Rational method for 3D manufacturing tolerancing synthesis based on the TTRS approach “R3DMTSyn”

Computers in Industry 62 (2011) 541–554

Rational method for 3D manufacturing tolerancing synthesis based on theTTRS approach ‘‘R3DMTSyn’’

Karim Jaballi a,b,*, Alain Bellacicco b, Jamel Louati a, Alain Riviere b, Mohamed Haddar a

a Unit of Mechanics, Modelling and Manufacturing, National Engineering School of Sfax, Department of Mechanical Engineering, University of the South, Sfax BP 1173, Tunisiab Laboratoire d’Ingenierie des Structures Mecaniques et des Materiaux, Institut Superieur de Mecanique de Paris (SUPMECA), 3 rue Fernand Hainaut, 93407 SAINT OUEN Cedex,

Paris, France

A R T I C L E I N F O

Article history:

Received 27 April 2009

Received in revised form 17 November 2010

Accepted 11 February 2011

Available online 10 March 2011

Keywords:

TTRS

MGDE

Functional specifications

Manufacturing specifications

SPIDER GRAPH

Tolerance transfer

Relative positioning parameters

A B S T R A C T

During the production of new part in an industrial environment, it results in a high percentage of scrap if

manufacturing planning is not carried out properly. One of the major factors responsible for this

phenomenon is tolerance synthesis. In this paper, we deal with tolerance synthesis, and especially

tolerance type identified after transfer. An algorithmic Rational method for 3D Manufacturing

Tolerancing Synthesis (R3DMTSyn), which is based on the use of the Technologically and Topologically

Related Surface (TTRS) rules, is developed. The TTRS concept helps to generate only the necessary

manufacturing specification needed to guarantee the respect of the functional specification studied.

The manufacturing project is modeled by a graphical representation called the SPIDER GRAPH. With

the SPIDER GRAPH, all active surfaces can be detected (machined and positioning surfaces), so it is

possible to identify the location of the functional surfaces used in each functional specification. The

construction or the determination of the tolerancing torsor, belonging to each active surface, contributes

to the selection of the adequate case of associations. A semantic study is already done to identify all

possible combinations or associations needed to locate surfaces during each phases.

Finally, referring to the developed TTRS_Cars_Process and in every phase, one or more manufacturing

specifications are generated until finishing the treatment of all surfaces (surface belonging to the

functional condition, or intermediate surfaces).

� 2011 Elsevier B.V. All rights reserved.

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1. Introduction

Geometric Dimensioning and Tolerancing (GD&T) as defined bythe standard, ISO 1101 [1] or ASME Y14.5.1M-1994 [2] and ASMEY14.5M-1994 [3], provides many unique and beneficial concepts indefining part tolerances. The GD&T System allows the designer tospecify three-dimensional tolerance zones for locating. Most of thecompanies are looking of using the Geometric Product Specifica-tion standards to improve the quality of their products.

The tolerance specifications assigned on a component governthe variability in the functional performance of the component, theaccuracy and the effort required by the various manufacturingprocesses to manufacture the component. Thus, tolerancesinfluence a number of factors such as design function, manufactur-ing conditions, component assembly, and serviceability. Nowa-days, a concurrent engineering is the only alternative for manycompanies to guarantee its competitiveness on the market [4].

* Corresponding author at: Department of Mechanical Engineering, National

Engineering School of Sfax, University of the South, Sfax BP 1173, Tunisia.

Tel.: +216 21 29 37 69.

E-mail addresses: [email protected], [email protected] (K. Jaballi).

0166-3615/$ – see front matter � 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.compind.2011.02.003

2. Literature

2.1. 1D methods

Until now the unidirectional (1D) methods are used in theindustries, the most popular is the DL method developed byAnselmetti Bourdet [5]. With the DL method, each functionalrequirement is decomposed into machining dimensions andtolerance. The graphical representation adopted to model themachined and positioning surfaces in the same graph facilitates theworks. As we indicated, the treatment is done along one direction.

Ping developed [6–8] the digraphic method and the reversedimensional chains. The key concepts in the digraphic method arethe three directed trees, the reverse and forward dimensionalchains. Using these trees, a reverse dimensional chain is introducedas follows: if an arc (an X) in the X tree is ‘added’ to the Y tree, thena unique loop can be obtained, being called a reverse dimensionalchain. This reverse dimensional chain can also be represented as alinear mathematical program. Those equations are represented in amatrix format. Whatever the size or datum surface change of adesign dimension in a product is, all manufacturing dimensionscan respond.

Notation

CAS 6-6_PRIS_PL (i) 6: 6th class of the TTRS type: planar

surface; (ii) 6–6: intersection of the 6th column and

the 6th line of Table 4; (iii) PRIS: the combination of

the two planar surfaces gives a prismatic associa-

tion; (iv) PL: the nature of the tolerance zone

CAS 6-6_PLA_PL ‘‘same item 1, ii and iv’’ and PLA: the

combination of the two planar surfaces gives a

planar association

MGDE Minimum Geometrical Reference Element

IT value of the tolerance zone

nap number of angular relative positioning parameters

SP1 primary positioning surface

SP2 secondary positioning surface

SP3 tertiary positioning surface

SRi reference surfaces belonging to the functional

specification

SR reference surface

STi toleranced surfaces belonging to the functional

specification

ST toleranced surfaces generated after transfer

SUd last machined surface

SUdT the last machined surface and taken as toleranced

TT tolerance torsor

TTRSt toleranced feature

TTRSR1 primary reference feature

TTRSR2 secondary reference feature

TTRSR3 tertiary reference feature

TTRS_CBMj manufacturing specification developed with

the respect of the TTRS rules, in the phase j

TTRS_MEPj TTRS association formed by the positioning

surfaces in the phase j

TZ Tolerance zone

X, Y, Z the local coordinate system

U, V, W the local basis

u, v, w displacements in the local basis

a, b, g rotations in the local basis

Class elementary surface

Subscripts

T, t related to the toleranced surface (target surface)

R related to the reference surface

1, 2 and 3 order of surface

j related to the order of machined phase

K. Jaballi et al. / Computers in Industry 62 (2011) 541–554542

Using the 1D method contributes to the development oftolerance chain which can be translated into dimensionalspecifications or simple position specification. This type ofspecification does not take into account the orientation of thepart or workpiece, and cannot be used to solve a specificationwhich presents a Datum Reference Frame (DRF) formed by morethan one surface.

2.2. 2D and 3D methods

Bourdet and Ballot [9] used Small Displacement Torsor (SDT) tomodel the geometric behavior of a mechanism for Computer AidedTolerancing (CAT). With this concept it is possible to simulate amachined phase like an intermediate situation or machined phase.

The SDT concept is used by many other authors in CADresearches. Ballot [10] and Thibaut [11] modeled the interactionbetween the parts of a mechanism in order to predict the variationsof position and orientation of these parts in a three-dimensional(3D) space.

This concept was extended to model the machined phases.Tichadou [12,13] formalized the geometrical defect of the entiremachine with the development of the 3D manufacturing chain,compared its result with the DL and affirmed that with the SDT theorientation effect is taken into account in his studies.

The Manufactured Machined Part (MMP) model developed byVignat [14,15] is based on the generation of a virtual manufacturedmodel. The deviations of the manufactured surfaces of the MMPare expressed in comparison with the nominal part, using thedefinition proposed by Bourdet and Thibaut. The MMP model iscombined with the virtual gauge and a signed GapGP measuringthe position of the MMP surface relative to the tolerance zonebounding the surfaces. This combination allows the verification ofthe functional specification. The verification of this combination ismade by a generic formula difficult to optimise:

MinCM;CH;CHP

DM;DH;LHPðMax

CGP

LGPðMin

jðGa pGP jÞÞÞ (1)

Le pivert [16] extended the SDT to an incertitude tensor tosimulate the manufacturing operation and to predict the cumula-tive defects.

Transfer of the tolerance zone (TZT) is developed by Anselmetti[17], in this method the degrees of freedom forbidden by thepositioning surfaces and those which let the machined surfaceinvariant, contributed to the choice of the manufacturing specifica-tion of the studied surface. This method can be considered as anextension of the DL due to the graphical representation adopted tomodelize the manufacturing project (phases) on 3 directions.

Jaballi developed [18] a 3D manufacturing tolerancing methodbased in CLIC rules [18,20] in which he considers a workpiecerelated to its holder as a mechanism. It is important to indicate thatCLIC approach is firstly developed to generate functional require-ment taking into account the initial functional gap [19].

Both, Anselmetti in [17] and Jaballi in [18] inspired from CLICapproach to generate the different steps to develop their approach.

2.3. TTRS approach

Tolerance specification is the activity of specifying tolerances,defining the tolerance types and related tolerance value. In order tocomputerize the tolerance representation, a lot of theoretically andmathematically possibilities are proposed in literature but they donot sufficiently comply with the international standards. With theTTRS approach, the tolerance specification method is carried out onthe basis of face associations between the different components inthe assembly either in CAD or in CAM activities.

Desrochers in [21,22] defined the notion of surfaces associa-tions in the following way:

� The concept is based on the notion of surfaces.� The association process is binary.� The association process is recursive.� Associations are guided by the kinematics loops in a mechanism.

Clement confirmed in [23] that the TTRS combinations areformed by even elementary surface and associated elementarysurface. Seven classes of TTRS surface were counted, and twenty-eight associations deduced of forty-four reclassifications TTRScombinations, but only six classes are kept in this study. The helicalclass of TTRS was not treated; it was taken as a cylindrical TTRS case.

Table 1Intrinsic characteristics of the simple and combined entities.

Association Cas MGDE elements Elementary surfaces Geometrical constraints to control Nature of the specification

Rotations Translations

Cas A-1 PT Spherical [0, 0, 0]T ½ui; vi;wi�T and or

Cas A-2 SL Cylindrical [0, bi, gi]T ½0; vi;wi�T and or

Cas A-3 PL Planar surface [ai, bi, 0]T ½0;0;wi�T or __

Cas A-4 PT[ SL Helical Non studied case Non studied case Non studied case

Cas A-5 PT[ SL Revolution [0, bi, gi]T ½ui; vi;wi�T or or

Cas A-6 SL[PL Prismatic [ai, bi, gi]T ½0; vi;wi�T or or __

Cas A-7 PT[ SL[PL Complete [ai, bi, gi]T ½ui; vi;wi�T or

K. Jaballi et al. / Computers in Industry 62 (2011) 541–554 543

In order to represent the real surfaces or the TTRS of the studiedsurface or combined surfaces, a theoretical reference calledMinimum geometric datum elements (MGDE) is used. It is easyto locate the MGDE than to locate the TTRS. The MGDE model thedegrees of invariance associated to a surface or TTRS. Every class ofTTRS has their intrinsic characteristics summarized in Table 1.

TTRS approach is the nearest representation to the standard ISOand ASME [24]. The various TTRS cards representation developedby Clement in [23] modeled all possibilities to locate two TTRS(simple or combined). Thus, the TTRS approach is a powerful toolbased on the mathematical definition of three-dimensional zonesestablished by geometrical tolerances. It is based on the hypothesisthat the position of real elements can be represented by means ofsmall ‘‘rigid body’’ displacements of the nominal ones [25].

In the following works, some modifications to those cards aremade, so only logical combination that can be used in manufactur-ing operation are kept. Those modifications took into account thetype of TZ that characterize each machined surface.

3. Rational method for 3D manufacturing tolerancingsynthesis based on the TTRS approach

3.1. Starting concepts

Before the development of the ‘‘R3DMTSyn’’ method, someconcepts used throughout the treatment of the functionalspecifications must be clarified. The most used reference datumconfigurations are gathered in Tables 2 and 3 and to eachconfiguration, the Class, the MGDE, and the TTRS were associated.

The natures of the Degrees Of Freedom (DOF) define clearly thenumber and the type of the positioning parameters of two TTRS.Gaunet in [26,27] gathered in Table 4 the number of angular andlinear positioning parameters between two TTRS. In this work,some modifications are introduced to the arrangement ofinformation gathered in Table 4 to make it suitable to themanufacturing surfaces associations.

In this work, the different formula developed by Gaunet will bethe basic rules to optimize the different manufacturing specifica-tion DRF, generated in each phase. (Those formulas will bementioned later.)

3.2. Steps to generate the manufacturing specifications

In the following sub-section, a general description will bepresented to describe how the R3DMTsyn allow rational genera-tion of geometrical specification according to the standard. In orderto clarify the different algorithms and rules, an example forvalidation will be presented in Section 4.

3.2.1. Intrinsic characteristics

The intrinsic characteristic are directly recopied on thedrawings of phase in which surfaces are carried out. There is notransfer in those cases [18]. Form tolerances are applicable tosingle (individual) features or elements of single features;therefore, form tolerances are not related to datum [2,3].

Example: Planar geometrical specification applied to thesurface number 2 shown in Fig. 4.

3.2.2. Step 1

In order to facilitate the treatment, the R3DMTSyn is also basedon a graphical representation called SPIDER GRAPH [18]. TheSPIDER GRAPH, shown in Fig. 6, describes the manufacturingprocess. This graphical representation takes into account thethree-dimensional aspect of all phases. Using the SPIDER GRAPH,the machined surfaces, the positioning surfaces, the tolerancedsurface and the reference surfaces belonging to the functionalspecification can be identified in location and characteristics (TZ,MGDE, and TT). The Section 4 shows how to draw a SPIDER GRAPHand how to use it.

3.2.3. Step 2

The second step verifies if there is a tolerance transfer of thestudied functional specification to generate the manufacturingspecification. The general algorithm shown in Fig. 1 traces the mainline of the ‘‘R3DMTSyn’’.

� Choose the functional specification and draw itsTTRS_GRAPH_CBE.� Draw the manufacturing project using the SPIDER GRAPH.� Identify the phase where the surfaces belonging to the functional

specification are manufactured.

The TEST 1 is considered as the pivot between two major Zones:

(1) If the TEST 1 = YesThe second zone is selected. In this zone, the manufacturingspecification is the same as the functional specificationstudied. However, the only difference is the value of thetolerance zone: ‘‘IT’’.

(2) But if the TEST 1 = NoThe first zone is selected. This zone permits the treatment ofan eventual transfer of the functional specification.

It is important to indicate that several links exist between thosetwo zones in order to treat all possible configurations of thesurfaces involved in the respect of the functional requirement.

Table 2Elementary elements.

MGDE TTRS associated Elements

PT R(3) Spherical class = 3

Z

Y

X

O

Center O

SL T(1) X R(1) Cylindrical class = 4 Axis of the cylinder

Z

Y

X

Simple « Axis D »

Z

Y

X

Coaxial « common axis D »

PL T(2) X R(1) Planar surface class = 3 Normal to the planar surface

Z

Y

X

« normal to the planar surface »

Z

Y

X

Planar surfaces shifted vertically «normal of one of the two planar

surfaces »

Z

Y

X

Coplanar planar surfaces «normal of one of the two planar surfaces »

K. Jaballi et al. / Computers in Industry 62 (2011) 541–554544

3.2.4. Step 3

If the answer to TEST1 = No, the treatment of the specification isexecuted by the zone 1 rules. This zone is modeled by thealgorithm shown in Fig. 2.

This algorithm is divided into three parts. The two first parts aredescribed in this third step and the last one is in the next fourthstep.

In the first part, all generated surfaces (intermediate surfaces),during the treatment of the functional surface, are accumulated. Itcan be either a toleranced or reference surface (STi, SRi, ST and SR).

With the second part of the algorithm, all manufacturingspecifications in each phase, without optimizing (reducing the

number of reference surfaces) the DRF, are generated. Thus, thefollowing stages describe how to proceed:(1) Select the last machined surface belonging to the functional

specification (all the surfaces were identified in the secondstep): SUd = ? The functional geometrical specification iscomposed by target surface, called also toleranced surface,and DRF. According to the ascendant nature of the R3DMTSyn,it is necessary to start with the last machined surface amongthose that constitute the functional requirement.

(2) Identify the MGDE of the last machined surface: MGDE(SUd) = ? The MGDE represent the invariants DOF thatcharacterize the studied surface.

Table 3Combined elements.

MGDE TTRS associated Elements

PT[ SL R(1) Revolution class = 5 Axis of revolution entity

Z

Y

X

Cylinder perpendicular to a planar surface « Axis D »

Z

Y

X

Axis of the cone D

SL[PL T(1) Prismatic class = 5

Z

Y

X

planar surface parallel to the line of prismatic surface

Two prismatic parallel surfaces «one of them can characterize the prismatic association »

PT[ SL[PL Complex class = 6

vw

u

Z

Y

X

α β

γ

Identity

K. Jaballi et al. / Computers in Industry 62 (2011) 541–554 545

(3) Build the Tolerance Torsor (TT) of SUd: the TT is the vectorialrepresentation of the MGDE.

(4) Identify the nature of tolerance zone (TZ) of the selectedsurface: according to the invariants DOFs nature, it is possibleto select the nature of the TZ (planar or cylindrical or spherical).

(5) Select the last phase where we found the SUd: Phasej.� Identification of the positioning system in this jth phase: the

R3DMTSyn is an ascendant method in which it is necessary tostart from the last machined surface. So, firstly, it is necessaryto study the target surface and its positioning surfaces (SP1,

Table 4Number of angular and linear positioning parameters during a manufacturing phase.

E

111 ), P, D(OTD1

11 ), P(D

RD1

11 ), D(OCD1

1)(D

GP1

1)(P

S O1

1)(O

E

222 ), P, D(O

6{3, 3} 5{3, 2} 5{2, 3} 4{2, 2} 3{2, 1} 3{0, 3}

TD2

22 ), P(D

D1==D2!5f3;2gelse!4f3;1g NO

D1==D2!4f2;2gelse!3f2;1g

D1==P1!3f2;1gelse!2f2;0g 2{0, 2}

RD2

22 ), D(O

D1 ¼ D2!4f2;2gD1==D2^D1 6¼D2!4f2;2gelse!4f1;3g

D1 ¼ D2!4f2;2gD1==D2^D1 6¼D2!3f2;1gelse!3f1;2g

D2? P1!3f2;1gelse!2f1;1g

O12D2!3f0;3gelse!2f0;2g

CD2

2)(D

D1 ¼ D2!4f2;2gD1==D2^D1 6¼D2!3f2;1gelse!2f1;1g

NOO12D2!2f0;2gelse!1f0;1g

GP2

2 )(P

D1? P1!2f2; 0gD2==P1!2f1;1gelse!1f1;0g

P1==P2!3f2;1gelse!1f1;0g 1{0, 1}

SO2

2)(O

NO NO O1 = O2!3{0, 3}

K. Jaballi et al. / Computers in Industry 62 (2011) 541–554546

SP2 and SP3). The equations of positioning system identifi-cation are:

ðTTRSðSP1Þ [TTRSðSP2ÞÞ [TTRSðSP3ÞÞ ¼ ? (2.a)

ðMGDEðSP1Þ [MGDEðSP2ÞÞ [MGDEðSP3ÞÞ ¼ ? (2.b)

� Validation of the manufacturing DRF formed by thepositioning surfaces using the following equation systemdeveloped by Gaunet in [27]: the DRF will be consistent (notover defined) when the rules modeled by the equations ofmanufacturing DRF validation ((3.a) and (3.b)) are satisfied.

ClassðTTRStÞ<ClassðTTRSR1 [TTRSR2Þ (3.a)

ClassðTTRSR1 [TTRSR2Þ<ClassððTTRSR1 [TTRSR2Þ [TTRSR3Þ (3.b)

In Tables 2 and 3 each TTRS class is mentioned below the TTRSnature.

By the end of this third step, manufacturing specifications witha consistent and complete DRF are developed.

The next step deals with the DRF optimizing.

3.2.5. Step 4

Before starting the third part of the algorithm shown in Fig. 3,the tolerance type must be identified, whether it is a position ororientation specification. This identification is possible by referringto the tolerance torsor ‘‘TT’’ form. In each TT, the nature ofparameters and its number contribute to identifying what sub-case can be suitable to the situation.

Example: the TT of a tolerance zone formed by the shift of aplanar surface according to its normal shown in Table 5.

The manufacturing specification DRF optimizing consists on thereduction of the number of the reference surfaces. In order toclarify the treatment of this forth step, an intermediate algorithmshown in Fig. 3 is made.

To be sure that the DRF is not over-defined, the followingequation developed by Gaunet [27] must be verified:

� For the position specification, the over-defined position specifi-cation test is:

ClassðTTRSt [ TTRSR1Þ<ClassðTTRSt [ ðTTRSR1 [TTRSR2ÞÞ (4.a)

ClassðTTRSt [ ðTTRSR1 [ TTRSR2ÞÞ<ClassðTTRSt [ððTTRSR1 [ TTRSR2Þ [ TTRSR3ÞÞ (4.b)

Fig. 1. General algorithm.

K. Jaballi et al. / Computers in Industry 62 (2011) 541–554 547

� For the orientation specification, the over-defined orientationspecification test is:

napðTTRSt [TTRSR1Þ<napðTTRSt [ ðTTRSR1 [TTRSR2ÞÞ (5.a)

napðTTRSt [ ðTTRSR1 [ TTRSR2ÞÞ<napðTTRSt [ððTTRSR1 [TTRSR2Þ [ TTRSR3ÞÞ (5.b)

Note: number of angular relative positioning parameters (nap).

The algorithm represented by Fig. 3 draws the different testswhich contributes to the selection of only necessary

manufacturing reference surfaces, involved in the respect of thefunctional requirement.

In a manufactured process, the nature of the TZ plays asignificant role in the choice of the DRF. Thus, each TZconfiguration must be clearly specified. So, in order to put theright dimension in the drawing process, the appropriate TTRS_Pro-cess_Card has to be selected.

Using TTRS associations, this work present an exhaustive studyin which all possible TTRS manufacturing associations are gatheredin Table 4.

Clement in [23] presented the twenty-eight configurationcorresponding to the forty-four re-classifications cases. To those

Fig. 2. Algorithm of transfer.

K. Jaballi et al. / Computers in Industry 62 (2011) 541–554548

Fig. 3. Algorithm of optimization.

Table 5Example of toleranced planar surface characteristics: TZ, MGDE, tolerance torsor.

Tolerance zone ‘‘TZ’’ MGDE Tolerance torsor ‘‘TT’’

Volume obtained by shift of a plan

Planar P

t

x→

y→

z→

γβ

Pu

u000bg

26666664

37777775ðO;~X;~Y ;~ZÞ

M2 P

K. Jaballi et al. / Computers in Industry 62 (2011) 541–554 549

cases, several modifications are made to keep only the logicalassociation needed in manufacturing situations.

Example 1A planar surface cannot be located by a cylindrical surface.

However, the opposite case is possible if the tolerance zone of theST is obtained by the shift of a virtual planar surface on both sidesof the axis with the same distance (IT/2).

Example 2If the machined surface is cylindrical surface, the TZ can have

two configurations:

Fig. 4. Example of functional specification.

� The axis of the hole can be constrained between 2 virtual planarsurfaces.� The axis of the hole can be constrained in a virtual cylindrical zone.

Each configuration is represented in a different way.

4. Example for validation

4.1. Functional definition drawing

In order to generate a manufacturing tolerance, a functionalspecification is studied according to the standards. The resultsfound by the ‘‘R3DMTSyn’’ are compared to others, developed inthe literature (Figs. 4 and 5).

4.2. Treatment of the functional specification

4.2.1. Step 1 and step 2

First, in order to translate the meaning of the functionalrequirement, the TTRS_GRAPH_CBE is drawn. Using the SPIDERGRAPH, each surface belonging to the functional specification islocated (Fig. 6)

Fig. 5. TTRS_GRAPH_CBE.

Fig. 6. SPIDER GRAPH.

K. Jaballi et al. / Computers in Industry 62 (2011) 541–554550

Just to clarify this graphical representation, the SPIDER GRAPHis formed by:

� The ring models a machining phase. On each ring, only activesurfaces exist.� The positioning surfaces are represented by hexagons, in which

the small lines indicate the positioning system type.� The machined surfaces are represented in simple circles.

The manufacturing project is modeled for this example in Fig. 6.

4.2.2. Step 3

The negative answer to the ‘‘TEST 1’’ led to Zone 1. This zone ismodeled by the following stages:

(1) Identification of surfaces belonging to the functional specifica-tion selected:i. STi = 13: Is the functional toleranced surface (initial target

surface).ii. SRi = 2, 9: Are the reference surfaces that constitute the

functional reference frame.

Fig. 7. Nature of the final manufacturing specification in phase 40.

Fig. 8. (a) Dimensioning_Card CAS 2-6_PRIS_PL. (b) Tolerancing_Card CAS 2-

6_PRIS_PL.

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(2) Select the last machined surface belonging to the functionalspecification (all the surfaces were identified in the Step 2):‘‘SUd = 13’’ (one of the surfaces of the functional geometricalrequirement being studied).

(3) Identify the MGDE of the last machined surface: MGDE(13) = PLAN.

(4) Build the TT of the specified surface (Table 5):

TTð13Þ ¼

u000bg

26666664

37777775ðO;~U;~V ; ~WÞ

M2 P (6)

In this case a relative coordinate system is adopted. Theidentification of the invariants DOF nature allows theconstruction of the TT.

(5) Identify the tolerance zone type of the selected surface TZ = PL.(6) Select the last phase where we found the SUd: Phase j = 40.� Identification of the positioning system in this phase j:

ðTTRSð4Þ [ TTRSð7ÞÞ [TTRSð14ÞÞ ¼ Identity (7.a)

� The association between the first, second and third position-ing surface contributes to the generation of a complex TTRS,called also Identity TTRS, that forbidden all possible DOFs.

ðMGDEð4Þ [MGDEð7ÞÞ [MGDEð14ÞÞ ¼ PL[ SL[PT (7.b)

� Validation of the manufacturing DRF formed by thepositioning surfaces using Eqs. (4.a) and (4.b).

The planar surface Class is according its DOFs to be limited:

ClassðTTRSð13ÞÞ ¼ 3 (8.a)

� The association between 2 planar surfaces builds a prismaticsurface which allows only a translation along the intersectionline of TTRS(4) [ TTRS(7):

ClassðTTRSð4Þ [TTRSð7ÞÞ ¼ 5 (8.b)

� The association between 3 planar surfaces builds a complexfeature which forbids any displacements:

ClassððTTRSð4Þ [ SATTð7ÞÞ [TTRSð14ÞÞ ¼ 6 (8.c)

So, the DRF is consistent.After the Step 3, it is possible to optimize the DRF developed or

develop the next manufacturing specification as described by thealgorithm presented in Fig. 2. (In this example, the treatment of thesame specification developed in this phase 40 continues, even ifthere are other SUd in this same phase (Part 2).)

4.2.3. Step 4

In the last third Step, the manufacturing specificationsgenerated are characterized by a complex DRF. So the TTRS(13)is located with the following complex DRF‘‘(TTRS(4)[SATT(7))[TTRS(14)’’. The use of the following equationcontributes to the optimization of the DRF, if it is possible:

ClassðTTRS13 [TTRS4Þ ¼ 5 (9.a)

ClassðTTRS13 [ ðTTRS4 [TTRS7ÞÞ ¼ 6 (9.b)

ClassðTTRS13 [ ððTTRS4 [ TTRS7Þ [TTRS14ÞÞ ¼ 6 (9.c)

In this case, the TTRS14 can be eliminated.The resulting manufacturing DRF is formed by

TTRS(4) [ SATT(7). Fig. 7 shows the graphical representation of

the manufacturing specification finally generated, in which allimportant information are identified:

(1) Phasej = OP 40(2) SUdT = 13(3) SR = 4/7

In order to guarantee the respect of the standard, thecorresponding case to the combination between SUdT and itsDRF (Table 6a) is CAS 2-6 (Fig. 8(a) and (b))

The sub-case of the CAS 2-6, represented by Table 6b, hasseveral configurations, according to the value to be allocated to theangle a2 0;p=2½ � (Table 6c):

� If a 6¼ 0 and a 6¼ p/2: the value of the dimensions b and c and ofthe angle a must be assigned.

Table 6aLogical reclassification between toleranced surfaces and their DRF.

SR ST

Complex 1 {E} Prismatic 2 {TD1} Revolution 3 {RD1} Cylindrical 5 {CD1} Planar 6 {GP1} Spherical 7 {SO1}

Complex 1 (E) CAS 1-1 CAS 1-2 CAS 1-3 CAS 1-5 CAS 1-6 CAS 1-7

Prismatic 2 {TD2} CAS 2-2 NO CAS 2-5 CAS 2-6 CAS 2-7

Revolution 3 {RD2} CAS 3-3 CAS 3-5 CAS 3-6 CAS 3-7

Cylindrical 5 {CD2} CAS 5-5 NO CAS 5-7

Planar 6 {GP2} CAS 6-5 CAS 6-6 CAS 6-7

Spherical 7 {SO2} NO NO CAS 7-7

Table 6bSub-case of TTRS_Process_Card of the CAS 2-6.

CAS 2-6

If there is a linear parameter: position specification

CAS 2-6_PRIS_PL

If there isn’t a linear parameter: orientation specification

CAS 2-6_PRIS_PL

Table 6cSub-case of TTRS_Process_Card of the CAS 6-6.

CAS 6-6

If the number ofangular parameter >1:CAS 6-6_PRIS_PL

If the number ofangular parameter is not >1:CAS 6-6_PLA_PL

If there is a linear parameter:

Position specification

If there isn’t a linear parameter:

Parallelism specification

If2πα ≠ : angularity If

2πα = : perpendicularity

K. Jaballi et al. / Computers in Industry 62 (2011) 541–554552

Fig. 10. Manufacturing project OP40.

Fig. 9. Manufacturing project OP30.

K. Jaballi et al. / Computers in Industry 62 (2011) 541–554 553

� If a = p/2: the value of the dimensions b and c and must beassigned. But the value of the angle a can be ignored.� If a = 0: the value of c becomes zero.

4.2.4. Totalities of the manufacturing specification generated

After the forth Step, all surfaces remained and also the newinjected surfaces must be identified. The same work is done untilthe accomplishment of the total accumulated surfaces.

Finally in order to guarantee the respect of the functionalspecification, three manufacturing specifications are generated:

� The location of the surface 9 by the DRF (G/H) in the phase 30.� The location of the surface 2 by the DRF (G) in the phase 30.� The location of the surface 13 by the DRF (G/H) in the phase 40.

According to the Dimensionning_Card (Fig. 8(a)) and theTolerancing_Card (Fig. 8(b)) developed to model all variousmanufacturing situation, it is possible to:

� Identify the phases involved.� Detect all surfaces that contribute to the respect of the functional

specification.� Identify the needed tolerance type: position or orientation.

Figs. 9 and 10 show the final manufacturing project with theappropriate tolerancing.

5. Discussion and conclusion

The ‘‘R3DMTSyn’’ are able to generate automatically amanufacturing GD&T while ensuring the respect of the standard.

Until now the problem of the inclined planar surface, treatedby this method, raises problems to manufacturer. Using

‘‘R3DMTSyn’’, it is possible to quickly generate an exhaustivestandard 3D tolerance type. This type of problem is alsotreated by a lot of authors using mathematical methods. Toresolve these mathematical equations, several authors adoptsuppositions on their models to simplify the resolution of theproblem.

There is a great similarity between the result found by the TZTapproach [17] and those developed by R3DMTSyn approach. Themain benefit of this method is the possibility to affect automati-cally a standard dimension once a database of Process cards isaccessible.

The needs of a prompt answer and a logical decision prove thatthe development of a digital tool is very useful and so urgentlyneeded during the stages of the manufacturing. The TTRS approachused in this work has three powerful aspects:

� Mathematical taxonomy of Euclidean surfaces modeled byMGDE.� The topological and technological related surface provides a

methodology to manage complex parts by composing theirsimple elements while assuring the coherence of the geometricaltolerancing method.� The mathematical definition of the tolerance zones established

by geometrical tolerances in 3D space.

The main lines of our method are:

(1) The identification of the TTRS classes of each surfaces belongingto every functional specification.

(2) The representation of the manufacturing project under theshape of a SPIDER GRAPH.

(3) The identification of the phases of manufacturing of everysurface belonging to the functional specification.

(4) Using an ascendant approach, the adequate manufacturingspecifications are produced in every phase.

(5) Optimization of the reference system of the generatedspecifications using TTRS rules.

(6) Development of process cards with the respect to thestandard.

The TTRS presents an exhaustive approach of all possibleassociation between simple or combined surfaces. Contrary to thework developed by Clement in [23], in this paper only logicalmanufacturing surfaces associations are selected taking intoaccount all constraint type during all machining phases. Thedifferent R3DMTSy rules allow the identification of the rationalway to select even the target surface or its references (it can beelementary or combined).

6. Perspective

‘‘R3DMTSyn’’ is an algorithmic method, which can beestablished in CAD-CAM software. This method assists theengineers to simulate the feasibility of both design andmanufacturing process. Using the ‘‘R3DMTSyn’’ method, we areable to treat an eventual modification manufacturing project. Theprocess card, that models each manufacturing phase, can beconsidered as an intermediate functional drawing, so it is possibleto select the supplier offering the most concurrent cost for eachoperation.

Acknowledgements

The authors are grateful to the anonymous referees for givinguseful comments to improve the manuscript.

K. Jaballi et al. / Computers in Industry 62 (2011) 541–554554

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Dr. Eng. Jaballi Karim received his PhD degree in

Mechanical Engineering in 2009 from Ecole Centrale

Paris in France and National School of Engineers of Sfax

in Tunisia. He is currently an assistant professor at the

National School of Engineers of Tunis (Tunisia). His

research interests include production system modeling,

manufacturing tolerancing, GPS, CAD-CAM software.

Mr. Bellacicco Alain is a teacher aggregated in

mechanics since 1993. He is currently teaching at

Higher Institute of Mechanics of Paris (the Institut

Superieur de Mecanique de Paris; Supmeca-France). He

is a judicial expert; he participates in the design of

special machines. His research interests include pro-

duction system modeling, tolerancing, CAD systems.

Dr. Prof. Louati Jamel his PhD degree in mechanical

engineering in 1986 from Ecole Nationale Superieure

d’Arts et Metiers Paris-France (ENSAM Paris). He is

currently a professor at the National School of

Engineers of Sfax (Tunisia). His research interests

include production system modeling, manufacturing

tolerancing, dynamic behavior of machine tool and

mechatronic system.

Dr. Prof. Riviere Alain received his PhD in 1993 from

Ecole Centrale Paris – France. He is currently the Head

of the Higher Institute of Mechanics of Paris (Institut

Superieur de Mecanique de Paris; Supemca-France). He

published books dealing with the functional tree-

dimensional tolerancing. His research interests include

CAD-CAM, Mechatronic system.

Dr. Prof. Haddar Mohamed received his PhD in

mechanical engineering IN1991 from the Universite

de Technologie de compiegne (UTC-France). He is a

professor at the National School of Engineers in Sfax,

Tunisia. His research topics include structural dynamic

and mechanical systems, vibro-acoustics, fluid-struc-

ture interactions and machine tools modeling.