Functional tolerancing of complex mechanisms: Identification and specification of key parts

Computers & Industrial Engineering 49 (2005) 241–265 www.elsevier.com/locate/dsw

Functional tolerancing of complex mechanisms: Identification and specification of key parts H. Mejbria, B. Anselmettia,b,*, K. Mawussia,c a

b

LURPA, ENS de Cachan, 61 Avenue du Pre´sident Wilson, 94235 Cachan Cedex, France IUT de Cachan, Universite´ Paris-Sud, 9 Avenue de la Division Leclerc, 94234 Cachan Cedex, France c IUT de Saint Denis, Universite´ Paris 13, Place du 8 Mai 1945, 93206 Saint Denis, France Received 22 July 2003; revised 7 April 2005; accepted 26 April 2005

Abstract The functional tolerancing process for complex mechanisms needs the study of the behavior of mechanical joints. The use of the Computer-Aided Design system helps in carrying out this task. This paper presents an effective new approach for decomposing a global geometric functional requirement of the mechanism into geometric specifications defined on key components (parts and sub-assemblies). A recursive tolerancing method serves to identify and functionally specify the key components related to a tolerance-chain. A geometric variation model, based on the invariant degrees of freedom (DOFs) of the datum reference frames and tolerance zones, enables validating a datum reference system built on the positioning features of a component and then deducing the influential mechanical joints. Formalized simple rules based on the topology of parts have also been developed for validating a datum reference system. A geometric specification defined on a sub-assembly has been introduced as a new geometric functional requirement: this provides the designer with an effective recursive tool for ensuring the functional tolerancing of the entire assembly. The proposed approach is straightforward to implement and has been devised from the concept of standardized specifications, with the particularity that it is capable of treating cases in which the positioning features are not perpendicular. q 2005 Elsevier Ltd. All rights reserved. Keywords: Geometric functional requirement (GFR); Functional direction; Invariant DOFs; Key parts; Topological rules; Tolerance zone; Datum reference frame (DRF)

* Corresponding author. Tel.: C33 1 47 40 29 71; fax: C33 1 47 40 22 20. E-mail address: [email protected] (B. Anselmetti). 0360-8352/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2005.04.002

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Nomenclature GFRi SPi Cij uia uA o h.,.i

Geometric Functional Requirement number i Geometric specification of ending part relative to GFRi Geometric tolerance j of an underlying key part relative to GFRi Direction vector of the toleranced entity ia Direction vector of datum reference frame A Vector cross-product operator Vector product operators

1. Background 1.1. Introduction Manufacturing and quality engineers agree that the two key tests for a technical drawing are thoroughness and lack of ambiguity. A poor drawing can in fact serve to double manufacturing and inspection costs (Chiabert, Lombardi, & Orlando, 1998). Geometric dimensioning and tolerancing (GD&T) proves an effective tool for the designer in defining a functional, manufacturable and inspectable part. GD&T greatly enhances the clarity, uniformity and consistency in drawing specifications, thus providing product, process and quality engineers with the same ISO terminology. The functional tolerancing of a mechanical part depends on both its positioning scheme and function within the assembly. In a mechanical system, the number and variety of Geometric Functional Requirements (GFRs), along with the complexity of the interfaces between parts, require a designer to study the influence of mechanical joints. Each geometric functional requirement is defined between the functional surfaces of different parts, called herein ‘ending parts’. In order to respect such a requirement, key parts must be identified and specified. This paper proposes a method for identifying and specifying tolerances on influential parts based on an analysis of both part geometry and the mechanical joints between parts. According to the suggested approach, the search for a contact loop relative to a given GFR, starting from an ending part, makes it possible to directly test whether the geometric variations of joint surfaces affect the GFR. In other words, we seek to verify if these variations are displacing the ending surfaces. The identification of key parts requires a description of the mechanism (see Fig. 1), with this description based on the CAD model created by the designer and comprising two tasks, i.e.: † the extraction of geometric parameters from the nominal CAD model, and † the description of the positioning scheme of parts. The second task is based on the assembly process (see Fig. 1) and defined by the positioning table, which will be presented in this paper. The study of all GFRs yields the functional dimensions and tolerances of the mechanism; a system of inequalities can subsequently be determined (Anselmetti et al., 2003). This system includes tolerances and small variations in nominal dimensions, which serve as variables to be resolved under cost-related criteria, in order to optimize tolerances and calibrate the nominal CAD model (Anselmetti, Mejbri, & Mawussi, 2001).

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243

Adjusted CAD model

Assembly process Knowledge of the Functional probable defects requirements

Means of design

Need

Design the nominal model (CAD) «Designer »

Standards ISO

Geometric model

Define the relative positioning of components

Positioning tables

«Designer & manufacturer» Study of a GFR

Generate and specifykey components « Assembly and tolerancing processes »

Specified key parts

Allocate tolerances

Specified and adjusted model

«tolerance analysis and synthesis process»

Fig. 1. Functional tolerancing within the product design cycle.

1.2. Literature review To identify the key parts with respect to a given 3D tolerance-chain, many researchers have represented the structure of assemblies based on a graphical model. One such assembly has been described graphically in Lee and Gossard (1985). Three graphical levels have been considered in Teissandier, Coue´tard, and Ge´rard (1999). On the first level, an assembly is described with kinematic subsets, while the second level describes an assembly with parts and the third level an assembly with ideal surfaces (both manufactured and nominal surfaces). A graph contains a set of vertices and edges, with a vertex representing a sub-assembly, part or surface and an edge symbolizing a joint. A 3D tolerance-chain is illustrated with a surface graph by means of associating edges in series and/or in parallel. The research on key parts to satisfy a functional requirement is based on the specialist’s expertise and remains mostly intuitive. In a complex mechanism, the large number of components (subassemblies and parts) complicates this task. Similarly, in Ballu and Mathieu (1999), the authors considered two types of graphs in order to describe a mechanical assembly: a ‘graph of the links’, and a ‘graph of the potential key surfaces’. For a mechanism with many parts, Ballu and Mathieu proposed a reduction process that leads to a single joint between the two parts targeted by the requirement. This reduction is not always possible, and the approach cannot easily handle a complex tolerance-chain. None of the present CAD systems available on the market are able to perfectly control the geometric dependencies existing between mechanism parts (Teissandier et al., 1999; Turner, 1993a). Turner proposed a ‘Relative Positioning Operator’, which creates a series of geometric relations between parts in the form of positioning constraints (Sodhi & Turner, 1994; Turner, 1990); he performed a linearization of the relative displacements between two parts. This step allowed for the creation of assembly models illustrated by relational graphs. The ‘Relative Positioning Operator’ (Sodhi & Turner, 1994), coupled with ‘Feasibility spaces’ (Turner, 1993b), gives rise to a 3D assembly dimension-chain.

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Another approach, which detects chains during the assembly sequence, has been indicated in Mantripragada and Whitney (1998); the authors have shown how a given assembly design enables defining these chains as the basis for designing assemblies first and parts next. They also proposed a directed acyclic graph that describes the whole assembly. Teissandier et al. (1999) consider that this method has not been sufficiently built upon the concept of standardized specifications. To make a mechanism assembly feasible, geometric Key Characteristics (KCs) are detected during the design process (Whitney, Mantripragada, & Adams, 1999). These authors forwarded a theory to lend support to assembly design. This theory shows how kinematically-constrained assemblies may be unambiguously designed so as to satisfy geometric requirements. In pursuit of the same objective, a simple method for analyzing constraints was presented in Whitney and Adams (2001). To identify a contact loop, the problem of joint surface influence has been partially introduced for a simple mechanism in Anselmetti and Mawussi (2002). The mechanism is considered to be assembled and the authors seek to detect influential surfaces in order to meet a precision requirement. In all of these approaches, the geometric variation of a geometric entity has been controlled and limited by geometric tolerances. The representation of geometric tolerances is based on the concept of Datum Reference Frame (DRF). According to the applicable standards (ASME, 1994a; ISO 1101, 1983), a datum reference frame is either a theoretically-exact geometric entity, such as a plane, or the resolved geometry of a size feature (the axis of a cylinder, the center-point of a sphere, the middle plane of a slot, etc.). DRF specifications depend both on the precise function of the part within the assembly and on the geometry and quality of real part faces (Chiabert et al., 1998). Geometric tolerances related to a datum reference frame are of three sorts: position, orientation, and run-out. Form and profile tolerances are not usually related to DRFs. A valid datum reference system may be formed using combinations of points, lines and planes. A datum reference system is composed of 1–3 datum features specified as primary, secondary and tertiary datum, respectively. The generation of a DRF depends on the type of positioning features, their mutual position and orientation, and their precedence (Kanikjan, Shah, & Davidson, 2001). In some instances, a datum reference frame may serve to define an incomplete coordinate system for locating and orienting tolerance zones, thus freeing certain translation or rotation transformations. 1.3. Proposed approach This paper will present a new functional tolerancing approach: a recursive tolerancing method allows identifying and specifying key parts. We will decompose a global functional requirement, as defined over the whole mechanism, into specifications on parts and on sub-assemblies. Each geometric specification on a sub-assembly, identified and toleranced as a part, will be considered as a new functional requirement defined on this sub-assembly and must be further developed; this stage provides the designer with an efficient recursive tolerancing approach. Topological rules help generate a tolerance-chain with respect to a specific geometric functional requirement. For each functional requirement with a tolerance-chain, a graph can be automatically built from the positioning tables by application of the recursive tolerancing approach (Mejbri 2004). This graph exclusively encompasses all of the key parts exerting effects upon the characteristic value of the studied requirement. Our approach thus avoids having to represent the entire mechanism on the same graph before searching for the key parts.

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2. Problem data 2.1. Geometric functional requirements of the mechanism A geometric functional requirement is a characteristic defined between ending geometric entities (plane, line, point, etc.). In general, this characteristic constitutes a distance or angle between two ending entities, yet can also be a location or orientation of an ending entity, with respect to a datum reference system formed by other ending entities. A GFR can be graphically defined on an assembly by means of ISO standards (Anselmetti, Thibault, & Mawussi, 2002) (see Fig. 2). Each GFR is controlled according to an analysis direction, which depends on the targeted geometric characteristic to be controlled. For example, if the requirement were a distance between two parallel ending surfaces, then the analysis direction would be the common normal direction to these two surfaces. Such a distance depends on the variation in both position and orientation of each ending entity, with respect to their nominal position. Geometric variations of joint surfaces induce a displacement of these ending entities; this displacement must be limited and controlled in order to satisfy the requirement value. A geometric functional entity must remain within a tolerance zone that features a functional direction u. Should the functional entity be a plane, then its functional direction would be given by its direction vector u, which consists of the normal to the plane. If the ending surface were a cylinder, then its functional direction would be all directions ui normal to its axis (see Fig. 3). This direction depends on the type of geometric entity (plane, cylinder, etc.). An example of a sub-assembly has been displayed in Fig. 2 with three geometric functional requirements GFR1, GFR2 and GFR3. The screw s is used here to link parts b and p. Plate p is positioned on three underlying parts b, c and a. The datum reference system A/B/C is built on the base b of the sub-assembly, which ensures the positioning of this sub-assembly in the external

Fig. 2. Example of a sub-assembly with three GFRs.

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∆

Functional directions 6p 5p

Ending entity: a cylinder ∅ 0.05

A B C

Ending entity: a plane

0.05 A

Fig. 3. Characterization of tolerance zones.

world (Mejbri, Anselmetti, & Mawussi, 2003). As an example, GFR2 signifies that the axis of cylinder 5p must lie within a small cylinder (diameter :Z0.05 mm) perpendicular to datum reference frame A and positioned with respect to the secondary and tertiary datum reference frames B and C with reference dimensions. The shape of the tolerance zone depends on both the specified entity and the geometric tolerance. Fig. 3 shows the tolerance zones and functional directions defined for ending geometric entities 5p and 6p. In order to satisfy GFR values, the ending geometric entity must lie within the associated tolerance zone. The displacement of this ending entity is caused by its own geometric variation, the variation of key joint surfaces, and the clearances in mechanical joints. The influence of the geometric variation in joint surfaces has been modeled and computed for a mechanism in Anselmetti, Mejbri, and Mawussi (2003). A geometric variation in joint surfaces that does not act to modify the GFR value implies that these surfaces do not affect the studied characteristic, an implication that allows determining the inactive mechanical joints defined by the positioning tables. 2.2. Part positioning model 2.2.1. Positioning features The elementary joints between parts are composed of positioning features. A non-exhaustive set of technological features is presented in Fig. 4 (Anselmetti et al., 2002). 2.2.2. Positioning table The positioning table concept has been presented and discussed in Anselmetti (1998). The quality of the interfaces between parts ensures meeting positioning requirements of a mechanism as well as the correct positioning of parts. In defining a mechanism, the designer analyzes the failures capable of being generated by geometric defects of the joint surfaces between parts. To formalize his intentions, the designer then uses the positioning table for each component (see Fig. 5b) so as to clearly indicate the positioning surfaces in contact, the preponderance order of surfaces (defined in a similar fashion to the notion of a datum reference system precedence order given in the ISO standards (ISO/CD 5459.1, 1981; ISO/CD 5459.2, 1981), along with the type of kinematic link assured and, if necessary, the calibration techniques to be employed during the mechanism assembly. This positioning table is used for

H. Mejbri et al. / Computers & Industrial Engineering 49 (2005) 241–265 Plane

Surface features

Parallel Planes

Coplanar planes

Fitting features

Cone

3D free surface

Extruded surface

Threading

Set of parallel cylinders

Symmetrical parallel planes

Sphere ( θ >180˚)

Cylinder

247

Extruded fitting surface

θ

Coaxial cylinders

(if θ<180˚, the sphere must be considered as a free surface)

Fig. 4. Positioning features.

the purpose of defining the relative positioning of components since each part or sub-assembly possesses its own positioning table. This tool avoids having to represent the entire mechanism on the same graph before searching for the key parts relative to a geometric functional requirement.

3. Identification and specification of key parts 3.1. Graphic representation of part positioning For each GFR of the mechanism defined within a given configuration, a weighted graph representing just the key parts may be built, starting at the ending part and extending until the base (see Fig. 5a). A vertex represents a component (part or sub-assembly), while an arc represents a mechanical joint Tertiary joint

p

Ending part

Part or block

plate 1

2

c

3

a

Code

plane

cylinder

cylinder

1p

2p

3p

contact clearance The base

b

b

b

1b

1c

Primary Secondary

(a) relative positioning of parts

Config. Author

p

clearance 2a Tertiary

(b) positioning table of the plate

Fig. 5. Definition of part positioning within the mechanism.

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H. Mejbri et al. / Computers & Industrial Engineering 49 (2005) 241–265 (GFR1)

(GFR2)

p

0.05 A

p

p

?

?

b

(GFR3)

∅ 0.05 A B C

∅ 0.05 A B C

c

a

b

b

?

b

The base

?

c

a

b

b

?

b

The base

?

c

a

b

b

The base

Fig. 6. Mechanical joints to be tested.

between two components. The arc converges towards the vertex depicting the underlying component listed in the fifth row of the positioning table. When a component is positioned on more than one underlying component, the number of arcs is equal to the number of these underlying components. The arcs, stemming from a component positioned on several underlying components, represent the mechanical joints in the underlying components and are classified, from left to right, as: the primary joint, secondary joint and tertiary joint. The weight on the arc yields the preponderance order of the particular joint; this weight corresponds to the order of the associated column of the TMP table. If just one joint lies between two components, then the associated arc will not be weighted. The positioning of parts within the mechanism shown in Fig. 2 has been defined in Fig. 5. In a complex assembly structure, many components feature more than one mechanical joint with underlying components. In many cases, the geometric variation of some joint surfaces does not induce any displacement of the ending geometric entities. The dominant mechanical joint is always influential and moreover the associated underlying part is always represented as a vertex on the graph. The secondary and tertiary mechanical joints (see Fig. 6) must be tested to determine whether they are influential (active) or not on the ending entity. If a joint is found to be not influential (inactive), then the associated underlying components will not be generated on the graph depicting the key parts. 3.2. Study of the influence of joint surfaces In this section of the paper, we will propose a model of mechanical joint influence based on both an analysis of the geometric tolerance specified on geometric entities and a validation of the datum reference system built on positioning surfaces. We will demonstrate how the validation of a datum reference system serves to implicitly determine the influential underlying parts. 3.2.1. Tolerancing of the ending part Typical candidate geometric tolerances that may be specified on the geometric features have been given in Fig. 7. The specified orientation and position symbols are, by default, the parallelism and location, respectively; they rely upon the mutual orientation between specified entity and DRFs. In Fig. 8a, a complete datum reference system has been built on the positioning features of plate p, based on the third row of the positioning table and in respect of the precedence order. The ending entities are specified in position with regard to this datum reference system. The secondary and tertiary DRFs reflect a resolved geometry of a size feature (the axis of a cylinder). They are referenced at the Least

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249

Fig. 7. Candidate geometric tolerances.

Material Condition (LMC). Three geometric specifications SP1, SP2 and SP3, associated respectively to GFR1, GFR2 and GFR3, are presented in Fig. 8b. An analysis of the tolerance zone for each geometric tolerance will then help validate the datum reference system; this validation approach is based both on knowledge of the function of each datum in the position and on the orientation of the tolerance zone. 3.2.2. Geometric variation modeling The directions of translation or rotation, during which the DRF geometry remains unchanged after performing the transformation, are called invariants. For example, if we rotate an infinite plane around its normal or translate it in-plane, its geometry will not be altered. An infinite plane is invariant therefore ∅ t1 D E Part or block

plate

p

plane

cylinder

1p

4p

L

(SP1 ⇐ GFR1)

t3

F

L

D E

L

5p

3p

F

F clearance

contact 1b

F

30

E

6p

cylinder

2p

D

20

Config. Author

Code

L

∅ t2 D E

1c

Primary Secondary

2a

Tertiary

1p

3p

E

D

2p (a) Definition of DRFs on the positioning features

(b) Geometric tolerancing of the ending entities

Fig. 8. Geometric tolerances defined on the plate.

(SP2 ⇐ GFR2)

L

F

L

(SP3 ⇐ GFR3)

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in one rotational and two translational directions within a Cartesian system, thereby leaving one translational and two rotational directions free. A Degree Of Freedom (DOF) model has been implemented: each geometric entity has three translational (x, y, z) and three rotational (a, b, g) DOFs before becoming constrained. Several researchers have independently devised the so-called ‘degree-of-freedom approach’ to modeling tolerances (Cle´ment et al., 1991; Guilford & Turner, 1993; Kandikjan et al., 2001; Rivest, Cle´ment, & Morel, 1994; Salomons, Jonge Poernik, Slooten, Houten, & Kals, 1995; Yan, 1995; Zhang, 1992). All combinations of datum features can be assigned to one of the six combinations of point, line and plane. This approach is known as the TTRS (Cle´ment et al., 1991; Cle´ment, Rivie`re, Serre´, & Valade, 1997; Rivest et al., 1994) approach, especially within the European tolerance research community. The concept of DOFs has also found its way into the new ASME Y14.5.1 tolerance standard (ASME, 1994b). According to our approach, the invariant DOFs of both a datum reference frame and the tolerance zone constitute important information, which helps validate a datum reference system built on positioning features. 3.2.3. Validation of a datum reference system The datum reference system built for each geometric tolerance on plate p (see Fig. 8b) must be validated. The datum reference frames are the plane 1p (primary DRF), the cylinder 2p (secondary DRF) and the cylinder 3p (tertiary DRF). For SP1, the tolerance zone of the specified cylinder 4p is constrained by the three datum reference frames D/E/F. Part a constrains the rotation of plate p around the axis of frame E. For SP2 however, the tolerance zone of the specified cylinder 5p is constrained by the primary and secondary datum reference frames D/E. The tertiary frame F can in fact be eliminated given that the rotation of plate p around the direction of frame E maintains cylinder 5p invariant. The primary and secondary DRFs form a new basic frame, which displays an invariant rotation around the line DDE (see Fig. 9), as both this line and the axis of the cylindrical tolerance zone are nominally coaxial (DDEhD5p). This rotational freedom then corresponds to an invariant DOF of the specified tolerance zone for cylinder 5p. The frame D/E is thus sufficient to position and orient this tolerance zone. The tolerance zone of the specified plane 6p is constrained solely by the primary datum reference frame D in that the invariant DOFs of this frame ∅ t2

(SP2)

D E

L

F

L

t3

30

u5p F

E E

D E

L

F

L

u5p

u4p u5p

(SP3)

u6p

∆5p

D

uE

D

u6p

u6p

uF

uD D

∆DE

∆DE ≡ ∆5p

u D parallel to u6p

Fig. 9. Validation of a datum reference system.

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251

Geometric specification (input data) Invariants DOFs of the tolerance zone A is a valid DRF ?

A - B is a valid DRF ?

A - B - C is a valid DRF ?

∅t yes

ye s

ye s

No (incomplete system)

A

B

C

∅t

A

∅t

A

B

∅t

A

B

C

∅t

A

B

C

Fig. 10. Algorithm for determining the valid datum reference system.

correspond to invariant DOFs of the planar tolerance zone of plane 6p (see Fig. 9). The invariant translations and rotations of the tolerance zones, along with the associated DRFs, are symbolized by arrows in Fig. 9. The determination of the valid datum reference system is given by the algorithm in Fig. 10. Topological rules for validating a datum reference system will be detailed in Section 4.3. If the complete datum reference system were not able to constrain the tolerance zone, then the datum reference system would not constrain a degree of freedom. This result could produce an error in the definition of either the geometric specification or the datum reference system; this degree of freedom could also be included by the studied function. 3.2.4. Inactive underlying parts Once the invariant DOFs of the tolerance zone relative to a geometric tolerance are known, the datum reference system can be validated. This validation step allows determining the inactive joints. By admitting that the primary joint is always an influential joint, the secondary joint and/or tertiary joint may thus be inactive. The principle for testing the influence of a joint is as follows: If the primary DRF is valid (it would be sufficient to position and orient the tolerance zone of the specified entity), then retain only the associated primary joint; Else if the new basic frame formed by the primary and secondary DRFs is valid, then the tertiary joint is inactive; Else, retain the complete datum reference system and all associated joints. Application of this approach to geometric specifications SP1, SP2 and SP3 (Fig. 8b) relative to requirements GFR1, GFR2 and GFR3, respectively has enabled determining the non-influential joints. (A non-influential joint implies that the associated underlying part is not a key part with regard to the studied GFR.) The underlying parts b, c and a are key for GFR1 because the three associated datum reference frames D/E/F are necessary and sufficient to constrain the corresponding tolerance zone (Fig. 11a). Underlying parts b and c are key for GFR2 because the two associated datum reference frames D/E are necessary and sufficient to constrain the corresponding tolerance zone (Fig. 11b).

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H. Mejbri et al. / Computers & Industrial Engineering 49 (2005) 241–265 ∅ 0.05 p

b

A B

c

a

b

b

(a)

∅ 0.05 A B (GFR2)

C

(GFR1)

p

0. 05 A

C p

(GFR3)

c

The base

b

b

(b)

The base

b

The base

(c)

Fig. 11. Inactive joints of plate p and its underlying parts.

Underlying part b is the only key for GFR3 because the primary datum reference frame D is necessary and sufficient to constrain the corresponding tolerance zone (Fig. 11c). The graph of key parts related to GFR1, GFR2 and GFR3 has been given in Fig. 11. In practice, a simple graph that includes just the key parts can be directly constructed for each GFR. We have already shown that the DOF model serves to remove inactive joints. The general characterization of tolerance zones encompasses the study of mechanical joint influence within a complex assembly.

4. Formalization of rules 4.1. Objective A general characterization of tolerance zones for each type of geometric entity, as well as the possible geometric variations, will be presented in this section. This approach is named ‘vectorial representation of tolerance zone’ (Anselmetti, Mawussi, & Mejbri, 2005) The formalization of a simple set of rules based on the topology of parts will now be developed. 4.2. General characterization of tolerance zones The invariant DOFs of the tolerance zone of a geometric entity depend on both the entity itself (plane, line or axis, point, surface) and the geometric tolerance defined by ISO standards. The specified plane 4a in Fig. 12 exhibits two constrained rotational DOFs around two perpendicular in-plane lines. The normal of this plane u4a is controlled by the angular deviation with respect to its nominal position. The corresponding type of tolerance zone has been denoted ou4a, where o indicates that the orientation is constrained around a line orthogonal to the normal direction of the plane. The specified plane 3a however has one constrained rotational DOF around an in-plane line given by the direction vector: wZuAou3a. The rotation of this plane is constrained only around w. The corresponding type of tolerance zone is denoted rw. The invariant translations and rotations of the tolerance zones are symbolized by arrows (see Fig. 12c). In the case of a complex 3D surface, the ISO standards do not provide the means for describing with precision the admissible degrees of freedom on either the positioning or the tolerance zone orientation

H. Mejbri et al. / Computers & Industrial Engineering 49 (2005) 241–265 u4a

u4a ou4a

t1 A 4a

253

4a

ua4 u3a u3a u3a

t2 A a

rw

w 3a uA

3a

A

(a) Geometric tolerances of entities.

(b) Characterization of the tolerance zones.

(c) Invariant DOFs of the tolerance zones.

Fig. 12. Characterization of tolerance zones.

with respect to the nominal position. To every geometric tolerance on a 3D surface, we have added comments specifying the function of this surface in order to satisfy the functional need. Fig. 13 shows two geometric specifications on the same surface. The first consists of limiting the form defect of the surface, which implicitly means that all tolerance zone degrees of freedom are indeed authorized. This geometric variation type is denoted ‘f’, which indicates that the tolerance zone is floating. The second geometric specification limits just the orientation of the surface in comparison with the nominal position, as defined by the specified datum reference system A/B/C. This geometric variation type is denoted ‘t’, which stands for the fact that the tolerance zone is free to translate within the three-dimensional space. A 3D surface can feature all combinations of translations and rotations, with such combinations depending on the function of the surface in the assembly. For example, if a 3D surface were authorized in simple translation for a functional direction u, then the geometric variation would be denoted mu. Moreover, if a surface were authorized in rotation around a line with direction vector u, then the geometric variation would be denoted au. Characterization of the variation of a geometric entity depends on its function within the assembly. For example, a plane specified in location has its normal vector u as the functional direction. In some u Form

Control of the orientation t1

t2 A

B

C

u

f

t t2 C B

A

Fig. 13. Characterization of a functional 3D surface.

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instances however, this plane features a constrained rotation DOF around an in-plane axis, in which case the actual functional direction of the plane is the direction of this in-plane axis. The general characterization of geometric variation of the main geometric entities considered herein is presented in Fig. 14. The geometric entity, geometric specification, type of associated tolerance zone, an illustration, plus the invariant rotations and translations of the tolerance zone are all given in this figure. This characterization serves to define the function of the geometric entity and helps validate the datum reference system, established for the part related to this particular entity. 4.3. General rules for validating a datum reference system Application of the datum reference system validation algorithm (Fig. 10) requires some general topological rules. The rules developed have been defined by a topological constraint on geometric parameters between a toleranced entity and the positioning entities on which the studied datum reference system A/B/C has been built. In order to test the validation, we will begin with the primary frame A. Then, if the last frame is not sufficient, we will reclassify both primary frame A and secondary frame B so as to obtain a new frame A/B. Should this reclassified new frame not be sufficient, it will be reclassified with the next frame C to form system A/B/C. The appropriate rule would then be applied to the datum reference frame (first row in the table of Fig. 15: planar, cylindrical, prismatic, revolution, spherical or complete datum reference frame) in order to determine whether it is sufficient to constrain the tolerance zone associated with the given geometric specification (first column in Fig. 15). If sufficiency is obtained, then it becomes a valid datum reference system. Without any compromise to the generality of our approach, Figs. 15 and 16 set forth the topological rules related respectively to the orientation specifications and location specifications of use in validating a DRF. The geometric specification, the invariant rotations and translations of the related tolerance zone, and the type of datum reference frame, along with the associated topological constraint between geometric parameters to be satisfied, are all defined in these figures. If the specified constraint is not satisfied for the primary DRF, then both the primary and secondary DRFs must be reclassified in order to obtain a new frame. This new frame would then be tested, and so forth and so on. In Figs. 15 and 16, the entry ‘yes’ indicates that the corresponding DRF is always valid for the considered tolerance zone, while ‘no’ means that the DRF is not valid and we must consider both this DRF and the next one (in the order of preponderance) built on the part to obtain a new frame, whose validation is then to be tested. Four examples have been given below for easy use of these proposed topological rules: Example 1: The geometric specification is the location of a plane exhibiting u as a functional direction (first specification in the Column 1 list) and the reclassified datum reference frame is planar with v as a functional direction (Column 2). If the two planes are parallel (uZGv), then this datum reference frame is sufficient (see Fig. 15). Example 2: The geometric specification is the location of a point F within a cylindrical tolerance zone exhibiting u as the axis direction (fourth specification in the Column 1 list) and the reclassified DRF is cylindrical with v as the axis direction and line of revolution D (Column 3). If (F 2 D and uZGv), then this DRF is sufficient (Fig. 15). Example 3: The geometric specification is the location of the center F of a sphere (sixth specification in the Column 1 list) and the reclassified DRF is spherical with center O (Column 6). If

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Entity

Type of the Geometric specification tolerance zone

InvariantDOFs of the tolerance zone

Illustration

u

u t

pu pu u

u ou

Plane

ou t

ru u ∅t

ru

u

u

u

cu cu

t

u

pu

u

Axis (or line)

pu u

u ∅t

ou ou

t

ru

t

u

u

ru u

u

pu

pu u

u Point

∅t

cu cu

s∅t

s s

u

u With comments

surface

t

pu, ou, mu, ru, au, f, t , z

mu

au

*

z

* all degrees of freedom are constrained.

Fig. 14. General characterization of geometric variations.

255

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Fig. 15. Topological rules relative to a location specification for validating a DRF.

Fig. 16. Topological rules relative to an orientation specification for validating a DRF.

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(FhO: the identical point), the DRF is thus sufficient (Fig. 15). Example 4: The geometric specification is the orientation of the axis of a cylinder with axis direction u (third specification in the Column 1 list) and the reclassified DRF is planar with functional direction v (Column 2). If (uZGv), then this DRF is sufficient (Fig. 16).

5. Functional tolerancing with a recursive approach 5.1. Operating principle We have already discussed how to validate a datum reference system using simple topological rules between geometric parameters with plate p being specified. The underlying key surfaces must now be functionally specified for each GFR. The geometric specification of an underlying key surface is based on how it functions within the assembly. A geometric tolerance of an underlying surface features a tolerance zone that must be defined with respect to the datum reference system built on the base b. The underlying part then becomes an ending part, upon which the same approach developed in Section 3 is applied, which implies that a geometric tolerance on the underlying key part is considered as a new GFR to study. This approach provides the designer with a recursive tolerancing method for ensuring the identification and geometric specification of key parts until reaching the mechanism base. 5.2. Tolerancing of underlying key parts Each datum reference frame of plate p displays a function in the positioning and orientation of the tolerance zone of the ending entity. This function allows specifying the associated underlying key surface. This particular entity must be functionally specified by constraining its influential orientation and/or location with regard to the complete datum reference system A/B/C of the mechanism (see Fig. 17). Subsequently, this datum reference system may be validated using the algorithm given in Fig. 10. For example, the invariant DOFs of the tolerance zone for the specified cylinder 5p of plate p have helped us first validate a datum reference system built on the positioning features of the plate and then determine key parts b and c. These same invariant DOFs have enabled reducing each datum reference

Fig. 17. Geometric tolerances of the underlying key entities relative to GFR2.

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Fig. 18. Specification of the influential underlying entities.

frame (D and E) by freeing its constrained DOFs that correspond to the invariant DOFs of the specified cylinder 5p (see Fig. 18). The new constrained DOFs of this datum reference frame then serve to specify the type of influential defects (location and/or orientation) on the underlying key surface. In Fig. 18, the tolerance zone of cylinder 5p is of the type cu5p (see Fig. 14 for notational meanings). The valid DRFs are, in order of precedence, the plane 1p of type puD, and the cylinder 2p of type cuE. The dominant plane 1p contains one constrained translational DOF and two constrained rotational DOFs. The constrained translational DOF of this plane is freed in the direction of its normal vector uD, oriented parallel to u5p, since it corresponds to an invariant DOF from the tolerance zone of cylinder 5p (with uD and u5p being parallel). The preponderant datum reference frame then assumes the type ouD instead of puD. For this reason, the associated underlying entity 1b is constrained only in orientation (see C21, Fig. 17). The secondary datum remains of the same type cuE, which makes the associated underlying entity 1c constrained in location (see C22, Fig. 17). Inasmuch as the orientation has already been constrained by the preponderant frame D, the secondary frame E does not constrain the orientation of entity 5p, which dictates why the associated underlying entity 1c is not constrained in orientation. In conclusion, the primary frame D constrains the two rotations of the cylinder 5p tolerance zone around two axes perpendicular to its direction vector; frame E however constrains the two translations of this tolerance zone in plane 1p. This illustration reveals that invariant DOFs are useful in focusing on just the influential defects (location and/or orientation) of the associated underlying key entities. 5.3. The recursive tolerancing method Each of the geometric tolerances defined on an underlying part, herein denoted Cij relative to the GFRi, is considered as a new geometric functional requirement defined on the mechanism and features its own small graph that includes the key parts. The two geometric specifications C21 and C22 (Fig. 17) are defined on the underlying key surfaces 1b and 1c and perceived as two new geometric functional requirements on the sub-assembly formed by parts b and c. Starting from the ending part, this approach provides the designer with a recursive tolerancing method for identifying and specifying the key parts relative to a given GFR. This approach enables controlling solely the influential defects in the right functional direction.

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Fig. 19. Functional tolerancing of underlying entities.

In Fig. 19, we have shown the result of the functional tolerancing of underlying entities for all geometric functional requirements GFRi (as defined in Fig. 2).

6. Application to an industrial product In order to illustrate the recursive functional tolerancing approach, we have employed the example of the industrial engine assembly proposed in Fig. 20a. The tolerancing of such a mechanism requires the two following steps: 1 Tolerancing of joint surfaces between each of the two parts to ensure respect of their assembly requirement (i.e. use of the maximum material condition—MMC). The joint surfaces are defined in the TMP table. This step comprises two distinct tasks: (a) tolerancing of joint surfaces to guarantee the assembly of each set of two parts, and (b) definition of the mechanism assembly conditions; 2 Tolerancing of the mechanism for a precision requirement, e.g.: maximum clearance requirement or maximum deviation requirement (use of least material condition—LMC). Our approach has presented the second of these steps. The mechanism assembly has been assumed guaranteed with a set of assembly specifications that are not represented on the following drawings. The objective here is to study the GFR requirement defined by the designer: minimum distance to be respected between the piston p and the part h (see Fig. 20). This GFR is defined within the configuration corresponding to the case where the piston is in the upper position (uZ908), with the base of the assembly being the body b. In this assembly, a block (pCa), constituted by the piston p and the axis a, has been considered given that the piston cannot be positioned without the axis and vice versa. The loop pertaining to the GFR, which relates the two ending components (pCa) and h, passes through this block.

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Fig. 20. Industrial engine with a GFR.

We have applied our approach to separately study the positioning branch related to each ending part until reaching the base b. The first positioning branch includes components (pCa), h, r, c and b, while the second positioning branch includes components h and b (see Fig. 20b). In the positioning branch relative to the block (Fig. 21b), the ending surface belonging to the piston is specified in a position with respect to the datum reference system A/B built on the positioning features; the geometric specification is SP1 (Fig. 21a). The valid datum reference system is hence A/B. The

Fig. 21. Study of the positioning branch related to the ending component (pCa).

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primary underlying part 1h is specified in orientation C12 with respect to a datum reference system built on the base b of the assembly. Since this datum reference system attached to the base has not been defined, the specified orientation tolerance, by default, is a parallelism. The right symbol (:, //, t) of the orientation tolerance depends upon the mutual orientation of the toleranced entity 1h and the DRFs of the base. The position of this cylinder 1h is not influential. The second underlying surface 1r is specified in its position within the functional direction given by the direction of the axis of cylinder 1h.. This geometric entity is influential in the indicated direction; for this reason, the modifier : has not been used in tolerance specification C11 (for this GFR). The corresponding tolerance zone is planar and of the type pu. On the other hand, a study of the positioning branch related to part h (Fig. 22b) consists of locating the ending surface 2h with respect to the datum reference system C/D/E built on the positioning features (see the positioning table in Fig. 22b); the geometric specification is SP2 (Fig. 22a) and the valid datum reference system is C. For this positioning branch, the only influential defect is the position of the primary underlying surface 2b (see C21, Fig. 22a). The orientation of this surface 2b will be influential on geometric specification C12 (see C112, Fig. 24a). Our recursive functional tolerancing approach serves to develop the two geometric specifications C11 and C12 as new GFRs defined on the sub-assembly that includes parts r, c and b (Fig. 23). The key parts of the two preceding new GFRs are given in Fig. 23b.

Fig. 22. Study of the positioning branch related to ending part h.

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Fig. 23. Study of the geometric specifications C11 and C12.

The recursive tolerancing approach has also been applied to detect and functionally tolerance key parts until reaching the base b. The study of the positioning branch related to ending part (pCa) has yielded the two geometric specifications C112 and C122 (see Fig. 24a), while the study of the positioning branch related to ending part h has produced C21 (see Fig. 24a). These geometric specifications of the base are defined with respect to a reference system on the base that still needs to be defined. For the choice of reference system on the base, many solutions may be proposed; one such solution has been presented in Fig. 24b, where: A datum reference A is defined on the joint surfaces between the crankshaft c and the body b. A datum reference system B/C is defined on the joint surfaces between parts h and b; the datum reference B is specified on plane 2b and reference C on cylinder 3b.

Fig. 24. Functional tolerancing of the engine body.

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Let M be the point on axis C positioned at a distance d1 mm to axis A. We now define the line D passing by point M and parallel to axis A, where D is the piston axis when it lies on top (uZ908). This D axis can be considered as a datum reference for defining specifications on the body (Fig. 24b). Specifications C122, C112 and C21 (Fig. 24a) have been respected. Axes A and D have been carefully held parallel and co-planar.

7. Conclusion and outlook This paper has presented an effective approach for identifying and functionally specifying the key components (parts and sub-assemblies) related to a GFR defined within a complex mechanism. The main results from this work are as follows: † A recursive functional tolerancing method has been developed for decomposing a global GFR of the mechanism into geometric specifications defined on the key components. The functional tolerancing of a component depends on both the positioning scheme and how it functions in the assembly. † A geometric variation model based on the invariant DOFs of datum reference frames and tolerance zones has enabled validating a datum reference system built upon the positioning features. † A formalization of general rules based on the topology of parts has been proposed in order to validate a datum reference system and then to deduce the key underlying entities. † This approach allows generating geometric specifications exclusively in the functional directions. A geometric specification in a sub-assembly has been developed as a new GFR, which provides the designer with an efficient tool for ensuring the functional tolerancing of the entire assembly. † The approach developed displays the particularity that it treats cases in which the positioning features are not perpendicular. † Our approach avoids having to graphically represent the entire mechanism before searching for key components. Moreover, this approach is straightforward to implement and has been elaborated from the concept of ISO standardized specifications, and then validated on many industrial mechanisms. Certain rules must still be formalized in order to automatically generate geometric specifications and extend the field of application to increasingly-complicated mechanisms. To quantify tolerance values for each GFR, a resultant inequality relating the value of the GFR with tolerances is to be automatically generated. Resolving the system of inequalities related to all GFRs enables obtaining optimum tolerance values.

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