[Psychology] Mechanical Assemblies Phần 5 pptx

The process of assembly is not just of fastening parts together but should be thought of as a process that first defines the location of parts using the mates and then reinforces their l

FIGURE 8-1 The Stapler, Its Liaison Diagram (left), and Two Key Characteristics (right). Several irrelevant liaisons have been colored gray because they play no role in positioning parts to deliver either KC.

FIGURE 8-2 The KCs of the Stapler Shown Separately with the Liaisons That Deliver Them. Irrelevant liaisons are not shown.

the process for creating it and thus only indirectly defines

the assembly

We will also define two types of assembly joints, called

mates and contacts: Mates pass dimensional constraint

from part to part, while contacts merely provide support,

reinforcement, or partial constraint along axes that do not

involve delivery of a KC Some joints act as mates along

some degrees of freedom and as contacts along others

Symbols for each of these types of joints will be

intro-duced We will then present the scope of the DFC in

as-sembly planning using several examples

Finally, we will see that the DFC contains all the

in-formation needed to carry out a variation analysis of the

KC it delivers This fact links the scheme by which the

parts are located in space to the sources of variation in

their locations

To visualize the ideas to be presented in this

chap-ter, we again turn to the desktop stapler In this chapchap-ter,

we will learn how to characterize the liaisons of an

as-sembly as delivery chains for key characteristics This is

illustrated in Figure 8-1, where some of the liaisons are

shown in gray to denote that they play no role in KC

delivery It is further emphasized in Figure 8-2, where

each KC chain is shown separately and the irrelevant

li-aisons are omitted altogether The stapler also illustrates

the difference between mates and contacts The ence is illustrated in Figure 8-3 All these concepts will bemade concrete in this chapter and related to their under-lying mathematical representations introduced in earlierchapters

differ-FIGURE 8-3 Illustrating the Difference Between a Mate and a Contact. The mate provides constraint for the staples

by establishing their position relative to the end of the carrier The pusher and staples share a contact, which reinforces or stabilizes the stapler-carrier mate In the vocabulary of Chap- ter 4, the staples are properly constrained along the axis of the carrier Note that the contact is colored gray, indicating that it does not participate in KC delivery.

B.C SUMMARY OF THE METHOD FOR DESIGNING ASSEMBLIES 213

8.B HISTORY AND RELATED WORK

Assemblies have been modeled systematically by [Lee

and Gossard], [Sodhi and Turner], [Srikanth and Turner],

and [Roy et al.] and others Such methods are intended to

capture relative part location and function, and they

en-able linkage of design to functional analysis methods like

kinematics, dynamics, and, in some cases, tolerances

Al-most all of them need detailed descriptions of parts to start

with, in order to apply their techniques [Gui and Mantyla]

applies a function-oriented structure model to visualize

as-semblies and represents them in varying levels of detail

In this book, we have not attempted to model assemblies

functionally Our work begins at the point where the

func-tional requirements have been established and there is at

least a concept sketch

Top-down design of assemblies emphasizes the shift in

focus from managing design of individual parts to

man-aging the design of the entire assembly in terms of

me-chanical "interfaces" between parts We saw in Chapter

4 that [Smith] proposes eliminating or at least

minimiz-ing critical interfaces, rather than part-count reduction, in

the structural assembly of aircraft as a means of

reduc-ing costs He emphasizes that, at every location in the

assembly structure, there should only be one controlling

element that defines location, and everything else should

be designed to "drape to fit." In our terms, the controlling

element is a mate and the joints that drape to fit are

con-tacts [Muske] describes the application of dimensional

management techniques on 747 fuselage sections He

de-scribes a top-down design methodology to systematically

translate key characteristics to critical features on parts

and then to choose consistent assembly and fabrication

methods These and other papers by practitioners indicate

that several of the ideas to be presented here are already in

use in some form but that there is a need for a theoretical

foundation for top-down design of assemblies

Academic researchers have generated portions of this

foundation [Shah and Rogers] proposes an attributed

graph model to interactively allocate tolerances, perform

tolerance analysis, and validate dimensioning and

toler-ancing schemes at the part level This model defines chains

of dimensional relationships between different features on

a part and can be used to detect over- and sioning (analogous to over- and underconstraint) of parts.[Wang and Ozsoy] provides a method for automaticallygenerating tolerance chains based on assembly features inone dimensional assemblies [Shalon et al.] shows how toanalyze complex assemblies, including detecting incon-sistent tolerancing datums, by adding coordinate frames

underdimen-to assembly features and propagating the underdimen-tolerances bymeans of 4 x 4 matrices [Zhang and Porchet] presentsthe oriented functional relationship graph, which is sim-ilar to the DFC, including the idea of a root node, prop-agation of location, checking of constraints, and prop-agation of tolerances A similar approach is reported

in [Tsai and Cutkosky] and [Soderberg and son] The DFC is an extension of these ideas, empha-sizing the concept of designing assemblies by designingthe DFC first, then defining the interfaces between parts

Johannes-at an abstract level, and finally providing detailed partgeometry

CAD today bountifully supports design of individualparts It thus tends to encourage premature definition ofpart geometry, allowing designers to skip systematic con-sideration of part-part relationships Most textbooks onengineering design also concentrate on design of machineelements (i.e., parts) rather than assemblies

Current CAD systems provide only rudimentary sembly modeling capabilities once part geometry exists,but these capabilities basically simulate an assembly draw-ing Most often the dimensional relations that are explic-itly defined to build an assembly model in CAD are thosemost convenient to construct the CAD model and are notnecessarily the ones that need to be controlled for properfunctioning of the assembly What is missing is a way torepresent and display the designer's strategy for locatingthe parts with respect to each other, which amounts to theunderlying structure of dimensional references and mutualconstraint between parts The DFC is intended to capturethis logic and to give designers a way to think clearly aboutthat logic and how to implement it

as-8.C SUMMARY OF THE METHOD FOR DESIGNING ASSEMBLIES

Ideally, the design of a complex assembly starts by a

general description of the top-level requirements in the

form of KCs for the whole assembly These requirements

are then systematically formalized and flowed down tosubassemblies and finally down to individual parts Theassembly designer's task is to create a plan for delivering

each KC To do this, he or she defines a DFC for each KC,

showing how the parts in each DFC will be given their

desired nominal locations in space This is equivalent to

properly constraining each part During these early stages

of design, the designer has to do the following:

Systematically relate the identified KCs to important

datums on subassemblies, parts, and fixtures at the

various assembly levels from parts to subassemblies

to the final assembly

Design consistent dimensional and tolerance

rela-tionships or locating schemes among elements of the

assembly so as to deliver these KC relationships

Identify assembly procedures that best deliver the

KCs repeatedly without driving the costs too high

These major elements of the assembly design process

are implemented by establishing three basic kinds of

in-formation about an assembly:

"Location responsibility": Which parts or fixtures

lo-cate which other parts

Constraint: Which degrees of freedom of a part are

constrained by which surfaces on which features on

which other parts or fixtures, including checking for

inappropriate over- or underconstraint

Variation: How much uncertainty there is in the

lo-cation of each of the parts relative to some base part

or fixture which represents the reference dimension

The design process comprises two steps: nominal

de-sign and variation dede-sign The nominal dede-sign phase

cre-ates the constraint structure described above, by using the

concepts in Chapter 4, and assuming that the parts and

their features are rigid and have nominal size, shape, and

location The variation design phase comprises making the

DFC robust against variations away from nominal

dimen-sions, plus checking each DFC using traditional tolerance

analysis, as described in Chapters 5 and 6, to determine if

each KC can be delivered A KC, as described in

Chap-ter 2, is said to be "delivered" when the required geometric

relationship is achieved within some specified tolerance an

acceptable percent of the time

The DFC provides a way to define a competent nominal

assembly Nominal means that the assembly has all its

di-mensions at their ideal values and that there is no variation

Competent means that the assembly is capable of properly

constraining all its parts, that all its KCs have been

identi-fied, and that a way to deliver each KC has been provided

We will see below that these elements of "competency"are all related to each other and that they are really differ-ent ways of saying the same thing Furthermore, they can

be addressed using the nominal dimensions Once we aresure that the nominal design is competent, we can exam-ine it for its vulnerability to variation Portions of this stepare included in conventional tolerance analysis, but it willbecome clear that we mean much more than that.The method is capable of describing assemblies that arebuilt simply by joining parts as well as those that are builtusing fixtures In either case, the participating elements(parts and fixtures) are linked by the DFC and its un-derlying constraint scheme A typical assembly sequencebuilds the DFC beginning at its root or datum referenceand working its way out to the KCs Sequences that "buildthe DFC" are a very small subset of the feasible sequencesfound by methods described in Chapter 7 When DFCs arefound to be deficient during the design process, it oftenemerges that a different assembly sequence is associatedwith an alternate DFC design This fact links assemblysequence analysis to assembly design, variation buildup,and assembly process planning

The method also provides guidance in the surprisinglycommon situation in which there are more KCs than thedegrees of freedom of the assembly can deliver indepen-

dently This situation is called KC conflict We will see

that KC conflict can be detected using the methods ofconstraint evaluation presented in Chapter 4

In this method, parts3 are merely frameworks that holdassembly features, while assembly features are the linksthat establish the desired state of constraint among adja-cent parts, leading to the achievement of the assembly-level geometric relationships The DFC is an abstractversion of this framework, providing a kind of skeletonfor the assembly

The mathematical foundation of the method is the 4 x 4transform and Screw Theory, which are used to describethe three-dimensional locations of parts and features, todetermine the degrees of freedom constrained by indi-vidual features, and to check for proper constraint whenparts are joined by sets of features These elements of themethod were presented in Chapters 3 and 4

3 Here, we mean parts considered only from the point of view of their membership in the assembly, not as, for example, carriers of load or liquids, barriers against heat flow, and so on These factors comprise significant requirements on parts that must be considered as part of their design.

8.D DEFINITION OF A DFC 215

An important conclusion from this method is that most

of the information required to support it can be stored

as text Very little detailed geometry is needed, and its

use is isolated to a few steps in the process and a few

places on the parts This is important because it reflects

the fact that the most important steps in designing an

as-sembly comprise establishing connectivity and constraint,

not defining geometry This, in turn, is important because

it provides a route to representing assembly information

more abstractly, richly, and compactly than is permitted

by geometry alone This, in turn, provides a language and

other constructs for capturing this information as a ral part of the design process, avoiding the need to dis-cover it by analyzing geometry, as many CAD systems dotoday

natu-A corollary is that the method describes steps that mand the careful definition of a data and decision recordthat constitutes declaration of the consistent design intentfor the assembly This record can be used to judge the ad-equacy of the design as well as to manage its realization

de-up and down the sde-upply chain and debug that realization

on the factory floor and in the field

8.D DEFINITION OF A DFC

8.D.1 The DFC Is a Graph of

Constraint Relationships

A datum flow chain is a directed acyclic graphical

repre-sentation of an assembly with nodes representing the parts

and arcs representing mates between them "Directed"

means that there are arrows on the arcs "Acyclic" means

that there are no cycles in the graph; that is, there are no

paths in the graph that follow the arrows and return to the

start of the path Loops or cycles in a DFC would mean

that a part locates itself once the entire cycle is traversed,

and hence are not permitted Every node represents a part

or a fixture, and every arc transfers dimensional constraint

along one or more degrees of freedom from the node at

the tail to that at the head Each arc has an associated

4 x 4 transformation matrix that represents

mathemati-cally where the part at the head of the arc is located with

respect to the part at the tail of the arc A DFC has only

one root node that has no arcs directed toward it, which

represents the item from which the locating scheme

be-gins This could be either a carefully chosen base part or

a fixture A DFC can be a single chain of nodes or it can

branch and converge For example, if two assembled parts

together constrain a third part, the DFC branches in order

to enter each of the first two parts and converges again on

the third part

Figure 8-4 shows a simple liaison diagram and

associ-ated DFC In this DFC, part A is the root It completely

locates parts B and C Parts A and C together locate part

D A thought question at the end of the chapter asks the

reader to define some assembly features that are able to

accomplish this locating scheme

FIGURE 8-4 A Simple Liaison Diagram and Datum Flow Chain. The liaison diagram (left) shows which parts are con- nected to each other The DFC (right) shows how they are connected and constrained Each arc is labeled with the de- grees of freedom it constrains or the names of those degrees

of freedom in any convenient coordinate system This DFC

is intended to deliver a KC between parts A and D The KC

is indicated by the double line next to the arrow No mation is given regarding which degrees of freedom are of interest in this KC.

infor-Every arc in a DFC is labeled to show which degrees offreedom it constrains, which depends on the type of matingconditions it represents The sum of the unique degrees offreedom constrained by all the incoming4 arcs to a node

in a DFC should be equal to six (less if there are somekinematic properties in the assembly or designed matingconditions such as bearings or slip joints which can ac-commodate some amount of predetermined motion; more

if locked-in stress is necessary such as in preloaded ings) This is equivalent to saying that each part should

bear-be properly constrained, except for cases where over- orunderconstraint is necessary for a desired function

4 Arcs that are "incoming" to a node are defined as arcs whose arrows point toward the node.

A DFC is similar in many ways to an electric circuit

diagram A circuit diagram defines a connection structure

or network that has many properties of its own,

indepen-dent of the resistors, capacitors, and other individual

cir-cuit elements It has a unique ground or reference voltage

Many operating characteristics of the circuit can be

cal-culated from its graphical properties, such as spanning

trees and independent loops Both the nominal operating

behavior and the sensitivity to component variations can

be calculated from the circuit We will see that many of

these properties of electric circuits are shared by DFCs,

including their ability to set the agenda for design and

analysis

8.D.2 Nominal Design and

Variation Design

The DFC represents the designer's intent concerning how

the parts will obtain their locations in space in all six

de-grees of freedom Each KC will have its own DFC, and

thus each DFC is responsible for delivering its KC If the

parts are perfect, then the KC will be delivered perfectly

If they are not, then a variation analysis like those in

Chap-ter 6 must be undertaken Variation in parts passes from

part to part along the DFC and accumulates to determine

the variation in the KC Thus the DFC acts as a tolerance

chain that guides the designer in finding all the variations

that contribute to each KC It is not necessary to perform

a separate analysis to find the tolerance chain in order to

carry out the variation analysis of a KC

8.D.3 Assumptions for the DFC Method

The following assumptions are made to model the

assem-bly process using a DFC:

1 All parts in the assembly are assumed rigid Hence

each part is completely located once its position

and orientation in three dimensional space are

determined

2 Each assembly operation completely locates the part

being assembled with respect to previously

assem-bled parts or an assembly fixture Only after the part

is completely located is it fastened to the remaining

parts in the assembly

Assumption 1 states that each part is considered to befully constrained once three translations and three rota-tions are established If an assembly, such as a preloadedpair of ball bearings, must contain locked-in stress in order

to deliver its KCs, the parts should still be sensibly strained and located kinematically first, and then a planshould be included for imposing the overconstraint in thedesired way, starting from the unstressed state If flexibleparts are included in an assembly, they should be assumedrigid first, and a sensible locating plan should be designedfor them on that basis Modifications to this plan may benecessary to support them against sagging under gravity

con-or other effects of flexibility that might cause some oftheir features to deviate from their desired locations in theassembly

Assumption 2 is included in order to rationalize theassembly process and to make incomplete DFCs makesense An incomplete DFC represents a partially com-pleted assembly If the parts in a partially completed as-sembly are not completely constrained by each other or

by fixtures, it is not reasonable to expect that they will

be in a proper condition for receipt of subsequent parts,in-process measurements, transport, or other actions thatmay require an incomplete assembly to be dimension-ally coherent and robust This assumption enables us tocritique alternate assembly sequences, as explained inSection 8.K

8.D.4 The Role of Assembly Features

in a DFC

The DFC comprises design intent for the purpose of ing the parts but it does not say how the parts will be lo-cated Providing location means providing constraint Weknow from the foregoing chapters that assembly featuresare the vehicles we use to apply constraint between parts.Thus the next step after defining the DFC is to choose fea-tures to provide the constraint Once features have beendeclared, we can calculate the nominal locations of all theparts by chaining their 4 x 4 transforms together, and wecan check for over- or underconstraint, using methods thatare by now familiar

locat-In order to be precise about our locating scheme, ever, we need to distinguish two kinds of feature joints:mates and contacts These are the subject of the nextsection

how-8.E MATES AND CONTACTS 217

8.E MATES AND CONTACTS

A typical part in an assembly has multiple joints with other

parts in the assembly Not all of these joints transfer

lo-cational and dimensional constraint, and it is essential to

distinguish the ones that do from the ones that are

redun-dant location-wise and merely provide support or strength

We define the joints that establish constraint and

dimen-sional relationships between parts as mates, while joints

that merely support and fasten the part once it is located

are called contacts Hence mates are directly associated

with the KCs for the assembly because they define the

resulting spatial assembly relationships and dimensions

The DFC therefore defines a chain of mates between the

parts If we recall that the liaison diagram includes all the

joints between the parts, then it is clear that the DFC is a

subset of the liaison diagram The process of assembly is

not just of fastening parts together but should be thought

of as a process that first defines the location of parts using

the mates and then reinforces their location, if necessary,

using contacts

8.E.1 Examples of DFCs

This section uses some simple examples to illustrate how

to draw a DFC starting from the KC(s) The first example

is assembly of an automobile wheel to an axle The second

is assembly of three simple sheet metal parts Both

exam-ples illustrate the difference between mates and contacts

8.E.1.a Wheel and Axle

Consider Figure 8-5, a simplified automobile axle and

wheel The axle hub includes a rim plus four studs The

wheel contains a round opening in the center, plus four

holes, larger than the studs, centered around this opening

When the wheel is mounted to the hub, the opening fits

snugly over the rim and the studs protrude through the

holes, ready for the nuts to be installed

The designer's goal for this design is to achieve

dy-namic balance and a smooth ride The KCs he has chosen

to achieve this goal are as follows:

Make the wheel concentric with the axle shaft's

fea-in the coordfea-inate frame fea-in which the wheel's fea-inertia trix is diagonal, and the opening must be centered on thisframe.5 In our terms, the hub face and rim constitute matefeatures, as do the wheel plane and opening The studs andtheir holes constitute contacts They play no role in achiev-ing the KCs They merely keep the wheel from falling off

ma-Of course, this is important and we could have called it a

KC, but achieving it does not depend on how the parts volved are geometrically located The important constraintrelationships between the axle and wheel are completelydetermined by the mate features already defined

in-A DFC for the wheel and hub is shown in Figure 8-6 Itrepresents mates as graph arcs with arrows on them as well

as a number indicating how many degrees of freedom arelocated by the mate Contacts are shown as dashed lines.All the important features are defined, and their roles inestablishing constraint relationships and KCs are shown

5 Small errors in the wheel features are inevitable due to the dictability of the mass distribution of the rubber tire These are re- moved by dynamically balancing the wheel using small lead weights.

unpre-FIGURE 8-5 A Wheel and Axle Illustrating the Difference Between Mates and Contacts. The dimensional and con- straint relationships between the wheel and axle are estab- lished by the mate between the wheel's opening and the axle's rim, as well as by the mate between the planar face of the wheel and the planar face of the hub All other interfaces between these parts provide no constraint and are contacts.

FIGURE 8-6 DFC and KCs for the Wheel and Axle in

Fig-ure 8-5 Top: The simplest representation of the DFC for this

assembly consists of two nodes representing the parts, a set

of parallel lines representing a KG, and one arrow with the

number 5 on it, indicating that the axle has a mate with the

wheel that defines 5 of its degrees of freedom. Bottom: A

little more detail (adapted from [Zhang and Porchet]) reveals

that the KG can be decomposed into two separate KCs and

that different features on the parts are involved in delivering

them The features on the axle and wheel are related in

differ-ent ways The hub and rim on the axle each have mates with

the opening and plane on the wheel, respectively Together,

these features define 5 of the wheel's six degrees of freedom

and all the KCs The joint between the studs and holes is a

dashed line, indicating that it is a contact When the nuts are

tightened onto the studs, the sixth degree of freedom is

fas-tened, but its exact value is not of interest to us There is no

KC on this dimension The studs fit easily into oversize holes,

and any orientation of the wheel within the stud-hole

clear-ance is acceptable.

Note that one of these datum features is the axle's

cen-terline This is not a piece of geometry itself Calling it a

feature is, however, perfectly consistent with GD&T

Figure 8-7 expands the DFC for the assembly to show

all the necessary features on each part and their relative

location requirements The symbolic blobs in Figure 8-6

representing the two parts, with their four black dots

rep-resenting the important features, have been expanded to

show the perpendicularity and concentricity relationships

between the features Also shown is a possible

simpli-fied statement of these requirements for the axle using the

symbols of GD&T as discussed in Chapter 5 Figure 8-5,

Figure 8-6, and Figure 8-7 present together a simple

exam-ple of definition of assembly requirements, their capture

as KCs, the definition of DFCs to deliver these KCs, the

identification of feature-to-feature relationships between

the parts that create the necessary mates, and finally

def-inition of the resulting requirements on mutual feature

relationships inside one of the parts of this assembly It

FIGURE 8-7 DFC with Features and Their Required tual Locations Inside the Parts. Above is an expanded view

Mu-of the assembly in symbolic form It shows all the interpart lationships between features These features play essential roles in delivering the axle-wheel assembly's KCs Below is a possible simplified rendition of a GD&T specification for real- izing the necessary feature-to-feature relationships inside one

re-of the parts The interpart relationships express the ments that the hub must be perpendicular to the axle shaft's centerline and that the rim must be located with respect to the centerline, both within some tolerances The circle on which the studs lie must also be located with respect to the shaft centerline, but a larger tolerance is allowed The root of the DFC in the axle's centerline is also the A datum for the axle.should be clear from these figures that the DFC represents

require-a continuous chrequire-ain not only between prequire-arts but inside them

as well The only difference between the arcs of a DFCbetween parts and the arcs inside a part is that only materelationships exist inside parts Contact relationships existonly between parts

An alternate design for joining these parts is commonlyused It dispenses with the rim and its mating opening anduses five studs and holes instead The nuts have generouschamfers on them where they engage chamfered holes inthe wheel A thought question at the end of the chapterasks the reader to compare this alternate design with theone described here

8.E MATES AND CONTACTS 219

TABLE 8-1 Distinguishing Mates and Contacts

Full six dof constrained

No dof constrained

Some dof constrained

along a KC

Yes No

Yes along KC directions

No Yes

Yes along non-KC directions

Square peg in square hole Nuts attaching wheel to axle hub Rim on axle hub; slip joint in sheet metal

FIGURE 8-8 An Assembly with a Mate and a Contact.

The KC is the overall length L of the assembly In the

direc-tion of the KC, the A-B joint provides locadirec-tion and constraint,

but the B-C joint does not It simply joins B and C and will

do so as long as overlap dimension b is large enough.

8.E.1.b Sheet Metal Parts

Figure 8-8 above shows three simplified sheet metal

au-tomobile body parts Between them they have two joints,

namely, one butt joint called a mate and one slip joint

called a contact.6 The KC is the overall length L of the

assembly The slip joint can be adjusted in the direction

oftheKC

If we consider this to be a full three-dimensional

as-sembly, then it is obviously underconstrained, and neither

of the joints would then be called a mate However, if we

consider the KC, which specifies one dimension only, then

we could argue that the joint between A and B is a mate

because it constrains the part-to-part relationship in a

di-rection that contributes to delivery of the KC Similarly,

we could argue that the joint between B and C is a contact

because it does not provide such constraint

However, the B-C joint clearly does provide constraint

in the direction normal to the planes of the parts Why then

call it a contact? The reason is that there is no KC specified

in that direction to which this joint makes a contribution

This leads us to a rule, namely that every assembly must be

properly constrained (up to the limit where function may

require some unconstrained degrees of freedom) but not

every joint that provides constraint in some direction(s)

6 Butt joints and slip joints were introduced in Chapter 6 In the auto

industry, the butt joints are called coach joints.

has to be a mate Underconstrained assemblies need help

to achieve proper constraint beyond what the joints selves can provide As we will see below, fixtures are usu-ally used to provide the missing constraint Typically, theparts will have joints with the fixtures at these points andthe DFC will pass through these part-fixture joints, caus-ing us to call them mates

them-Table 8-1 combines these definitions Later in this ter we will use the name "hybrid mate-contact" to refer tojoints that provide incomplete constraint and which act asmates along the directions they constrain In terms of thedefinitions used in Chapter 4, joints that provide full sixdegree of freedom (dof) constraint play the role of "loca-tors" while joints that provide no constraint play the role

chap-of "effectors."

8.E.2 Formal Definition of Mate and Contact

Generalizing on Table 8-1, we can categorize all joints tween parts as shown in Figure 8-9 This figure makes use

be-of the concepts be-of wrench space and twist space introduced

in Chapter 4 It permits us to examine a joint cally, surface contact by surface contact, to determine thefunction of each surface contact in the assembly

systemati-The categorization in Figure 8-9 can be applied tojoints or to fundamental surface-to-surface contacts asdiscussed in Chapter 4 For example, Figure 8-10 reviewsthe cylinder-plane contact and shows its twist space andwrench space Constraint and variation occur only alongthe directions in the wrench space

8.E.3 Discussion

Explicit identification and definition of the mates in anassembly is an integral part of assembly design and is aprerequisite to assembly process planning and variationanalysis The choice of which joints will be mates andwhich ones will be contacts is made by the designer at theconceptual design stage

Example Contact?

Mate?

Function

FIGURE 8-9 Categories of Joints Between Parts. Some joints are mates while others are contacts Within each mate is a twist space and a wrench space Constraint behavior characteristic of a mate occurs in its wrench space Adjustment behavior (typically asso- ciated with contacts) can occur in its twist space Joints where this occurs are called hy- brid mate-contacts.

FIGURE 8-10 Twist Space (a) and Wrench Space (b) for the Cylinder-Plane Surface Contact.

8.F TYPE 1 AND TYPE 2 ASSEMBLIES EXAMPLE 221

When defining the DFC, the designer must define

ex-plicitly the surfaces or reference axes on mating features

which are intended to carry dimensional constraint to the

mating part This approach makes it unnecessary, even

counterproductive, to construct algorithms that "identify"

tolerance chains or loops, since the DFC equips the

de-signer to define them purposefully as a main objective of

assembly design On the other hand, defining the DFC and

its implementing features prepares the designer to carryout the steps of GD&T or some other systematic toleranc-ing scheme for each part, as illustrated by the example inFigure 8-5 through Figure 8-7

We turn next to the distinction between two types of semblies, called Type 1 and Type 2 The DFCs for these,and the strategies used to achieve their KCs, are quitedifferent

as-8.R TYPE 1 AND TYPE 2 ASSEMBLIES EXAMPLE

To clarify our approach to designing assemblies, we need

to distinguish between two kinds of assemblies, which we

call Type 1 and Type 2 Type 1 assemblies are constrained

completely by feature relations between their parts Type 2

assemblies are underconstrained by their features and need

fixtures or measurements to add the missing constraint

We will illustrate the difference with an example from the

automobile industry

Figure 8-11 shows a simplified car floor pan.7 This

as-sembly consists of three stamped sheet metal parts The

KC is the overall width of the car, which is nominally of

dimension L The design shown in the figure consists of

parts with flanges that are spot-welded together to form

butt or coach joints On the right in Figure 8-11 is the

liaison diagram for this assembly, showing the KC as a

double line joining parts A and C Parts A and C contain

the features that must be a distance L apart in order to

deliver the KC

The way this assembly has been designed, each part

lo-cates the adjacent part in the left-right direction by means

of a flange, a short piece of metal that is intended to be

per-pendicular to the plane of the part This flange is formed

by stamping the part from flat stock The flanges are

typ-ically spot welded together As discussed in Chapter 6,

when such a part is stamped, there is some uncertainty

in the bend radii at each end The result of this is that the

overall width of the part from flange to flange is uncertain

Figure 8-12 shows a DFC for this assembly Because

each part locates the adjacent part, we say that it has a

mate with that part We indicate this with arrows between

the parts in the DFC Figure 8-12 can be read to say: "Part

A locates part B and part B locates part C The KC is a

geometric relationship between part A and part C." Note

7 This example was provided by Robert Bonner and James

D'Arkangelo of Ford Motor Company.

that we can trace a chain of mates from one end of the KC

to the other Note, too, that the flange joints completelyconstrain the adjacent parts along this chain On this ba-sis, we say that this assembly is a Type 1 The direction

of the chain, as well as the designation of part A as theroot, is arbitrary A feasible assembly sequence for thisassembly is

1 Mate parts A and B;

2 Mate parts B and C

All of the foregoing, together with the DFC, comprisethe documentation of the design intent for this simpleassembly

Figure 8-13 shows an alternate design for this bly It differs from that shown in Figure 8-11 in that there

assem-is a contact between part B and part C The designer hasproposed this design because he predicts that the sizes ofthe parts measured between the flanges will not be accu-rate enough to ensure delivery of the KC He knows that

only the overall width L matters, so he has shown parts B

and C joined by a slip joint This joint can be adjusted so

that width L will be achieved.

However, this design differs fundamentally from theoriginal A candidate DFC appears in Figure 8-14 ThisDFC does not contain a chain of mates from one end ofthe KC to the other In fact, we can see that part B andpart C do not constrain each other in the direction of the

KC These two facts tell us that this is a Type 2 assemblyand that we need a fixture or measurement to provide themissing constraint

Figure 8-15 shows a candidate fixture designed to move the under-constraint from this assembly, while Fig-ure 8-16 shows the DFC that applies to the assembly whenthis fixture is used A number of points are worth noticing.First, it is now possible to trace a chain of mates throughthe DFC from one end of the KC to the other, although this

re-FIGURE 8-11 Example Simplified Car Floor Pan Left: Top and side views of a three-part sheet metal car floor pan These are U- or channel-shaped parts stamped from flat stock The KC for this assembly is its overall width L Right: The liaison diagram and the KC.

FIGURE 8-12 Datum Flow Chain for the Assembly in Figure 8-11. Part A locates part B while part B locates part C The KC is a geometric relation- ship between A and C.

FIGURE 8-13 Alternate Design for Car Floor Pan The

KC is the same as in Figure 8-11, but in this design there

is a slip joint contact between part B and part C.

FIGURE 8-14 Proposed DFC for the Assembly in

Fig-ure 8-13. The mate is shown by an arrow as in Figure 8-12,

while the contact is shown as a dashed line In this DFC it

is not possible to trace a chain of mates from one end of the

KC to the other This DFC does not completely constrain the

parts It is therefore not capable of delivering the KC.

FIGURE 8-15 Fixture for Providing Constraint for Parts B

andC.

chain, unlike that in Figure 8-12, not only passes throughparts but also passes through the fixture Indeed, whereasparts B and C have a contact with each other, they havemates with the fixture The fixture provides the missingconstraint in the direction of the KC via these mates Wecan read Figure 8-16 to say: "The fixture locates parts Band C, while part B locates part A." All the methods welearned in Chapter 4 about assessing the adequacy of con-straint can be used on feature mates between parts andfixtures, just as they can on feature mates between parts

In this case, such an analysis will reveal that the fixture

is free to constrain parts B and C because the contact tween these parts applies no constraint of its own in thedirection of interest.8 If this contact were a mate, thenthere would be overconstraint in this fixture design.The assembly process implied by Figure 8-13,Figure 8-15, and Figure 8-16 is as follows:

be-1 Place parts B and C in the fixture and weld themtogether

2 Weld part A to part B, completing the assembly

No other assembly sequence is possible using the ture in Figure 8-15

fix-It may appear that we are finished, but in fact we are not.There is an alternate way to remove the under-constraintfrom this assembly It is shown in Figure 8-17 The cor-responding DFC is shown in Figure 8-18 We can readFigure 8-18 to say: "The fixture locates parts A and C,while part A locates part B." Again, we can trace a chain

of mates in this DFC from one end of the KC to the other,and again it passes through the fixture The assembly

8 Remember that this is a one-dimensional example, so only the right dimension matters The KC is measured in this direction, and the mates between parts, and between parts and fixtures, apply con- straint only in this direction.

left-8.F TYPE 1 AND TYPE 2 ASSEMBLIES EXAMPLE 223

FIGURE 8-16 Improved DFC for the Assembly in Figure 8-13 Using the Fixture in Figure 8-15. In this DFC, it

is possible to trace a chain of mates from one end of the KC to the other.

However, this chain passes through the fixture.

FIGURE 8-17 Alternate Fixture Design for the Assembly

in Figure 8-13. If this fixture is used, then part A is welded

to part B first using the mate The weld is indicated as the fat

gray line Then the subassembly of A and B is placed in the

fixture on top of part C The parts are pushed firmly against

the ends of the fixture to create the A-F and C-F mates, and

finally the contact is fastened Alternately, all the parts can

be placed in the fixture at once as long as the A-B, B-C

fastening sequence is followed.

process implied by Figure 8-13, Figure 8-17, and

Fig-ure 8-18 is as follows:

1 Mate part A and part B

2 Put the A-B subassembly in the fixture and join

part C to part B

Alternately, do the following:

1 Place all the parts in the fixture

2 Weld part A to part B, then weld parts B and C

together

No other sequences are possible using this fixture, and

only one joining sequence for the parts has a chance of

delivering the KC

Are the two assembly strategies, fixtures, and DFCsfor this Type 2 assembly shown in Figure 8-16 and Fig-ure 8-18 equivalent? Let us recall the reason why the de-signer chose to investigate a Type 2 in the first place Hewanted the ability to adjust one of the joints (he choseB-C) so as to improve the likelihood of delivering the

KC We are not done comparing the design alternatives,including the Type 1 in Figure 8-12, until we examine, atleast in principle, the variation that could result from each

so that we can compare their ability to deliver the KC.First, examine the DFC in Figure 8-12 The variation

in the KC arises from the combination of the individualpart variations Since we know that stamped flanges couldcontain variation affecting the size of the part, we knowthat this design is vulnerable to this kind of error Second,examine the design in Figure 8-16 Without performing adetailed variation analysis, we can see that it could sufferfrom the same difficulty as the design in Figure 8-12 be-cause variation from the stamping of Part A could still be

a factor In addition, there is going to be some variationdue to the construction of the fixture Finally, examine thedesign in Figure 8-18 Here, the KC is completely underthe control of the fixture, and fixture variation will be theonly contributor While we will not conduct a full variationanalysis, it is a good bet that the third design will have theleast variation A thought question at the end of the chapterasks the reader to analyze this situation quantitatively.The designs in Figure 8-12 and Figure 8-16 both suf-fer from error due to stamping the flanges, although thetotal variation is larger in Figure 8-12 Only the design inFigure 8-18 really eliminates error due to flange stamp-ing The reason it can while that in Figure 8-16 cannot

is a fundamental one that we will make into a rule Thisrule states that in an assembly where a part is connected

to others, or to fixtures, by both mates and contacts, theincoming mates should be fully fastened before any ofthe contacts are fastened.9 The reason is that the incomingmates define the location of the part If a contact is fastenedbefore all the incoming mates are fastened, then the partwill be positioned at least in part by the one to which it has

a contact The contact has thus been given a role it doesnot have the capability to handle, namely to provide loca-tion for another part When the remaining mate(s) is/arefastened, there are two possibilities First, the variation

9 In Figure 8-16, part C's contact with part B is made before its mate with the fixture In Figure 8-18, this sequence is reversed, and the B-C contact is made after C acquires its mate with the fixture.

FIGURE 8-18 DFC for the Assembly in

Figure 8-13 Using the Fixture in

Fig-ure 8-17.

in the KC will be larger than it would have been if all

the mates were fastened first Second, an overconstrained

situation could result

We can make another observation from this example

that will hold for others: In a Type 1 assembly, the overall

variation depends directly on the variation in the

individ-ual parts Furthermore, the assembly sequence does not

matter; every assembly sequence will give the same final

variation in the assembly In a Type 2 assembly, the overall

variation depends not only on the parts, but also on tures or, more generally, on the assembly process Equiv-alently, we can say that different Type 2 assemblies are infact different assembly processes, with different fixtures,different assembly sequences, and different final varia-tion in the assembly, even though they assemble the sameparts Type 1 assemblies may thus be called part-driven,while Type 2 assemblies are called assembly process-driven

fix-8.G KC CONFLICT AND ITS RELATION TO ASSEMBLY SEQUENCE

AND KC PRIORITIES

As mentioned in Chapter 2, a single assembly can often

have several KCs associated with it Because each

assem-bly has a limited number of mates and assemassem-bly steps, it

is possible that achievement of the KCs cannot be

guaran-teed independently Multiple KCs in the same assembly

can be classified as follows:

Independent: The delivery chains of the KCs share

no degrees of freedom of any mates The variation

in each KC is completely independent of the

varia-tion in every other KC For example, in Figure 8-6,

the concentricity KC and the perpendicularity KC

arise from different degrees of freedom and feature

surfaces, and follow separate DFC delivery chains

Correlated: The delivery chains share some degrees

of freedom Variation in these degrees of freedom

will affect all KCs that share them However, there

is still some opportunity to improve the variation of

each KC without degrading the variation of the

oth-ers In Figure 8-2, the variation of each KC in the

stapler is lower-bounded by the carrier's variation

However, the probability of each KC achieving its

tolerance also depends on variation in other parts

that are not shared The correlation may in some

cases be broken by such means as providing an

ad-justment or resorting to selective assembly in one

or more of the legs of the DFC that are not shared

([Goldenshteyn]) But these are serious redesigns and

are often unavailable

Conflicting: The KC delivery chains share so many

degrees of freedom that attempts to improve one KC

will always degrade another; or the probability of

achieving one KCs tolerance requirement will always

be lower than the probability of achieving another

KC conflict can arise in two ways In one situation,there is no remedy short of drastic redesign of the parts,while in the other, the conflict can be resolved by choosinganother assembly sequence

1 The DFCs for different KCs share so many arcs thatany adjustments or statistical error accumulationswill be identical or additive, preventing indepen-dent achievement of the KCs This is illustrated inFigure 8-19 There is no possibility in such situa-tions of relieving the problem by choosing a differ-ent assembly sequence Instead, one must choose

a priority for the KCs, and the one that is finishedfirst in the assembly sequence is the one that willhave the higher probability of being achieved, or

is the one that may be given tighter tolerances.This situation may occur in Type 1 or Type 2 as-semblies Here, too, redesign of the parts to permitadjustments or selective assembly may relieve thesituation

2 Due to the requirement that each subassembly becompletely constrained, some KCs may be impos-sible to adjust into achievement because the avail-able degrees of freedom were "used up" during priorassembly steps Some assembly sequences permitindependent achievement of the KCs, while oth-ers do not This is illustrated in Figure 8-20 and

is discussed in connection with aircraft assembly inSection 8.1.3 This situation occurs only in Type 2assemblies

An indicator that KC conflict could arise is the case wheremore than one KC chain is completed at the same assem-bly step ([Arora]) This is illustrated for car doors later inthis chapter in Figure 8-46

8.G KC CONFLICT AND ITS RELATION TO ASSEMBLY SEQUENCE AND KC PRIORITIES 225

KC 2 is favored if tolerance on KC 1 = tolerance on KC 2

Or, same probability of KC delivery requires

tolerance on KC 1 > tolerance on KC 2

KC 1 is favored if tolerance on KC 1 = tolerance on KC 2

Or, same probability of KC delivery requires tolerance on KC 2 > tolerance on KC 1

FIGURE 8-19 Example of KC Conflict. This example is similar to that in Figure 8-13 with the addition of a second KC (Pr (KC-\) means probability of achieving KC-\ sp-\ means error in fixture F1. KC-\u means upper specification limit on KC-\.

Other notation is to be interpreted similarly.) There is a chain of mates from one side of each KC to the other side, but these chains contain arcs that are part of both chains Since there is only one contact by which to adjust two KCs into compliance, one is bound to be achieved with lower probability than the other, or else one must be given looser tolerances than the other This problem exists in both of the assembly sequences shown here.

FIGURE 8-20 Example of KC Sensitivity to Assembly Sequence. The parts in Figure 8-19 have been rearranged so that there are two contacts in the assembly and no mates Now there are in principle enough degrees of freedom to adjust both KCs into compliance but, if the wrong assembly sequence is used (process 1), one of these degrees of freedom will be used

up before any adjustment can be made, rendering the situation similar to that in Figure 8-19 In process 1 the chains of mates connecting the ends of the KCs share some arcs, whereas in process 2 the chains are independent.

Next Page

226 8 THE DATUM FLOW CHAIN

[Hu] points out that errors in car body assembly can

be traced to their sources by observing whether the

er-rors are correlated at the final assembly level,

subassem-bly levels, or not at any level A correlation occurs when

measurements relating an entire group of parts to a

ref-erence location all show errors in the same direction or

errors of similar magnitude and direction For example,

8.H EXAMPLE TYPE 1 ASSEMBLIES

correlations at the assembly level imply that an entire assembly was built correctly but was installed in the finalassembly incorrectly In this case, there may be no need

sub-to seek error sources at the subassembly level or below.The situation in Figure 8-19 occurs in practice, as il-lustrated in Section 8.1.1, which discusses assembly of cardoors

The fan motor is part of a low-cost table fan It is shown

in Figure 8-21

The motor consists of four main parts, plus fasteners:

the stator, the rotor, and front and rear end housings, as

shown in Figure 8-22 Four long screws hold the

assem-bly together The rotor shaft runs in solid oil-impregnated

self-aligning bronze bearings mounted in the housings

Self-aligning means that the bearings can wobble slightly

about two axes normal to the bearing axis They can do

this because their outer shape is spherical and they mount

in spherical pockets pressed into the housings The axial

location of the rotor with respect to the stator is adjusted

by selecting the right number of spacers and putting them

on the shaft before assembling it to the end housings

Figure 8-23 shows the DFC for the fan motor The KC

relates the rotor and the stator Actually, two dimensions

must be controlled, namely, the axial and radial

relation-ships between rotor and stator that are discussed in the

caption of Figure 8-22 These are called out explicitly in

Figure 8-24, which identifies at least schematically the

features inside each part that play roles in delivering each

KC or controlling each degree of freedom in the assembly

Figure 8-27 is a similarly detailed DFC for the rotor

Important features on the end housings and rotor

con-vey dimensional relationships between these parts Details

10 This section makes use of report material prepared by MIT

stu-dents Cesar Bocanegra, Winston Fan, Sascha Haffner, Yogesh Joshi,

Tsz-Sin Siu, and Carlos Tapia.

about how they are constructed are in Figure 8-25 In spite

of the apparently casual way these features are formed,they are able to provide the necessary accuracy

Several points about the fan motor are worth ing First, the self-aligning bearings in the end housingsprovide both position and angular location for the rotor.The design is symmetric, so we could have chosen eitherhousing and its bearing as the root of the DFC Once we

mention-pick one, we say that its bearing provides X, Y, and Z

location Without the other housing in place, however, the

shaft can wobble about 9 X and 9 y For this reason, we

note on the DFC that the other housing provides angular

alignment about X and Y.

Second, the cast raised bevel features used to align thehousings to the stator strictly speaking create an overcon-strained situation unless a small amount of clearance isprovided Cast-in features are not very accurate, however,

so interference could occur some of the time The signer probably felt that any excess material on the bevelswould be crushed when the screws were tightened and thatthe variation, if any, would be smaller than the tolerancesought on radial centering

de-Third, the self-aligning feature of the bearings preventsoverconstraint from developing between the housings, thestator, and the rotor

Finally, we can easily see from the detailed DFCs inFigure 8-24 and Figure 8-27 how to choose datum fea-tures and the dimensioning scheme for locating the fea-tures of each part so that each part will be able to carryits branch of the DFC For example, we must carefullymake the rotor so that the core is the right diameter andthat the outer-diameter surface is concentric with the shaftcenterline Similarly, we must center the bearings in thehousings with respect to the raised bevels These featuresare important for delivering the axial and radial alignmentKCs necessary for the efficient operation of the motor Athought question at the end of the chapter asks the reader

to consider the rear housing in detail

In this section we will look at some Type 1 assemblies and

learn a few more things about using the DFC: a fan motor,

a front wheel drive automobile transmission, a Cuisinart

food processor, the pump impeller discussed in Chapter 7,

and a machined part assembly for an automobile called a

throttle body

8.H.1 Fan Motor 10

8.H EXAMPLE TYPE 1 ASSEMBLIES 227

FIGURE 8-21 Small Fan Motor Left: The parts—stator and windings, rotor, front and rear housings, plus screws, washers,

and spacers Right: The motor assembled (Photos by the author.)

FIGURE 8-22 Schematic of Fan Motor (S, stator; FH, front housing; RH, rear housing; R, rotor.) Left: Assembled Right:

Showing front housing and rear housing slid away from stator along shaft The partially obscured spheres on the shaft

repre-sent self-aligning bronze bearings The KCs are the correct axial (Z) and radial (X and Y) positions of the rotor with respect

to the stator These are important for the efficiency of the motor To achieve these KCs, the edges of the rotor core must be opposite the edges of the stator, and the outside diameter of the rotor core must be centered radially with respect to the inside diameter of the stator The screws that join the housings and the stator are not shown Thin thrust washers lie on the rotor shaft between the rotor and each housing The correct number of these is selected to just barely fill the axial (Z) gap and center the rotor axial ly.

The discussion about the rotor and housings can begeneralized to an important rule: The chain of feature con-straints within a part is, or should be, a little DFC of itsown, obeying all the rules of a DFC.'! It should be a subset

of the whole assembly's DFC for that KC This rule ments the top-down nature of this approach to design ofassemblies and creates the starting point for detailed de-sign, dimensioning, and tolerancing of individual parts sothat they will play their desired role in delivery of the KCs

imple-8.H.2 Automobile Transmission

FIGURE 8-23 DFC for Fan Motor This DFC delivers the

KC that requires the rotor to be centered, radially and axially,

inside the stator The rear housing aligns the front housing

radially and axially via the stator It aligns the rotor radially by

means of a spherical self-aligning bearing The front housing

has a similar bearing Together, the front and rear housings

locate the rotor Cast-in raised bevels on the rear and front

housings mate to holes in the stator These bevels and holes

are visible in Figure 8-22 and details of their construction

are shown in Figure 8-25 The bevels can also be seen in

Figure 8-26.

Automobile transmissions are complex assemblies prising a die-cast case, a number of planetary gear sets,shafts, and subassemblies called clutches The general

com-"Recall that the features inside a rigid part are always properly constrained with respect to each other For this reason, all feature relations within a part will be mates, never contacts.

FIGURE 8-24 Detailed DFC for the Fan tor This DFC defines two separate KCs and in-

Mo-cludes details of the individual features in each part that are involved in locating the parts with respect to each other and delivering each KC.

A separate DFC can be drawn for each KC A thought question at the end of the chapter asks the reader to do this.

FIGURE 8-25 Detail of Stator Construction.

The stators are built by stacking laminae over pins through the rivet holes The first two and last two laminae have extra large holes at the four corners These enlarged holes are the locating features on the stator that accept cast raised bevels on the front and rear end housings for the purpose of aligning the housings and the stator The screws that fasten these three parts together play no role in locating them because they are smaller in diameter than the holes they pass through The cast raised bevels can be seen in Figure 8-26.

FIGURE 8-26 Detail of Cast Raised Bevel on Motor

Housing (Photo by the author.)

layout of a front wheel drive transmission consists of two

parallel shafts, one concentric with the engine's crankshaft

and the other offset to one side that carries the output power

to the differential and the wheels The clutches are used

to immobilize rings, planets, or suns of different planetary

gear sets, thereby causing the transmission to have a ferent gear ratio The clutches in turn are activated by pis-tons powered by oil pressure provided by an oil pump atone end of the transmission A transfer chain carries powerfrom the input side to the output side These parts and theirrelationships are shown in Figure 8-28 and Figure 8-29.The internal moving parts of the transmission, consist-ing of the rotating clutches and transfer chain hub, make up

dif-a stdif-ack thdif-at must fit between the bottom, formed by the oilpump, and the top, formed by the bell housing Elements

of this stack include layers of clutch plates made of metalwith friction material bound to them Since the thickness

of the friction material is difficult to control, the height ofthis stack is quite uncertain To allow for this uncertainty,the opening between the oil pump and the bell housing ismade deliberately large, and the space is filled by a selectthrust washer The assembly process, here greatly simpli-fied, involves joining the oil pump to the case, inserting

8.H EXAMPLE TYPE 1 ASSEMBLIES 229

a thrust washer for the rotating clutches to thrust against,

inserting the rotating clutches and transfer chain hub, and

measuring the empty space between the top of the case and

FIGURE 8-27 Details of Rotor Construction Top: Two

key dimensions of the rotor are the length L of the core and

the core's radius r In addition, the outer surface of the core

must be concentric with the shaft These two requirements

can be expressed as local DFCs inside the rotor Two such

DFCs are shown at the bottom in this figure together with the

KCs that they deliver.

FIGURE 8-28 Cross-Sectional View

of a Typical Front Wheel Drive mission Input power comes in from

Trans-the engine to Trans-the central shaft It passes through the gears and rotating clutches, then to the transfer chain, and finally through the differential and out

to the wheels The oil pump provides hydraulic power to activate the pistons that operate the clutches to change gears.

the top of the hub Another thrust washer of the correctthickness is selected and placed on top of the hub, and thebell housing is installed on top, closing the case

A problem with this assembly sequence is that the casealso contains a band clutch that must be installed in thecase from the oil pump end before the oil pump is attached

to the case This is a wide sheet of spring steel with tion material on the inside It wraps around the outside ofthe rotating clutches and can stop them from rotating if

fric-it is pulled tight around their outer diameter This actionprovides an additional gear ratio The assembly problemarises because this band is not perfectly circular when it isinstalled It could protrude into the region where the rotat-ing clutches are to be inserted If it does, then the rotatingclutches could collide with it during assembly, strippingoff the friction material or doing worse damage

The author and his Draper colleagues attempted toavoid this problem by choosing a different assembly se-quence This sequence builds the transmission upsidedown from the one described above It starts with the bellhousing, then places a thrust washer on it, then places thetransfer chain hub and rotating clutches on the washer Theempty space is measured between the bottom of the rotat-ing clutches and the case at the oil pump end A washer

of the correct thickness is selected and inserted, the bandclutch is inserted, and the case is closed up by insertingthe oil pump Since the band clutch is inserted after therotating clutches and in full view of the operator, damage

is avoided

FIGURE 8-29 Exploded View Cross Section of a Typical Front Wheel Drive Transmission The parts of the transmis-

sion are shown slightly separated in the vertical direction.

Unfortunately, this alternate sequence cannot be used

The reason is that the two thrust washers are not equivalent

This can be seen by examining Figure 8-29 or Figure 8-30

in detail The oil pump feeds oil to the individual

rotat-ing clutches through circumferential grooves in its central

post Each of these grooves must line up axially

(verti-cally in the figures) with a corresponding groove on the

inner diameter of the rotating clutches If the grooves are

misaligned axially, high-pressure oil will be fed to morethan one piston at the same time It is then possible thatthe transmission will shift into the wrong gear or that itwill try to be in two gears at once This could cause roughshifting or even serious damage to the transmission

We can describe this situation using our vocabulary andsymbols as follows The alignment of the grooves is ob-viously an important KC for this assembly We can create

a DFC for this KC by tracing a path from the face of theoil pump through intermediate parts and features to each

of the grooves, as shown in Figure 8-30 and Figure 8-31.This DFC clearly passes through the thrust washer at theoil pump end If we selected this washer based on theheight of the rotating clutch stack, we would have no abil-ity to control its size for the purpose of achieving the KC Atolerance analysis of this DFC would reveal unacceptablevariation in oil groove alignment

The select thrust washer at the bell housing end is notinvolved in delivering a KC Its job is merely to fill emptyspace It does not control the location of any part or fea-ture It is appropriate to say that it is involved in a contact.However, the thrust washer at the oil pump end must beinvolved in a mate because it is in the chain that controlsthe location of the rotating clutches with respect to the oilpump For this reason, it is part of a DFC

In terms of constraint and degree of freedom sis, we can say that the rotating clutches have only one

analy-FIGURE 8-30 Detail of Oil Pump and Rotating Clutches

Showing Alignment of Oil Grooves These grooves guide

high-pressure oil from the oil pump to the pistons in the

ro-tating clutches Alignment of the oil grooves on the oil pump

and on the rotating clutches is the KC.

8.H EXAMPLE TYPE 1 ASSEMBLIES 231

FIGURE 8-31 DFC to Align Oil Grooves in the Oil Pump Hub and the Rotating Clutches This DFC

starts at the face of the oil pump that mates to the case and follows two paths One path leads to the oil grooves on the oil pump post while the other path leads to the oil grooves on the rotating clutches.

On the way, the second path passes through the thrust washer The lower thrust washer participates

in a mate between the rotating clutches and the oil pump, while the upper thrust washer participates in

a contact between the transfer chain hub and the bell housing.

axial degree of freedom, and if we locate it using a select

washer, we no longer have that degree of freedom

avail-able to us to ensure that the oil grooves line up Our only

alternative would be to provide some means to adjust the

oil pump axially with respect to the case or adjust the hub

axially inside the oil pump, but either would probably be

too expensive and prone to oil leaks

Finally, once we have identified the washer at the bell

housing end as a contact and the washer at the oil pump

end as a mate, then we can invoke the rule that says "make

the mates before the contacts" to give us a clue that the

bell housing end washer must be the last part installed in

the internal stack prior to closing the case

This example shows, among other things, that we can

use the DFC to describe situations that involve selective

assembly It also shows that we can seek alternate

assem-bly sequences to solve assemassem-bly problems, but it may

oc-cur that the alternate sequence is unavailable We will

en-counter this problem again and again, reinforcing the idea

that assembly sequence analysis is an essential element in

design of the delivery strategy for the KCs

8.H.3 Cuisinart12

12 This section makes use of report material prepared by MIT students

Chris Anthony, Cristen Baca, Eric Cahill, Gennadiy Goldenshteyn,

and Amy Rabatin.

driven by an electric motor through a planetary gear train.The sun gear is on the motor shaft, while the ring gear isheld stationary by the top frame The three planet gearsdrive the shaft that turns the knife

The DFC of interest to us here is the one that alignsthe sun gear on the motor shaft to the center of the com-bined pitch circles of the three planet gears Misalignmentmeans a noisy unit that will wear out rapidly One couldimagine delivering this KC either of two ways One waywould prescribe a mate between planets and sun and acontact between the motor and the top frame Assemblywould consist of carefully establishing the mate and thenfastening the contact The other possibility is to establish

a mate between the motor and the top frame and a contact

FIGURE 8-32 A Cuisinart Food Processor (Photo by the

author Drawing by the students.)

Figure 8-32 shows a Cuisinart food processor Figure 8-33

shows an exploded view and names the parts while

Fig-ure 8-34 shows the DFCs of interest The rotating blade is

between the motor gear and the planets Even though the

second chain is longer and subject to several uncertainties

due to the presence of the motor mount gaskets, this is the

way the designer intended the assembly to work The

gas-kets fit tightly over the ends of the posts on the top frame

and into slots in the motor brackets There is no way to

adjust the motor's position relative to the gears There is a

little running clearance between sun and planets and lots

a user's point of view

This example shows that certain recommended designpractices can be captured in DFCs and employed over andover in different situations

8.H.4 Pump Impeller

The pump impeller assembly was discussed in Chapter 7.There we saw that one assembly sequence presented a highprobability of assembly problems due to loose tolerances

on the features used for fixturing A different sequence,fixtures, and fixturing features had to be used We canuse the DFC to represent the different sequences and atthe same time we can learn a little more about the DFCmethod

Figure 8-35 shows the two processes Figure 8-36shows DFC representations for these two processes Thesediagrams show unambiguously how the two processes dif-fer A thought question at the end of the chapter asks thereader to think about this assembly, especially the design

of fixture 2

This example shows that we can use the DFC to lyze assembly processes as well as assemblies, and it alsoshows that we can describe part mating criteria as well asassembly quality with KCs

ana-FIGURE 8-34 DFCs for the Cuisinart.

Two DFCs are shown: One provides correct clearance between the blade and the bowl, and the other provides proper alignment of the planetary gear system Especially important is the relation between the motor (sun) gear and the planetary gears.

FIGURE 8-33 Exploded View of the Cuisinart.

8.H EXAMPLE TYPE 1 ASSEMBLIES 233

FIGURE 8-35 Comparison of Two Assembly Processes for the Pump Impeller Left: The original process Right: The

improved process.

FIGURE 8-36 DFCs for the Two Assembly Processes Shown in Figure 8-35 Top: DFC for the original process, drawn

to describe the assembly of the impeller to the shaft This DFC is intended deliver the KC shown, which describes the part mating criteria that avoid wedging and collision with the part chamfer This DFC is unable to deliver the KC a high enough

percent of the time Bottom: DFC(s) for the improved process At the left is the first phase, which joins the shaft to the bottom

washer The KC for this step describes the conditions for successful shaft-washer assembly This process is relatively easy because thread mating operations are relatively tolerant of angular error At the right is the second phase, which joins the impeller to the shaft The KC for this step describes the conditions for shaft-impeller assembly This operation is by far the most difficult since the clearance is so small The DFC shows that the bottom washer plays no role in this process.

8.H.5 Throttle Body

A throttle body mounts to the intake manifold of an

auto-mobile engine and controls air flow to the engine A photo

of a typical throttle body is in Figure 8-37, while

draw-ings of the main parts appear in Figure 8-38 At the left

end of the shaft is a cam and lever to which is attached

the cable from the accelerator pedal Also at this end is a

return spring that closes off the air flow when the pedal

is released At the right end of the shaft, mounted to the

bore, is a potentiometer called the throttle position sensor

that reports shaft angle to the engine control computer

Midway along the shaft is mounted a disk that serves as

the air flow control device

The main KC for this device is that the disk close

tightly within the bore, while subsidiary KCs are that

the disk not stick in the open or closed positions See

Figure 8-39 for details about how the disk fits to the shaft,and see Figure 8-40 for details of how the disk fits in thebore

We will consider two ways of designing the throttlebody to deliver these KCs They appear in Figure 8-41.The bore locates the shaft in five degrees of freedom, withthe remaining degree of freedom being rotation about the

X axis to provide the operating motion of opening and

closing the air flow passage The bore and the shaft share

in locating the disk In the DFC on the left, the disk isfastened to the shaft by means of screws that pass throughclearance holes in the disk The screws have a contact withthe disk In the DFC on the right, accurate location of thedisk inside the bore is sought by means of locating pinsthat mate the disk to the shaft A thought question at theend of the chapter asks the reader to compare these twodesigns

FIGURE 8-37 Photograph of Throttle Body (Photo by the

author.)

FIGURE 8-38 Throttle Body Parts and Assembled.

(Drawing prepared by Stephen Rhee.)

FIGURE 8-39 Detail of Shaft and Disk The shaft has a

recess into which the disk fits with a little clearance in the X

direction The screws go through clearance holes in the disk and into threaded holes in the shaft.

FIGURE 8-40 Detail of Disk in Throttle Body Bore This

view is along the X or shaft axis On the left, the disk is in

the closed position On the right, the disk is open, and the closed position is shown in light gray Note that the disk in this view is not a rectangle but is a parallelogram so that its skewed edges conform to the inside diameter of the bore when the disk is closed at an angle that is not perpendicular

to the bore's axis An important KC of this assembly is that the disk fit tightly inside the bore.

8.1 EXAMPLE TYPE 2 ASSEMBLIES 235

FIGURE 8-41 Two Possible DFCs for the Throttle Body Left: The DFC for the design shown in Figure 8-38 Right: An

alternate design All the KCs regarding how the disk fits to the bore without sticking are condensed into one KC symbol.

8.I EXAMPLE TYPE 2 ASSEMBLIES

In this section we will look at some Type 2 assemblies

These assemblies cannot be built merely by joining their

mating features because some of them provide

insuffi-cient constraint There are several possible reasons for

this Most revolve around the fact that it may be

impos-sible or uneconomical to make the parts with sufficient

accuracy or repeatability to deliver their KCs as Type Is

Sheet metal assemblies are commonly of this type, but a

number of machined parts assemblies fall into this class

as well Car doors and aircraft assemblies are both made

of flexible parts, and assembly using fixtures is common

Below we consider one example of each

8.1.1 Car Doors

We considered car doors in Chapter 2, where we noted

that they typically have two conflicting KCs Figure 8-42

shows a typical car door assembly process, while ure 8-43 repeats a figure from that chapter, showing thetwo KCs and a diagram that we now recognize as a DFC.This DFC assumes that there are features on the innerpanel that permit it to completely locate the outer panel,

Fig-as well Fig-as features on the car body that permit it to pletely locate the door via complete location of the hinges

com-If only it were so! In fact, no one tries to make car doorsthis way because, as we showed in Chapter 6, the toler-ances on gaps and flushness at the assembly level are toosmall, on the order of ±2 mm or less, while tolerances onthe parts are nearly as large (±1.5 mm or so)

In fact, fixtures are needed to support the process that

is used to build the subassembly, place the hinges erly, and install the door onto the car body Figure 8-44and Figure 8-45 show two possible DFCs for this processthat include fixtures One of them appears to achieve both

prop-FIGURE 8-42 Typical Car Door Assembly Process Left: The door is made by joining an outer panel and an inner panel.

Right: The subassembly of door inner and door outer plus hinges and latch bar is ready to be attached to the car body At the

subassembly level, the hinges are used to adjust the door in the in/out and up/down positions At the final assembly level, the hinges are used to adjust the fore/aft (and possibly the up/down) position Other strategies are possible.

FIGURE 8-43 Car Door, Its Two KCs, and a DFC This DFC imagines making a door and installing it as a Type 1 assembly.

Also shown is a detail of how the hinge interfaces to the door inner panel and the car body.

KCs independently but is in fact impossible by today's

methods The other suffers from KC conflict but is used

anyway for lack of a better alternative

In each of these door assembly processes, we can see

that the fixtures are unconstrained by the joints between

the parts, which are contacts or at least have unconstrained

degrees of freedom in the directions controlled by the

fix-tures A thought question at the end of the chapter asks the

reader to label the arcs of these DFCs with explicit degree

of freedom notations Also, in Figure 8-45, there are not

enough degrees of freedom using fixture F2 or F2' alone

to achieve both KCs independently A thought question

FIGURE 8-45 Second Candidate DFC for Car Doors.

This DFC first uses fixture F1 to make a subassembly of door inner and door outer plus hinges There are then two possi- bilities for step 2 Either fixture F2 achieves the weather seal

KC or fixture F2' achieves the appearance KC In either case, the KC that is not directly controlled is achieved with larger tolerances or lower probability.

at the end of the chapter asks the reader to use the twistmatrix intersection algorithm to prove this

Figure 8-46 uses the notation of Chapter 7 to scribe the alternate assembly sequences for the car doors.Note that several apparently feasible sequences are in factunavailable once we take the KCs and constraints intoaccount

de-8.I.2 Ford and GM Door Methods

In this section we consider in some detail methods of taching doors to cars used on some models of cars at GMand Ford, respectively These are examples of widely dif-fering methods used by different car manufacturers They

at-FIGURE 8-44 First Candidate DFC for Car Doors This

DFC starts by installing the door inner panel (Dl) to the car

body, using the hinges to achieve the weather seal KC

Fix-ture F1 is used for this step Then fixFix-ture F2 is used to

as-semble the door outer panel (DO) to the door inner panel

in such a way as to achieve the appearance KC

Unfortu-nately, this assembly sequence is impossible using today's

door construction methods.

8.1 EXAM RLE TYPE 2 ASSEMBLIES 237

FIGURE 8-46 Alternate son Sequences for Car Doors.

Liai-The liaison diagram is at the lower right The achievement

of each KC is indicated by a shaded square next to a state in the liaison sequence diagram Sequences that travel down the left side of the diagram appear able to achieve the KCs one at

a time These sequences, fortunately, are impossible The surviving sequences that travel down the right side of the di- agram necessarily achieve the KCs simultaneously at the last step.

un-help us illustrate additional properties of DFCs and un-help us

understand the behavior of the hybrid mate-contact type

of joint We do not know the variations of all the fixtures

and parts in these processes, so we will not make any

judg-ment concerning which one is better The discussion that

follows has the flavor of analysis because we are reverse

engineering existing processes If we were designing new

processes, then everything we are depicting would be

de-termined by the designers of the parts and fixtures as they

sought to determine the design intent of the assembly and

deliver the KCs No reverse engineering would or should

be necessary

Figure 8-47 shows schematically the two methods we

will discuss (The reader may also refer to Figures 2-9

through 2-12 for additional views of these processes.)

Each method comprises two stages In the first stage,

hinges are attached to a door subassembly consisting of

door inner and door outer In the second phase, this

sub-assembly is attached to the car In both cases the KCs

are as discussed above Both KCs are affected by how

the door's position with respect to the body varies in all

three directions: up/down, in/out, and fore/aft However,

each company's method delivers these KCs in different

ways Furthermore, we need to look carefully at each of

the three directions in order to see all the differences,

check for constraint violations, and calculate accumulated

variation

Note that each KC could have different tolerances in

each direction, making these distinctions important This

point is true in general: any KC may be defined and eranced in one or a few directions and be undefined oruntoleranced in others

tol-The GM method applies the hinges to the door whilegripping the door on the outer panel Not only are thehinges attached in this setup, but a locator cone is attached

to each hinge The hinge mounting machine positions andfastens each hinge in the in/out and up/down directions,and then it places and fastens the locator cone on each freehinge flap carefully in the up/down and fore/aft directions.The door is then attached to the car by mating the locatorcones in a hole and slot compound feature set vertically onthe body just forward of the door opening Screws fastenthe hinges to the frame These hinges come apart at thepivot, permitting the door to be removed after painting sothat the door and car final assembly processes can occurindependently The doors are rehung by reconnecting thehinges

In the Ford process, the hinge mounting machine mates

to the door's inner panel It locates the hinges in the in/outand up/down directions, as does the GM hinge mountingmachine A moveable fixture then is used to pick up thedoor by mating to the door's outer panel using a hole andslot feature set shown in Figure 8-48 An operator car-ries the door to the car using this fixture and mates thefixture's two locator pins to a hole and slot compoundfeature on the body The hole is in the body just ahead ofthe door opening while the slot is in the body just behindthe door opening Screws fasten the hinges to the body

remounted accurately later because the hinge flaps can be separated from each other and rejoined repeatibly Hinges contain all the features needed to locate the door to the car body The hinge mounting fixture is responsible for placing the hinges accurately on the door as well as attaching a cone locator accurately to each hinge The door is aligned to the body by mating the cone locators to features on the body, (b) The Ford method leaves the doors on once they are attached to the body Hinges contain the features necessary to locate the door in the in/out direction However, the door mounting fixture contains the features necessary to position the door in the up/down and fore/aft directions As the screws are tightened the door slides in/out along the pins on the mounting fixture.

FIGURE 8-48 Photo of How Ford Door

Mounting Fixture Mates to Door Left:

A pin in the fixture mates to a hole at the front of the door where the side-view mir-

ror will be attached later Right: Another

pin mates to a horizontal slot at the rear

of the door where the handle will be tached later (Photo by the author Used

at-by permission of Ford Motor Company.)

238

8.1 EXAMPLE TYPE 2 ASSEMBLIES 239

FIGURE 8-50 Chains for Determining Constraint and Assembly-Level Variation for the Appearance KC in the

GM Method The chain (traced by the heavy line) comprises

properly constrained items that link one end of the KC to the other end All three directions that could contribute variation follow the same chain Fixture F1 and the door inner to door outer joint play no role in this KC and thus are shown in gray Because all directions are portrayed in this figure, these joints are shown in their original hybrid form without distinguishing which directions play a role in which KC.

These hinges do not come apart, and the doors stay

at-tached to the car from that point on

Let us consider each of these processes in detail, first

the GM process, then the Ford process The DFC for the

GM process is shown in Figure 8-49.13 It is drawn, like

all other DFCs, as a chain of mates from one end of a

KC to the other It shows that two fixtures are involved,

one for joining door inner to door outer, and the other

for attaching the hinges (and the locator cones which are

not shown separately) to the door No fixture is shown for

attaching the door to the body because the hinges and

loca-tor cones provide all the location constraints Note in this

figure that, in addition to the arrow indicating a mate and

a dashed line indicating a contact, there is an arrow with

a dashed line This indicates a hybrid mate-contact This

symbol is necessary in order to describe carefully how the

door subassembly is made, and it will play an additional

role when we consider the Ford process

The shape of door inner constrains the location of door

outer with respect to door inner in the in/out direction, but

fixture Fl constrains door inner with respect to door outer

I3 ln this figure, there is an arrow from the hinge on the door to

the body of the car, implying that the door or the hinge locates the

body Obviously, the body is big and heavy while the door is

rela-tively small and light, so in a physical sense the door cannot change

the position of the body Thus it may not make sense physically to

have the arrow point from the hinge to the body The reader may

reverse this arrow if it makes him or her feel better, but the nature of

the chain as a whole will not change.

in the up/down and fore/aft directions Thus each ship is a mate in one direction and a contact in another

relation-As long as neither tries to constrain a direction that is strained by the other, there is no problem What is impor-tant to us is that variation in the in/out direction is governed

con-by the process capability of stamping while variation inthe other two directions includes stamping variation plusthat contributed by fixture Fl In order to determine whichdirections are mates and which are contacts in these hy-brid joints, we need to consider the different directionsseparately and carefully

The question before us is to determine how each rection is constrained and how variation will accumulatealong the chain of mates that joins each end of the KC anddelivers it To accomplish this, we need to identify eachfeature in the chain for each direction and the internal sur-faces of each feature that affect the chain Figure 8-50 andFigure 8-51 do this for the GM process Figure 8-50 showsthat all directions of the appearance KC are delivered inthe same way Figure 8-51 shows that the weather seal KC

di-is delivered differently in the different directions Each brid joint is shown as a mate for the direction or directions

hy-it constrains and to which hy-it contributes variation, and as

a contact for the directions in which it plays no role

Figure 8-52 shows the DFC for the Ford door process.This process differs from the GM process in several ways,

as discussed above The hinge mounting machine F2 terfaces with the door inner panel, and a door mountingtool F3 aligns the door in two directions with respect to

in-FIGURE 8-49 DFC for GM Door Process Fixture F1 joins

door inner to door outer Hinge mounting machine F2 locates

the hinges and cone locators with respect to the door outer.

Each dashed line arrow indicates a hybrid mate-contact

rela-tionship in which some directions are constrained and act as

mates while others are contacts and provide no constraint.

Door inner locates door outer in the in/out direction, while

fixture F1 locates these parts with respect to each other in

the up/down and fore/aft directions Variations in these

di-rections will be different because the fixture is present along

some directions but not others.

Next Page

240 8 THE DATUM FLOW CHAIN

FIGURE 8-51 Chains for Determining Constraint and Assembly-Level Variation for the Weather Seal KC in the GM

Method Left: The in/out direction's chain follows the hinge mounting process and includes the stamping variation in the door

inner panel's relationship to the outer panel Fixture F1 plays no role in this direction so it appears gray with contact

relation-ships to these panels Right: The up/down and fore/aft directions' chain includes both F1 and F2 The door inner to door outer

relationship plays no role and is shown as a contact.

FIGURE 8-52 DFC for the Ford Door Process Fixture F1

joins door inner to door outer Hinge mounting fixture F2

lo-cates the hinges with respect to door inner Door mounting

fixture F3 locates the door to the body with respect to door

outer In this process, the hinge is located with respect to the

door inner panel, and a separate fixture aligns the door to the

body The hinge to body, door inner to door outer, F3 to door

and body, and F1 to door relationships are shown as hybrids

because they act as mates in some directions and contacts

in others.

the body Several joints are shown as hybrid mate-contacts

because they are mates in some directions and contacts in

others A detailed direction-by-direction drawing is

nec-essary to define which direction is which

Figure 8-53 and Figure 8-54 show how each KC is

delivered in each direction in the Ford process

This example has shown how to decompose a

multi-direction KC into its components in a convenient

co-ordinate frame and then to draw the DFC and

iden-tify constrained directions separately for each coordinate

direction Once this is done, the variation contributed by

each feature in each chain in each direction can be

com-puted and combined into a unified 4 x 4 matrix model of

the entire KC delivery process We can identify arcs in

in-dividual directions that are claimed by more than one KC

chain, such as (a) F2-hinge-door-inner in the in/out

direc-tion of the Ford process and (b) F2-hinge-door-outer in alldirections of the GM process Then a rational discussionmay be conducted to determine the effect of giving one

or another of these coupled KCs priority Different jointschemes, assembly sequences, and fixture designs can beconsidered as part of the KC delivery design process

8.I.3 Aircraft Final Body Join

Most large aircraft fuselages are assembled using fixturesthat are even larger These fixtures are made of aluminum,

as are the aircraft themselves, to equalize induced expansion or contraction The tolerances sought

temperature-on such aircraft are challenging As a result, the fixturescost a great deal to design, build, and keep exactly theright shape The description that follows is generic but issimilar to that used by major airframe manufacturers

A simplified version of this process is shown in ure 8-55 It creates a full 360° fuselage tube, ready to bejoined to another one Typical individual sections of large

Fig-aircraft are about 40 feet long and 12 to 24 feet in diameter.

Figure 8-56 shows the DFC for controlling the diameterand circumference, including contributions by suppliers.Figure 8-57 shows the joining process for two of thesetubes Typical aircraft have between two and four suchjoints, depending on the length of the aircraft

The DFC(s) required to achieve the diameter and cumference KCs are shown in Figure 8-56

cir-Figure 8-57 shows all the KCs that are sought duringfinal body join It is easy to see that there are more than can

be individually adjusted or given independent tolerances,inasmuch as the two sections are practically rigid Themost important KC is structural, requiring minimum edge