- Email: [email protected]
Optimization of a workpiece considering production requirements
Computers in Industry 21 (1993) 23-34 Elsevier
23
Applications
Optimization of a workpiece considering production requirements B. Anselmetti
a n d P. B o u r d e t
Laboratoire Unit ersitaire de Recherche en Production Automatism;e, Ecole Normale Sup~;rieure de Uachan, France
Received July 14, 1992; accepted August 4. 1992 As a contribution to the future of integrated CAD/CAM systems, this paper proposes a procedure to include the constraints of manufacturing in the dimensioning and the tolerancing of a workpiece. This algorithm is well adapted to consider the unilateral conditions extracted from the functional dimensioning and from the machining requirements in order to obtain a part model in medium dimensions. A rule for choosing dimensions which must be respected in machining is proposed. The system was implemented on an IBM RISC 6(10(Istation and a simple example is presented.
C A D / C A M systems have a l r e a d y greatly a i d e d the designing of w o r k p i e c e s , n a m e l y with kinematic analysis a n d the verifications of the resist a n c e of w o r k p i e c e s . V a r i o u s m e t h o d s define tole r a n c e limits a c c e p t a b l e to the m e c h a n i s m [1-3]. Some a u t h o r s have i n t r o d u c e d the e n g i n e e r i n g of quality a n d the loss function [4], using for exampie the work of T a g u c h i [5]. W e now p r o p o s e a new tool to c o n s i d e r the realizable accuracy of the m a n u f a c t u r i n g process
and the constraints of machining. W i t h this m e t h o d , we are able to optimize a g e o m e t r i c a l m o d e l which m i n i m i z e s the m a n u f a c t u r i n g cost (for a given process). T h e s h a p e of the w o r k p i e c e must be c o m p l e t e l y d e f i n e d with a p p r o x i m a t e d i m e n s i o n s in o r d e r to c h o o s e the m a n u f a c t u r i n g process. T h e definition d r a w i n g must display the functional d i m e n s i o n i n g and t o l e r a n c i n g (with u n i l a t e r a l or b i l a t e r a l tolerances). T h e s e conditions give the constraints of mating, guiding and m e c h a n i c a l resistance for a m e c h a n i s m . T h e basic sizes of the w o r k p i e c e can be c a l c u l a t e d to d e t e r mine the w o r k p i e c e m o d e l in m e d i u m d i m e n sions. F o r e x a m p l e , this could be used for calculation of the tool p a t h s or for the definition of the s t a m p i n g shape. T h e p r e s e n t e d c o m p u t i n g m e t h o d is g e n e r a l e n o u g h to be a d a p t e d to m a n y needs: - feasibility study of a m a n u f a c t u r i n g process with given m a c h i n e - t o o l s ; - choice of m a c h i n e - t o o l s a c c o r d i n g to the necessary accuracy; - accuracy of a raw shape; - t o l e r a n c i n g of the set-ups, etc. T h e result also gives the m a n u f a c t u r i n g d i m e n sioning a n d tolerancing, the a d j u s t m e n t limits and the definition of the Statistical Process Control b o u n d a r y .
Correspondence to: Dr. B. Anselmetti, Laboratoire Universi-
1.2. S t a t e - o f - t h e - a r t
taire de Recherche en Production Automatis~e, Ecole Normale Sup6rieure de Cachan, 61 avenue du Pdt Wilson, 942235 Caehan Cedex, France.
T h e m e t h o d used analyses the f u n c t i o n a l dim e n s i o n s of the definition d r a w i n g a n d the re-
Keywords: Optimization; Automation; Dimensioning; Toler-
ancing; CAD/CAM; Production; Machining dimensioning; Computer-aided process planning
1. I n t r o d u c t i o n
1.1. A i m s
0166-3615/93/$06.00 ~ 1993 Elsevier Science Publishers B.V. All rights reserved
24
Applications
Computers in Industry
20 mini 2mini ~ - I ~ of oil)
F" ~
-0 ep 1,6-0,06
1
20maxi ...,, 11maxiI 9ma,, I
~,.~,, 21,84maxih.,.' .,,
21"
~1~ (
...,7,a2mi
(2mini) /
1mma
a=118+3
omini ,
i "J I I %
'1mini U U
" e p 12"0'12 ( O r r l i n i ) ~ *
I I I 1mini + 4,1/tg 11s/2
2miniJ 111,5mini.IJ ' 'r
L2mini
I L r l,6mini
J13,6___H3,2)| ~
Fig. 1. Workpiece in its mechanism and the definition drawing.
quirements of production. Remy-Vincent and Schneider [6] have carefully analysed the functions of workpieces. So, we know that a definition drawing is always made up of independent unilateral tolerances and maximum material size conditions (see Fig. 1). With their data, it is very difficult to build a basic m o d e l for the C A D / C A M system. The manufacturing process is fixed. So, we propose to distribute the tolerances on each phase of machining and on the raw shape. For a long time, the traditional tolerance chain theory has
B. Anselmetti is a lecturer in the Department of Mechanical Engineering
of ENS Cachan, France. He obtained his Doctorate from Institut Sup~.rieur des Mat6riaux et de la Construction M~canique. His research interests are in the area of manufacturing methods. He is in charge of a researchteam computer-aided process planning in turning.
P. Bourdet is Professor, Head of the Department of Mechanical Engineering, and Director of the Laboratoire Universitaire de Recherche en Production Automatis6e of ENS Cachan, France. He has authored and coauthored over 50 research papers. He is a member of the CIRP and he has special interest in metrology and the designing of advanced manufacturing systems.
been used to achieve this aim. In 1973, Bourdet proposed the AI model based on the variabilities of positions of tooled surfaces and reference surfaces on each machine-tool [7,8]. Using a definition drawing which describes a part with medium dimensions and tolerances, the algorithm distributes the tolerances. Much software has been developed [9]. In 1988, Duret proposed a method using AI model, based on graph representation and introduced the evolutionary manufacturing dimensioning [10]. In 1989, Lehtihet et al. analysed various conditions with a linear algebraic formulation and generated the solution with the TOLC()N software [11]. C'ATAP[12] iS another statistical approach for tolerancing and also gives a new model for the part. Schneider and RemyVincent [13] define the completed mechanism simultaneously considering functional requirements and accuracy of manufacturing processes. In this article, the AI model is used with a new calculation method, because the current algorithm only distributes the tolerances (not the dimensions) and needs centered tolerances. Also, it does not work with unilateral tolerances from functional and machining requirements.
2. Study and definitions
2.1. Approach The AI model of Bourdet is used with a new algorithm which generalizes the optimization for
B. Anselrnetti, P. Bourdet / Workpiece optimization
Computers in Industry
,'/ "/// /
unilateral requirements and it may be possible to introduce production cost as a p a r a m e t e r in the algorithm.
2.2. Data and results The workpiece is described by the definition drawing (NF E04 550) and its manufacturing process. Tolerances cannot be modified. Dimensions of workpieces for each phase will be optimized with different criteria. If some tolerances are too small, we can conclude that manufacturing is impossible with the proposed process.
2.3. Definition of a dimension for the simulation model Usually, the principle of independence or the envelope principle (NFE 04 561 = ISO 8015) precisely defines the dimension of a workpiece. The calculation of tolerance transfer requires one common axis for the dimensions, in order to add simple scalars. So, with our approach, the dimension is also a specific model. We neglect form and location deviations, and we consider the dimensions on one single axis. We assume that one distance is given between two surfaces of a real workpiece. This distance is defined on the same axis for all dimensions. The tolerance zone is either a closed interval (bilateral tolerances) or a half-open interval (unilateral tolerances), and must contain the distance of each workpiece including measured deviations.
2.4. General definitions In this article, we use this following terminology: Definition drawing (used by the workshop). Description of the shape of the workpiece and set of requirements in accordance to NF E04 550. Machining operation. Machining carried out by one tool to obtain one shape on the workpiece. Sub-phase. An outline of operations which are carried out without removing the workpiece from its set-up and without moving the whole workpiece and set-up to another machine-tool. Phase. Sequence of sub-phases made on one production cell or one production area. Production area. Set of machine-tools, managed by one adjuster, which carry out the unbroken
25
intersectiono n active activeintersections
~ t l v e
gauge
Fig. 2. Active surfaces.
machining of a workpiece. The workpieces are machined on all machine-tools in the same order, so the settings are conserved and not independent for the different operations. Examples: The milling of the two sides of a T-nut is carried out with two sub-phases when we turn it in the set-up (no variability between the machining of the two sides). - Each stage of a transfer-line system corresponds to a sub-phase. Actit,e surfaces [10]. In a phase, active surfaces are those that are in contact with the set-up, are being machined or are being probed. (The operator can control the relative position of two active surfaces.) Three rules complete this definition: - The axis of two active surfaces is active. - The intersection of two active surfaces is active. - The gauge (of a cone) of an active surface is active. In the example of Fig. 2, fixture surface 1 is active (its position is defined by the gripper). The jaws make the position of axis 2 concentric. Axis 2 is also active. But cylinder 3 is not active, because the variability of its diameter modifies the position of the external surface with regard to the reference of machining. Machining dimension. This tolerance defines a requirement between two active surfaces in the same phase. -
2.5. Comments Manufacturing is done by many different machine-tools. Each phase has an adjuster in charge. The workpieces can be mixed between two production areas. So, the settings must be independent. For each functional or production requirement, the dimension transfer gives only one machined dimension. Symmetry tolerances are a
26
Applications
combination of 2, 3 or 4 requirements and so give that many machining dimensions. Each adjuster must produce workpieces which respect the different requirements, whatever the dimensions of workpieces in previous phases, but reject bad workpieces.
3. Model of simulated machining
3.1. Rules of decomposition of requirements" in machining dimensions Each requirement must be decomposed (transfered) into machining dimensions and tolerances. With this aim, a chain-transfer graph can be analysed [10,11]. As a matter of fact, with the notion of active surfaces, the decomposition rule is very simple and no complex algorithm is required. If Cii is a dimension between the surfaces i and j, there are three cases: (i) Both surfaces i and j are active in the same phase: Cf~iis the machining dimension. (ii) Surfaces i and j are not active in the same phase; so, the surfaces i and j are not machined in the same phase. If i is the last surface tooled and k the fixture surface in this phase, then surface i is set up in reference with k. The condition C~i is decomposed into the machining dimension Cfik and a new requirement Ckj which we must decompose with the same rule again. (iii) One of the surfaces i or j is not an active surface (axis of symmetry or intersection). It is necessary to decompose the requirement Cij into many conditions built using active surfaces. For example, if i is the axis of both surfaces g and k, then Cij =
Chj ) ,
where Cgj and Chj must be analysed with the same rule. The last part of this rule is fundamental because with the definition of active surfaces, it gives an automatic method for symmetry specifications or axis localizations.
3.2. Proposed Al model
Computers m Industry
For a given sub-phase, the positions of all surfaces are described by the tolerance fields AI defined by Bourdet [16]. The following notations are used (see Fig. 3): All surfaces are numbered from left to right, beginning with number 1 (raw surfaces, roughing surfaces, axis and surfaces used by functional and machining requirements). ~l~ is the field which contains each surface i of all workpieces (for all batch productions); Alz represents the variability of machining and adjusting. Al~' is the position field for fixture surface k in sub-phase n. Li is the position of the axis of the field ~l~ with regard to a reference attached to the machine-tool (one reference for each phase). For each phase, the machining tolerance can be analysed by the 2xl [8]: (a) Machining dimension between two machining surfaces i and j: Cf/i~× = L, -- L i + ½(AIi + A ) , Cfijmi n --
Lj - L , - ½(AIi + Ali).
(b) Machining dimension between a machining surface i and a fixture surface k in phase n: Cfkima x =
L i - L k + ½(A//+ AI~!).
Cfkimm = L i - L k - ½( A[ i + Al~!)~
In order to superpose the different references of all phases, we can intermingle the axis of the field of setting of the surface with the axis of the field of machining of this surface. We can choose the common reference origin on the axis of the surface the most on the left ( L ~ = 0 and L ~ > 0 for i > 1).
3.3. Introduction of manufacturing constraints Manufacturing constraints are very varied: Either the minimal or maximal thickness of chips can be specified for good tooling. (e.g. Cfki -~ Cmax).
The length of clamping surfaces can be assigned. The field AIi must be greater than the accuracy of the process (e.g. AIi > 0.05). Certain tools are available with standard tolerances. For example, if a groove between surfaces i and j is made with a tool (COmm and COmax), we have two inequalities: COma x ~ L j - L i 4- ½ ( A I i 4- A I i ) ~ C o .....
The method analyses the random dimensional dispersions of machining and positioning [7,14,15].
COmi n ~ t j -
L i - le( Al i + Alj) ~ Comm.
Computers in Industry
B. Anselmetti, P. Bourdet / Workpiece optimization
PHASE 10 9
If the location uses a reference surface k, we have: Lj - L i = V ,
L k -- L i = V ' ,
Al i + AI k <~t, At5
A l i + Al k <~t / 2 ,
Lj - L i = V ,
,!5 L9
Fig, 3. Simulation model.
27
AI~ <~t,
AIj <~t.
The simplex method can solve this problem with an objective function based on minimal cost for tooling, adjustment and raw material. 3.5. Objecti~'e function
Raw tolerances can be written (for example, if i and j are raw surfaces, Ali + Ali < 1). The standard dimensions for a raw material can be defined: steel bar toleranced by the dimension 60 hl0 gives:
-
-
L i - L i + (AIi + A/j)/2 L i - L i - (AIi +
-----6 0 ,
A/i)/2 = 59.88.
To limit the tool flexion, we can restrict the variation of thickness as a function of the finishing accuracy. If i is the finishing surface and j the roughing surface, the condition is AIi + Alj < 10 AIi. These manufacturing requirements nmst be respected only during the machining (not on the final component shape). 3.4. Set o f inequalities The problem formulation gives a linear matrix, with three types of conditions: >/, ~<, = . For each condition x, the inequality has the form: ~exiL,
+ ( F__,Axi
EexiLi + (Ea,,
&li) >~ Cx min A/i) ~< C,~ max,
where L i and Ali are unknown (>/0) and exi are 0; 0.5; - 0 . 5 ; + 1; - 1... For example: Cf8,1o rain ~< 13.6 gives
This function use the parameters L i and Ali. The adjustment cost depends on Ali. The raw material and tooling cost depend on L i. 3.5.1. Adjustment cost Cheikh and McGoldrick [17] propose an estimation of the cost according to the manufacturing process and the tolerances of the definition drawing. To optimize the calculation, we detail this cost function for each adjustment. The estimation of the adjustment time of a tool yields the graph of Fig. 4. The adjustment time depends on the quality of the surfaces. With this first graph, the different adjustment cost can be estimated with a "difficulty factor of adjustment" based on the value which necessitates 5 minutes of adjustment: k = 1: positioning of a tooled surface ( A I = 0.05); k = 10: realization of a raw surface (Al = 0.5); k = 1: positioning of a workpiece on a machining surface (AI = 0.05); k = 8; positioning of a workpiece on a raw surface (Al = 0.4). The adjustment time is t = f ( A 1 / k ) . Figure 4 gives for k = 1, in the interval M1M2, a slot: - 5 / 0 . 3 . The cost function is C = Ct - ( 5 / 0 . 3 × C H / n ) AI~,
L t o - L ~ - 0.5 A l ~ - 0.5 All0 ~< 13.6. If i is the symmetry axis of surfaces g and h, there is a new condition: Li
=
(Lg+Lh)/2.
adjustment brae
slot-20/0.3 f Mo slot -10/0,3
5mln
~
slot-5/0,3 slot -2.5/Q.3
If i and j are hole axis with location tolerances (nominal distance V and tolerance t) there are three conditions: Lj - L i = V,
AI~ <~t,
0,02 ~,
0,1
0,2
0,3
Alj <~t. Fig. 4. Graph of adjustment time.
slol 1.2SZ0.3
0.4
Al(mm)
28
Applications
Computers in lndustrv 15mini
I,
,1 1
II
I
Fig. 5. Influence of
Ali
Ill '°+''+''+°''' +°'''' on the raw material dimensions.
form partition of tolerances. Therefore, the algorithm does not give a good solution directly. There are probably better methods with nonlinear cost functions (MATLAB or [11]). We have fitted the algorithm proposed by Bourdet [16] to the simplex method, with iterations and a modification of variables: A I i = k i A + d i,
where CH is the hourly machining cost and n is the number of workpieces tooled between two adjustments. This simple graph gives good results. 3.5.2. R a w material cost
The raw material cost is given by the volume of raw material. It depends on the position of the raw surfaces. The cost function is of the form:
Cm = E 6 i m i L i , where m is the raw material cost by millimetre (e i = 0 for a machining surface, e~ = - 1 if the normal of the surface is on the left, eg = + 1 on the right). The length of the raw part depends considerably on the fields A l i (see Fig. 5).
where k i is a factor of difficulty, A > 0 and d i >/0). To obtain a uniform partition, we must find the biggest A possible. To achieve this, the coefficient of A in the cost function is multiplied by 2. The simplex method gives a first value of A. The AIi that are totally constrained are identified by a result d i = 0 . In this aim, we add a new condition d i <~ 0.005. After calculation, the determined values AI i = k i A are substracted from the set of inequalities. The simplex method is repeated again up to the end. Any inversed constraints (tool and raw conditions) demand one other change of variables: AI i = ~l i + d i + k i A .
3.5. 3. Tooling cost
For example, the inequality
The tooling cost depends on the length of tooling, the cutting conditions and the hourly cost of machining. The cost function is:
Lj - L i - 0.5 A l i - 0.5 A t i <~ Cn, m
C,=
Euis(Ls-Li
gives four conditions: 3l,/> 0;
).
In the example of Fig. 6, surface 2 can be positioned within a field of 5 ram. For the rotational workpiece, the turning cost will minimize the length 2-3. On the other hand, the manufacturing cost is independent of the drilling position. The influence of the manufacturing cost is smaller than the adjustment cost. Its effect is to minimize any lengths when a large field adjustment is still possible. 3.6. Calculation
The simplex method used demands a linear objective function, and therefore gives a nonuni-
Oli >>.0;
d, < 0.005;
dj < 0.005:
and Lj -- L~ - 0.50l i - 0.50l s = Cmi,,. (In the cost function, the factors of Oli are zero, because their value cannot be minimized.)
4. Application to an example 4.1. Data
The definition drawing of the example workpiece is given in Fig. 1. The surfaces are num-
!
Definition d r a w m 10rnini
10mini
10mini
lOmini
[
'
,
I 25rrlini Cu=u23
(L3-L2)
Cu=cte
Fig, 6. Influence of the manufacturing cost.
2
34
5
6
'8
Fig. 7. Numbering of surfaces.
10 111:;!
B. Anselmetti, P. Bourdet / Workpiece optimization
Computers in Industry
12
Table 1 Machining process
PROCESS.,
Phase
Fixture surface
Tooled surface
Tool
Machine
10 10 20 20 20 20 20 20 30 30
12 11 2 2 2 2 2 2 11 11
2 6 3 7 8 7 10 11 4 5
1 1 1 2 3 4 4 5 1 1
NC lathe NC lathe NC lathe NC lathe NC lathe NC lathe NC lathe NC lathe Drilling Drilling
bered as shown in Fig. 7. The machining process is represented in Fig. 8 and in Table 1. Drawing requirements are shown in Table 2. They must be completed by implicit conditions which are not defined on the drawing; these are marked by an asterisk (*) in Table 2. Table 3 shows the decomposition in machining dimensions (M = maximum, m = minimum).
34
5
6
78
910
29
1112
I I 15
'1 Illi PPHH12~ ' 1~ PH30 ~ "17[~I [
II "~
1"1
J~
II
..... . J'J" ~T[ x .°°ledsur~.... ....
II
Fig. 8. Representation of the process,
System after change of variables
The transformations are: AI 8 = A + d 8 and bllo = A + d l o ( k = 1). The condition Cf8,1o now gives:
L10 - L 8 + 0.5d 8 + 0.5dlo + A ~< 14. The new system is given in Fig. 9.
Objective function.
Before the change of variables, the objective function is the sum of the three cost functions analysed in section 3.5. The following array gives the coefficient of each variable: L2
L3
L4
Ls
L6
L7
0.00
- 0.04
0.00
0.00
0.08
- 0.05
4.2. Set of inequalities
Ls -0.04 AllO
L9 0.00 A120
Llo 0.00 A/30
L11 0.04 AI 1
LI2 1.10 Al 2
Initial system.
-2.6"7
-2.67
-2.67
-2.67
-2.67
~l 3
~l 4
AI 5
AI 6
AI 7
-2.67
-2.67
-2.67
-2.67
-2.67
The unknown values are L~, AIi and Al/. The first line of the matrix shows requirement No. 2 (Table 3). The constraint Cfs,lomax ~< 14 gives: Llo - L 8 + 0.5 AI 8 + 0.5 Allo ~< 14.
Al 8
Al 9
All0
Alll
Al12
-2.67
-2.67
-2.67
-2.67
-2.67
Table 2 Drawing requirements No, 1
2 3 *4 5 6 7 8 9 10 11 12 13 14 15 16 "17 * 18
Phase Phase
Left surface
Right surface
Medium value
Tolerance
Drawing Drawing Drawing Drawing Drawing Drawing Drawing Drawing Drawing Drawing Drawing Drawing PH 10 PH 10 PH 20 PH 20 PH 30 PH 30
8 2 4 5 6 7 8 9 10 2 5 6
10 3 6 6 8 8 9 10 11 6 6 10
13.80
0.400
1 1
2 2
1.000
11 11 3 5
12 12 4 7
1.000
* Implicit requirement, not defined on the drawing.
Min. value
Max. value
3.612 11.000 2.000 7.820 2.000 11.500 1.600 2.000 20.000 9,000 21.840 2.500 2.500 0.000 7.231
30
Applications
Computers in btdust~
Table 3 Decomposition in machining dimensions No:
Phase 30
Drawing requirements 1 2 3 4 411ml 5 511ml 6 7 8 9 10 11 12 511M1 13
Phase 20 810ml 810M1 2 3ml 211M-1 211M-1 2 8ml 7 8ml 8 9ml 910ml 10 11 m 1 211m-I 210M1
Phase 10 14 15
2/1M-I 211m -1
411M -1 511ml
,- 13.600 4, 14.00tl -;- 3.612 -;~ 11.0(XI ~ 2.000 7.820 5, 2.000 11.500 1.6011 2.000 20.001) ~: 9.00{) 21.840
2 6ml 2 6ml 2 6M--I
2 6MI 2 6M1 2 6m--t
Raw Raw PH 10 PH I0 PH 10 PH 10 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 30 PH 30 PH 30 PH 30
1 12m 1 1 12M1
>1.1t0// ~<2.500
212ml 212Ml
1.000 < 2.500
311ml 711M -1
Table 4 Set of machining dimensions 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Raw
212M -1 2 12m - 1
Phase 20 16 17 Phase 30 18 19
Phase 10
0.000 >~ 7.231
The Cf(1, 12)M = 43.730 Cf(1,121m = 42.695 Cf(2, 6)M = 16.441 Cf(2, 6)m = 16.298 Cf(2, 12)M = 41.695 Cf(2, 12)m = 41.230 Cf(2, 3)m = 3.612 Cf(2, 81m = 24.261 Cf(2, 10)M = 38.138 Cf(2, l l ) M = 40.230 Cf(2, 11 )m = 40.046 Cf(3, l l ) m = 35.933 Cf(7, 8)m = 2.000 Cf(7, l l ) M = 18.376 Cf(8, 9)m = 11.500 Cf(8, 10)M = 13.784 Cf(8, 10)m = 13.600 Cf(9, 10)m = 1.600 Cf(10, l l ) m = 2.000 Cf(4, l l ) M = 35.933 Cf(4, l l ) m = 34.933 Cf(5, l l ) M = 32.604 Cf(5, 11)m = 31.604
change
and creates
of variables
transforms
AIi
4.3. Solution T h e s o l u t i o n o f t h e s y s t e m is: Fixture: A l to = 0 . 4 1 4 ,
into
A with a coefficient of -231.75.
Al 2° =
0.092,
A13° = 0 . 5 0 0 .
Tooling: Al I = 0.517,
AI 2 = 0.052,
Al 3 = 0.500,
Al 4 = 0.500,
Al 5 = 0.500,
~ l 6 = 0.092,
Al 7 = 0.500,
Al s = 0.092,
Al 9 = 0.500,
Allo = 0.092,
Al~l = 0 . 0 9 2 ,
All2
= 0.517.
P o s i t i o n o f s u r f a c e s ( L l = 0): L 4 = 6.455,
L 2 = 1.750,
L 3 = 5.658,
L 5 = 9.784,
L 6 = 18.120,
L 7 = 23.807,
L s = 26.104,
L 9 = 37.900,
Lit = 39.796,
Lll = 41.888,
Llz
= 43.212.
di
.0
.0
L3
.04
L2
0
.0
L4
.0 1.0
.0
.0
.0
L5
.0
.0 1.0
.0 .0 .0 .0 .O
. £'
- £
.0 .0 .C' .0 .C'
.0
L4 .0 .0 .0 .0 .0 .0 .0 .0 -0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .O .0 .0 1.0 .0
L3 .0 .0 .0 .0 .0 .0 .0 .0 0 0 0 0 0 0 0 0 0 .0 .0 .0 .0 .0 1.0 .0 .0
.'3 .0 .0 .0 1.'3,
L2 .0 rl.D .0 .0 1.0 .0 .0 .0 .0 •0 .0 .0 .0 .0 .O .0 .0 .0 .0 .0 .0 .0 1.0 .O .0
.08
L6
- 1.0
.O
.0
.0 .C, .0 .~ .0
. £'
L5 .0 .0 -1.0 0 0 0 0 0 0 0 0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 -1.0
.05
L7
.0
.0
.0
.O .0 .0 .0 .0
1 . C'
L6 .0 1.0 1.0 -I.0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 1.0
.04
L8
1.0
.0
.0
1.0 .0 .0 .0 .0
. 8
L7 0 0 0 0 0 O 0 0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .O .0 .0 .0 .,0
.0
L9
.0
.0
.0
.0
LI0
1.O -i.0 .3 .0 .0
1 . 0
L8 1.0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 .0 .,3 .0
.0
.0 .0
.0
.0
LI2
. '3
.'9
1.~,
.6 .,3, .0 1. ) .C,
. C'
LII 0 .0 .0 .0 .0 1.0 .0 .C' .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0
i.i0
.0 .0 1.C' -1.0 .0
- 0
LI0 1.0 .0 .0 1.0 .0 .0 .0 .0 -0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 .0 .0 .0
LII .04
3
.0 1.0 1.0 .'3 .0
. O
L9 .0 .0 .0 .0 .'3 0 0 0 0 0 0 0 0 0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 . ,:3
. C'
.5
2.6 ~
dl 2°
. C'
.2
. :
.5
.5
.0
.0
.£,
.0
.0
.0
. ,:',
.5
.0 -.5
.0
.0
.0
2.67
dl 3°
. ~
.~
. ~
.5
.0
.~
.2
.,C'
.,J
.0
. ,1:'
.O
. '3'
.0
. ,0
dl 2
2.6 ~
dl 3
FUNCTION
.0
.0
.0
.,3
. ,3
.0
.E
. 'J
.£'
.u
0
.0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 .0 .0 .0 .0 .0 .0 .0 .0
dl 3
.0
0 0 0 0 0 0 0 0 0
dl
• u
2.6 ~ - 2 . 6 7
dl I
. ,;,
.0
.0
- . 5
.0
.O
.0
.0
. ,2'
.C, .0 . ,%,
.0
.0
.0
.0
.0 .0 .0 .5 .0 .0 .0 .0 1.0 .0 .0 .0 .0 .0 .0 .0 .0
.0 .5 .0 .0 .0 .0 .0 0 0 0 0 0 0 0 0 0 .0 .0
dll .0
dl 3° .0
OBJECTIVE
. )
.~
.~
.;,
., . 7,
.~
• E
. , :'
. '}'
. ~ . ~
• 3
<
. t
.0
• 0
. 2
. ,)
.5 .E
.0
.0
.0 .g
.0
. ,,3
0
.0 .5 .5 .3 .5 .0 1.0 .0 .0 .0 .0 .0 .C, .0 .0 .,)
.0
.0 .0 .0 .0 .0 .0
0 0 0 5 5
dl ~° d] 2° 0 .0
.0
2.6 ?
d[ 4
.0
-.5
.0
.0
.0
.0
. ,3'
.0
0
. 5
.0
.0
.0
0
0 0 0 0 0 0 0
d14
Fig. 9. Set of inequalities and objective function.
2.6-
dl I'
.
.
,.,,
LI2 .0 .0 .0 .0 .0 1.0 .0 .C' .0 .0 .0 .£ .0 .0 .0 .C .C .0
2.67
dl 5
. 5
.0 .;
.0
.O
.0
.0
.0
. 0
.5
. 0
. ':3
.0
.0
.0 .0 .5 .0 .C' .0 .0 .0 .0 .0 .0 .0 .0 1.0 .0 .0 .0 .0 .0 P'
dl 5
. 5
2.6 ~
Jl 6
. ,',,
." .'
. f.,
.C
.(
. ,{
.r
.=
. 5
. C:
.C'
.C
.0 .5 ,5 .5 .C, .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 .0 .0 .0 .C' ¢,
dl 6
.0
.0
. ,2; .,3 .,2,
2.E ~
dl-
. ']
0 .'J ~,
,3
.
.E
3
.'$
. 0
.) .0 .0
.
. '3'
.':3
. 0
.0
.0
.0
.0 .0 .0 .0 .'0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 l.O .0 .0
dl 9
2.67
dl~
.£
0 .0
- . ~_
.(:'
.0
. '3
.5
.O
. ~
0
dl~ .5 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 .n .0 .C'
.
. ~
.0
. ~
. '~'
. ~
.n
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
dl 7
• ?'
•
. '
. ~ .,2 .'7
.~
.0
.
2.67
dl 9
.
.0
. '2
. "
.fi
.C:'
.5 .0 .0 .5 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .C, .0 1.0 .C
.
dllo
d111
.
-2.6
dllo
.£
. ~
. [ .2 .=
.')
.7
]'
• <
.E
. ~
. '$
.,:)
.0
.0 .0 .5 .0 .q .5 .0 .C .0 .0 .0 .0 .0 .0 .C, . '[:' .0 .0 .C' 1.o
dl12
.
'
A
]
)
-2.1
cond
. 820
>-
231.~5
A
.231
->. '"3' 0
2£ . ,30> 1 . 0 0 r', ->l .OOC'
>1.6'00
u...... uu ->11 . 5C'C' > ~
_> -
->12.600 >-2 . 6 1 2 ->1 1 . O 0 C' ->2 . OOC'
<.'301
-<14.000 -<20.000 <9.000 -<21.840 -<2.500 <2.500 -<.001 -<.001 -<.001 -<.001 -<.001 -<.001 <-.001 -<.001 -<.001 -<. 001 -<.001 <.001 -<.001 -<.001
2.67
dll2
- 2 . ;
i .7 1~.5 :.c
l.v
1.7'
.]
2
I .7; 1 . '2' 3 . 2' 2 .3
3
1.0 1.0 3.0 2.0 14.5 5.5 0 0 0 0 0 0 0 0 0 0 0 0 '3 '3'
2.6 ~
Jill
. '-
.
. .~
.-:
.
.
. 7
. '
[
1.
.0 .0 .0 .0 .5 .0 .0 .3 .0 .0 .0 .0 .0 .( _3 . :[:' .3' _? .0 .(
e~
R
t
32
Applications
Computers in Industr)
Each machining tolerance can be calculated with these values: Cfijmi n
=
1
Oefinilion drawing
L j -- L i -- I ( A I i 4- a l j ) ,
2
"
Cfijma
x =
Lj - L i + l(Al i +
Alj).
The results are correct and can be used in production. For this example, we carried out two experiments: (i) The raw material cost is very important for the length of the block of raw material: with zero cost: Cr, w = 47.621 + 0.517; with cost 0 . 5 F / m m : Craw = 44.029 + 0.517; with cost 1 F / m m : Cra w = 43.212 _+ 0.517.
SIMULATION MODEL (Cf)
IZ
PH2£
41,465 33,428
~
_
30±0,2
[~
22,057_. I
II II
I 40,138
I
~
1,6mini
L_ £%118_ 1;oL --
=>residue0,36 onCf2,3
36,150 438,046
3,908
qFql
25±0,02
41 462 +1,084
t~
6_
40,138±0,092 38,138maxi ~l >1. 2mini 24,261mini I
•
30±0,02 . .
Cc 1,12 = 43,213_+0,518
71
_IL
.
,314=0,02
~,=,~
VERIFICATION D I M E N S I O N S (Cc)
NC DATA (position on Z axis)
41,462+0,232
j /
.
L3.,o I
(ii) The tooling cost has no influence, because the adjustment cost defines small tolerance fields and there is no more freedom for the position of the surfaces. In other cases, a normal tooling cost gives a better solution but the results is quite insensitive to the value of the cost, which can be very inaccurate.
Cf 1,12 = 43~213+0~518
#H10
~~ ) ,M330.02
Fig. 10. Non-independent adjustment.
4. 4. Analysis of results
RAW
AI2=0,02
~1:o t---- J--~o- ~
Machrnlo nns g ~ 20±0,02 dpmensi Extendedmachining dimensions
Results are presented in Table 4.
3 25_+0,02
~ " All 30= 0 02
~,MoOe,
30±0,2
20±0,02
l
" ll~r
2~
_ _IZI ~_Ll~_o_l~_
_
,~.~o,2
~.933 re,o(-
;.I
iq[
~'1
35,433
~H30 ___.
4
35,433d:0,5
, ~,,69
~
"1
11 m m ~
--
_
Fig. 11.
11
Computersin Industry 5. Using
the
results
5.1. Feasibility of the manufacturing process A system w i t h o u t solution m e a n s t h a t the constraints are i n c o m p a t i b l e . T h e definition d r a w i n g a n d the m a n u f a c t u r i n g p r o c e s s m u s t b e b e t t e r analysed. In the o p p o s i t e case, we can s e a r c h for m a c h i n e - t o o l s which have t h e a d e q u a t e m a c h i n ing capability [18].
5.2. Updating of the model T h e m o d e l gives the L i values which define the a v e r a g e p o s i t i o n of the surfaces. T h e workp i e c e m o d e l m u s t b e a c t u a l i z e d to g e n e r a t e the tool p a t h a n d the N C d a t a with the C A D / C A M system (see Fig. 11, c o l u m n 2).
5.3. Extension of machining dimensions 5.3.1. Problem T h e analysis of the m a c h i n i n g d i m e n s i o n s Cf in T a b l e 3 shows that t h e r e a r e c o n d i t i o n s which a r e fulfilled within t h e r e q u i r e d margin, leaving a residue. This r e s i d u e m a y b e u s e d for t h e adjustments. F i g u r e 10 gives an e x a m p l e . T h e exact c o n s t r a i n t for the m i d d l e of the w o r k p i e c e is 30 _+ 0.2, b u t t h e Al m o d e l , b a s e d on t h e i n d e p e n d a n c e o f surfaces, i m p o s e d Cf23 = 30 _+ 0.02. O n l y Cf12 and Cf34 a r e c o n s t r a i n e d by the c o n d i t i o n 20 + 0.02 a n d 25 _+ 0.02. T h e machining d i m e n s i o n Cf23 has a r e s i d u e of 0.36. So, Cf23 = 30 +_ 0.4 again.
B. Anselmetti, P. Bourdet / Workpieceoptimization
33
Table 5 Extended machining dimensions (see Fig. 1l, column 1) 1 2 3 4 5 6 7 8 9 10 11 12 13 14
RAW RAW PH 10 PH 10 PH 10 PH 10 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20
15 16
PH 20 PH 20
17 18 19 20 21 22 23
PH 20 PH 20 PH 20 PH 30 PH 30 PH 30 PH 30
Cf(1, 12)M = 43.730 Cf(1, 12)m = 42.695 Cf(2, 6)M = 16.441 Cf(2, 6)m = 16.298 Cf(2, 12)M = 41.695 Cf(2, 12)m = 41.230 Cf(2, 3)m = 3.612 Cf(2, 8)m = 24.261 Cf(2, 10)M = 38.138 Cf(2, ll)M = 40.230 Cf(2, ll)m = 40.046 Cf(3, ll)m = 35.933 Cf(7, 8)m = 2.001t Cf(7, ll)M = 21.375 * (18.376 condition 19) Cf(8, 9)m = 11.500 Cf(8, 10)M = 14.000 * (13.784 condition 2) Cf(8, 10)m = 13.600 Cf(9, 10)m = 1.600 Cf(10, ll)m = 2.001/ Cf(4, ll)M = 35.933 Cf(4, ll)m = 34.933 Cf(5, ll)M = 32.604 Cf(5, ll)m = 28.6116 * 131.61)4 conditions 5 and 19)
* Extended tolerances
5.3.3. Operations phases restrictions This d o c u m e n t gives all i n f o r m a t i o n for the m a c h i n i n g (clamping, process, cutting c o n d i t i o n ) a n d the m a c h i n i n g d i m e n s i o n s a n d tolerances. So, a m a c h i n e tool is well a d j u s t e d if all m a c h i n ing w o r k p i e c e s r e s p e c t the m a c h i n i n g t o l e r a n c e s .
5.4. Extension of the method
5.3.2. Calculation of extended machining dimensions
5.4.1. Adjustment, cerification and statistical process control
T h e analysis of T a b l e 3 d e t e r m i n e s the r e s i d u e for e a c h condition. W h e n the r e s i d u e is zero, all m a c h i n i n g d i m e n s i o n s which d e p e n d on this condition a r e c o n s t r a i n e d . N o n - z e r o r e s i d u e s m a y be d i s t r i b u t e d on t h e u n c o n s t r a i n e d m a c h i n i n g dimensions, b e g i n n i n g with t h e smallest increase. R e s u l t s a r e shown in T a b l e 5. T h e m a c h i n i n g t o l e r a n c e s which w e r e e x t e n d e d are m a r k e d by an asterisk ( * ) (see Fig. 11, c o l u m n 1). T h e extension of t h e m a c h i n i n g d i m e n s i o n s facilitates t h e a d j u s t m e n t a n d can r e d u c e any verification t o l e r a n c e s in o t h e r phases. F o r this r e a s o n , the extension m u s t be a choice for the planner.
T h e result of the s i m u l a t i o n gives a m o d e l o f the b e h a v i o u r of m a n u f a c t u r i n g a n d the tolerances for each phase. This i n f o r m a t i o n m a y be u s e d to d e f i n e the a d j u s t m e n t t o l e r a n c e s with d i f f e r e n t a p p r o a c h e s [19] (see Fig. 11, c o l u m n 3): - a d j u s t m e n t with i n d e p e n d e n t tools; - c o n t i n u o u s c o n t r o l o r s a m p l i n g control; - analysis o f tool wearing. T h e b o u n d a r y of S P C (statistical p r o c e s s control) m a y b e d e f i n e d [18].
5.4.2. Numerically controlled machine-tool This m e t h o d can i n t e g r a t e specific p r o c e s s e s in N C machining, for e x a m p l e to i n c o r p o r a t e the
34
Applications
two-dimensional [14,19].
( bmputer~ in lndust~'
cases (cones, chambers
etc.)
5.4.3. Statistical method Machining variations come from adjustments (constant for a batch, but variable between two batches) and random dispersions. Only the random variations can be added with statistical methods. In the opposite hypothesis [20] it would be possible that all combinations of good workpieces given by the same production batch, cannot be correctly assembled. This is not acceptable. This is why we propose the following solution: AI = Ar + Aa, where AI is the field of tolerance, AI is the adjustment field, Aa is the dispersion. To use the statistical method, it is necessary to know the spread of each dispersion. The inequality is now: Lj
_ L i + ½ ( A r i + A r y ) <~ COmax _ 21~ / A a i2 + Aa~ .
6. Conclusion The proposed calculation method has given logical results for many real examples. The economic consequences are important because, with this approach, designers can draw up minimal tender specifications which allow the widest tolerances for manufacturing. This analysis updates the geometric model of the workpiece which can be used directly by NC machining and measuring equipment. To employ this technique efficiently, it will now be integrated into our computer-aided process planning system. The calculation method proposed has given consistent results for many real examples.
References [1] P. Clozel, "M6ca Master: outil de conception m6canique et de cotation 3D pour les bureaux d'6tudes", Strucome, 1990. [2] A. Clement, A. Desrochers and A. Rivi~re, "Theory and practice of 3D tolerancing", CIRP Int. Work. Seminar on Computer-aided Tolerancing, Pennsylvania State University, 1991.
[3] J. Stinson and P. Urbaniak, "Calcul et optimisation des tol6rances", Actes MICAD 92, Paris, 1992. [4] P.T. Jessup, "The value of continuing improvement". Int. Communications Conf. ICC 85. June 1985. [5] "Initiation h l'engineering de la qualitC', Institut des methodes Taguchi, May 1989. [6] J. R6my-Vincent and F. Schneider, " A p r o p o s de la cotation fonctionnelle", M6moire bibliographique, DEA de Production Automatis6e, Universit6 de Nancy 1, 1990. [7] P. Bourdet, "Chaines de cotes de fabrication: le module", L, ingdnieur et le technieien de l'enseignement technique, December 1973. [8] P. Bourdet, J. R6my-Vincent and F. Schneider, "Tolerance analysis in manufacturing", CIRP Int. Work. Seminar on Computer-aided Tolerancing, Pennsylvania State University, 1991. [9] D. Fainguelernt, R. Weill and P. Bourdet, "Computer aided tolerancing and dimensioning in process planning", Ann. CIRP, Vol. 35, No. 1, 1986. [10] D. Duret, Cotation et M~thodes de Contr61e en Frabrication Mdcanique, Collection Introduction a I'art de l'ing6nieur, Editions Augustin. [11] E.A. Lehtihet, N.U. Gunasena and I. Ham, "Element of computer-aided tolerance analysis", Int. Conf. on C A D / CAM and AMT, CIRP Sessions, Jerusalem, December 1989. [12] K. Panshal, S. Raman and P.S. Pulat, "Computer-aided tolerance assignment procedure (CATAP) for design dimensioning", Int. J. Prod. Res., Vol., 30, No. 3, 1992. [13] F. Schneider and J. R6my-Vincent, "D6finition des ensembles m6caniques: Condition d'obtention de cotations optimales", 23rd Int. Seminar on Production Systems, Nancy, June 1991. [t4] B. Anselmetti, "Simulation d'usinage bidimensionnelle sur un exemple en tournage en commande num6rique", Rev. Mec. Mater. Electr., No. 398, March-April 1983. [15] C. Maria, La qualitd des produits industriels, Edition Dunod, 1991. [16] P. Bourdet, "Cha]nes de cotes de fabrication: le mode op6ratoire", L'ingdnieur et le technicien de l'enseignement technique, December 1973. [17] A. Cheikh and P. McGoldrick, "Functional tolerancing in CAD/CAM", Int. Conf. on C A D ~ C A M and AMT, CIRP Sessions, Jerusalem, December 1989. [18] M. Vigier, Pratique de la maitrise des procFdds-MSP, Let 6ditions d'organisation, 1989. [19] B. Anselmetti and V. Anselmeni, "La fabrication, une d6marche globale", Rev. Technol. Formation, Nos. 27, 29, 30, 32, 1989-1990. [20] E.A. Lehtihet, N.U. Gunasena and I. Ham, "An update --Statistical tolerance control methods and computations", CIRP Int. Work. Seminar on Computer-aided Tolerancing, Pennsylvania State University, 1991. [21] P. Bourdet, "Introduction ~ la conception automatique de gammes d'usinage", La gamme automatique en usinage, Edition Hermes, 1990. [22] P. Souvay, La statistique, outil de la quaKtd, Edition Afnor Gestion, 1986.
23
Applications
Optimization of a workpiece considering production requirements B. Anselmetti
a n d P. B o u r d e t
Laboratoire Unit ersitaire de Recherche en Production Automatism;e, Ecole Normale Sup~;rieure de Uachan, France
Received July 14, 1992; accepted August 4. 1992 As a contribution to the future of integrated CAD/CAM systems, this paper proposes a procedure to include the constraints of manufacturing in the dimensioning and the tolerancing of a workpiece. This algorithm is well adapted to consider the unilateral conditions extracted from the functional dimensioning and from the machining requirements in order to obtain a part model in medium dimensions. A rule for choosing dimensions which must be respected in machining is proposed. The system was implemented on an IBM RISC 6(10(Istation and a simple example is presented.
C A D / C A M systems have a l r e a d y greatly a i d e d the designing of w o r k p i e c e s , n a m e l y with kinematic analysis a n d the verifications of the resist a n c e of w o r k p i e c e s . V a r i o u s m e t h o d s define tole r a n c e limits a c c e p t a b l e to the m e c h a n i s m [1-3]. Some a u t h o r s have i n t r o d u c e d the e n g i n e e r i n g of quality a n d the loss function [4], using for exampie the work of T a g u c h i [5]. W e now p r o p o s e a new tool to c o n s i d e r the realizable accuracy of the m a n u f a c t u r i n g process
and the constraints of machining. W i t h this m e t h o d , we are able to optimize a g e o m e t r i c a l m o d e l which m i n i m i z e s the m a n u f a c t u r i n g cost (for a given process). T h e s h a p e of the w o r k p i e c e must be c o m p l e t e l y d e f i n e d with a p p r o x i m a t e d i m e n s i o n s in o r d e r to c h o o s e the m a n u f a c t u r i n g process. T h e definition d r a w i n g must display the functional d i m e n s i o n i n g and t o l e r a n c i n g (with u n i l a t e r a l or b i l a t e r a l tolerances). T h e s e conditions give the constraints of mating, guiding and m e c h a n i c a l resistance for a m e c h a n i s m . T h e basic sizes of the w o r k p i e c e can be c a l c u l a t e d to d e t e r mine the w o r k p i e c e m o d e l in m e d i u m d i m e n sions. F o r e x a m p l e , this could be used for calculation of the tool p a t h s or for the definition of the s t a m p i n g shape. T h e p r e s e n t e d c o m p u t i n g m e t h o d is g e n e r a l e n o u g h to be a d a p t e d to m a n y needs: - feasibility study of a m a n u f a c t u r i n g process with given m a c h i n e - t o o l s ; - choice of m a c h i n e - t o o l s a c c o r d i n g to the necessary accuracy; - accuracy of a raw shape; - t o l e r a n c i n g of the set-ups, etc. T h e result also gives the m a n u f a c t u r i n g d i m e n sioning a n d tolerancing, the a d j u s t m e n t limits and the definition of the Statistical Process Control b o u n d a r y .
Correspondence to: Dr. B. Anselmetti, Laboratoire Universi-
1.2. S t a t e - o f - t h e - a r t
taire de Recherche en Production Automatis~e, Ecole Normale Sup6rieure de Cachan, 61 avenue du Pdt Wilson, 942235 Caehan Cedex, France.
T h e m e t h o d used analyses the f u n c t i o n a l dim e n s i o n s of the definition d r a w i n g a n d the re-
Keywords: Optimization; Automation; Dimensioning; Toler-
ancing; CAD/CAM; Production; Machining dimensioning; Computer-aided process planning
1. I n t r o d u c t i o n
1.1. A i m s
0166-3615/93/$06.00 ~ 1993 Elsevier Science Publishers B.V. All rights reserved
24
Applications
Computers in Industry
20 mini 2mini ~ - I ~ of oil)
F" ~
-0 ep 1,6-0,06
1
20maxi ...,, 11maxiI 9ma,, I
~,.~,, 21,84maxih.,.' .,,
21"
~1~ (
...,7,a2mi
(2mini) /
1mma
a=118+3
omini ,
i "J I I %
'1mini U U
" e p 12"0'12 ( O r r l i n i ) ~ *
I I I 1mini + 4,1/tg 11s/2
2miniJ 111,5mini.IJ ' 'r
L2mini
I L r l,6mini
J13,6___H3,2)| ~
Fig. 1. Workpiece in its mechanism and the definition drawing.
quirements of production. Remy-Vincent and Schneider [6] have carefully analysed the functions of workpieces. So, we know that a definition drawing is always made up of independent unilateral tolerances and maximum material size conditions (see Fig. 1). With their data, it is very difficult to build a basic m o d e l for the C A D / C A M system. The manufacturing process is fixed. So, we propose to distribute the tolerances on each phase of machining and on the raw shape. For a long time, the traditional tolerance chain theory has
B. Anselmetti is a lecturer in the Department of Mechanical Engineering
of ENS Cachan, France. He obtained his Doctorate from Institut Sup~.rieur des Mat6riaux et de la Construction M~canique. His research interests are in the area of manufacturing methods. He is in charge of a researchteam computer-aided process planning in turning.
P. Bourdet is Professor, Head of the Department of Mechanical Engineering, and Director of the Laboratoire Universitaire de Recherche en Production Automatis6e of ENS Cachan, France. He has authored and coauthored over 50 research papers. He is a member of the CIRP and he has special interest in metrology and the designing of advanced manufacturing systems.
been used to achieve this aim. In 1973, Bourdet proposed the AI model based on the variabilities of positions of tooled surfaces and reference surfaces on each machine-tool [7,8]. Using a definition drawing which describes a part with medium dimensions and tolerances, the algorithm distributes the tolerances. Much software has been developed [9]. In 1988, Duret proposed a method using AI model, based on graph representation and introduced the evolutionary manufacturing dimensioning [10]. In 1989, Lehtihet et al. analysed various conditions with a linear algebraic formulation and generated the solution with the TOLC()N software [11]. C'ATAP[12] iS another statistical approach for tolerancing and also gives a new model for the part. Schneider and RemyVincent [13] define the completed mechanism simultaneously considering functional requirements and accuracy of manufacturing processes. In this article, the AI model is used with a new calculation method, because the current algorithm only distributes the tolerances (not the dimensions) and needs centered tolerances. Also, it does not work with unilateral tolerances from functional and machining requirements.
2. Study and definitions
2.1. Approach The AI model of Bourdet is used with a new algorithm which generalizes the optimization for
B. Anselrnetti, P. Bourdet / Workpiece optimization
Computers in Industry
,'/ "/// /
unilateral requirements and it may be possible to introduce production cost as a p a r a m e t e r in the algorithm.
2.2. Data and results The workpiece is described by the definition drawing (NF E04 550) and its manufacturing process. Tolerances cannot be modified. Dimensions of workpieces for each phase will be optimized with different criteria. If some tolerances are too small, we can conclude that manufacturing is impossible with the proposed process.
2.3. Definition of a dimension for the simulation model Usually, the principle of independence or the envelope principle (NFE 04 561 = ISO 8015) precisely defines the dimension of a workpiece. The calculation of tolerance transfer requires one common axis for the dimensions, in order to add simple scalars. So, with our approach, the dimension is also a specific model. We neglect form and location deviations, and we consider the dimensions on one single axis. We assume that one distance is given between two surfaces of a real workpiece. This distance is defined on the same axis for all dimensions. The tolerance zone is either a closed interval (bilateral tolerances) or a half-open interval (unilateral tolerances), and must contain the distance of each workpiece including measured deviations.
2.4. General definitions In this article, we use this following terminology: Definition drawing (used by the workshop). Description of the shape of the workpiece and set of requirements in accordance to NF E04 550. Machining operation. Machining carried out by one tool to obtain one shape on the workpiece. Sub-phase. An outline of operations which are carried out without removing the workpiece from its set-up and without moving the whole workpiece and set-up to another machine-tool. Phase. Sequence of sub-phases made on one production cell or one production area. Production area. Set of machine-tools, managed by one adjuster, which carry out the unbroken
25
intersectiono n active activeintersections
~ t l v e
gauge
Fig. 2. Active surfaces.
machining of a workpiece. The workpieces are machined on all machine-tools in the same order, so the settings are conserved and not independent for the different operations. Examples: The milling of the two sides of a T-nut is carried out with two sub-phases when we turn it in the set-up (no variability between the machining of the two sides). - Each stage of a transfer-line system corresponds to a sub-phase. Actit,e surfaces [10]. In a phase, active surfaces are those that are in contact with the set-up, are being machined or are being probed. (The operator can control the relative position of two active surfaces.) Three rules complete this definition: - The axis of two active surfaces is active. - The intersection of two active surfaces is active. - The gauge (of a cone) of an active surface is active. In the example of Fig. 2, fixture surface 1 is active (its position is defined by the gripper). The jaws make the position of axis 2 concentric. Axis 2 is also active. But cylinder 3 is not active, because the variability of its diameter modifies the position of the external surface with regard to the reference of machining. Machining dimension. This tolerance defines a requirement between two active surfaces in the same phase. -
2.5. Comments Manufacturing is done by many different machine-tools. Each phase has an adjuster in charge. The workpieces can be mixed between two production areas. So, the settings must be independent. For each functional or production requirement, the dimension transfer gives only one machined dimension. Symmetry tolerances are a
26
Applications
combination of 2, 3 or 4 requirements and so give that many machining dimensions. Each adjuster must produce workpieces which respect the different requirements, whatever the dimensions of workpieces in previous phases, but reject bad workpieces.
3. Model of simulated machining
3.1. Rules of decomposition of requirements" in machining dimensions Each requirement must be decomposed (transfered) into machining dimensions and tolerances. With this aim, a chain-transfer graph can be analysed [10,11]. As a matter of fact, with the notion of active surfaces, the decomposition rule is very simple and no complex algorithm is required. If Cii is a dimension between the surfaces i and j, there are three cases: (i) Both surfaces i and j are active in the same phase: Cf~iis the machining dimension. (ii) Surfaces i and j are not active in the same phase; so, the surfaces i and j are not machined in the same phase. If i is the last surface tooled and k the fixture surface in this phase, then surface i is set up in reference with k. The condition C~i is decomposed into the machining dimension Cfik and a new requirement Ckj which we must decompose with the same rule again. (iii) One of the surfaces i or j is not an active surface (axis of symmetry or intersection). It is necessary to decompose the requirement Cij into many conditions built using active surfaces. For example, if i is the axis of both surfaces g and k, then Cij =
Chj ) ,
where Cgj and Chj must be analysed with the same rule. The last part of this rule is fundamental because with the definition of active surfaces, it gives an automatic method for symmetry specifications or axis localizations.
3.2. Proposed Al model
Computers m Industry
For a given sub-phase, the positions of all surfaces are described by the tolerance fields AI defined by Bourdet [16]. The following notations are used (see Fig. 3): All surfaces are numbered from left to right, beginning with number 1 (raw surfaces, roughing surfaces, axis and surfaces used by functional and machining requirements). ~l~ is the field which contains each surface i of all workpieces (for all batch productions); Alz represents the variability of machining and adjusting. Al~' is the position field for fixture surface k in sub-phase n. Li is the position of the axis of the field ~l~ with regard to a reference attached to the machine-tool (one reference for each phase). For each phase, the machining tolerance can be analysed by the 2xl [8]: (a) Machining dimension between two machining surfaces i and j: Cf/i~× = L, -- L i + ½(AIi + A ) , Cfijmi n --
Lj - L , - ½(AIi + Ali).
(b) Machining dimension between a machining surface i and a fixture surface k in phase n: Cfkima x =
L i - L k + ½(A//+ AI~!).
Cfkimm = L i - L k - ½( A[ i + Al~!)~
In order to superpose the different references of all phases, we can intermingle the axis of the field of setting of the surface with the axis of the field of machining of this surface. We can choose the common reference origin on the axis of the surface the most on the left ( L ~ = 0 and L ~ > 0 for i > 1).
3.3. Introduction of manufacturing constraints Manufacturing constraints are very varied: Either the minimal or maximal thickness of chips can be specified for good tooling. (e.g. Cfki -~ Cmax).
The length of clamping surfaces can be assigned. The field AIi must be greater than the accuracy of the process (e.g. AIi > 0.05). Certain tools are available with standard tolerances. For example, if a groove between surfaces i and j is made with a tool (COmm and COmax), we have two inequalities: COma x ~ L j - L i 4- ½ ( A I i 4- A I i ) ~ C o .....
The method analyses the random dimensional dispersions of machining and positioning [7,14,15].
COmi n ~ t j -
L i - le( Al i + Alj) ~ Comm.
Computers in Industry
B. Anselmetti, P. Bourdet / Workpiece optimization
PHASE 10 9
If the location uses a reference surface k, we have: Lj - L i = V ,
L k -- L i = V ' ,
Al i + AI k <~t, At5
A l i + Al k <~t / 2 ,
Lj - L i = V ,
,!5 L9
Fig, 3. Simulation model.
27
AI~ <~t,
AIj <~t.
The simplex method can solve this problem with an objective function based on minimal cost for tooling, adjustment and raw material. 3.5. Objecti~'e function
Raw tolerances can be written (for example, if i and j are raw surfaces, Ali + Ali < 1). The standard dimensions for a raw material can be defined: steel bar toleranced by the dimension 60 hl0 gives:
-
-
L i - L i + (AIi + A/j)/2 L i - L i - (AIi +
-----6 0 ,
A/i)/2 = 59.88.
To limit the tool flexion, we can restrict the variation of thickness as a function of the finishing accuracy. If i is the finishing surface and j the roughing surface, the condition is AIi + Alj < 10 AIi. These manufacturing requirements nmst be respected only during the machining (not on the final component shape). 3.4. Set o f inequalities The problem formulation gives a linear matrix, with three types of conditions: >/, ~<, = . For each condition x, the inequality has the form: ~exiL,
+ ( F__,Axi
EexiLi + (Ea,,
&li) >~ Cx min A/i) ~< C,~ max,
where L i and Ali are unknown (>/0) and exi are 0; 0.5; - 0 . 5 ; + 1; - 1... For example: Cf8,1o rain ~< 13.6 gives
This function use the parameters L i and Ali. The adjustment cost depends on Ali. The raw material and tooling cost depend on L i. 3.5.1. Adjustment cost Cheikh and McGoldrick [17] propose an estimation of the cost according to the manufacturing process and the tolerances of the definition drawing. To optimize the calculation, we detail this cost function for each adjustment. The estimation of the adjustment time of a tool yields the graph of Fig. 4. The adjustment time depends on the quality of the surfaces. With this first graph, the different adjustment cost can be estimated with a "difficulty factor of adjustment" based on the value which necessitates 5 minutes of adjustment: k = 1: positioning of a tooled surface ( A I = 0.05); k = 10: realization of a raw surface (Al = 0.5); k = 1: positioning of a workpiece on a machining surface (AI = 0.05); k = 8; positioning of a workpiece on a raw surface (Al = 0.4). The adjustment time is t = f ( A 1 / k ) . Figure 4 gives for k = 1, in the interval M1M2, a slot: - 5 / 0 . 3 . The cost function is C = Ct - ( 5 / 0 . 3 × C H / n ) AI~,
L t o - L ~ - 0.5 A l ~ - 0.5 All0 ~< 13.6. If i is the symmetry axis of surfaces g and h, there is a new condition: Li
=
(Lg+Lh)/2.
adjustment brae
slot-20/0.3 f Mo slot -10/0,3
5mln
~
slot-5/0,3 slot -2.5/Q.3
If i and j are hole axis with location tolerances (nominal distance V and tolerance t) there are three conditions: Lj - L i = V,
AI~ <~t,
0,02 ~,
0,1
0,2
0,3
Alj <~t. Fig. 4. Graph of adjustment time.
slol 1.2SZ0.3
0.4
Al(mm)
28
Applications
Computers in lndustrv 15mini
I,
,1 1
II
I
Fig. 5. Influence of
Ali
Ill '°+''+''+°''' +°'''' on the raw material dimensions.
form partition of tolerances. Therefore, the algorithm does not give a good solution directly. There are probably better methods with nonlinear cost functions (MATLAB or [11]). We have fitted the algorithm proposed by Bourdet [16] to the simplex method, with iterations and a modification of variables: A I i = k i A + d i,
where CH is the hourly machining cost and n is the number of workpieces tooled between two adjustments. This simple graph gives good results. 3.5.2. R a w material cost
The raw material cost is given by the volume of raw material. It depends on the position of the raw surfaces. The cost function is of the form:
Cm = E 6 i m i L i , where m is the raw material cost by millimetre (e i = 0 for a machining surface, e~ = - 1 if the normal of the surface is on the left, eg = + 1 on the right). The length of the raw part depends considerably on the fields A l i (see Fig. 5).
where k i is a factor of difficulty, A > 0 and d i >/0). To obtain a uniform partition, we must find the biggest A possible. To achieve this, the coefficient of A in the cost function is multiplied by 2. The simplex method gives a first value of A. The AIi that are totally constrained are identified by a result d i = 0 . In this aim, we add a new condition d i <~ 0.005. After calculation, the determined values AI i = k i A are substracted from the set of inequalities. The simplex method is repeated again up to the end. Any inversed constraints (tool and raw conditions) demand one other change of variables: AI i = ~l i + d i + k i A .
3.5. 3. Tooling cost
For example, the inequality
The tooling cost depends on the length of tooling, the cutting conditions and the hourly cost of machining. The cost function is:
Lj - L i - 0.5 A l i - 0.5 A t i <~ Cn, m
C,=
Euis(Ls-Li
gives four conditions: 3l,/> 0;
).
In the example of Fig. 6, surface 2 can be positioned within a field of 5 ram. For the rotational workpiece, the turning cost will minimize the length 2-3. On the other hand, the manufacturing cost is independent of the drilling position. The influence of the manufacturing cost is smaller than the adjustment cost. Its effect is to minimize any lengths when a large field adjustment is still possible. 3.6. Calculation
The simplex method used demands a linear objective function, and therefore gives a nonuni-
Oli >>.0;
d, < 0.005;
dj < 0.005:
and Lj -- L~ - 0.50l i - 0.50l s = Cmi,,. (In the cost function, the factors of Oli are zero, because their value cannot be minimized.)
4. Application to an example 4.1. Data
The definition drawing of the example workpiece is given in Fig. 1. The surfaces are num-
!
Definition d r a w m 10rnini
10mini
10mini
lOmini
[
'
,
I 25rrlini Cu=u23
(L3-L2)
Cu=cte
Fig, 6. Influence of the manufacturing cost.
2
34
5
6
'8
Fig. 7. Numbering of surfaces.
10 111:;!
B. Anselmetti, P. Bourdet / Workpiece optimization
Computers in Industry
12
Table 1 Machining process
PROCESS.,
Phase
Fixture surface
Tooled surface
Tool
Machine
10 10 20 20 20 20 20 20 30 30
12 11 2 2 2 2 2 2 11 11
2 6 3 7 8 7 10 11 4 5
1 1 1 2 3 4 4 5 1 1
NC lathe NC lathe NC lathe NC lathe NC lathe NC lathe NC lathe NC lathe Drilling Drilling
bered as shown in Fig. 7. The machining process is represented in Fig. 8 and in Table 1. Drawing requirements are shown in Table 2. They must be completed by implicit conditions which are not defined on the drawing; these are marked by an asterisk (*) in Table 2. Table 3 shows the decomposition in machining dimensions (M = maximum, m = minimum).
34
5
6
78
910
29
1112
I I 15
'1 Illi PPHH12~ ' 1~ PH30 ~ "17[~I [
II "~
1"1
J~
II
..... . J'J" ~T[ x .°°ledsur~.... ....
II
Fig. 8. Representation of the process,
System after change of variables
The transformations are: AI 8 = A + d 8 and bllo = A + d l o ( k = 1). The condition Cf8,1o now gives:
L10 - L 8 + 0.5d 8 + 0.5dlo + A ~< 14. The new system is given in Fig. 9.
Objective function.
Before the change of variables, the objective function is the sum of the three cost functions analysed in section 3.5. The following array gives the coefficient of each variable: L2
L3
L4
Ls
L6
L7
0.00
- 0.04
0.00
0.00
0.08
- 0.05
4.2. Set of inequalities
Ls -0.04 AllO
L9 0.00 A120
Llo 0.00 A/30
L11 0.04 AI 1
LI2 1.10 Al 2
Initial system.
-2.6"7
-2.67
-2.67
-2.67
-2.67
~l 3
~l 4
AI 5
AI 6
AI 7
-2.67
-2.67
-2.67
-2.67
-2.67
The unknown values are L~, AIi and Al/. The first line of the matrix shows requirement No. 2 (Table 3). The constraint Cfs,lomax ~< 14 gives: Llo - L 8 + 0.5 AI 8 + 0.5 Allo ~< 14.
Al 8
Al 9
All0
Alll
Al12
-2.67
-2.67
-2.67
-2.67
-2.67
Table 2 Drawing requirements No, 1
2 3 *4 5 6 7 8 9 10 11 12 13 14 15 16 "17 * 18
Phase Phase
Left surface
Right surface
Medium value
Tolerance
Drawing Drawing Drawing Drawing Drawing Drawing Drawing Drawing Drawing Drawing Drawing Drawing PH 10 PH 10 PH 20 PH 20 PH 30 PH 30
8 2 4 5 6 7 8 9 10 2 5 6
10 3 6 6 8 8 9 10 11 6 6 10
13.80
0.400
1 1
2 2
1.000
11 11 3 5
12 12 4 7
1.000
* Implicit requirement, not defined on the drawing.
Min. value
Max. value
3.612 11.000 2.000 7.820 2.000 11.500 1.600 2.000 20.000 9,000 21.840 2.500 2.500 0.000 7.231
30
Applications
Computers in btdust~
Table 3 Decomposition in machining dimensions No:
Phase 30
Drawing requirements 1 2 3 4 411ml 5 511ml 6 7 8 9 10 11 12 511M1 13
Phase 20 810ml 810M1 2 3ml 211M-1 211M-1 2 8ml 7 8ml 8 9ml 910ml 10 11 m 1 211m-I 210M1
Phase 10 14 15
2/1M-I 211m -1
411M -1 511ml
,- 13.600 4, 14.00tl -;- 3.612 -;~ 11.0(XI ~ 2.000 7.820 5, 2.000 11.500 1.6011 2.000 20.001) ~: 9.00{) 21.840
2 6ml 2 6ml 2 6M--I
2 6MI 2 6M1 2 6m--t
Raw Raw PH 10 PH I0 PH 10 PH 10 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 30 PH 30 PH 30 PH 30
1 12m 1 1 12M1
>1.1t0// ~<2.500
212ml 212Ml
1.000 < 2.500
311ml 711M -1
Table 4 Set of machining dimensions 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Raw
212M -1 2 12m - 1
Phase 20 16 17 Phase 30 18 19
Phase 10
0.000 >~ 7.231
The Cf(1, 12)M = 43.730 Cf(1,121m = 42.695 Cf(2, 6)M = 16.441 Cf(2, 6)m = 16.298 Cf(2, 12)M = 41.695 Cf(2, 12)m = 41.230 Cf(2, 3)m = 3.612 Cf(2, 81m = 24.261 Cf(2, 10)M = 38.138 Cf(2, l l ) M = 40.230 Cf(2, 11 )m = 40.046 Cf(3, l l ) m = 35.933 Cf(7, 8)m = 2.000 Cf(7, l l ) M = 18.376 Cf(8, 9)m = 11.500 Cf(8, 10)M = 13.784 Cf(8, 10)m = 13.600 Cf(9, 10)m = 1.600 Cf(10, l l ) m = 2.000 Cf(4, l l ) M = 35.933 Cf(4, l l ) m = 34.933 Cf(5, l l ) M = 32.604 Cf(5, 11)m = 31.604
change
and creates
of variables
transforms
AIi
4.3. Solution T h e s o l u t i o n o f t h e s y s t e m is: Fixture: A l to = 0 . 4 1 4 ,
into
A with a coefficient of -231.75.
Al 2° =
0.092,
A13° = 0 . 5 0 0 .
Tooling: Al I = 0.517,
AI 2 = 0.052,
Al 3 = 0.500,
Al 4 = 0.500,
Al 5 = 0.500,
~ l 6 = 0.092,
Al 7 = 0.500,
Al s = 0.092,
Al 9 = 0.500,
Allo = 0.092,
Al~l = 0 . 0 9 2 ,
All2
= 0.517.
P o s i t i o n o f s u r f a c e s ( L l = 0): L 4 = 6.455,
L 2 = 1.750,
L 3 = 5.658,
L 5 = 9.784,
L 6 = 18.120,
L 7 = 23.807,
L s = 26.104,
L 9 = 37.900,
Lit = 39.796,
Lll = 41.888,
Llz
= 43.212.
di
.0
.0
L3
.04
L2
0
.0
L4
.0 1.0
.0
.0
.0
L5
.0
.0 1.0
.0 .0 .0 .0 .O
. £'
- £
.0 .0 .C' .0 .C'
.0
L4 .0 .0 .0 .0 .0 .0 .0 .0 -0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .O .0 .0 1.0 .0
L3 .0 .0 .0 .0 .0 .0 .0 .0 0 0 0 0 0 0 0 0 0 .0 .0 .0 .0 .0 1.0 .0 .0
.'3 .0 .0 .0 1.'3,
L2 .0 rl.D .0 .0 1.0 .0 .0 .0 .0 •0 .0 .0 .0 .0 .O .0 .0 .0 .0 .0 .0 .0 1.0 .O .0
.08
L6
- 1.0
.O
.0
.0 .C, .0 .~ .0
. £'
L5 .0 .0 -1.0 0 0 0 0 0 0 0 0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 -1.0
.05
L7
.0
.0
.0
.O .0 .0 .0 .0
1 . C'
L6 .0 1.0 1.0 -I.0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 1.0
.04
L8
1.0
.0
.0
1.0 .0 .0 .0 .0
. 8
L7 0 0 0 0 0 O 0 0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .O .0 .0 .0 .,0
.0
L9
.0
.0
.0
.0
LI0
1.O -i.0 .3 .0 .0
1 . 0
L8 1.0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 .0 .,3 .0
.0
.0 .0
.0
.0
LI2
. '3
.'9
1.~,
.6 .,3, .0 1. ) .C,
. C'
LII 0 .0 .0 .0 .0 1.0 .0 .C' .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0
i.i0
.0 .0 1.C' -1.0 .0
- 0
LI0 1.0 .0 .0 1.0 .0 .0 .0 .0 -0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 .0 .0 .0
LII .04
3
.0 1.0 1.0 .'3 .0
. O
L9 .0 .0 .0 .0 .'3 0 0 0 0 0 0 0 0 0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 . ,:3
. C'
.5
2.6 ~
dl 2°
. C'
.2
. :
.5
.5
.0
.0
.£,
.0
.0
.0
. ,:',
.5
.0 -.5
.0
.0
.0
2.67
dl 3°
. ~
.~
. ~
.5
.0
.~
.2
.,C'
.,J
.0
. ,1:'
.O
. '3'
.0
. ,0
dl 2
2.6 ~
dl 3
FUNCTION
.0
.0
.0
.,3
. ,3
.0
.E
. 'J
.£'
.u
0
.0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 .0 .0 .0 .0 .0 .0 .0 .0
dl 3
.0
0 0 0 0 0 0 0 0 0
dl
• u
2.6 ~ - 2 . 6 7
dl I
. ,;,
.0
.0
- . 5
.0
.O
.0
.0
. ,2'
.C, .0 . ,%,
.0
.0
.0
.0
.0 .0 .0 .5 .0 .0 .0 .0 1.0 .0 .0 .0 .0 .0 .0 .0 .0
.0 .5 .0 .0 .0 .0 .0 0 0 0 0 0 0 0 0 0 .0 .0
dll .0
dl 3° .0
OBJECTIVE
. )
.~
.~
.;,
., . 7,
.~
• E
. , :'
. '}'
. ~ . ~
• 3
<
. t
.0
• 0
. 2
. ,)
.5 .E
.0
.0
.0 .g
.0
. ,,3
0
.0 .5 .5 .3 .5 .0 1.0 .0 .0 .0 .0 .0 .C, .0 .0 .,)
.0
.0 .0 .0 .0 .0 .0
0 0 0 5 5
dl ~° d] 2° 0 .0
.0
2.6 ?
d[ 4
.0
-.5
.0
.0
.0
.0
. ,3'
.0
0
. 5
.0
.0
.0
0
0 0 0 0 0 0 0
d14
Fig. 9. Set of inequalities and objective function.
2.6-
dl I'
.
.
,.,,
LI2 .0 .0 .0 .0 .0 1.0 .0 .C' .0 .0 .0 .£ .0 .0 .0 .C .C .0
2.67
dl 5
. 5
.0 .;
.0
.O
.0
.0
.0
. 0
.5
. 0
. ':3
.0
.0
.0 .0 .5 .0 .C' .0 .0 .0 .0 .0 .0 .0 .0 1.0 .0 .0 .0 .0 .0 P'
dl 5
. 5
2.6 ~
Jl 6
. ,',,
." .'
. f.,
.C
.(
. ,{
.r
.=
. 5
. C:
.C'
.C
.0 .5 ,5 .5 .C, .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 .0 .0 .0 .C' ¢,
dl 6
.0
.0
. ,2; .,3 .,2,
2.E ~
dl-
. ']
0 .'J ~,
,3
.
.E
3
.'$
. 0
.) .0 .0
.
. '3'
.':3
. 0
.0
.0
.0
.0 .0 .0 .0 .'0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 l.O .0 .0
dl 9
2.67
dl~
.£
0 .0
- . ~_
.(:'
.0
. '3
.5
.O
. ~
0
dl~ .5 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 1.0 .n .0 .C'
.
. ~
.0
. ~
. '~'
. ~
.n
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
dl 7
• ?'
•
. '
. ~ .,2 .'7
.~
.0
.
2.67
dl 9
.
.0
. '2
. "
.fi
.C:'
.5 .0 .0 .5 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .C, .0 1.0 .C
.
dllo
d111
.
-2.6
dllo
.£
. ~
. [ .2 .=
.')
.7
]'
• <
.E
. ~
. '$
.,:)
.0
.0 .0 .5 .0 .q .5 .0 .C .0 .0 .0 .0 .0 .0 .C, . '[:' .0 .0 .C' 1.o
dl12
.
'
A
]
)
-2.1
cond
. 820
>-
231.~5
A
.231
->. '"3' 0
2£ . ,30> 1 . 0 0 r', ->l .OOC'
>1.6'00
u...... uu ->11 . 5C'C' > ~
_> -
->12.600 >-2 . 6 1 2 ->1 1 . O 0 C' ->2 . OOC'
<.'301
-<14.000 -<20.000 <9.000 -<21.840 -<2.500 <2.500 -<.001 -<.001 -<.001 -<.001 -<.001 -<.001 <-.001 -<.001 -<.001 -<. 001 -<.001 <.001 -<.001 -<.001
2.67
dll2
- 2 . ;
i .7 1~.5 :.c
l.v
1.7'
.]
2
I .7; 1 . '2' 3 . 2' 2 .3
3
1.0 1.0 3.0 2.0 14.5 5.5 0 0 0 0 0 0 0 0 0 0 0 0 '3 '3'
2.6 ~
Jill
. '-
.
. .~
.-:
.
.
. 7
. '
[
1.
.0 .0 .0 .0 .5 .0 .0 .3 .0 .0 .0 .0 .0 .( _3 . :[:' .3' _? .0 .(
e~
R
t
32
Applications
Computers in Industr)
Each machining tolerance can be calculated with these values: Cfijmi n
=
1
Oefinilion drawing
L j -- L i -- I ( A I i 4- a l j ) ,
2
"
Cfijma
x =
Lj - L i + l(Al i +
Alj).
The results are correct and can be used in production. For this example, we carried out two experiments: (i) The raw material cost is very important for the length of the block of raw material: with zero cost: Cr, w = 47.621 + 0.517; with cost 0 . 5 F / m m : Craw = 44.029 + 0.517; with cost 1 F / m m : Cra w = 43.212 _+ 0.517.
SIMULATION MODEL (Cf)
IZ
PH2£
41,465 33,428
~
_
30±0,2
[~
22,057_. I
II II
I 40,138
I
~
1,6mini
L_ £%118_ 1;oL --
=>residue0,36 onCf2,3
36,150 438,046
3,908
qFql
25±0,02
41 462 +1,084
t~
6_
40,138±0,092 38,138maxi ~l >1. 2mini 24,261mini I
•
30±0,02 . .
Cc 1,12 = 43,213_+0,518
71
_IL
.
,314=0,02
~,=,~
VERIFICATION D I M E N S I O N S (Cc)
NC DATA (position on Z axis)
41,462+0,232
j /
.
L3.,o I
(ii) The tooling cost has no influence, because the adjustment cost defines small tolerance fields and there is no more freedom for the position of the surfaces. In other cases, a normal tooling cost gives a better solution but the results is quite insensitive to the value of the cost, which can be very inaccurate.
Cf 1,12 = 43~213+0~518
#H10
~~ ) ,M330.02
Fig. 10. Non-independent adjustment.
4. 4. Analysis of results
RAW
AI2=0,02
~1:o t---- J--~o- ~
Machrnlo nns g ~ 20±0,02 dpmensi Extendedmachining dimensions
Results are presented in Table 4.
3 25_+0,02
~ " All 30= 0 02
~,MoOe,
30±0,2
20±0,02
l
" ll~r
2~
_ _IZI ~_Ll~_o_l~_
_
,~.~o,2
~.933 re,o(-
;.I
iq[
~'1
35,433
~H30 ___.
4
35,433d:0,5
, ~,,69
~
"1
11 m m ~
--
_
Fig. 11.
11
Computersin Industry 5. Using
the
results
5.1. Feasibility of the manufacturing process A system w i t h o u t solution m e a n s t h a t the constraints are i n c o m p a t i b l e . T h e definition d r a w i n g a n d the m a n u f a c t u r i n g p r o c e s s m u s t b e b e t t e r analysed. In the o p p o s i t e case, we can s e a r c h for m a c h i n e - t o o l s which have t h e a d e q u a t e m a c h i n ing capability [18].
5.2. Updating of the model T h e m o d e l gives the L i values which define the a v e r a g e p o s i t i o n of the surfaces. T h e workp i e c e m o d e l m u s t b e a c t u a l i z e d to g e n e r a t e the tool p a t h a n d the N C d a t a with the C A D / C A M system (see Fig. 11, c o l u m n 2).
5.3. Extension of machining dimensions 5.3.1. Problem T h e analysis of the m a c h i n i n g d i m e n s i o n s Cf in T a b l e 3 shows that t h e r e a r e c o n d i t i o n s which a r e fulfilled within t h e r e q u i r e d margin, leaving a residue. This r e s i d u e m a y b e u s e d for t h e adjustments. F i g u r e 10 gives an e x a m p l e . T h e exact c o n s t r a i n t for the m i d d l e of the w o r k p i e c e is 30 _+ 0.2, b u t t h e Al m o d e l , b a s e d on t h e i n d e p e n d a n c e o f surfaces, i m p o s e d Cf23 = 30 _+ 0.02. O n l y Cf12 and Cf34 a r e c o n s t r a i n e d by the c o n d i t i o n 20 + 0.02 a n d 25 _+ 0.02. T h e machining d i m e n s i o n Cf23 has a r e s i d u e of 0.36. So, Cf23 = 30 +_ 0.4 again.
B. Anselmetti, P. Bourdet / Workpieceoptimization
33
Table 5 Extended machining dimensions (see Fig. 1l, column 1) 1 2 3 4 5 6 7 8 9 10 11 12 13 14
RAW RAW PH 10 PH 10 PH 10 PH 10 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20 PH 20
15 16
PH 20 PH 20
17 18 19 20 21 22 23
PH 20 PH 20 PH 20 PH 30 PH 30 PH 30 PH 30
Cf(1, 12)M = 43.730 Cf(1, 12)m = 42.695 Cf(2, 6)M = 16.441 Cf(2, 6)m = 16.298 Cf(2, 12)M = 41.695 Cf(2, 12)m = 41.230 Cf(2, 3)m = 3.612 Cf(2, 8)m = 24.261 Cf(2, 10)M = 38.138 Cf(2, ll)M = 40.230 Cf(2, ll)m = 40.046 Cf(3, ll)m = 35.933 Cf(7, 8)m = 2.001t Cf(7, ll)M = 21.375 * (18.376 condition 19) Cf(8, 9)m = 11.500 Cf(8, 10)M = 14.000 * (13.784 condition 2) Cf(8, 10)m = 13.600 Cf(9, 10)m = 1.600 Cf(10, ll)m = 2.001/ Cf(4, ll)M = 35.933 Cf(4, ll)m = 34.933 Cf(5, ll)M = 32.604 Cf(5, ll)m = 28.6116 * 131.61)4 conditions 5 and 19)
* Extended tolerances
5.3.3. Operations phases restrictions This d o c u m e n t gives all i n f o r m a t i o n for the m a c h i n i n g (clamping, process, cutting c o n d i t i o n ) a n d the m a c h i n i n g d i m e n s i o n s a n d tolerances. So, a m a c h i n e tool is well a d j u s t e d if all m a c h i n ing w o r k p i e c e s r e s p e c t the m a c h i n i n g t o l e r a n c e s .
5.4. Extension of the method
5.3.2. Calculation of extended machining dimensions
5.4.1. Adjustment, cerification and statistical process control
T h e analysis of T a b l e 3 d e t e r m i n e s the r e s i d u e for e a c h condition. W h e n the r e s i d u e is zero, all m a c h i n i n g d i m e n s i o n s which d e p e n d on this condition a r e c o n s t r a i n e d . N o n - z e r o r e s i d u e s m a y be d i s t r i b u t e d on t h e u n c o n s t r a i n e d m a c h i n i n g dimensions, b e g i n n i n g with t h e smallest increase. R e s u l t s a r e shown in T a b l e 5. T h e m a c h i n i n g t o l e r a n c e s which w e r e e x t e n d e d are m a r k e d by an asterisk ( * ) (see Fig. 11, c o l u m n 1). T h e extension of t h e m a c h i n i n g d i m e n s i o n s facilitates t h e a d j u s t m e n t a n d can r e d u c e any verification t o l e r a n c e s in o t h e r phases. F o r this r e a s o n , the extension m u s t be a choice for the planner.
T h e result of the s i m u l a t i o n gives a m o d e l o f the b e h a v i o u r of m a n u f a c t u r i n g a n d the tolerances for each phase. This i n f o r m a t i o n m a y be u s e d to d e f i n e the a d j u s t m e n t t o l e r a n c e s with d i f f e r e n t a p p r o a c h e s [19] (see Fig. 11, c o l u m n 3): - a d j u s t m e n t with i n d e p e n d e n t tools; - c o n t i n u o u s c o n t r o l o r s a m p l i n g control; - analysis o f tool wearing. T h e b o u n d a r y of S P C (statistical p r o c e s s control) m a y b e d e f i n e d [18].
5.4.2. Numerically controlled machine-tool This m e t h o d can i n t e g r a t e specific p r o c e s s e s in N C machining, for e x a m p l e to i n c o r p o r a t e the
34
Applications
two-dimensional [14,19].
( bmputer~ in lndust~'
cases (cones, chambers
etc.)
5.4.3. Statistical method Machining variations come from adjustments (constant for a batch, but variable between two batches) and random dispersions. Only the random variations can be added with statistical methods. In the opposite hypothesis [20] it would be possible that all combinations of good workpieces given by the same production batch, cannot be correctly assembled. This is not acceptable. This is why we propose the following solution: AI = Ar + Aa, where AI is the field of tolerance, AI is the adjustment field, Aa is the dispersion. To use the statistical method, it is necessary to know the spread of each dispersion. The inequality is now: Lj
_ L i + ½ ( A r i + A r y ) <~ COmax _ 21~ / A a i2 + Aa~ .
6. Conclusion The proposed calculation method has given logical results for many real examples. The economic consequences are important because, with this approach, designers can draw up minimal tender specifications which allow the widest tolerances for manufacturing. This analysis updates the geometric model of the workpiece which can be used directly by NC machining and measuring equipment. To employ this technique efficiently, it will now be integrated into our computer-aided process planning system. The calculation method proposed has given consistent results for many real examples.
References [1] P. Clozel, "M6ca Master: outil de conception m6canique et de cotation 3D pour les bureaux d'6tudes", Strucome, 1990. [2] A. Clement, A. Desrochers and A. Rivi~re, "Theory and practice of 3D tolerancing", CIRP Int. Work. Seminar on Computer-aided Tolerancing, Pennsylvania State University, 1991.
[3] J. Stinson and P. Urbaniak, "Calcul et optimisation des tol6rances", Actes MICAD 92, Paris, 1992. [4] P.T. Jessup, "The value of continuing improvement". Int. Communications Conf. ICC 85. June 1985. [5] "Initiation h l'engineering de la qualitC', Institut des methodes Taguchi, May 1989. [6] J. R6my-Vincent and F. Schneider, " A p r o p o s de la cotation fonctionnelle", M6moire bibliographique, DEA de Production Automatis6e, Universit6 de Nancy 1, 1990. [7] P. Bourdet, "Chaines de cotes de fabrication: le module", L, ingdnieur et le technieien de l'enseignement technique, December 1973. [8] P. Bourdet, J. R6my-Vincent and F. Schneider, "Tolerance analysis in manufacturing", CIRP Int. Work. Seminar on Computer-aided Tolerancing, Pennsylvania State University, 1991. [9] D. Fainguelernt, R. Weill and P. Bourdet, "Computer aided tolerancing and dimensioning in process planning", Ann. CIRP, Vol. 35, No. 1, 1986. [10] D. Duret, Cotation et M~thodes de Contr61e en Frabrication Mdcanique, Collection Introduction a I'art de l'ing6nieur, Editions Augustin. [11] E.A. Lehtihet, N.U. Gunasena and I. Ham, "Element of computer-aided tolerance analysis", Int. Conf. on C A D / CAM and AMT, CIRP Sessions, Jerusalem, December 1989. [12] K. Panshal, S. Raman and P.S. Pulat, "Computer-aided tolerance assignment procedure (CATAP) for design dimensioning", Int. J. Prod. Res., Vol., 30, No. 3, 1992. [13] F. Schneider and J. R6my-Vincent, "D6finition des ensembles m6caniques: Condition d'obtention de cotations optimales", 23rd Int. Seminar on Production Systems, Nancy, June 1991. [t4] B. Anselmetti, "Simulation d'usinage bidimensionnelle sur un exemple en tournage en commande num6rique", Rev. Mec. Mater. Electr., No. 398, March-April 1983. [15] C. Maria, La qualitd des produits industriels, Edition Dunod, 1991. [16] P. Bourdet, "Cha]nes de cotes de fabrication: le mode op6ratoire", L'ingdnieur et le technicien de l'enseignement technique, December 1973. [17] A. Cheikh and P. McGoldrick, "Functional tolerancing in CAD/CAM", Int. Conf. on C A D ~ C A M and AMT, CIRP Sessions, Jerusalem, December 1989. [18] M. Vigier, Pratique de la maitrise des procFdds-MSP, Let 6ditions d'organisation, 1989. [19] B. Anselmetti and V. Anselmeni, "La fabrication, une d6marche globale", Rev. Technol. Formation, Nos. 27, 29, 30, 32, 1989-1990. [20] E.A. Lehtihet, N.U. Gunasena and I. Ham, "An update --Statistical tolerance control methods and computations", CIRP Int. Work. Seminar on Computer-aided Tolerancing, Pennsylvania State University, 1991. [21] P. Bourdet, "Introduction ~ la conception automatique de gammes d'usinage", La gamme automatique en usinage, Edition Hermes, 1990. [22] P. Souvay, La statistique, outil de la quaKtd, Edition Afnor Gestion, 1986.