principles of selforganization and carry out synthesis of an adequate object model with the use of minimum information apriori. The basic theoretical postulate of MDHG theory is the hypothesis on the existence of a single optimum complexity mode1,which may be found by sorting of a great pretender models number accordanly to the set of exterinal criterion. In so doim? the apriori information required by the investigator is minimized. Hence,the information apriori is minimized. In carrying out tool life investigation for the end milling process,it is consedered a s the output valuepa while among the input values Xi are the depth(a? and the width (b) of cut feed per minute (Sm),spindle revolution per minutejn) ,and the diameter of an end milling cutter(D).Table 1 shows the matrix of initial data,which has been obtained the manufacturing statistical data basis,while machininy the D16T aluminium allorv.Mathematica1 expectations are presented in Table 1 as of tool life T well as its meaX souare deviation 6, ; it is assumed that the random variable Tav has the normal distribution. The most suitable structure class in the data identification when studying the end milling process turned out to be the Kolmogorov-Gabor polinomials class.In the Keneral case the MDHG algorithms contain the generator of various complexity models and the selecting block for the best decisions,working in the following 1ink:generation selection z qeneration + selection + In realizing this link,fundamental principles of self-organization are used such as external addition and selecting decision freedom. The external addition principle states to indicate the optimum model some criterion reach their minimum with a gradual increase in the complexity of a mathematical model. The essense of the selectinq decision freedom for the cutting process consists of certain set of model selection for the external criterion, to en sure the future choice possibility for the whole relationships. The initial data are divided in two sequences according to MDHG algorithm named teaching and checking.The teaching sequences is used for evaluating the model coefficients only and the checking sequences ie used for calculating the external criterion. The multilayers algorithm has been applied in the proposed case.The matrix X(N,M) and the vector TaV(N) qiven into theTables 1 have been used as the input data for simulating tool life relationships.There =e,N-P’T-number of the tool life experiments;M-5-number of the non-conti ous input variables (cutter conditions n,a,b,a, D r The ensemble of possible arguments desired data model Z o has been built usiw matrix X(B,P).It can be the members of a secondration linear polynomial o r thirdratio polynomia1,etc. According previous experience the enssemble of possible aguments desired data m del Zo has been chosen as: zo b,ci,n,D,Sm, 1, 1, -, D Sm b a n The multilayers selection procedure of the optimal model including following steps has been reali-
...
-
The regular criterion has being calculated using checking sequence to find out the every model error
j.BA+l where is MA the sequence lewth, Tij End Tij-experimental and calculated tool life data;Tav tool life arithmetic mean got on checking sequence.Al1 L models are selected to find out the best F solution of the first layer accbr ~p to-gpe regu$F criterion ,T minimum d Bi: ( 3 ) T1 /%n, , Tn2
-
-
-
,...
Model structures was given by means of the knsemble:
Such is the Gay the quantity of tde ‘bestmodels equal F are selected on first laver. The second layer. ?he enssemble of model -pretenders of the second layer is calculated as a straieht product:
The ensseible Z consists of F*L members determining the possible godel structure of the aecond layer. The best F models in sense ofaB are selected analo-
-
where is L‘”( i-l,F) redesignated members of the i enssemble Znl, and its possible to be executed as: (1)
-
(1)
LJ ( i I( j, i- l , F 1 Li 4 2 ) So as more than one best solution Tnl ( 1 -1,g) of the second layer can be found for the same Zn, and F- const, taking in account b’:L Zn(2) (8) The orderliness according to A B of the enssemble of the members of the first and second layers are possible to write down from (71
-
-, -
I
ard_--. The first
layer. The ens&mbb31d2po;sibL1; layer .Z1 Z
zl-
2,
-{
-
,...,
b,a,n,D,Sm,-,b
arguments of the first is like Z o :
Pi
The given equationsbve shown the procedure of the third layer is like the procedure of the second layer.So,it’s possible all needed expressions for an arbitary S ( S > l ) layer to write down: 2s
‘n(s-1)
*
‘0
+$I
1, 0 1 n , * All types of t e equation coefficients havk beeus-aoi + ali’zi,where is i l,L timated: ( 1 ) L 3 M; zi E 2, and aol- free equation members ali- equation coefficients
30
I J L. F models are selected on the second layer. Third layer
(S-1)
Li
(S-2)
(S)
bpi ... z“i , i
*
~
. )
1,P
-the permissible horizontal force for machine-tool: x' 'D.er: . the feed: S m . m i n 4 S m S Sm.m,x (with stepless control)
-
................................................
.
AS
,...(z g
22'
EZo
The maximum possible model structure can be exDressed by algorithm with layers number S - L according to qiven eouations.The algorithm selects the best six models (F.6) according to the CriterionaB-min. The r u l e to stop the aleorithm is fol1owing:the optimal minimum model is selected according to comhined criterion A B and no: ao-bo , no is the free member displace no
--
-
k , l i Sm E [Smi , ] the rotational speed :
nmin' . < n S n max n E [ n j , .i-l,kl] I
-
Dp =G Drouqh
-
T1506.9,+ 40432000- 36.554n D-n D
+
0.7436n a
186.77b D
-0.0013;6.S: !he value of an error in the calculation accordinq to
the above formula does not exceed 14% over the whole region of the given tool life values. Thus,the cutting process relationships are required for manufacture,quickly realize from its statistical data.To carry out all necessary investigations the mathematical model of the end millins: p r o cess has been determined and presented for D16T aluminium alloys as follows: Optimization criterion: Tp min
-
-
Relationships. T- 506.g1+P0432000-36*454n + -0 0.0013664D en D a 186.77b D ?x = 15.37 8029.2/bSm-91.199/ab+0.012911aSm+
-
--
-
;
desing of workpiece :
80
ment;bo-calculated from checkim sequence free member ao- calculated from teaching sequence free member. In the result of the ontimum tool life end milling model synthesis,the best model is obtained at the last row of selection:
(with step control);
DP
-
2.( bl.sin V n 2 -
Dmax
6 +
Vmin 2
-sin-
1
o - = 'rough whereymin is the tion within b is the 4 is the in the
%in
=S rmaxs minimum angle of edqe conjuga-
the contour; stock for finish cutter; maximum stock for finish cutinner angle; ter;
The vector of optimizinn variables:
8- I%
z, zm,
:L311J
The dependence
of
Tp-& cutter geometry are il-
lustrated in Fig. l. and in Fig.2.
cutting conditions
To solve the problem, the method of structural optimization is used at the first stage,while at the second stage there is used the nonlinear optimization with the Lagrange's multipliers. The problem of nonlinear programing with constrains is transformed into nonlinear task without constrains by penalty function in followiw f o r m [ 31:
+ 8~a.a7/02+io.427.10-~n/a+i 47.12a/n;
-
Py .0.22657+9.8166*Sm/n+16. 679-
48.079/aSm
- .
for the end milling process:
-1 5.381 1 0-5DSm-22. 1 19/aSm-10.573' 10-3Sm/b;
Pz- -2.098+1,696627ab/D+O.O3755961S~/n +
401 14. f6 7 Sm* b
Constrains; cost per piece: jciSu,< c=,-workpiece accuracy.Def6ction of cutter:
-
y-
'
24 bEI - {3 [I4 -(L-b)4-4b(L-b)3-4X[L3-(L-b)3]+
+ X4]
C
, where
6;
L is the cutter extension
(the distance from the spindle end to the cutter edpd E is the elasticity modulus; I is the moment of cutter inertia; X is the point in which the cutter deflection is invectigated; b,is workpiece tolerance. Surface roughness of the machined su face: ( 5 ) R, -0.91246 0.63257/a2 +47.787*1O-gnb 1.1663b + $5.726 41.152*104D
-
ninq variables; w1 VW2 *w3PW4Sw5S w6 Sw7'w8 -Lagrangets multipliers. Yeaking variables transform the inequality type constrains into its equality type.Fina1 integrated method decision of end milling optimization on the second stage is found out from nonlinear system:
IaF 7
q
-
O
-
+
Sm 0 2141Sm +n
W m
a
c
Ramxi
- the cutting tool resistance: mEcI I
PO
[I,],where
9OOD tar tangent force; a machine tool power: Nef
-
Nem.
P, is the cut-
2
.kn > Ncut
Ncut-v. Po/6120, where Nem is the power of the machine-tool drive;
9
11
is the efficiency; is the overload coefficient of the machinetool drive;
The decision optimizes one of the first stage variable and the other variables are optimized in iterative loop of all procedures.The example of the interated optimization end milling of a work iece 2 ?Fig.3,4) uade of the aluminium alloy D?&!.Table gives the data on the main parameters of the end milling process,which are set on the basis of standard dato,while Table 3 shows the data obtained after inteqrated optimization.Comparison of the results shows the end milling process by integrated optimization reduces productive time at 1.5 times even while machining the simple shape pocket. Acknowlednement The authors wish to thanks Kiss Vnukova T.M. for typinr: and Mr.Tikot V. for his kind help. References 1.W.Eversheim.D.Gelauer.H.Wesch.RWTH Aachen/D.and C. Ravignani;Milano/Itily. ltFunhamentalsof Determination of Cutting Time and Cutting Costs f o r Multi-Tool Optimization",CIRP/vo1.30./2/1981.
31
2 .\I Ever . she im ,W. Koniq ,W. Schwauborn, H.We sch,L. Dammer
,
Computer Aided Planning and Optimization of Cutting Data.Time and Cost.30/1/,1981,pp.409-413. 3.David M. Himmelblau.Applied Nonlinear F'rogramming. McGraw-Hill Book Company.1972.pp.337-340. 4.A.ZompiIR.Levi"Tool Life Distribution in Drocess Optimization Parameters Identification and Applica
24
tiona",CIRP/~.28/~/1979~pp.371-377. 5.Ivanenko A.G.Heuristic Self-Orzanization in Problems of Engineering Cybernetics. "Av tomatica" ,v. 16, 1970. 6.B.N. Colding'@The IJachining Froductivity Pountain and Its Tall of Optimum ?roductivity."Technical Paper Society of hlanufacturinr: Enqineers".MM80-215. 1980.p~.18. 7.Gabor D. The Proper Priorities of Science and Tech nology.Universitg Southhamuton,Enqland,l972. Table 1 N/N D n a b Sm Tav GT mm rev/min mm mm mm/min min
x l x
2 1045
40 40 20 40 40 20
1.
2.
3. 4.
5. 6. 7.
1045 1045 1045 1046 1400 730
56
60 56
11. 12. 13. 14.
19.
60 36 36 30 30 20 20 28 60
20. 21. 22. 23. 24. 25. 26. 27.
60 1045 60 1045 50 730 30 730 50 730 730 30 30 730 60 368
16.
17. 18.
730
1045 1045 1045
1045 1045 1045 1045 1045
x
x
4
70
2
78 78 78 78
55 4
3 2 5.75 13.8 2 2 1 1 4.5 60 30
10 35 20
36 14 36 36 30 30
4
45
40 40 10
28
37
25 3 3 42
30 25 20 18 20
5
12
9 6 3
Y
?2 ;;:3-
50 300 150 250 250 175 220 250 330 200 300 240 240 300 280 300 300 250 270 270 270 300 400 320 300 300 220
30 30
2 6 29.3 2
730 730
10.
15.
3
1045 12.3
60
8.
9.
x
21 18 15
479 308 200 600 120 600 900 840 1200 762 704
568 488 900 360 600 450 95 210
1044 852 536 1250 1260 1925
cutier
2 30 6 +.
2
u
20 6
1
k
2
30 6
6
c 20 6
1
+q
V k
30 b
,I F $20 V
6
3
1
%
n a rev
b
@lhelh?l
45 groove 45 plane
t8heli.x*
120 800 30 160 800 16,4 145 800 1Q 150 1250 1 150 1250 22 120 800 30 160 800 17.4 150 800 17.4 150 1250 1 150 1250 120 800 30 200 800 10
45 a l o w edge 45 alonn edge Tp 18.1
45 45 45 45 43
in corner groove plane along edge
45 in corn% 45 groove 45 Dlane
he lixv* T -6.3 p whelhll
150 1250 1 45 along edge 150 1250 22 45 in corner Tpg2*58 C Tp-28.11
32
from end millcutter
0.9
0.7 0.5
0.3
Fig.2.Dependence
of Tp from cutting conditions.
ODtinnrm Version Table 3 n a b Tspe of cutter rev Notes movement passes min min mm mm min llhelix,l 120 710 40 45 groove 4 :80 710 20.2 45 plane 180 710 242 45 along edge Fig.4.
Table 2 Type of Notes movement
groove 160 800 174 45 plane
,
1.1
Fb.3.Standard version. D rCUt num er
Dependence of Tp geometry. I
Fig. 1.
65.84 33.21 21.59 47.86 22.81 54.04 76.78 78.42 94.61 56.14 65.25 75.21 93.14 98.32 75.55 94.24 87.67 19.19 29.03 106.41 92.77 10.48 89.56 84.2 91.1
D rCUt number S
mm mm
2
$
20 6
k
40 5
4
g-20 6
1
2
10 1 IEO 18 $40 5 160 710 24 45 550 1600 1 150 1600 3.7 120 710 40 180 710 2$ 180 710 2 W 550 1600 QY
45
groove plane *@helix** along edne along edge in corner Tp = 5.7
45
zroove
45
45 plane **helix" 45 alonn edne 45 alonq edge-
-
Summary: Inteqrated method of end milling has been developed to optimize both cutter path,type of tools,number of cuts and cutting conditions.Method of Data Handlinq by Croups(MDHG) was used to sot up a model for complex non-homoqeneoa svstems with minimum data applied.Machining analytical introduction carrying from manufacturing process data directly with reliable accuracy was got by MDHG For creating technolorical process variables relationships now used manufacturtlig statistics taking in account the shop production conditions.Two staqes iterative mode interacting was carried out for optimization process: 1- the process structure has been optimized by cutter path,type of tools and number of cuts;2- using nonlinear uroqramminq the generalized LagranKe Multiplier Method for cutting condition outimization.Increase shop production rate has been confirmed by the method proposed. A feature inherent in designing production processes consists in a great number of alternative methods of machinina: a workpiece.Since it seems impossible to evaluate all these methods,computer means are widely used for constructing ontimum production processes which in turn requires the development of a mathematic model to describe adequately the process being designed and efficient procedure of its optimization [I ,2,4.6). The authors offer the procedure of integrated end milling o3timization process to permit oqtimization both of machininq conditions including cutter path and cuttinq tool qeometry.The process efficiency on the expensive equipment as mschininq centres with such statement can be increased by several times without any additional production expence.The development of the procedure of inteqrated optimization permits to qet the optimum selection of cutting conditions, required diameter of a cutters and its psthes in the machining grooves and pockets of complex confi,qurations. At present these parameters are specified separately,at best by varying one or two parameters,which does not allow to reveal all the reserves intrinsic in the milling process. The value of productive time T is selected as the objective function according P12] :
+ Tir.rouf Tir.rough * min9where Tj,rough is the machining time for the j-th roughing pass;^,,^ is the time for tool edqe preparation and chanqing;Trough is the tool life of the r o u q h i n q b ter;Tiefin is the machining time for the i-th finishinq pass;Tfin is the tool life of the finishing cutter;Taux is the auxiliary time;Tirafin is the idle time of the finishing cutter;Tir.rough is the idle time of the roughing cutter. In accordance with the objective function T the problem of intenrated optimization of the p&d milling process should be solved in two staqes with the use of the search methods of the iteration natu& The following parameters are selected as initial data for the integrated optimization of the end milling process: part drawing,drawing of workpiece,processing data bank whose bases contain the information on the cutting tools,machine-tools.The first stage starts with the check of the rough milling necessity. The rough cutter diameter,Drough,is limited by the dimensions and the inner angles of milling contour. The maximum deoth of cut a in the machiniq of in< 0.25 Drough. ner angles should not e x c e w aThe selection of an optimum diameter and geometry of a rough cutter is carried out according to tool base data using the gradient method [3]. Determination of milling depth, a, and width,b, presents a significant difficulty since they are vadable along the complicated surface of the workpiece profile in the general case.Considering stated above, the method has been develoned for reduced depth of milline.According to the method the nonuniform djstribution of the stock in the depth direction is substituted for the uniform distribution thus the volume of metal removed remains constant.The defined values of b and a, are optimized during the process of solving the problem. bring the optimization process the necessety of +
+
Annals of the CIRP Vol. 33/1/1984
rough machining in qeneral becomes clear.The criterion of the machining model ishT which is defined 8s difference between T (withoutP utilization of P1 rough milling cutter)and T (with utilization of p2 rough milling cutter). The process of cutter path optimization within the contour is carried out in accordance with the optimization criterion maximum production rate. Usually the optimization criterion the length of the end milling cutter path is chosaHowever,according to the production time criterion it has been proved the cutter path length could not be objective fcrt ction.Selection of T as the end milling cutter oriP herion willallowes to consider not only length of path,but the condition of machining as wel1;i.e. the modes of milling.At the first stage cutter diameter, geometry,its path and depth of cut is selected. For the second staqe of inteqrated optimization the mathematic model of milling process is presented
x
I
x
(2.W
)
Y t Y (X,Z,W) F -F(Y,Z) 9 min < R* 2 ma=, where X=X(X, ,X2. ,Xn) is the vector of optimizi variables(cutting speed and feed),Z.Z(Z ,Z Z? are input parameters (mechanical and ~he?mal-an8-physicalproperties of work and tool materials,kind of workpiece maintenaxe, data of machine tool and tooling);Y-Y(y, ,y2,...,y , )is the intermediate vector of process parameters(cutting force,roup;hness,machining accuracy,temperature,tool life etc.);F-F(f f f is the objective function vector(produc4jog'cost %nd rate);W-W(wl ,w2,. ,w$ is the vector of random parameters(mechanica1 property work iece and tool material change8,shape nonuniformityp. pmax, ie a certain number gmind machininq parameters constraints(feeds and speeds for a given machine too1,end cutters available at the shop etc.).Consequently,it is necessary to determine the vector X of optimizinq variables within permissible region. In the general case the mathematic model of the end milling process is represented by a system of= linear equations.To obtain necessary relationships of the process parameters,long-term laboratory studies are qenerally carried out. In addition to considerable time expenditures, large number of hiqhly skilled experts and siqnificant expenses,such data are introduced into the manufacture with a big delay.Though the same materials have been machined at the factories for many years and numerous data are available,the needed relationships have not yet been obtained because of their technique processing absence.A certain acceleration of this process materials machining had been achieved by implementing the experiments planning.However,the experiments carried out in accordance to a given plan do not allow to embrace the whole cutting conditions applicating field.Their extreme values are not always practicable or even fuefilled. At that time an enormous experience accumulated in the industry the first received the material for machininq,can be up to now evaluated only from the qualitative standpoint,not allowing for strategies. For example,to obtain a machining process relationships,long-term tool life experiments have been conducted and required expenditures a lot of work materials especially in the case of machinin ferrous allows as alumi~um,brass,bronzeetc.ke"?;nd
,...,
...,
-
..
-
29
of planning experiments predetermines a type of relationships and degree of its nonlipearfty. Therefore,it-isnot quite efficient to carry out an active experiment,because the relationships obtained durirq the laboratory experiment conditions can sinnificantlv differ from its in particular manufactGring. At the same time rather sood informative base there are in manufacture.HOwever,the lack of sufficientlv effective for a lone time led to absence of relationships from manufacturing data. For this reason,there is pro osed a Method of Data Handlinq by Groupa(MDHC) [5,79, allowing to process numerous statistical manufacture data and to obtain the required relationship8 as tool life,surface quality,etc.It provides for the Dossibility of taking into account specific features of each branch of industry,factory,workshop,and rapidly apulied the most effective cutting tool and its machining conditions.The MDHG is based on the principles of selforganization and carry out synthesis of an adequate object model with the use of minimum information apriori. The basic theoretical postulate of MDHG theory is the hypothesis on the existence of a single optimum complexity mode1,which may be found by sorting of a great pretender models number accordanly to the set of exterinal criterion. In so doim? the apriori information required by the investigator is minimized. Hence,the information apriori is minimized. In carrying out tool life investigation for the end milling process,it is consedered a s the output valuepa while among the input values Xi are the depth(a? and the width (b) of cut feed per minute (Sm),spindle revolution per minutejn) ,and the diameter of an end milling cutter(D).Table 1 shows the matrix of initial data,which has been obtained the manufacturing statistical data basis,while machininy the D16T aluminium allorv.Mathematica1 expectations are presented in Table 1 as of tool life T well as its meaX souare deviation 6, ; it is assumed that the random variable Tav has the normal distribution. The most suitable structure class in the data identification when studying the end milling process turned out to be the Kolmogorov-Gabor polinomials class.In the Keneral case the MDHG algorithms contain the generator of various complexity models and the selecting block for the best decisions,working in the following 1ink:generation selection z qeneration + selection + In realizing this link,fundamental principles of self-organization are used such as external addition and selecting decision freedom. The external addition principle states to indicate the optimum model some criterion reach their minimum with a gradual increase in the complexity of a mathematical model. The essense of the selectinq decision freedom for the cutting process consists of certain set of model selection for the external criterion, to en sure the future choice possibility for the whole relationships. The initial data are divided in two sequences according to MDHG algorithm named teaching and checking.The teaching sequences is used for evaluating the model coefficients only and the checking sequences ie used for calculating the external criterion. The multilayers algorithm has been applied in the proposed case.The matrix X(N,M) and the vector TaV(N) qiven into theTables 1 have been used as the input data for simulating tool life relationships.There =e,N-P’T-number of the tool life experiments;M-5-number of the non-conti ous input variables (cutter conditions n,a,b,a, D r The ensemble of possible arguments desired data model Z o has been built usiw matrix X(B,P).It can be the members of a secondration linear polynomial o r thirdratio polynomia1,etc. According previous experience the enssemble of possible aguments desired data m del Zo has been chosen as: zo b,ci,n,D,Sm, 1, 1, -, D Sm b a n The multilayers selection procedure of the optimal model including following steps has been reali-
...
-
The regular criterion has being calculated using checking sequence to find out the every model error
j.BA+l where is MA the sequence lewth, Tij End Tij-experimental and calculated tool life data;Tav tool life arithmetic mean got on checking sequence.Al1 L models are selected to find out the best F solution of the first layer accbr ~p to-gpe regu$F criterion ,T minimum d Bi: ( 3 ) T1 /%n, , Tn2
-
-
-
,...
Model structures was given by means of the knsemble:
Such is the Gay the quantity of tde ‘bestmodels equal F are selected on first laver. The second layer. ?he enssemble of model -pretenders of the second layer is calculated as a straieht product:
The ensseible Z consists of F*L members determining the possible godel structure of the aecond layer. The best F models in sense ofaB are selected analo-
-
where is L‘”( i-l,F) redesignated members of the i enssemble Znl, and its possible to be executed as: (1)
-
(1)
LJ ( i I( j, i- l , F 1 Li 4 2 ) So as more than one best solution Tnl ( 1 -1,g) of the second layer can be found for the same Zn, and F- const, taking in account b’:L Zn(2) (8) The orderliness according to A B of the enssemble of the members of the first and second layers are possible to write down from (71
-
-, -
I
ard_--. The first
layer. The ens&mbb31d2po;sibL1; layer .Z1 Z
zl-
2,
-{
-
,...,
b,a,n,D,Sm,-,b
arguments of the first is like Z o :
Pi
The given equationsbve shown the procedure of the third layer is like the procedure of the second layer.So,it’s possible all needed expressions for an arbitary S ( S > l ) layer to write down: 2s
‘n(s-1)
*
‘0
+$I
1, 0 1 n , * All types of t e equation coefficients havk beeus-aoi + ali’zi,where is i l,L timated: ( 1 ) L 3 M; zi E 2, and aol- free equation members ali- equation coefficients
30
I J L. F models are selected on the second layer. Third layer
(S-1)
Li
(S-2)
(S)
bpi ... z“i , i
*
~
. )
1,P
-the permissible horizontal force for machine-tool: x' 'D.er: . the feed: S m . m i n 4 S m S Sm.m,x (with stepless control)
-
................................................
.
AS
,...(z g
22'
EZo
The maximum possible model structure can be exDressed by algorithm with layers number S - L according to qiven eouations.The algorithm selects the best six models (F.6) according to the CriterionaB-min. The r u l e to stop the aleorithm is fol1owing:the optimal minimum model is selected according to comhined criterion A B and no: ao-bo , no is the free member displace no
--
-
k , l i Sm E [Smi , ] the rotational speed :
nmin' . < n S n max n E [ n j , .i-l,kl] I
-
Dp =G Drouqh
-
T1506.9,+ 40432000- 36.554n D-n D
+
0.7436n a
186.77b D
-0.0013;6.S: !he value of an error in the calculation accordinq to
the above formula does not exceed 14% over the whole region of the given tool life values. Thus,the cutting process relationships are required for manufacture,quickly realize from its statistical data.To carry out all necessary investigations the mathematical model of the end millins: p r o cess has been determined and presented for D16T aluminium alloys as follows: Optimization criterion: Tp min
-
-
Relationships. T- 506.g1+P0432000-36*454n + -0 0.0013664D en D a 186.77b D ?x = 15.37 8029.2/bSm-91.199/ab+0.012911aSm+
-
--
-
;
desing of workpiece :
80
ment;bo-calculated from checkim sequence free member ao- calculated from teaching sequence free member. In the result of the ontimum tool life end milling model synthesis,the best model is obtained at the last row of selection:
(with step control);
DP
-
2.( bl.sin V n 2 -
Dmax
6 +
Vmin 2
-sin-
1
o - = 'rough whereymin is the tion within b is the 4 is the in the
%in
=S rmaxs minimum angle of edqe conjuga-
the contour; stock for finish cutter; maximum stock for finish cutinner angle; ter;
The vector of optimizinn variables:
8- I%
z, zm,
:L311J
The dependence
of
Tp-& cutter geometry are il-
lustrated in Fig. l. and in Fig.2.
cutting conditions
To solve the problem, the method of structural optimization is used at the first stage,while at the second stage there is used the nonlinear optimization with the Lagrange's multipliers. The problem of nonlinear programing with constrains is transformed into nonlinear task without constrains by penalty function in followiw f o r m [ 31:
+ 8~a.a7/02+io.427.10-~n/a+i 47.12a/n;
-
Py .0.22657+9.8166*Sm/n+16. 679-
48.079/aSm
- .
for the end milling process:
-1 5.381 1 0-5DSm-22. 1 19/aSm-10.573' 10-3Sm/b;
Pz- -2.098+1,696627ab/D+O.O3755961S~/n +
401 14. f6 7 Sm* b
Constrains; cost per piece: jciSu,< c=,-workpiece accuracy.Def6ction of cutter:
-
y-
'
24 bEI - {3 [I4 -(L-b)4-4b(L-b)3-4X[L3-(L-b)3]+
+ X4]
C
, where
6;
L is the cutter extension
(the distance from the spindle end to the cutter edpd E is the elasticity modulus; I is the moment of cutter inertia; X is the point in which the cutter deflection is invectigated; b,is workpiece tolerance. Surface roughness of the machined su face: ( 5 ) R, -0.91246 0.63257/a2 +47.787*1O-gnb 1.1663b + $5.726 41.152*104D
-
ninq variables; w1 VW2 *w3PW4Sw5S w6 Sw7'w8 -Lagrangets multipliers. Yeaking variables transform the inequality type constrains into its equality type.Fina1 integrated method decision of end milling optimization on the second stage is found out from nonlinear system:
IaF 7
q
-
O
-
+
Sm 0 2141Sm +n
W m
a
c
Ramxi
- the cutting tool resistance: mEcI I
PO
[I,],where
9OOD tar tangent force; a machine tool power: Nef
-
Nem.
P, is the cut-
2
.kn > Ncut
Ncut-v. Po/6120, where Nem is the power of the machine-tool drive;
9
11
is the efficiency; is the overload coefficient of the machinetool drive;
The decision optimizes one of the first stage variable and the other variables are optimized in iterative loop of all procedures.The example of the interated optimization end milling of a work iece 2 ?Fig.3,4) uade of the aluminium alloy D?&!.Table gives the data on the main parameters of the end milling process,which are set on the basis of standard dato,while Table 3 shows the data obtained after inteqrated optimization.Comparison of the results shows the end milling process by integrated optimization reduces productive time at 1.5 times even while machining the simple shape pocket. Acknowlednement The authors wish to thanks Kiss Vnukova T.M. for typinr: and Mr.Tikot V. for his kind help. References 1.W.Eversheim.D.Gelauer.H.Wesch.RWTH Aachen/D.and C. Ravignani;Milano/Itily. ltFunhamentalsof Determination of Cutting Time and Cutting Costs f o r Multi-Tool Optimization",CIRP/vo1.30./2/1981.
31
2 .\I Ever . she im ,W. Koniq ,W. Schwauborn, H.We sch,L. Dammer
,
Computer Aided Planning and Optimization of Cutting Data.Time and Cost.30/1/,1981,pp.409-413. 3.David M. Himmelblau.Applied Nonlinear F'rogramming. McGraw-Hill Book Company.1972.pp.337-340. 4.A.ZompiIR.Levi"Tool Life Distribution in Drocess Optimization Parameters Identification and Applica
24
tiona",CIRP/~.28/~/1979~pp.371-377. 5.Ivanenko A.G.Heuristic Self-Orzanization in Problems of Engineering Cybernetics. "Av tomatica" ,v. 16, 1970. 6.B.N. Colding'@The IJachining Froductivity Pountain and Its Tall of Optimum ?roductivity."Technical Paper Society of hlanufacturinr: Enqineers".MM80-215. 1980.p~.18. 7.Gabor D. The Proper Priorities of Science and Tech nology.Universitg Southhamuton,Enqland,l972. Table 1 N/N D n a b Sm Tav GT mm rev/min mm mm mm/min min
x l x
2 1045
40 40 20 40 40 20
1.
2.
3. 4.
5. 6. 7.
1045 1045 1045 1046 1400 730
56
60 56
11. 12. 13. 14.
19.
60 36 36 30 30 20 20 28 60
20. 21. 22. 23. 24. 25. 26. 27.
60 1045 60 1045 50 730 30 730 50 730 730 30 30 730 60 368
16.
17. 18.
730
1045 1045 1045
1045 1045 1045 1045 1045
x
x
4
70
2
78 78 78 78
55 4
3 2 5.75 13.8 2 2 1 1 4.5 60 30
10 35 20
36 14 36 36 30 30
4
45
40 40 10
28
37
25 3 3 42
30 25 20 18 20
5
12
9 6 3
Y
?2 ;;:3-
50 300 150 250 250 175 220 250 330 200 300 240 240 300 280 300 300 250 270 270 270 300 400 320 300 300 220
30 30
2 6 29.3 2
730 730
10.
15.
3
1045 12.3
60
8.
9.
x
21 18 15
479 308 200 600 120 600 900 840 1200 762 704
568 488 900 360 600 450 95 210
1044 852 536 1250 1260 1925
cutier
2 30 6 +.
2
u
20 6
1
k
2
30 6
6
c 20 6
1
+q
V k
30 b
,I F $20 V
6
3
1
%
n a rev
b
@lhelh?l
45 groove 45 plane
t8heli.x*
120 800 30 160 800 16,4 145 800 1Q 150 1250 1 150 1250 22 120 800 30 160 800 17.4 150 800 17.4 150 1250 1 150 1250 120 800 30 200 800 10
45 a l o w edge 45 alonn edge Tp 18.1
45 45 45 45 43
in corner groove plane along edge
45 in corn% 45 groove 45 Dlane
he lixv* T -6.3 p whelhll
150 1250 1 45 along edge 150 1250 22 45 in corner Tpg2*58 C Tp-28.11
32
from end millcutter
0.9
0.7 0.5
0.3
Fig.2.Dependence
of Tp from cutting conditions.
ODtinnrm Version Table 3 n a b Tspe of cutter rev Notes movement passes min min mm mm min llhelix,l 120 710 40 45 groove 4 :80 710 20.2 45 plane 180 710 242 45 along edge Fig.4.
Table 2 Type of Notes movement
groove 160 800 174 45 plane
,
1.1
Fb.3.Standard version. D rCUt num er
Dependence of Tp geometry. I
Fig. 1.
65.84 33.21 21.59 47.86 22.81 54.04 76.78 78.42 94.61 56.14 65.25 75.21 93.14 98.32 75.55 94.24 87.67 19.19 29.03 106.41 92.77 10.48 89.56 84.2 91.1
D rCUt number S
mm mm
2
$
20 6
k
40 5
4
g-20 6
1
2
10 1 IEO 18 $40 5 160 710 24 45 550 1600 1 150 1600 3.7 120 710 40 180 710 2$ 180 710 2 W 550 1600 QY
45
groove plane *@helix** along edne along edge in corner Tp = 5.7
45
zroove
45
45 plane **helix" 45 alonn edne 45 alonq edge-
-