functional dimensioning and tolerancing software for ... integrated... · functional dimensioning...

Functional dimensioning and tolerancing software forconcurrent engineering applications

M.N. Islam*

Department of Mechanical Engineering, Pohang University of Science and Technology, San 31 Hyoja-dong,

Nam-gu, Pohang, Kyungbuk 790-784, South Korea

Received 18 September 2002; accepted 13 September 2003

Abstract

This paper describes the development of a prototype software package for solving functional dimensioning and tolerancing

(FD&T) problems in a Concurrent Engineering environment. It provides a systematic way of converting functional requirements

of a product into dimensional specifications by means of the following steps: firstly, the relationships necessary for solving

FD&T problems are represented in a matrix form, known as functional requirements/dimensions (FR/D) matrix. Secondly, the

values of dimensions and tolerances are then determined by satisfying all these relationships represented in a FR/D matrix by

applying a comprehensive strategy which includes: tolerance allocation strategies for different types of FD&T problems and for

determining an optimum solution order for coupled functional equations. The prototype software is evaluated by its potential

users, and the results indicate that it can be an effective computer-based tool for solving FD&T problems in a CE environment.

# 2003 Elsevier B.V. All rights reserved.

Keywords: Functional dimensioning and tolerancing; Concurrent engineering; Tolerance allocation

1. Introduction

Functional Dimensioning and Tolerancing (FD&T)

is a concept widely used for specifying dimensions

and tolerances of the component parts and sub-assem-

blies of a product according to their functional

requirements. These functional requirements arise

from all life cycle issues, such as manufacturing,

assembly and inspection. Concurrent Engineering

(CE) is an engineering and management philosophy,

which also deals with the life cycle issues of a product.

CE is based on the idea of carrying out as many stages

of product development concurrently as possible,

rather than in a sequential order. It calls for the

formation of a cross-functional product development

team, which includes people from a wide range of

departments, such as: product planning, design, man-

ufacture, assembly, quality assurance, marketing,

sales and finance.

Dimensions and tolerances influence almost all

aspects of product development which are of interest

to CE team members who consider all the life cycle

issues of a product during its design stage. Therefore, a

CE approach will be ideal for selection of dimensions

and tolerances through applications of FD&T metho-

dology. Furthermore, FD&T can serve as a common

link between all members of the CE team; hence it can

enhance the CE team performance [1,2]. It is also

argued in [3] that CE offers the best option for finding

the values of dimensions and tolerances using informal

optimization methods because the data required for

Computers in Industry 54 (2004) 169–190

* Tel.: þ82-54-279-8639; fax: þ82-54-279-5899.

E-mail address: [email protected] (M.N. Islam).

0166-3615/$ – see front matter # 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.compind.2003.09.006

application of formal optimization methods are often

not available at the early design stage.

However, for the successful implementation of

FD&T principles in a CE environment, appropriate

tools are needed. The development of such a tool is the

main objective of this project. It will be a computer-

based tool to be used by product development teams

for quick ‘what-if’ analyses for evaluation of different

design alternatives and their influences on dimensions

and tolerances.

2. Review of existing FD&T tools

The first commercially available 3D tolerance ana-

lysis software package VSA was introduced by Varia-

tion Systems Analysis Inc.1 in 1982 [4]. At present

there are a number of other tolerance calculation

software packages available in the market, such as

CETOL from Sigmetrix [6], DCS from Dimensional

Control Systems [7], Mechanical Advantage from

Cognition Corporation [8], Analytix from Saltire Soft-

ware [9], VALISYS from Tecnomatix [10], Crystal

Ball from Decision Engineering [11] and CATS-1D

XL from ADCATS [12].

VSA, CETOL and DCS are the three most popular

brands of the commercially available tolerance calcu-

lation packages. VSA and DCS were initially sup-

ported by the big three automotive manufacturers (i.e.

General Motors, Ford and Chrysler) and are widely

used in automotive industry. CETOL, previously

known as TI/TOL, was backed by Texas Instruments

and based on research carried out at Brigham Young

University’s ADCATS program. Its focus has been

mechanical aspects of electronically based products.

Detailed description of these packages and their

capabilities can be found on their respective websites;

only briefs are given below:

� VSA is 3D tolerance analysis software package

available as modules within a number of leading

CAD systems. It has a module for checking

parts’ Geometric Dimensioning and Tolerancing

(GD&T) schemes. It allows two types of simula-

tions for tolerance analysis: Monte Carlo simula-

tions and High-Low-Median (HLM) simulations.

Monte Carlo simulation method provides distribu-

tion curve and quality parameters of the total output

variation. In graphic mode it can handle three2 types

of distributions of the input tolerances: actual,

normal and Pearson. HLM analysis is based on

variance analysis technique and is used for deter-

mining the percent contribution of each input fac-

tors to the total output variation.

� CETOL is a 3D vector assembly modeling package

fully integrated with Pro/ENGINEER; a CATIA

version is due out this year. It has the capability

to apply three tolerance accumulation models:

Worst case (WC), root sum square (RSS) and mean

shift based on Six Sigma analysis. It can handle

normal and non-normal distribution of input toler-

ances. It does not require any simulation and pro-

vides three outputs, viz. distribution curve and

quality parameters of the total output, contribution

plot and sensitivity plot.

� DCS is a tolerance simulation software package

integrated with Mechanical Desktop CAD package.

It consists of two modules: 1-DCS for solving 1D

problems and 3DCS for solving multi-dimensional

problems. It has the capability to apply three tol-

erance accumulation models: WC, RSS and Monte

Carlo simulation. Monte Carlo simulation can use

different types of distributions for input variables. It

can also take actual part data and use it by direct

loading or selecting the best curve to fit the data.

� Mechanical Advantage comes as a module within

Mechanical Advantage CAD package. It is a fea-

ture-based variational geometry package which can

perform WC, RSS and Monte Carlo analysis.

� Analtix is a mechanism design and analysis soft-

ware package which runs on Microsoft Windows. It

has a tolerance analysis module for tolerance stack-

up calculations using RSS or WC accumulation

models. It is capable of performing a percent con-

tribution analysis.

� VALISYS is a 3D assembly stack-up analysis soft-

ware package which uses Monte Carlo simulation

technique for variation analysis. It is fully inte-

grated with Unigraphics CAD system and has a

module for GD&T syntax checking.

� Crystal Ball is a Windows Excel spreadsheet add-on

software and performs tolerance and risk analysis

1 Recently purchased by Electronic Data Systems (EDS) [5]. 2 Eight in text mode.

170 M.N. Islam / Computers in Industry 54 (2004) 169–190

through Monte Carlo simulation technique; the

distribution gallery contains 12 different distribu-

tions.

� CATS-1D XL is 1D tolerance stack-up analysis

package which runs on Windows Excel. It supports

four stack-up analysis models: WC, RSS, Six Sigma

and Measured Data; available free on ADCATS

website [12].

From this review, it is noted that the emphasis of these

packages has been on tolerance analysis and as such, are

not very helpful for FD&T which primarily deals with

tolerance allocation problems. These packages often

apply statistical tolerancing theories (Monte Carlo

simulation found to be the most popular one) for finding

the combined effect of the input tolerances (part toler-

ances) on the assembly tolerance. In principle they

could be used for ‘what-if’ analysis. However, gener-

ally they lack any intervention strategy. They are sui-

table for verifying the results once the part tolerances

and their distributions are known, but generally are not

suitable for use in a CE environment, where the prime

objective is to allocate part tolerances based on func-

tional requirements of the product. Moreover, to the

best of the author’s knowledge, none of the commer-

cially available software includes any strategy for

handling coupled loop equations, i.e. when two or more

functional equations are related to each other.

3. FD&T methodology

Although FD&T plays an important role in product

design it lacks a comprehensive systematic methodol-

ogy. Such a methodology is considered necessary for

the application of FD&T concepts into the CE envir-

onment. Therefore, a new methodology has been

developed [13–17] and the prototype software is based

on this. The main features of the methodology are:

� Establishment of clear links between the customer’s

and stakeholders’ needs of a product and its func-

tional requirements. These links will assist the CE

team to understand the product requirements and

how they are to be satisfied through dimensional

aspects of a design.

� Graphical representation of all relationships neces-

sary for solving FD&T problems in a matrix form

known as a Functional Requirements/Dimensions

(FR/D) matrix. This graphical representation will

provide the total picture of the FD&T problem and

assist in considering different controllable vari-

ables, such as dimensions, tolerances, and manu-

facturing processes, concurrently.

� Development of a tolerance allocation strategy

suitable for a CE environment. It considers manu-

facturing, assembly, and inspection requirements in

the selection of dimensions and tolerances. It also

applies informal cost optimization methods, thus

the need for cost-tolerance data is avoided.

� Development of a strategy for grouping and solving

coupled functional equations. This will improve the

efficiency of the solution search.

� Development of a comprehensive strategy for finding

the values of dimensions and tolerances of a product

with the help of a FR/D matrix. A number of com-

monly used cost minimization strategies such as the

use of standard parts, the use of preferred sizes, the

use of preferred fits and others, have been incorpo-

rated into the proposed methodology.

� Compilation of a database, which includes data on

preferred sizes, preferred fits, process capability

tolerances and others, required for solving FD&T

problems.

4. Conceptual design of the software

The conceptual design of the proposed software is

illustrated in Fig. 1. It contains three major parts: (i)

main program, (ii) solution library and (iii) database.

4.1. Main program

The functions of the main program (MatrixSolver)

are: (i) to put all relationships necessary for solving

FD&T problems in a matrix form; (ii) separating

FD&T problems into groups; (iii) finding an optimum

solution order of a group of coupled loop equations;

and (iv) storing and cross-checking the solution

results. The flow-diagram of MatrixSolver is illu-

strated in Fig. 2.

4.1.1. Module for formation of FR/D matrix

The centerpiece of the prototype software is the

FR/D matrix (Fig. 3). It is a multiple relationship

matrix, which makes it possible to visually represent

M.N. Islam / Computers in Industry 54 (2004) 169–190 171

the interaction of a number of interrelated sets of items

viz. functional requirements, dimensions and toler-

ances, and process capability tolerances. Different

types of relationships illustrated in the FR/D matrix

(Fig. 3) are described below. Numbers shown in boxes

refer to relationship types.

1. Along each row dimensions and tolerances are

tied to a functional requirement. This relationship

is expressed by a loop equation and is controlled

by the target value, which imposes performance

constraints on the selection of dimensions and

tolerances.

2. Each chosen tolerance must satisfy the process

capability requirements. This type of relationship

imposes manufacturing process constraints on the

choice of tolerances.

3. All tolerances viz. size and geometric tolerances,

and surface texture parameters of a feature are

controlled by the process for manufacturing that

feature and should be compatible.

4. All dimensions, both functional and non-func-

tional, within a part are related through the parts

process plan and these relationships have profound

effects on the manufacturing cost of the product.

5. If one or more dimensions are shared by two or

more functional equations then the functional

equations are called coupled functional equations.

Coupled functional requirements should be solved

simultaneously.

Besides relationships among functional require-

ments, dimensions and tolerances, and process para-

meters, the FR/D matrix illustrates some other

information viz. target values, relationships (loop

equations), and types of functional requirement and

target differences. The target difference is the differ-

ence between the value calculated by functional con-

strains and the specified target value. The target

difference column will be helpful during the search

for a solution of the FR/D matrix.

4.1.2. Module for separating FD&T problems

into groups

The management of the FR/D matrix becomes more

difficult as the number of functional requirements

grow. However, it is anticipated that not all functional

equations will be related to each other. Therefore, it is

proposed to consider related functional equations as a

group which allows the CE team to concentrate on a

particular area in the FR/D matrix without losing sight

of the overall picture. This grouping is based on the

interrelationships between the functional equations

(relationship type 5 illustrated in Fig. 3).

Main Program (MatrixSolver) Database

Preferred Sizes Table Preferred Tolerances Table

Preferred Fit TableLimits and Fits Table

Standard TolerancesProcess Capability Data

Solution Library

Module for Solving Fitting Feature Problems (FitSolver)

Module for Solving LengthDimension Problems (LengthSolver)

Module for Solving MixedType Problems (MixSolver)

Module for Solving SurfaceTexture Problems (SurfaceSolver)

Module for Formation of FR/D Matrix

Module for Separating FD&T Problems into Groups

Module for Finding Optimum Solution Order

Module for Sorting and Crosschecking Solutions

Fig. 1. Conceptual design of the prototype software.


Enter number of FRs

Enter FRs with their IDs Module for Separating FRs

into Groups

Enter number of parts

Select a FR

Enter part names

Module for Finding Optimum

Revise? Solution Order

No

Enter relationships between

FRs and parts

Enter details of relationships

between FRs and parts Module of Storing and

Cr osschecking Solutions

Enter type of FR

Enter relationships and

their target values

Preliminary FR/D matrix

Yes

Complete FR/D matrix

A

START

A coupled FR?

Yes

Solved all FRsin the group?

No Solved all FRs in FR/D

matrix?

STOP

Yes

Solution Library

Solved FR

No

Enter proposed tolerance values (if desired)

Enter proposed manufacturing processes and their capability

details

Enter purchased parts details

A

No

Solved FR/D matrix

Fig. 2. Flow diagram of MatrixSolver.


Fig. 3. Types of relationships illustrated in FR/D matrix.

17

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In Table 1, FD&T problems of a product are repre-

sented in a simplified FR/D matrix which contains

eight functional requirements and 19 dimensions.

Further analysis of this matrix reveals the following

inter-relationships between the functional require-

ments:

On the basis of the above relationships the FR/D

matrix is separated into four groups (Table 2). These

groups can be further classified into two types: (i)

where the number of functional equations is 1, i.e. the

group consists of one independent functional equation

only (Group nos. 1 and 2). and (ii) where the number

of functional equations is more than 1, i.e. the group

consists of a number of coupled functional equations

(Group nos. 3 and 4). Details of the algorithm used for

separating FD&T problems into groups are given in

[13].

4.1.3. Module for finding optimum solution order

When solving coupled functional equations the

target values for all the functional equations involved

have to be satisfied simultaneously, but the question

arises in what order should the equations be solved?

The solution order plays an important role in the

solution search because the values of tolerances deter-

mined in one functional equation are considered as

non-negotiable in subsequent equations. The proto-

type software is equipped with a module which pro-

vides an optimum solution order for a group of

coupled functional equations taking into account sev-

eral of factors such as the number of functional

equations involved, the nature of the relationships

and the number of independent and coupled dimen-

sions in each functional equation.

Table 1

Simplified FR/D matrix

Functional

requirements

Dimensions

A B C D E F G H J K L M N O P Q R S T

FR1 � � �FR2 � � � �FR3 � � � � �FR4 � �FR5 � � � �FR6 � �FR7 � �FR8 � � �

Dimension C: FR1 FR3 FR5

Dimension D: FR1 FR3 FR5

Dimension M: FR3 FR8

Dimension O: FR2 FR6

Table 2

Grouping of functional requirements

Functional

requirements

Dimensions Group no.

A B C D E F G H J K L M N O P Q R S T

FR4 � � 1

FR7 � � 2

FR1 � � � 3

FR3 � � � � � 3

FR5 � � � � 3

FR8 � � � 3

FR2 � � � � 4

FR6 � � 4


This strategy is based on the following observa-

tions:

1. When functional equations are solved as coupled,

the number of choices for selection of values of

tolerances decreases gradually and decisions,

made in earlier equation, restrict the choice of

decisions in subsequent equations. It is proposed

to generate all the possible combinations of

solution sequences and then select the solution

sequence with minimum restricting effect of

coupled variables.

2. The functional equations with a greater number of

variables are easier to solve. This can be explained

by the fact that tolerance allocation is an economic

trade-off and the greater the number of variables

in a functional equation, the greater the flexibility.

Therefore, a functional equation with the lowest

number of independent variables will get pre-

ference over functional equations with a greater

number of independent variables.

Detailed treatment of this strategy can be found in

[17].

4.1.4. Module for storing and cross-checking

solutions

This module allows the user to save the project data

between each session. It is also applied for checking

whether all relationships illustrated in FR/D matrix in

Fig. 3 are fulfilled.

4.2. Solution library for different types of

FD&T problems

Different types of problems require different solu-

tion strategies. Therefore, it is proposed to build a

solution library for dealing with different FD&T

problems common to most engineering designs. How-

ever, due to the diverse range of problems involved, it

is not possible to include solution procedures for all

types of FD&T problems in the proposed software.

Therefore, at this stage, only four types of FD&T

problems are included, viz. fitting feature problems,

length dimension problems, mixed type of problems

(i.e. combination of length dimension problem and

fitting feature problem), and surface texture problems.

Here it should be noted that if desired, the solution

library can be expanded without interfering with the

proposed overall strategy. The strategies for solving

different types of FD&T problems adopted in the

prototype software are primarily from Gladman [18]

which are modified to suit computer applications.

4.2.1. Strategy for solving fitting feature problems

The fitting feature problem is one of the most

frequently encountered type of FD&T problems in

engineering design. It consists of an assembly between

two parts with cylindrical features: (i) one part with an

internal cylindrical feature, by convention referred to

as a hole, and (ii) another part with an external

cylindrical feature by convention referred to as a shaft.

The prototype software has a module (FitSolver) for

dealing with fitting feature problems. The flow dia-

gram of FitSolver is illustrated in Fig. 4. The solution

procedure for fitting feature problems depends on the

form in which the target values are specified. How-

ever, the first few steps involving selection of basic

sizes and selection of a fit system in the solution search

are the same for all cases as illustrated in Fig. 4. The

prototype software also provides two options for the

selection of a fit (i) select a fit and then select a suitable

manufacturing process and (ii) select a manufacturing

process and then find a suitable fit.

The target value of a fitting feature problem is

usually specified in one of the following forms:

1. general description (e.g. close running fit);

2. minimum and maximum clearance (Cmin, Cmax);

3. mean fit and variation (MF � VAR);

4. specification (e.g. H7f7).

The first form for specifying the target values of a

fitting feature problem is a general description of the

fit. This description expresses the type of fit according

to its function, e.g. precision fit, close running fit, etc.

In this case, the first step in the solution search is to

identify the fits suitable for that description from the

Preferred Fits Table. The final selection is made on the

basis of fit quality, i.e. normal, coarse, etc.

The second and the third form for specifying the

target values of a fitting feature problem are clearance

conditions. These two forms are equivalent. The

clearance conditions such minimum clearance (Cmin),

maximum clearance (Cmax), mean fit (MF) and varia-

tions (VAR) can be calculated using the following

formulas where LDH is lower deviation for the hole,

UDH is the upper deviation for the shaft, th is tolerance


value for the hole and ts is the tolerance value for the

shaft.

Cmin ¼ LDH þ UDS (1)

Cmax ¼ Cmin þ th þ ts (2)

MF ¼ 12ðCmin þ CmaxÞ (3)

VAR ¼ � 12ðth þ tsÞ (4)

Select target requirement type

Enter target requirements

Enter symbolic dimensions

Select fit system

Enter basic size

Fit description Fit specification

No

START

Final Results FitSolver

STOP

Yes Enter values of non-negotiable

BA

Fit/ProcessProcess/Fit

Proposed manuf. processes Select fit searching database

Select fit ranking criteria

Select a fit from fit ranking table

Select manufacturing processes

Select fit ranking criteria

Select a fit from fit ranking table

Tolerance evaluation

Fit/Process

Select fit option

Select manufac. processesTolerance evaluation

A

Proposed manuf. processes

Non-negotiabledimensions?

Target requirement type?

Select search strategy

Clearance conditions

Validate fit description

Select search strategy

STOP

Process/Fit

Vlidate fit Specification

Select manuf. processes

B


STOP


Fig. 4. Flow diagram of FitSolver.


For a hole-based fit,

LDH ¼ 0 and Cmin ¼ UDS (5)

For a shaft-based fit,

UDH ¼ 0 and Cmin ¼ LDH (6)

A new computer-based strategy has been developed

for selection of fits when clearance conditions are

given this consists of the following steps:

� Select fit selection criteria. There are differences of

opinion among designers about the use of particular

clearance conditions for fit selection [19]. The

prototype software package leaves the choice to

the user.

� Considering production costs, select a suitable fit

system, i.e. hole-based or shaft-based.

� Based on Eq. (5) or (6), select three shaft specifica-

tions for a hole (or three hole specifications for a

shaft) from the Limits and Fits Table [20] which are

nearest to the given value of Cmin.

� Vary hole and shaft IT grades between 5 and 12.

� For each combination calculate hole and shaft

tolerance values,

� Calculate variations from hole and shaft tolerance

values.

� Compare the calculated values with the given value

of variation.

� Pick combinations for which the differences are

least and rank them according to the criteria set by

the user.

� Compare each fit combination with preferred fits

stored in the database and show the findings.

� Illustrate the results in a tabular form known as Fit

Ranking Table.

The fourth form for expressing the target value of a

fit is fit specification. When the basic size is selected

the fit specification is sufficient to calculate the hole

and the shaft dimensions and tolerances using the

Limits and Fits Table [20].

Fitting features may contain non-negotiable dimen-

sions coming from purchased or existing parts. Exam-

ples of such component parts are: gears, pulleys, oil

seals, bearings, bushings, dowel pins, etc. FitSolver

can handle this type of fitting feature problems also.

In this case the designer has to specify only one

element of the fitting feature. In effect, non-negotiable

dimensions impose additional constraints on the solu-

tion search.

4.2.2. Strategy for solving length dimension

problems

Length dimension problem is another type of FD&T

problem frequently encountered in engineering

designs. It usually consists of two or more component

parts and deals with the dimensioning and tolerancing

of their length dimensions.3 In general, a functional

equation of a length dimension problem has the

following form:

ðZ � zÞ ¼Xn

i¼1

AiSiðXi � xiÞ (7)

where Z is the functional requirement (basic size), z

the functional requirement (size tolerance), Ai the sign

constants 1 or �1, Si the sensitivity factors, Xi the

functional dimensions (basic size), and xi is the func-

tional dimensions (size tolerance).

Applying the method of extremes to Eq. (7) it can be

shown that

Z ¼Xn

i¼1

AiSiXi (8)

and

z ¼Xn

i¼1

Sixi (9)

The prototype software has a module (LengthSol-

ver) for dealing with length dimension problems.

The flow chart of LengthSolver is shown in Fig. 5. It

uses Eq. (8) for selecting basic sizes. The usual

approach is to select all but one basic size, the

remaining one is calculated to satisfy Eq. (8). The

strategy is to select as many basic sizes as possible

from preferred sizes. Eq. (9), known as a tolerance

equation, is used for selection of individual tolerance

values. LengthSolver can also handle non-negotiable

dimensions.

For allocating the individual tolerance values a new

tolerance allocation strategy suitable for CE environ-

ment was applied. The strategy is based on a variation

3 Length dimensions are usually defined by distance between

two parallel planes.


of tolerance allocation by the process capability model

proposed by Farmer [21] detail of which can be found

in [16]. The main steps of this strategy are described

below:

� To begin, the CE team needs to consider the pro-

duction of each part and propose manufacturing

processes which they think are the most suitable and

least costly for the manufacture of each feature.

� The next step is to find the achievable process

capability tolerances of all the dimensions with

the proposed manufacturing processes. The team

may consult the Process Capability database and

determine the IT grade range for each proposed

manufacturing process under average conditions.

� Then the CE team will assess the difficulty of

producing the feature by considering factors which

they feel may contribute to the particular case.

Fig. 5. Flow diagram of LengthSolver.


On the basis of this assessment the CE team can

select the IT grade, which they think is achievable

� Based on the achievable IT grade and the selected

basic size the CE team is now able to calculate the

achievable process capability tolerance under real

conditions for each dimension and proposed man-

ufacturing process.

� The CE team selects the tolerance values equal to or

greater than the achievable process capability tol-

erance values for each dimension.

� It is proposed to enlarge the process capability

tolerances by safety factors to account for the

degree of uncertainty of process capability data

used.

� The CE team then selects tolerance values for each

dimension based on enlarged process capability

tolerance. If desired, the selected tolerance values

can be adjusted to standard values (preferred tol-

erances).

� The methodology also allows some or all of the

tolerance values be proposed by the CE team.4 In

search of an optimum solution these will have a

higher priority than other tolerances in the system

although all the proposed tolerance values must be

compatible with the proposed process capability

tolerances.

� The next step is to calculate the residual tolerance,

which is the difference between the available

tolerance band and the sum of all selected toler-

ance values. A negative value of residual tolerance

will indicate the target tolerance requirement has

not been met and some corrective action is neces-

sary.

� When a corrective action is necessary, the software

calculates the percentage change required for each

tolerance which shows the improvement necessary

to fulfill a target tolerance requirement if only one

process is altered. However, if necessary, the CE

team may decide to combine or change more than

one process rather than change only one.

� After deciding which process(es) is (are) to be

changed the CE team can recalculate the residual

tolerance and evaluate the outcome.

� The process is repeated until an acceptable solution

is found.

4.2.3. Strategy for solving mixed-type problems

Often fitting feature and length dimension problems

are combined. In this paper, these are termed mixed-

type problems. It is a length dimension problem where

one or more fitting feature problems are embedded

within the dimension loop. In general, mixed-type

problems have the following form:

ðZ � zÞ ¼Xp

i¼1

ðFFÞi þXq

i¼1

AiSiðXi � xiÞ (10)

where (FF)i is the length variations resulting from

pairs of fitting features, p the number of pair of fitting

features, q the number of length dimensions, Ai the

sign constants 1 or �1, Si the sensitivity factors, Xi the

functional dimensions (basic size), and xi the func-

tional dimensions (size tolerance).

For fitting features the choice of dimensions and

tolerances of the mating parts are restricted and the

proposed strategy assigns a higher priority in their

solution search over others. Thus, by solving fitting

features first, the mixed type of problem will be

converted into a length dimension problem. After

solving each fitting feature problem the target value

must be modified. For this modification the sign of the

clearance of each fitting feature has to be considered.

When a clearance is negative, i.e. interference fit, the

influence of that fit on the target value of the mixed-

type problem is neglected.

4.2.4. Strategy for solving surface texture problems

The solution of surface texture problems refers to

finding manufacturing processes which can be used

for achieving the specified Ra value. This is done by

consulting process capability databases containing

relationships between surface texture parameters

and manufacturing process. Examples of such rela-

tionships between surface texture parameters (Ra

values) and different machining processes are given

in Appendix C of AS1100.201-1992 [22].

4.3. Database for solving FD&T problems

The third major part of the prototype software is a

database containing information required for solving

4 Experienced designers tend to allocate tolerance values based

on past experience and while a formal method should always be

encouraged, there may be situations where previous experience will

prove to be beneficial.


FD&T problems such as: preferred sizes table; pre-

ferred tolerances table; preferred fits table; limits and

fits table; fit application table, standard tolerances, and

process capability. Data adopted in the prototype

software are indicative only and are based on cap-

abilities of different processes under average condi-

tions gathered from different sources [18–24] reported

in literature. The process capability database contains

data on the process capability tolerances expressed in

IT grades and process capability surface texture para-

meter Ra values expressed in microns. The database

can be accessed from any part of the program; how-

ever, it can be updated only at the beginning of the

program.

5. Software development

The prototype software, Functional Dimensioning

and Tolerancing (FDT), has been developed using

Microsoft Visual Basic (Version 4.0, Professional

Edition) software development package. It has a menu

system of five titles viz. File, Revise, Database,

Results, and Help. Navigation between forms is per-

formed through clicking the Confirm and Previous

buttons placed at the bottom of the screen. All data

entry points are protected against entering the wrong

type of data. Not only that, where applicable, entered

data is checked for any errors of logic, such as target

values where Cmin cannot be greater than Cmax.

FDT starts with a screen where the user has two

options: (i) solve a group of loop equations and (ii)

solve one loop equation only. The main program

(MatrixSolver) deals with the first option. When the

second option is selected it solves one loop equation

by calling a particular module from the solution

library. The advantage of solving one loop equation

without the use of the main program is that it will

obviously save time because it will not represent the

problem in a matrix form.

6. Solved examples

To demonstrate the effectiveness of the prototype

software the following two examples have been

solved. The steps involved in solving these two exam-

ples are described below.

6.1. Problem no. 1

This problem is taken from Chase and Greenwood

[25] first introduced by Fortini [26]. The target values

are changed to demonstrate the effectiveness of the

strategy (all dimensions are in mm).

Functional requirements:

� Shafts endplay: (Z � z) (clearance in Fig. 6).

� Target values: ðZ � zÞ ¼ 0:5 � 0:266.

Functional equation:

ðZ � zÞ ¼ �ðA � aÞ þ ðB � bÞ � ðC � cÞ þ ðD � dÞ� ðE � eÞ þ ðF � f Þ � ðG � gÞ (11)

Purchased parts:

� Retaining ring, ðA � aÞ ¼ ð1:283 � 0:019Þ.� Bearing, ðC � cÞ ¼ ð12:936 � 0:032Þ.� Bearing, ðG � gÞ ¼ ð12:936 � 0:032Þ.

6.1.1. Solution

This is a length dimension problem represented by a

single functional equation; therefore, the LengthSol-

ver module was used for the solution. This assembly

contains three purchased parts; their dimensions are

non-negotiable and were taken out of both basic size

and tolerance equations. Fig. 7 shows the initially the

CE team proposed turning operation with IT grade 9

and capability index 1.33 for manufacture of the

remaining features. However, this proposal is not

acceptable because of negative residual tolerance

(�0.0177 mm). Fig. 7 also reveals that the problem

could be rectified by changing the process used for

Fig. 6. Shaft and bearing assembly [25].


Fig. 7. Screen print showing the intervention strategy of LengthSolver.

Fig. 8. Screen print showing ‘what-if’ capability of LengthSolver.


producing dimensions B or E by 1 IT grade (24%

reduction), whereas for achieving the same goal the IT

grade for the process used for the manufacture of

dimension D or F had to be reduced by 3 (68%

reduction). On this basis the CE team decided to

change the manufacturing operation for producing

the dimension B to fine turning with IT grade 8 and

capability index 1.33. Fig. 8 illustrates that where the

proposed manufacturing processes are acceptable

(residual tolerance þ0.093 mm), some of the proposed

tolerance values (d and f) are not compatible with the

process capability tolerances. At this stage the CE

team was able to do a ‘what if’ analysis, by trying out

different scenarios and finally came up with an accep-

table solution. The final results from the LengthSolver

are given in Fig. 9.

6.2. Problem no. 2

Fig. 10 illustrates a belt drive unit dimensioning and

tolerancing problem adopter from Williams [27]. The

following functional requirements are to be met (all

dimensions are in mm):

FR1: Fit 1, 2: Cmin ¼ 0, Cmax ¼ 0:092.

FR2: Fit 3, 4: MF ¼ �0:018 � 0:017.

FR3: Fit 5, 6: should be a ‘normal running’ fit.

FR4: Fit 7, 8: Cmin ¼ 0, Cmax ¼ 0:092.

FR5: The end play of the spindle is not to be less

than 0.06 and is not to exceed 0.188.

FR6: The distance Z1 is to be controlled within

65:� 0:064 taking into account the end play.

FR7: The distance Z2 is to be between 14.08 and

13.92 taking account the end play.

Calculate the values of the dimensions and toler-

ances to be placed on the drawing to satisfy the above

functional requirements.

6.2.1. Solution

The loop equations are as follows (see Fig. 11):

FR1: Fitting feature problem with type 2 target

value.


value.


value.


value.

FR5: End play ¼ B � A.

FR6: Z1 ¼ B þ C

FR7: Z2 ¼ B � A þ D.

Fig. 9. Print showing final results of LengthSolver.


The initial choices of the CE team for manufacture

of the different features in the proposed design are

given in Table 3.

This problem consists of seven functional equations;

the problem was firstly represented in the matrix format

with the help of MatrixSolver. Then the functional

requirements were grouped into five groups according

to the interrelationships among the variables. The soft-

ware then provided an optimum solution order. The first

four functional requirements are independent func-

tional requirements and were solved first. As FR1 to

FR4 are fitting feature problems, FitSolver module was

used for finding their solutions. Fig. 12 illustrates a fit

ranking table used in the process of solving FR2. The

final results from MatrixSolver are depicted in Fig. 13.

Note that the solution order of functional requirements

in Group no. 5 were changed from FR5 > FR6 > FR7

to FR6 > FR5 > FR7. The effectiveness of the proto-

type software becomes more evident with an increase in

the number of functional equations with more compli-

cated interrelationships. However, such problems were

not chosen here due to space constraints.

7. Software evaluation

The primary characteristic of the CE concept is its

team approach to product development; the prototype

software was designed with this in mind. However,

the evaluating of software for team work is difficult

[28] because it is difficult to create the dynamics of

interactions between individual team members in the

laboratory; it is also time-consuming, because team

interactions usually unfold over a relatively long period

of time. Anticipating these difficulties, the prototype

software was not evaluated in a CE environment. The

two evaluation processes followed are described below.

7.1. Software evaluation against selected criteria

First the requirements of a FD&T tool suitable for

a CE environment were formulated by considering

Fig. 10. Belt drive unit [27].

Table 3

Initial choices of the CE for manufacture of belt drive unit

Part name Feature name Dimension symbol Design size Material Manufacturing process Difficulty level

Pulley Hole diameter D2 10.00 Cast iron – –

Bush External diameter D4 20.00 Bronze – –

Spindle Bearing diameter D6 12.00 Steel – –

Gear Hole diameter D8 10.00 Steel – –

Housing Hub depth A 36.00 Cast iron Turning Moderate

Spindle Bearing length B 36.00 Steel Turning Moderate

Pulley Hub offset C 29.00 Cast iron Turning Moderate

Gear Boss depth D 14.00 Steel Turning Moderate


various requirements for such a tool as reported

in the literature. Details of this establishment process

can be found in [13]. FDT was then evaluated

against these criteria and the evaluation results

are summarized in Table 4, with brief explanations in

subsequent paragraphs.

R1: The proposed methodology helps to describe or

quantify the functional requirements of the design

with the help of some existing CE tools, such as

QFD and tree diagram. However, this aspect of the

Fig. 11. Loop equations: belt drive unit.

Fig. 12. Screen print showing fit ranking table in FitSolver.


methodology has not been integrated into the pro-

totype software. Software packages are available for

QFD analysis, e.g. QFD/CAPTURE developed by

International Technical Group [32] and for construc-

tion of Tree Diagrams, e.g. Microsoft Visio [33].

R2: FDT helps to develop functional equations in a

structured way, although the actual equations are

not generated.

R3: FDT tries to find economic solutions by

applying an informal optimization strategy.

Fig. 13. Screen print showing final results from MatrixSolver.


R4: FDT considers assembly and inspection

requirements when specifying functional require-

ments in form of functional equations and

manufacturing requirements during finding a

solution for functional equations.

R5: FDT helps in decision making at multiple

stages of product development.

R6: FDT is easy to use and does not require

advanced technical knowledge.

R7: FDT is interactive and has an attractive

Graphic User Interface.

R8: FDT runs on Windows which is the most

widely used computer platform.

R9: FDT is easy to learn; many other usability

attributes are built into it.

R10: FDT has a structured methodology which

helps the user in solving FD&T problems through

step by step instructions.

R11: It can solve four commonly occurring types

of 1D, FD&T problems, viz. fitting feature

problems, length dimension problems, mixed-type

problems, and surface texture problems as well

as coupled loop equations for these types of

problems.

R12: The results obtained from the prototype

software were compared with the results obtained

through manual calculations and they matched

one another. This indicates that the results are

credible.

From the foregoing, its appears that the prototype

software has satisfied most of its requirements at a

satisfactory level and it has the potential to be an

effective tool for solving FD&T problems in a CE

environment. As the above evaluation was per-

formed by the developer there might be a perceived

bias in the evaluation findings. In this case, the

evaluation findings should be treated as the devel-

oper’s claims about the prototype and a survey of

the opinions of independent potential users was

undertaken.

7.2. Software evaluation by its potential users

The prototype software was evaluated by 15

volunteers chosen from the staff and students of

the School of Mechanical and Manufacturing Engi-

neering, The University of New South Wales, Aus-

tralia. Participants had varying degrees of familiarity

with FD&T problems. Evaluation sessions were run

within a lab environment with one participant at a

time. At the beginning of each session each partici-

pant was briefed on the evaluation procedure; cap-

abilities of the prototype software were demonstrated

by the evaluator solving some sample problems.

Participants were then individually asked to solve

some exercise problems and were monitored during

their use of the software and any difficulties faced by

them were noted as well any questions or additional

Table 4

Evaluation summary of the prototype software against selected criteria

Req. No Description Source Findings

R1 It should help the user to describe and quantify the functional requirements of the design [29] No

R2 It should help the user to develop functional equations [29] Yesa

R3 It should provide and economic solution to the functional equations [29,19] Yesb

R4 It should consider manufacturing, assembly, and inspection requirements in tolerance selection [29,30] Yes

R5 It should help in decision making at multiple stages of product development [30] Yes

R6 It should be suitable for team members from different tecchnical backgrounds [29,30] Yes

R7 It should be interactive [29] Yes

R8 It should run on a platform that is easily accessible [31] Yes

R9 It should be easy to learn [31] Yes

R10 It should be based on a structured methodology [31] Yes

R11 It should be applicatble to a wide range of products [31] Noc

R12 It should be provide credible results [31] Yes

a It helps but does not generate loop equations automatically.b It does this indirectly.c It can solve certain types of 1D problems and is capable of solving coupled requirements.


information they requested. At the completion of

the tasks by each participant a short interview was

then recorded.

About 2 h of conversation was recorded, transcribed

and then evaluated. Details of the evaluation can be

found in [13]. From the analysis it appears that except

for three participants all were very satisfied with

the performance of the software. One participant

expressed reservations about its practical use. Another

participant felt that due to the sequential nature of the

program, making changes might be difficult. Some of

the participants praised its capability of making a

‘what-if’ analysis and were impressed with the good

use of colors. Others found the ability to produce fast

results most useful.

The prototype software required a great amount of

manual input. This problem will be rectified once it is

interfaces with a CAD system for which collaboration

of the CAD supplier is required. The inability of the

prototype software to illustrate the whole matrix on a

single screen was identified as the single most con-

cerning issue throughout the evaluation process. How-

ever, this is a common problem in most computer

programs and could be solved by displaying the screen

on a wall. This solution is particularly useful in a CE

environment where the members of the CE team

would be able to get a ‘total’ picture of the problem

being solved.

8. Discussion

FDT is FD&T software in its true sense. It is based

on a structured FD&T methodology that leads the user

towards an optimum solution for common 1D dimen-

sioning and tolerancing problems. The emphasis of

most of the commercially available packages has been

tolerance analysis. Therefore, during FD&T the CE

team members are only able to test the validly of their

selection using these packages. Still, the tolerance

values should not be changed on the basis of simula-

tion results only, which analyses the assembly require-

ments only, whereas FD&T involves the fulfillment of

a number of other requirements, such as manufactur-

ing and inspection. Furthermore, the statistical data

required by these packages, such as distributions of

input tolerances, are often not available at an early

design stage.

The use of a matrix or spreadsheet format for

organizing dimensioning and tolerancing problems

is not new; designers have been using different

custom-made formats for years. Some of the commer-

cially available packages, e.g. CATS-1D XL, also use

such format. However, their representation is not

meaningful. The format used by FDT is unique and

it represents the complexity of the FD&T problem of

product.

FDT increases the efficiency of the solution

search process, firstly, by dividing the whole

FD&T problem of a product into groups and then

by providing solution orders for functional require-

ments in each group. In a real life problem the

number of requirements and the number of variables

will be high and a way of storing and managing all

this information is required. Any FD&T tool suita-

ble for the CE environment should provide all of

these facilities.

FD&T requires frequent use of different types of

data which is catered for in FDT in form of a database.

The accessibility of the database from any part of the

software is of enormous help to its user. The upgrading

facility of the process capability data to reflect the

capability of actual manufacturing processes being

used is another plus.

FDT is PC-based and has an attractive GUI which

makes it very easy to use. It is a much simpler tool than

those available commercially. It does not require a

high level of technical knowledge, thus is suitable for

all members of a CE team, especially the non-tech-

nical members. It produces fast results, indeed almost

instantly, after entering required input data. It provides

a ‘what-if’ analysis facility for rapid evaluation of

different design alternatives. A number of error avoid-

ance measures are built into the software which pre-

vents the user from making any error in the first place

and if any error is made its step back facility allows

easy correction. It allows the user to save the project

data so that the user can come back to it later. It gives

attractive print-outs (both in graphical and text form)

which are useful for presenting results.

9. Concluding remarks

Evaluation of the prototype software indicated

that it can be a useful tool for solving FD&T


problems, though its interface needs some improve-

ments. Although the prototype software was not

evaluated in a CE environment, the evaluations

results indicate that it could be an effective tool

for solving FD&T problems in a CE environment.

The evaluation results of the prototype software

presented in this paper are based on qualitative

evaluation only which highlighted some problems

areas. These problem areas should be further inves-

tigated through quantitative evaluation. Finally,

copies of the prototype software should be made

available to the members of the CE team for evalua-

tion, and the feedback should be incorporated into

further refinements.

Acknowledgements

The work presented in this paper was carried out

in the School of Mechanical and Manufacturing

Engineering, University of New South Wales,

Australia under the supervision of Dr. L. E. Farmer.

The author would like to thank Dr. Farmer for

his input and the Department of Employment, Train-

ing, and Youth Affairs, Government of Common-

wealth of Australia, for their financial support

through the provision of an Australian Postgraduate

Award.

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HTTP://WWW.VSA.COM

HTTP://WWW.VSA.COM

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HTTP://WWW.3DCS.COM

HTTP://WWW.3DCS.COM

HTTP://WWW.CI.COM/PRODUCTS/PRODUCTS.HTML

HTTP://WWW.CI.COM/PRODUCTS/PRODUCTS.HTML

HTTP://WWW.SALTIRE.COM/AX.HTML

HTTP://WWW.SALTIRE.COM/AX.HTML

HTTP://WWW.TECNOMATIX.COM/

HTTP://WWW.DECISIONENGINEERING.COM

HTTP://WWW.DECISIONENGINEERING.COM

HTTP://ADCATS.ET.BYU.EDU/HOME.HTML

HTTP://ADCATS.ET.BYU.EDU/HOME.HTML

[29] L.E. Farmer, C.A. Gladman, Tolerance technology—

computer based analysis, Annals CIRP 35 (1) (1986)

7–10.

[30] G. Sohlenius, Concurrent engineering, Keynote Paper at the

42nd General Assembly of CIRP, Annals of the CIRP 41 (2)

(1992) 645–655.

[31] Tool and Manufacturing Engineers Handbook, vol. VI,

Design for Manufacturability, 4th edition, SME, Dearborn,

MI, 1992.

[32] International Technical Group Inc. Home Page, http://

www.qfdcapture.com, (13/3/2003).

[33] Microsoft Visio Home Page, http://www.microsoft.com/

office/visio/, (13/3/2003).

M.N. Islam obtained his engineering

degree from the Higher Institute for

Mechanical and Electrical Engineering,

Varna, Bulgaria, his M.E. (Hons.) from

the University of Wollongong, Australia

and his Ph.D. from the University of New

South Wales, Australia. Currently he is

working as a Visiting Professor at the

Department of Mechanical Engineering,

Pohang University of Science and Tech-

nology, South Korea. His research inter-

ests include Dimensioning and Tolerancing, Concurrent Engineering

in Product Development and Machine Quality Monitoring.


HTTP://WWW.QFDCAPTURE.COM

HTTP://WWW.QFDCAPTURE.COM

HTTP://WWW.MICROSOFT.COM/OFFICE/VISIO/

HTTP://WWW.MICROSOFT.COM/OFFICE/VISIO/

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