Musa Rami Adnan

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UNIVERSITY OF CINCINNATI

_____________ , 20 _____

I,______________________________________________,

hereby submit this as part of the requirements for the

degree of:

________________________________________________

in:

________________________________________________

It is entitled:

________________________________________________

________________________________________________________________________________________________

________________________________________________

Approved by:

________________________________________________

________________________

________________________

________________________

August 30th 03

Rami A. Musa

Master's of Science

Industrial Engineering

Simulation-Based Tolerance Stackup

Analysis for Machining

Dr. Samuel Huang (Chair)

Dr. Richard Shell

Dr. Sam Anand


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Simulation-Based Tolerance Stackup Analysis in Machining

A thesis draft submitted to the

Division of Research and Advanced Studies

of the University of Cincinnati

in partial fulfillment of the

requirements for the degree of

MASTER OF SCIENCE

In Industrial Engineering

in the department of Mechanical, Industrial and Nuclear Engineering

of the College of Engineering

August, 2003

by

Rami A. Musa

Bachelor of Science in Mechanical Engineering

Jordan University of Science and Technology, 1999

Committee Chair: Dr. Samuel H. Huang


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I

Abstract

Dimensional and geometric tolerance can result from either process variation and/or process

stackup tolerance. Tolerance stackup (accumulation) is an important topic in machining that

is interrelated with tolerance control, tolerance allocation and setup planning. During

machining operations of a part, tolerance stackup is inevitable most of the time. Therefore,

tolerance stackup must be studied accurately and efficiently. In spite of this, traditional

methods for analyzing stackup (statistical and worst-case methods) have some drawbacks that

reduce their accuracies. These drawbacks are discussed in details. This study presents a novel

method for analyzing tolerance stackup in three dimensional-space by simulating machining

and inspection process using Monte Carlo simulation along with major manufacturing errors.

It overcomes the argued drawbacks in the traditional methods. Further, it is proved that both

the statistical and worst-case methods are conservative compared to the proposed one.

Therefore, simulation-based tolerance stackup analysis is more cost-effective as it gives more

chances to accept process plans that are usually precluded using the traditional ones. Three

illustrative examples are presented to compare the results of the simulation with the

traditional methods.


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Acknowledgments

First of all, I wish to offer my sincerest gratitude to my advisor; Dr. Samuel Huang who was

an outstanding advisor in all measures during my work with him. His professionalism,

knowledge and keenness inspired and taught me a lot.

True thanks to Dr. Sam Anand and Dr. Richard Shell for serving as committee members in

my thesis defense, words of encouragement and appraising my effort. Also, I would love to

thank and recognize my friends: Mohammad Hamdan, Mohammad Younis and Zain Dewaik,

who introduced and encouraged me all the way to go for my graduate study. Also, I would

like to extend my thanks to my colleague and friend Anshum Jain who contributed

significantly in conducting the experiment. Most prominently, my deepest gratefulness is to

my family for their encouragement and support. I always felt I am the luckiest person in the

world to have such a family; my late father, my loving mother and my brothers: Naji and

Husam.

This work has been gratefully sponsored by the National Science Foundation and thankfully

collaborated with Delphi Automotive Systems in Dayton, Ohio.


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Contents

1. INTRODUCTION 6

1.1BACKGROUND AND MOTIVATION 6

1.2OBJECTIVES OF THE RESEARCH 12

1.3THESIS ORGANIZATION 13

2. LITERATURE REVIEW 14

2.1BASIC CONCEPTS 14

2.2TOLERANCE STACKUP;DEFINITION AND APPLICATIONS 18

2.3TRADITIONAL ANALYTICAL TOLERANCE STACKUP ANALYSES 20

2.3.1WORST-CASE ANALYSIS 22

2.3.2STATISTICAL ANALYSIS 22

2.4TOLERANCE CHART 23

3. SIMULATION-BASED TOLERANCE STACKUP ANALYSIS 25

3.1SIMULATION ARCHITECTURE 25

3.1.1MONTE CARLO SIMULATION 29

3.1.3MANUFACTURING ERRORS 30

3.1.3.1ERRORCATEGORIES 30

3.1.3.2MACHINING ERROR(CUTTING TOOL DEVIATION) 34

3.1.3.3LOCATING/CLAMPING DEVIATION (FIXTURE UNIT ERROR) 35

3.1.3.4RAW PART ERROR 35

3.1.5ERRORSYNTHESIS (AGGREGATION) 35

3.2VIRTUAL INSPECTION 36


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3.2.1DATUM EVALUATION 36

3.2.2EVALUATION ALGORITHMS FORGD&T 41

3.3SAMPLE PLAN 43

3.4STOPPING (TERMINATING)CRITERIA 45

4. MANUFACTURING ERROR EVALUATION 48

4.1MACHINING ERROREVALUATION 48

4.2LOCATING/CLAMPING ERROR(FIXTURE UNIT ERROR)EVALUATION 51

4.3RAW PART ERROREVALUATION ALGORITHM 54

5. ILLUSTRATIVE EXAMPLES 58

5.1.EXAMPLE 1:TWO MACHINING OPERATIONS (WITHIN ONE SETUP) 58

5.2.EXAMPLE 2:FOURMACHINING OPERATIONS (IN THREE SETUPS) 60

5.3.EXAMPLE 3:ABS PART 61

6. CONCLUDING REMARKS AND RECOMMENDATIONS 68

6.1SUMMARY 68

6.2RECOMMENDATIONS FORFUTURE WORKS 69

BIBLIOGRAPHY 71


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List of Figures

Figure 1-1: One way clutch mechanism [Chase, Gao and Magleby (1994)].............................8

Figure 1-2: One way clutch mechanism vector loop [Chase, Gao and Magleby (1994)] .........8

Figure 1-3: 1-D assembly mechanism [Law (1995)] ...............................................................10

Figure 1-4: 2-D Closed vector loop for one way clutch mechanism [Chase, Gao and Magleby

(1991)]..............................................................................................................................10

Figure 1-5: 3-D Closed vector loop for crank slider mechanism [Chase, Gao and Magleby

(1991)]..............................................................................................................................10

Figure 1-6: Ideal process condition..........................................................................................13

Figure 2-1: With Cp=1, only 2700 part per million (PPM) defects are expected ....................17

Figure 2-2: Mean drift in processes .........................................................................................17

Figure 2-3: Typical setup planning approach ..........................................................................20

Figure 2-4: Dimension Chain of c, 2 links, 1D........................................................................21

Figure 2-5: Dimension Chain of c, 4 links, 1D........................................................................21

Figure 2-6: Example of tolerance chart [Xue and Ji (2002)] ...................................................24

Figure 3-1: Part representation by sample points ....................................................................26

Figure 3-2: System Architecture ..............................................................................................26

Figure 3-3: Monte Carlo Simulation (source:

http://www.ymp.gov/documents/ser_b/figures/chap4_2/f04-174.htm)...........................30

Figure 3-4: Error models used in this study.............................................................................33

Figure 3-5: Setup Error ............................................................................................................34

Figure 3-6: Combined effects of setup and machining errors..................................................34

Figure 3-7: Translated least-squares approach ........................................................................37

Figure 3-8: Candidate datum set approach ..............................................................................39

Figure 3-9: 2-D projection of the convex hull [Wilhelm (1998)]............................................40


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Figure 3-10: Non-rejected datum [Wilhelm (1998)] ...............................................................40

Figure 3-11: Sample points located in a square feature using different approaches ...............45

Figure 3-12: Sample point locations using random and low-discrepancy methods [Davis and

Martin (1998)]..................................................................................................................45

Figure 3-13: Benchmarked results at 1 billion iterations [Cvetko, Chase and Magleby (1998)]

..........................................................................................................................................47

Figure 4-1: Dial Indicator measuring machined surface..........................................................50

Figure 4-2: Fixture Unit with the workpiece ...........................................................................52

Figure 4-3: Part Surfaces .........................................................................................................53

Figure 4-4: Coordinate measuring machine (CMM)...............................................................56

Figure 4-5: Fixture unit with CMM probe...............................................................................56

Figure 5-1: Example 1 (design requirements and the machining sequence) ...........................58

Figure 5-2: Example 1 results ..................................................................................................60

Figure 5-3: Example 2 (design requirements...........................................................................61

Figure 5-4: Example 2 output (The simulation output for the distributions of the distances

between surfaces).............................................................................................................61

Figure 5-5: Example 3; ABS (Antiblock System) housing, Bosch (Source:

http://www.wzl.rwth-aachen.de/WM/SIMON/deliverables/DA0/DA0_02D.htm).........62

Figure 5-6: ABS dimensional requirements ............................................................................62

Figure 5-7: ABS part setup plan ..............................................................................................64

Figure 5-8: Tolerance Chart of ABS part ................................................................................65

Figure 5-9: Dimensions histogram using simulation...............................................................66

Figure 5-10: Progress of results with sample size increase .....................................................66

Figure 5-11: Rejection areas comparison when allocating concluding links tolerance using

worst case, statistical and simulation methods ................................................................67


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List of Tables

Table 2-1: Geometric Tolerances (ASME Y14.5M-1994)......................................................16

Table 3-1: Manufacturing Error Classification........................................................................31

Table 3-2: Manufacturing Error Models..................................................................................32

Table 3-3: Recommended sample size for different geometries [Henzold (1995)].................44

Table 4-1: Data Collection.......................................................................................................49

Table 4-2: Machining Error Data.............................................................................................50

Table 4-3: Data Collection.......................................................................................................54

Table 4-4: Variance comparison between simulation and experiment for smooth part ..........57

Table 4-5: Variance comparison between simulation and experiment for a rough part..........57

Table 5-1: Tolerance stackup evaluation comparison for example 1 ......................................60

Table 5-2: Tolerance analysis results for example 2 ...............................................................61

Table 5-3: Simulation results at 100,000 iterations .................................................................63

Table 5-4: Tolerance evaluation using the three approaches...................................................64

Table 5-5: Part per million (PPM) rejections comparison when allocating tolerance using

worst case, statistical and simulation methods ................................................................67


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1.Introduction

1.1 Background and Motivation

Tolerance is a common arguing point between design and manufacturing. Design engineer

tends to tighten the tolerance to meet functional requirements whereas production engineer

tends to loosen (relax) it to satisfy resource availability. Nevertheless, the most important

factor to be considered is the cost. Cost increases hysterically by tightening the tolerance.

However, since tolerance is inevitable as it is impossible to have perfectly accurate

machining, raw part, fixture unit and measurement machine, it has to be compromised by

different departments in the companies. Obviously, tolerance problem is kind of promoter for

concurrent engineering work among organization departments; namely: design,

manufacturing, customer service and management.

One serious problem in process planning is that some good plans (plans that lead to design

requirement satisfaction) could be rejected and some bad plans could be accepted due to

inaccurate traditional methods of evaluating tolerance stackup. Tolerance stackup can be

defined as the accumulation (or stackup) of errors when machining a part using different

operational datum than the ones specified in the blueprints. The two traditional methods used

nowadays to analyze tolerance stackup in machining are: worst-case and statistical methods.

These methods are believed to have major drawbacks that reduce the accuracy of tolerance

stackup evaluation. These drawbacks are:

1. Worst-case is exaggeratedly pessimistic in calculating tolerance stackup.2. Statistical analysis assumes independency between dimensions. Further, statistical

analysis assumes that the contributing links are normally distributed.


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3. Tolerance stack between features is preformed in one dimension; which does notrepresent the actual three-dimensional features of interest. 3-D simulation must be the

driving force behind the entire dimensional management process [Craig (1996)].

4. Manufacturing errors are not taken into account.5. Geometric tolerance stackup cannot be estimated. The stackup of geometric tolerance

was usually ignored [Lin and Zhang (2002)].

Additionally, we found that both of the traditional methods evaluate tolerance stackup

conservatively. In this work we developed a more accurate method for evaluating tolerance

stackup in machining that can lead to more cost-effective (less conservative) and/or less

tighter plans. Our method overcomes the above-mentioned drawbacks by simulating

machining and inspection processes along with major manufacturing errors using Monte

Carlo simulation. It will be shown in chapter 5 (illustrative example 3) that using our method

for stackup evaluation will result in much less rejects expectations per million parts compared

to the traditional methods using the same resources.

Machining Tolerance Stackup vs. Assembly Tolerance Stackup

Some research works have been done in assembly tolerance stackup using Monte Carlo

simulation in the literature. Although this seems quite close to our work here in tolerance

stackup for machining, there are exclusive differences between the two problems, their

formulations and applications. Component (part) and assembly designs are the two major

tasks in any design department. Component design provides a single component drawing that

include dimensions; and dimensional and geometric tolerances. Some examples of

components are: shaft, gear, pulley, etc. However, it is unlikely to have a component

functioning alone as there is a need to assemble it with other components. Assembly design


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studies the suitability of two or more components to meet machine functions. When

assembling parts together, there should be some manufacturing variation in the part that will

cause assembly tolerance stackup. Figures 1-1 and 1-2 show an example of an assembled one

way-clutch mechanism. The mechanism consists of: four rollers, a hub, four springs and an

outer ring. The objective of the tolerance analysis here is to study the effect of manufacturing

errors in component dimensions (a, e, c) on assembly dependent dimensions ( b,1 ).

Figure 1-1: One way clutch mechanism

[Chase, Gao and Magleby (1994)]

Figure 1-2: One way clutch

mechanism vector loop [Chase, Gao

and Magleby (1994)]

This problem has been studied extensively by Chase, Magleby and Gao in Brigham Young

University. They developed computer software (CATS) that applies their methods in

assembly tolerance analysis. Another system has been developed by Variation System

Analysis (VSA).

The first step in evaluating assembly tolerance stackup in the literature is to find an explicit

function of the dimension (tolerance) to be controlled in terms of the other components using


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trigonometric functions. The following are the explicit functions of the assembly dimensions

for the mechanism shown in figure 1-2 [Gao, Chase and Magleby (1997)]:

)(cos1

1ceca

+= (1.1)

22 )()( caceb ++= (1.2)

Some assembly tolerance stackup methods in the literature that assumes the availability of

explicit assembly functions are [Gao, Chase and Magleby (1997)]:

1. Linearization of the assembly function using Taylor series expansion,

2. Method of system moments,

3. Quadrature,

4. Monte Carlo simulation,

5. Reliability index,

6. Taguchi method.

Normally, it is very hard or even impossible to get explicit assembly equations for a typical

assembly mechanism. Vector-loop-based assembly models use vectors to represent the

dimensions in an assembly that can be used to find a set ofimplicit assembly equations.

Figures 1-3, 1-5 and 1-6 show examples of closed vector loops for 1-D, 2-D and 3-D

mechanisms.


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Figure 1-3: 1-D assembly

mechanism [Law (1995)]

Figure 1-4: 2-D Closed vector

loop for one way clutch

mechanism [Chase, Gao and

Magleby (1991)]

Figure 1-5: 3-D Closed vector

loop for crank slider mechanism

[Chase, Gao and Magleby (1991)]

The following are the governing assembly equations for the closed loop one-way-clutch

shown in figure 1-2 [Gao, Chase and Magleby (1997)]:

2121

11

11

901809090900

)cos()cos(0

)sin()sin(0

+=+++==

++==

+==

h

eccah

ecbh

y

x

(1.3)

From the third equation in the previous set of equations (1.3), it can be seen that == 21 .

This reduces the equations into two as follows [Gao, Chase and Magleby (1997)]:

)cos()cos(0

)sin()sin(0

++==

+==

eccah

ecbh

y

x(1.4)

It is apparent that it is very difficult to convert these equations into explicit form. The main

two methods available in the literature to solve this problem for implicit assembly functions

are:Direct Linearization Method(DLM) and Monte Carlo Simulation. First order Taylor


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series linearizes the assembly constraints in DLM to have a set of linear simultaneous

equations and then linear algebra is used to solve them. Afterwards, assembly tolerance

stackup are estimated using statistical or worst case methods. Monte Carlo simulation method

includes the following steps: (1) generate random variates for each variable in the assembly

constraints (2) Select appropriate nonlinear solvers to solve the constraints (3) fit the output

numbers with a distribution and get its parameters (first four moments: mean, variance,

skewness and kurtosis.) Chase, Gao and Magleby use Crystal Ballsoftware to solve the

problem. Crystal Ball is spreadsheet Monte Carlo Simulation software that can solve implicit

nonlinear simultaneous equations.

Gao, Chase and Magleby (1995) made a comparison between the two methods. It turned out

that the concern regarding the DLM is the accuracy and the concern regarding the Monte

Carlo simulation is the huge number of iterations needed to solve the problem.

Noteworthy, the following are the differences between using Monte Carlo simulation for

machining tolerance stackup analysis [Musa and Huang (2003)] and assembly tolerance

stackup analysis [Chase, Gao and Magleby]:

(1)Method. Machining tolerance stackup analysis simulates manufacturing variationswhereas assembly tolerance analysis simulates component variations because of

manufacturing variations. The two analyses are close in the sense that we are trying to

maintain a tolerance for a critical componentin the case of assembly tolerance

analysis and a concluding linkin the case of Part tolerance analysis.

(2)Objective. The objective of assembly tolerance analysis is to assign tolerances for allthe assembly components to maintain a specific tolerance for a critical component

whereas the objective of machining tolerance stackup analysis is to evaluate the


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goodness of a process plan and/or assign proper contributing link tolerances

(increasing and decreasing links) to maintain concluding link tolerance.

(3)Sequence. Assembly tolerance analysis comes after machining tolerance stackupanalysis.

(4)Independence. It is safe to say that mechanical components variations areindependent which is not the case for machined features in machining.

1.2 Objectives of the Research

Improving quality and reducing cycle time and cost are the main objectives for competitive

manufacturing these days. In other words, achieving minimum tolerance possible using the

available resources, reducing trial and error procedures and taking economical issues into

consideration can lead to the ideal process which all industries aim at (figure 1-6). These

objectives can be achieved partially by effectively controlling the tolerance in manufacturing.

Tolerance control involves controlling the tolerance stackup via proper choices of processes,

process sequence, and locating datums.

The objective of this study is to present a novel, less conservative and more accurate

evaluation method of tolerance stackup compared to the existed analytical ones (worst case

and statistical methods) in the literature. This method is based on simulating machining and

inspection process using Monte Carlo simulation along with major manufacturing errors.


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IDEAL PROCESS

TIGHTEST

TOLERANCE

POSSIBLE

ITERATIVE

DESIGN/

MANUFACTURE

AVOIDANCE

ECONOMICALLY

FEASIBLE

Figure 1-6: Ideal process condition

1.3 Thesis Organization

The thesis is divided into six chapters. It starts in chapter 1 with the introduction that explores

background of the problem, motivation and objectives of the work. Then, chapter 2 reviews

some basic concepts and terms that are commonly used in later chapters and discusses the

problem of tolerance stackup by defining it, presents the traditional analytical methods

available in the literature and discusses tolerance chart method. Monte Carlo simulation

based tolerance stackup method is described in chapter 3; in which simulation architecture,

manufacturing error categories and models, sample plan and stopping criteria for the

simulation are illustrated. Afterwards, experiment procedures, requirements and algorithms

for evaluating: machining, fixture unit and raw part errors are outlined in chapter 4. In

chapter 5, three illustrative examples are demonstrated and solved using the proposed method

and comparisons are made between the traditional methods and the proposed one. Finally,

concluding remarks and future work comments are addressed in chapter 6.


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2.Literature Review2.1 Basic Concepts

The following are some commonly-used terms and concepts in this thesis:

Feature: Any surface in the machined part (e.g. hole, slot, boss, tab).

Datum: It is a reference feature for machining and measurement.

Dimension: Dimension is the representation of feature size or its location.

Tolerance: The permissible amount of variability in geometry.Limit of size andplus-minus

tolerances are two methods used to specify tolerances. Limit of size means that an upper and

lower limit are given for a specific dimension. As for plus-minus tolerance, a nominal (target

value) followed by a plus-minus expression of a tolerance [Krolikowski (1998)].

Setup: The state of locating and clamping workpiece to be machined.

Fixture unit: A unit that is used to constrain the workpiece from movement during machining.

Size tolerancing (coordinate dimensioning and tolerancing) used to be the only approach for

dimensioning and tolerancing. In this approach, the dimension and its tolerance are

represented by the distance and its variation between two features or points. Although this

approach was found to be successful for many design cases, there were major shortcomings.

These shortcomings showed up because of the increased demand and need for high quality

products. The three main shortcomings are [Krolikowski (1998)]:

1. Coordinate dimensioning does not provide a clear relation between design,manufacturing and inspection, which could result in different interpretation in

manufacturing and inspecting a part.

2. It does not represent the tolerance zones properly in some cases. An example is thatfor a cylindrical feature, the tolerance zone is rectangular.


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3. The functional requirements (e.g. assembly) in manufacturing the part are not valid.This is because that the tolerance zone is fixed in size.

In order to remedy these shortcomings when using coordinate tolerancing, long written

comments have to be provided in the design drawings. More practically, Geometric

Dimensioning and Tolerancing(GD&T; ASME Y14.5M-1994) can be used and can solve all

the shortcomings efficiently by:

1. Obtaining clear instructions for inspection and manufacturing (by using the datumconcept).

2. Tolerance zone geometries can be rectangular, circular or cylindrical.3. Providing clear functional requirements of manufacturing a part by using material

condition modifiers (Maximum Material Condition (MMC), Least Material Condition

(LMC), and Regardless of Feature Size (RFS)).

Geometric tolerances include fourteen types of tolerances that are usually categorized into

five categories; namely: form, orientation, profile, location and runout. Form tolerances

include: flatness, straightness, cylindricity and circularity (roundness). Orientation tolerances

include: parallelism, angularity and perpendicularity. Profile includes: profile of a line and

profile of a surface. Runout tolerances include: circular runout and total runout. Finally,

location tolerances include: position, symmetry and concentricity. They can be further

classified into datum-dependent and datum-independent tolerances. Table 2-1 depicts all the

geometric tolerances, their symbols and their dependencies on datum.


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Table 2-1: Geometric Tolerances (ASME Y14.5M-1994)

Category Characteristic Symbol Datum Dependency

FlatnessStraightness

CylindricityForm

Roundness

Never

Parallelism

AngularityOrientation

Perpendicularity

Always

Profile of a line

Profile Profile of a surfaceSometimes

Position

SymmetryLocation

Concentricity

Always

Circular runoutRunout

Total runoutAlways

Process Capability: The process is considered capable if the process variability is equal or

less than the design specification (tolerance). Usually, it is represented by the Cp index which

is the ratio of design specifications (tolerance, T) to the process variability (6).

66

LSLUSLTCp

== (2.1)

The USL and LSL are the upper and lower specification limits. Referring to figure 2-1,

considering the design tolerance equals to 6 (Cp=1) implies that we are satisfied with about

2700 PPM rejects. Nevertheless, Cp index assumes that the process does not drift from the

mean (refer to figure 2-2). Six sigma quality strategy (developed by Motorola in 1980s)


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2.2 Tolerance Stackup; Definition and Applications

In general, tolerance results from bothprocess tolerance and the tolerance stackup

[Whybrew and Britton (1997)]. The latter is the accumulation (buildup) of error (tolerance) in

a dimension between features resulting from taking operational datums that are different from

the ones indicated in the design specifications. In other words, if the datum indicated in the

design drawings is the one used for locating and clamping, then a stackup-free dimension will

result and there will be no tolerance stackup in this specific dimension. Consequently,

tolerance analysis and tolerance control will not be necessary since the tolerance will depend

solely on the process capability [Huang (1995)]. However, in practice, due to economic

reasons and resource constraints, design datums are not always used as locating and clamping

datums. Therefore, some of the blueprint dimensions will be machined indirectly. Hence, in

most cases tolerance stackup is inevitable.

The way of machining a part determines the stackup in a dimension. There are three main

approaches for machining a part:

1. Chain machining(point-to-point machining): In this approach, the current machined

surface is used as a datum to machine the next surface. This will result in thegreatest

accumulation of tolerance.

2.Base-line: This is how parts are machined in a single setup using NC machines. In this

approach, the operational datum is fixed (zeroed) by the coordinate system in the NC

machine for each machining cut. Using this approach decreases the tolerance stackup.

3. Mixed of chain machining and base-line: This happens when parts are machined in

multiple setups.


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Tolerance allocation is a crucial step in setup planning. Figure 2-3 shows a typical approach

for designing a setup plan. It can be seen that tolerance stackup analysis and tolerance

allocation play important roles in setup planning. Thus, tolerance stackup behavior needs to

be studied carefully and analyzed accurately in order to generate cost-effective setup plans.

During the setup planning, in order to maintain the required tolerances provided in the

blueprints, proper choices of the contributing ones (increasing and decreasing tolerances)

must be made. Achieving this with simulation is possible if we think of the problem in an

opposite way. Rather than providing tolerances for the contributing tolerances to get the

concluding one, the required (concluding) tolerance is provided in order to get tolerances of

contributing ones. Simulation can be run a number of times for a range of the modeled

manufacturing error values to find what the tolerance for each case. A more general

application of this simulation is automating setup planning (tolerance allocation is part of

setup planning). Setup planning can be defined as the act of preparing instructions to machine

a part. Decisions usually taken by the setup planner are: proper datums, machined surfaces,

operations and sequence of operations. The input of the problem is: design requirements and

available resources (tools, machines and fixtures). Essentially, this is an optimization problem

that aims at decreasing: cost and tolerance stackup. Tolerance stackup is part of the cost of

material removal operation. Simulation can be used here to check the goodness of a given

setup plan by examining if the proposed plan leads to acceptable tolerances or not.


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Setup Plan

- Setup formation

- Datum selection

- Setup sequences

ToleranceStackup Analysis

ProcessTolerance

Analysis

Tolerance

Allocation

Feasible?

Feasible

UnconstrainedPlans

Constrained

Optimization

Optimal Setup

Plan

Yes

No

Figure 2-3: Typical setup planning approach

2.3 Traditional Analytical Tolerance Stackup Analyses

The general relation of a distance in the x, y and z space can be expressed as following [Lin

and Zhang (2001)]:

),,( kji zyxfd= (2.3)

Where:

xi: (i=1,,l) the component dimensions in the X-axis.

yi: (j=1,,m) the component dimension in the Y-axis.

zk: (k=1,,n) the component dimension in the Z-axis.


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Dimensionchain (sometimes called tolerance chain) is a closed loop of interrelated

dimensions. It consists of increasing, decreasing links and a single concluding link. In figures

2-4 and 2-5, linki is the increasing link, dis a decreasing linkand c is the concluding link.

Apparently, the concluding link c is the one whose tolerance is of interest and which is

produced indirectly. Increasing and decreasing links (both called contributing links) are the

ones that by increasing them, concluding link increases and decreases; respectively.

c

i

d

Operational datum

Machined surface

Figure 2-4: Dimension Chain of c, 2 links, 1DFigure 2-5: Dimension Chain of c, 4 links, 1D

The equation for evaluating the concluding link dimension is [Lin and Zhang (2001)]:

==

=m

k

k

l

j

j dic11

(2.4)

Where:

i: The summation of the increasing link dimensions.

d: The summation of the decreasing link dimensions.

j: increasing links index.

k: decreasing links index.

l: number of increasing links.


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m: number of decreasing links.

For figure 2-4, c can be found as:

dic = (2.5)

As for chain in figure 2-5, c can be found as:

)()( 2121 ddiic ++= (2.6)

2.3.1 Worst-Case Analysis

In worst-case method, the concluding dimensions tolerance c can be found as following:

==

+

=

m

k

k

k

l

j

j

j

dd

ci

i

cc

11

|||| (2.7)

Referring to figure 2-5 and equations (2.6 and 2.7), the deviation of the concluding link is:

2121 ddiic +++= (2.8)

2.3.2 Statistical Analysis

In statistical method, the concluding dimensions tolerance c can be found as following:

==

+

=

m

k

k

k

l

j

j

j

dd

ci

i

cc

1

2

1

2 )()( (2.9)

Here, the tolerance is considered as the difference between two or more independentrandom

variables (links) which is calculated by adding variances up. Referring to figure 2-5 and

equations (2.5 and 2.9), the deviation of the concluding link c is given by:

2

2

2

1

2

2

2

1 )()()()( ddiic +++= (2.10)


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of the chart) with the design requirements (blueprints). Process plan here satisfies the design

requirements as it can achieve the dimensions and tolerances sought.

Figure 2-6: Example of tolerance chart [Xue and Ji (2002)]


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3.Simulation-Based Tolerance Stackup Analysis

3.1 Simulation Architecture

Simulation is defined by Kelton (2002) as a board collection of methods and applications to

mimic the real world behavior. We need to tackle the problem of machining tolerance

stackup by simulating the inspection process, after simulating the machining process in terms

of material removal and manufacturing errors. Since manufacturing errors have random

characteristics that can take any probability distribution function (pdf), Monte Carlo

simulation will be the natural choice to solve this problem.

The idea of this simulation is to represent the features of interest by sample points (Figure 3-1

as an example). Then enough number of parts are then virtually machined according to the

intended material removal and the manufacturing errors and inspected according to the

standard CMM (coordinate measuring machine) inspection procedures by tracking the spatial

changes of the features. For more details about the simulation methodology and its

applications, readers should refer to reference [Liu and Huang (2001)]. Simulation is a proper

choice for this problem since other different types of errors can be incorporated in the model.

Furthermore, simulation is not restricted to normal error distributions only; rather, it can take

any probability distribution function (Normal, Uniform, Weibull, Triangular, etc) depending

on the actual error distribution.


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Figure 3-1: Part representation by sample points

Figure 3-2 is a flowchart that illustrates the general simulation system architecture we are

using in this study. The components of the flowchart are further explained as follows:

Setup Plan Sample PlanVirtual

Machining

Virtual

InspectionError Modeling

Feasibility?

End

YES

Terminate?Stopping criteria

Yes

NO

NO/setup planenhancement

Verification?

Validation?

NO

YES

NO

YES

Figure 3-2: System Architecture


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(1) Setup plan

The flowchart starts with a proposed setup plan for machining the part. Setup plan can be

defined as the instructions for machining a part in order to meet the design requirements by

choosing proper: setup formation, datum and operations sequence. The aim of this planning is

to develop the way of machining a part with the minimum cost and the least tolerance stackup

possible.

(2) Sample plan

In order to represent our parts, we use the same concept used in the coordinate metrology by

representing features by sample points in the space. Since manufacturing processes are far

from perfect, there is no way to yield 100% accurate parts. Therefore, we need to make

representative sample points for the features by choosing proper sample size and sample

point locations. This will be discussed more in details in section 3.3.

(3) Error modeling

Since our simulation is based on simulating manufacturing errors, we need to identify the

contributing error sources that shape up the features in the space. In our model, as it will be

shown later, we adopted the following error sources: (a) cutting tool deviation that includes:

workpiece-tool interaction and cutting tool repeatability and (b) setup error that includes:

fixture unit error and raw part inaccuracies. These errors were categorized (section 3.1.3) and

some evaluation procedures were developed in chapter 5 in case they are not available.

(4) Virtual machining


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Simulation starts from here by considering a virtual part, shaping its form and orientation in

the space by the sample points and keeping track of the changes of feature representation due

to material removals and manufacturing errors.

(5) Stopping criteria

Validity of the Monte Carlo results depends highly of the number of iterations executed.

Unfortunately, if the number of iterations (number of virtual parts here) is not big enough,

overly misleading results will show up. Therefore, there should be some metrics or criteria

that are used to determine the number of iterations (sample size of the virtual part batch) to

achieve certain accuracy. This will be discussed more in details in section 3.4.

(6) Virtual inspection

After collecting enough data (or sample points/dimensions), tolerances can be evaluated

using the standard methods. This usually includes: datum evaluation, dimensional and

geometric tolerance evaluation. This will be discussed more in details in section 3.2.

(7) Verification

It is the task of ensuring that the simulation is modeled properly. It is also known as

debuggingthe model. If a bug was found in the code, a review must be done from the start of

the code in the virtual machining part.

(8) Validation

It is the task of ensuring that the simulation model is close enough to the real world behavior.

This is mainly done by conducting real experiment that make physical machining and

inspection for the same part requirements and setup plan and then checking the closeness of


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the simulation results with the physical ones. As usual, the closeness can be checked by

making statistical inference tests (t and F tests). If a problem was caught at this stage, a

feedback will be given to the manufacturing error model to make another experiment to

check out the manufacturing errors or to lookup at any existed thing in the database.

(9) Feasibility

Simulation checks if the proposed plan is doable using the available resources and taking into

account the constraints.

3.1.1 Monte Carlo Simulation

Monte Carlo methods are numerical methods used to solve probabilistic and deterministic

problems by taking samples from contributing populations and plugging them in the

governing function of the system. Another definition is [Kalos and Whitlock (1986)]: a

numerical stochastic process; that is, it is a sequence of random events.

Monte Carlo Simulation can be further explained as follows: given input random variables

(X1, X2 XN) with their probability distribution functions (pdfs) and the governing function

that relates them with the output random variable Y=f(X1, X2 XN), approximate behavior of

the output random variable can be found. After enough number of simulation iterations,

distribution of the output random variable can be found (Refer to figure 3-3). Apparently,

increasing number of iterations increases the accuracy of the output. Sample size and point

locations and number of iterations are important parts to be determined when working with

Monte Carlo simulation.


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Figure 3-3: Monte Carlo Simulation (source: http://www.ymp.gov/documents/ser_b/figures/chap4_2/f04-

174.htm)

3.1.3 Manufacturing Errors

3.1.3.1 Error Categories

Researchers classified manufacturing errors according to different factors (refer to table 3-1).

These factors are:

1. Time. This classification accounts for the error variation with time. Quasi-static errors do

not change considerably (or change slowly) with time such as errors due to dead weights.

Dynamic errors change with time such as cutting tool wear error [Ramesh, Mannan and Poo

(2000)].

2. Randomness. According to this classification, error can be categorized as deterministic

and random errors. Deterministic errors do not have considerable random nature; rather, they

have deterministic dependent output on different independent input parameters such as


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cutting tool wear. On the other hand, random errors are the ones that change according to

specific probability distribution function (pdf) such as spindle repeatability.

3. Sources of errors. Geometric error sources represent the inaccuracy of surfaces moving

relative to each others. Furthermore, it is believed that it is the biggest contributor in

manufacturing inaccuracy [Ramesh, Mannan and Poo (2000)]. Thermal error accounts for

thermal deformation in the tool because of heat provided by cutting process, machine, people,

thermal memory (from previous environments) and cooling for the coolant. The third major

contributor to inaccuracy of machined part is the cutting-force induced errors that come from

the dynamic stiffness of all components of the machine tool.

4. Errors influence on geometric positions. This classification takes into account the effect

of the error on the finished part. Locating error accounts for the variation between the ideal

datum and the one after locating and clamping. And machining error accounts for the

variation between the ideal position of the machine tool and the actual one [Lin and Zhang

(2001); Huang (1995); Lin, Wang and Zhang (1997)]. This is the classification we adopted in

our model here.

Table 3-1: Manufacturing Error Classification

Factor Categories

Time- Quasi-static

- Dynamic

Randomness - Deterministic- Random

Sources of errors- Geometric

- Thermal- Cutting force-induced

Errors influence on geometricpositions

- Machining (Machine motion

error)- Fixture (Setup error)


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Since the final dimension is either related to one or two features, it is good enough to

consider errors existed in the features of interest. Generally, it is safe to say that

manufacturing inaccuracy can be owed only to two factors; that are: (1) cutting tool deviation

from its theoretical (ideal) path and/or (2) Setup errordue to locating, clamping and raw part

inaccuracy. In this thesis, sometimes cutting tool deviation is called machining error (3.1.3.2)

while setup error is sometimes called workpiece error. Setup error can be further divided into

locating/clamping error (3.1.3.3) and raw part error (3.1.3.4). Also, machining error can be

further divided into: cutting tool repeatability and tool-workpiece interaction (refer to figure

3-4).

Dimensional tolerance and most of the geometric tolerances are datum-related. Some of the

geometric tolerances are not datum-related as shown in table 2-1. In the case of datum-

unrelated tolerances (such as flatness, straightness, etc.), cutting tool deviation is enough to

consider whereas in the case of datum-related tolerances (such as dimensional tolerance,

angularity, etc.), cutting tool deviation and setup error must be both considered.

Error Models

Cutting Tool

Deviation

(Machining Error)

Setup Error

(Workpiece Error)

RepeatabilityTool-Workpiece

InteractionFixture Unit Error Raw Part Error

Figure 3-4: Error models used in this study


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Figures 3-5 and 3-6 depict setup and machining errors effects on the cutting process.

Referring to figure 3-5, workpiece coordinate system (WCS) deviation from the machine

coordinate system (MCS) causes removing material we do not intend to cut and avoiding

material removal we intend to cut. The inclination in the machined surface shown in figure 3-

6 is caused by the setup error whereas machined surface irregularities represent machining

error effect.

Figure 3-5: Setup ErrorFigure 3-6: Combined effects of setup and

machining errors

3.1.3.2 Machining Error (Cutting Tool Deviation)

This error accounts for cutting tool path deviation from its idea path. It is assumed in this

study that the deviation is limited to z-coordinate deviation as the cutting tool must travel in

parallel paths. Although, this assumption is valid for prismatic and rotational parts machining,

it is not valid for free-form (sculptured) part machining. This error can be further divided into:

cutting tool repeatability and tool-workpiece interaction error. In this work, we only

considered the tool-workpiece interaction error since the repeatability error is usually

negligible compared to the tool-workpiece interaction error.


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3.1.3.3 Locating/Clamping Deviation (Fixture Unit Error)

This error accounts for surface in the workpiece deviation from its ideal location due to

clamping and locating. It can be represented by six parameters; namely: translation in x, y

and z and rotation around x, y and z. It is one of the contributors to the setup error.

3.1.3.4 Raw Part Error

Raw part error accounts for part datum inaccuracy contribution to setup error. Part

inaccuracy is represented in our study by the flatness values of the primary, secondary and

tertiary datums. Raw part error and locating/clamping error together establish the setup error.

It can be represented by six parameters; namely: translation in x, y and z and rotation around

x, y and z.

3.1.5 Error Synthesis (Aggregation)

Although, there is a great amount in the literature about machining error modeling and its

compensation, very few researchers attacked the problem of synthesizing the error sources

for multi-operation machining in order to predict the quality of the finished part. This is

because of the complexity of the problem. Yao et al. (2002) developed a desktop virtual-

reality approach to represent the machining and measurement processes by including some

machining error sources in the model. Huang, Zhou and Shi (2002) studied the same problem


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analytically to determine root-causes of machined part inaccuracy. We argue here about the

need of Monte Carlo simulation use to solve this problem. [Liu and Huang (2001)] presents

the use of Monte Carlo simulation for dimensional accuracy prediction.

3.2 Virtual Inspection

Standardizing and developing accurate methods for evaluation tolerances and datums are

very important. Since different interpretations for the same data can result in different results,

standardizing is so important. Choosing accurate methods for evaluation is important because

if the method is not accurate enough, some good parts can be rejected and some bad parts can

be accepted. It was mentioned previously that tolerance can be categorized into datum-

dependent and datum-independent tolerances. Datum-dependent tolerances evaluation (such

as profile, runout, parallelism, etc) must include datum evaluation. The next two sections

present standard methods to evaluate tolerances when discrete data points for the machined

surface and/or the datum are available.

3.2.1 Datum Evaluation

It is an important task to evaluate the datum in order to find the tolerances related to this

datum. Generally, there are two approaches for evaluating the datum. These approaches are

presented and summarized in [Wilhelm et al. (1996 and1998)] as the following:


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Least squares (LS) approach: In this method, all the sampled points on a surface (datum)

found by the CMM are fitted and then the fitted plane is translated parallelly to the outmost

point from the material. This method is defined by ISO/WD 5459-3 as the following:

Location of the datum is defined for planar datums as the plane which is parallel to the least

squares plane and contains the extreme point of the extracted datum feature as measured

from the least squares associated line of the hill in the direction of the outward normal from

the material.

A procedural interpretation of this definition is as the following [Wilhelm et al. (1996, 1998)]

(refer to figure 3-7):

1. Form the 3D convex hull from the given points.2. For vertices on the convex hull which are on the surface of the datum feature, not

within the material of the workpiece, fit a least squares plane. These points are the

ones notified by rectangles in figure 3-7.

3. Translate the least squares in the direction of its surface normal away from thematerial of the workpiece until the furthest point.

Figure 3-7: Translated least-squares approach


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Figure 3-8: Candidate datum set approach

The ASME Y14.5 standard does not give the details for applying this procedure. Wilhelm et

al. (1996, 1998) proposed methods for evaluating the planar datums and feature of size (FOS).

Wilhelms (1998) procedure is as the following:

1. Construct the 3D convex hull for the sampled points. The convex hull consists of facets.

Each facet on the hull that is about the material side of the sampled points is an external set of

support.

2. Each facet is considered as the candidate datum P to be checked. A two dimensional

projection of the candidate facet is taken (refer to figure 3-9).


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3.2.2 Evaluation Algorithms for GD&T

In order to evaluate a tolerance that depends on a datum, both the datum and the surface must

be evaluated so and the associated tolerance is found accordingly. The allowable variation of

the tolerances in GD&T is based on the envelope principle. The entire surface shall lie

between two ideal envelope features [Zhang (1997)].

CMM data must be further interpreted to evaluate the geometric deviations mathematically.

Usually, this is done by using the least sum of distances fitting, Least Squares fitting (LS) or

the Minimum Zone fitting (MZ). All of them are optimization problems with different

objective functions.

Data fitting in metrology is defined generally by the following equation:

min

p

i

p

ip rnL

/11

=


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The fitted feature for the data points (x, y, z) is called thesubstitute feature and the geometric

deviation is evaluated as the difference between the maximum and the minimum distances

between the data points and the substitute surface multiplied by 2.

Another widely used method is the minimum zone approach method. The objective function

of this approach (also called two-sided minimax fitting) is given by equation (1) with p .

The resulting fit is strongly affected by the data outliers. The objective function turns to be as

shown in equation (3.2).

min (max |ei|) for ni 1 (3.2)

There is another approach called one-sided minimax fitting which is a constrained

minimization form of the minimum zone. This approach is used to measure the size of the

feature rather than measuring the form deviation. It has two forms, depending if the feature is

internal or external.

For external feature, the optimization problem is:


Subject to 0ie

For internal feature, the optimization problem is:



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Subject to 0ie

3.3 Sample Plan

Finding a proper sampling plan for the machined feature to be inspected is a crucial step in

coordinate metrology since the chosen points are considered as the only representative points

of the feature and the other points are overlooked.

In order to have an accurate strategy for sampling points to be measured in a feature, the

minimum sampling size and the best sampling point locations must be found out. In general,

the sampling of a machined feature depends on the machining capability, the part dimensions,

the surface topography, the required tolerance to be found and the accuracy level.

Unfortunately, machined features can never approach the perfect. An awkward solution for

inspecting feature will be by measuring as many points as possible to figure out the shape and

orientation in the space.

Finding the proper sample size (number of points to be inspected) is a major research topic in

the literature. Increasing the number of sample points leads to a more accurate evaluation but

increasing the sample size increases the inspection cost. There are some recommended sizes

for different feature geometries (refer to table 3-3). In table 3-3, the mathematical column

refers to the number of points needed to define the given geometry mathematically and the

recommended values are the ones recommended for measuring the features of the given

geometries. As was mentioned earlier, the more points taken, the more accurate the results


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are. Another method that can be used to evaluate the sample size is by checking the change of

the results by changing the sample size. Some researchers are proposing using the Shannon-

Nyquist theorem that is used to sample the signals. This theorem states that in order to have a

fair approximation to a wave (in our case is the feature topography), the sampling interval

must be at least double the frequency of the wave.

Table 3-3: Recommended sample size for different geometries [Henzold (1995)]

Feature

geometry Mathematical Recommended

Straight line 2 5Plane 3 9

Circle 3 7

Sphere 4 9

Cylinder 5 15

Cone 6 15

Concerning the sample point locations, the widely used approaches are: random, uniform

(equidistant), stratified sampling (randomized block or randomized grid), refer to figure 3-11.

In random sampling, the location of each point in the space has the same chance of being

chosen as then others. Uniform sampling distributes the points in the space with fixed

distance between them. Uniform distribution is believed to be very sensitive to periodic

variations in the machined feature. In stratified sampling, the feature is divided into blocks

and a number of sample points are chosen randomly inside each of block. Stratified

distribution has a better coverage of the feature than the random distribution approach. Some

researchers recommend distributions according to low-discrepancy sequences (examples of

these sequences are: Hammersley and Halton-Zaremba sequences) [Woo et al. (1995)]. These


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our problem here, the number of iterations represents the number of virtually machined part

to be inspected. There are two methods to find a proper amount of iterations (number of

virtually machined and inspected parts); sometimes called terminating criteria. Making

approximate statistical calculations to find the sample size is the first method. The second one

is by using empirical methods by considering a tolerance band.

Statistically, the minimum number of iterations can be calculated as follows. Suppose that

when running the simulation for no iterations, the half width (ho) of the confidence interval is

given by the following equation when sample standard deviation (so)is known:

o

ono

n

sth 2/1,1 =

(3.5)

When we want to achieve half confidence interval (h), then the number of termination

iterations can be calculated using the following equation:

2

2

2/1,12

h

stn on =

(3.6)

However, there is an apparent difficulty that the right hand side of the equation depends on a

prior knowledge of n. In order to overcome this problem, we can replace the t random

variable with standard normal critical values as shown in the following equation (this is valid

when the sample size is over 30).

2

2

2/12

h

szn

(3.7)


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An easier but different approximation is given by the following equation [Kelton, Sadowski

and Sadowski (2002)]:

2

2

h

hnn oo (3.8)

Cvetko, Chase and Magleby (1998) developed new metrics to evaluate their simulation. One

method presented in their paper is by benchmarking the results of the simulation for a big

sample size (e.g. 1 billion). The objective of benchmarking the results at such a big number is

to evaluate the performance of the simulation at different sample sizes. When there are no

change in the fist four moments (mean, variance, skewness and kurtosis), the sample size of

the number of iterations is chosen (refer to figure 3-13). From the figure, it can be seen that at

1 million iterations, the results are roughly accurate by 95%.

Figure 3-13: Benchmarked results at 1 billion iterations [Cvetko, Chase and Magleby (1998)]


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4.Manufacturing Error Evaluation

The objective of this chapter is to introduce the requirements and procedures of experiments

and algorithms for evaluating manufacturing error; which are the inputs of the simulation.

This will be required in case the manufacturing errors are not available. Afterwards, the error

distributions and their parameters (normal, uniform ) can be plugged in the simulation to

get the results. The discussed manufacturing errors here are machining and setup errors. As it

was previously mentioned, machining error can be further classified as: cutting tool

repeatability and cutting tool-workpiece interaction error. And setup error can be further

classified as: fixture unit and workpiece irregularities errors.

4.1 Machining Error Evaluation

If the machining error (cutting tool deviation) is not available, an experiment must be

conducted to evaluate it. The following are the requirements and the procedure of a proposed

experiment we developed to evaluate machining error for a specific CNC machine. This

experiment can be used to evaluate both the: cutting tool repeatability and cutting tool-

workpiece interaction error.

Requirements:

1. A prismatic Aluminum part.2. A fixture unit.3. CNC milling machine.4. Magnetic dial indicator.


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Figure 4-1: Dial Indicator measuring machined surface

Table 4-2: Machining Error Data

Trial/dial

indicator

measurement-

nominal height

1 2 30

1

2

30

6. Fit the data in table 4-2using a proper distribution by finding the first four moments (mean,

variance, skewness and kurtosis.)

Dial Indicator

Cutting Tool

Prismatic

Workpiece

Fixture Unit

Machine Table


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Figure 4-4: Coordinate measuring

machine (CMM)

Figure 4-5: Fixture unit with CMM probe

2. Evaluate primary, secondary and tertiary datum flatness values for the rough partusing CMM (input of the program).

3. Find the rotational and translational deviations for the rough part using the sameprocedure described in section 4.2.

The output of the program (simulation) was found to be consistent and close to the

experimental results. However, although the results found look fairly close and stable,

sometimes results show some very different behavior. In other words, sometimes results are

truly misleading. This can be justified by insufficient iterations of the simulation. A sample of

the program output and the experimental results are shown in tables 4-4 and 4-5. It is clear

that the simulation output and the experimental results are statistically the same since the p-

value is very high for all the cases except forx in table 4-5.

Table 4-4 data are for extremely smooth part (all flatness values are 0). And table 4-5 is for a

rough part. Therefore, the following null hypothesis cannot be rejected:

Probe

Workpiec

Fixture

Unit


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erimentsimulation

erimentsimulation

H

H

exp22

1

exp22

0

:

:

=(4.2)

Table 4-4: Variance comparison between simulation and experiment for smooth part

Rotational

parameters

Experimental

Results

n = 30 (df=29)

Simulation Results

m = 6117 (df=6116)

Statistic

F

p-value

x 6.761e-006 4.85448499311699e-006 1.39273 0.1569

y 1.925e-006 2.15862432856938e-006 0.891772 0.5963

z 3.945e-006 2.91489690848275e-006 1.353393 0.19544

Table 4-5: Variance comparison between simulation and experiment for a rough part

Rotationalparameters

ExperimentalResults

n = 30 (df=29)

Simulation Resultsm = 6207 (df=6206)

StatisticF

p-value

x 0.001119036304 0.00260209785491036 0.430051584 0.00014

y 0.000644855236 0.000613310910554736 1.051432846 0.78038

z 0.003803312241 0.00561939444032142 0.676818878 0.09514


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Machining error = N (=0, 2=0.038 m2) Rotational setup error =U (-0.003 m, 0.005 m) Translational setup error =U (-0.004o, 0.010o)

Figure 5-2 depicts the output of the simulation conducted for 500 virtual parts (500 iterations).

Notice that the variation of the concluding link was found to be less than the other links,

which contradicts to what were expected using traditional methods. According to traditional

methods of evaluating the tolerance stackup, tolerance of the concluding link (distance

between featuresf2 andf4) should be the summation of the two deviations of the other links in

the chain in the worst-case scenario and the square root of the sum of squares of the two

deviations in statistical analysis. Actually, getting such a lower variation in the concluding

link is justified in our point view since these two features are machined in the same setup.

There is no tolerance stackup in this case, as has been demonstrated in [Huang (1995)].

Machining error is the only error that causes the variation here. There is no contribution from

the setup error. Table 5-1 shows a comparison of the tolerance stackup evaluation using the

three methods.


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Figure 5-2: Example 1 results

Table 5-1: Tolerance stackup evaluation comparison for example 1

02 0.057648

04 0.052874

Worst Case Statistical Simulation

24 0.1105 0.0782 0.0126

5.2. Example 2: Four Machining Operations (In Three Setups)

The example is an extension of the first one (refer to figure 5-3). It involves four machining

operations with two changes of the machining datums, a total of three setups. Changing

datums will result in a tolerance chain. The errors included in the simulation of example 2 are

the same as that in example 1.

Figure 5-4 depicts the output of the simulation for 500 virtually machined parts (iterations).

Again, the results here do not agree or even close to either the worst-case or the statistical

method. Table 5-2 summarizes the results.


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Figure 5-5: Example 3; ABS (Antiblock System) housing, Bosch (Source: http://www.wzl.rwth-

aachen.de/WM/SIMON/deliverables/DA0/DA0_02D.htm)

Figure 5-6: ABS dimensional requirements

A simulation was conducted according to the setup plan described in figure 5-7. The plan

includes two setups. The part includes six surfaces of interest that are numbered from 1 to 6.

Milling is the process used to machine the surfaces. Hole drilling is not included in the setup

plan and simulation since it does not have an effect on the tolerance chain of concern. Figure

5-8 shows a tolerance chart of the part in order to predict tolerance stackup in the tolerance

chains. Here, we are interested in the dimension shown in line 8 in the tolerance chart as a

concluding link. The contributing links are shown in lines: 7, 4 and 1.


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As was mentioned earlier, determining the number of iterations for the simulation is a crucial

task in Monte Carlo simulation. Here, we benchmarked the results of the simulation at

100,000 to calculate the errors in the first two moments when having less sample size.

Actually, 10,000 iterations are considered large enough by most of Monte Carlo parishioners

[Cvetko, Chase and Magleby (1998)]. The first two moments (mean and variance) at 100,000

iterations are shown in table 5-4 for dimensions in lines: 1, 4, 7 and 8.

Table 5-3: Simulation results at 100,000 iterations

Mean Variance

L1 84.9983246659704 0.000113812363943454L4 59.9943255754447 0.000114461403303197

L7 49.9926265647646 0.000118204861229129

L8 74.9965883885753 0.000118607406206314

The following were the inputs to the simulation:

Flatness of the raw part: 0.05 mm (Flatness is considered to be representative for the raw part

error.)

Machining error (Cutting Tool deviation) ~ N (0, 0.00752)

Rotational setup error ~ U (-0.002, 0.005) degrees

Translational setup error ~ U (-0.0015, 0.005) mm


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RAW PART, MATERIAL REMOVAL: 15RAW PART, MATERIAL REMOVAL: 104 5 RAW PART, MATERIAL REMOVAL: 106

RAW PART, MATERIAL REMOVAL: 15

x

z

1,2,3

5

6y

4 x

z

RAW PART 1

x

z

y 1,2,3

x

z

4,5,6

1


1

5

6y

4 x

2

2

6y

2

z

45

3

3

4,5,6

y 1,2,3

x

5

4

y

z

1,2,3

x

FINAL PART

1

6

x

y

z

2

5 4

3


5

4

y

z

1,2,3

6

100

90

160

Figure 5-7: ABS part setup plan

A comparison between the results of simulation, worst-case and statistical methods in finding

concluding link tolerance stackup is shown in table 5-4. Again, Monte Carlo simulation

results were found to be less than both the traditional methods. The ratio of simulation

tolerance stackup to the worst case tolerance stackup was found to be 0.34 and the ratio of

simulation tolerance stackup to the statistical method tolerance stackup was found to be 0.59.

Table 5-4: Tolerance evaluation using the three approaches

Standard

DeviationTolerance=6

L1 0.010668288 0.064009727

L4 0.010698664 0.064191982

L7 0.010872206 0.065233235

L8 0.010890703 0.065344216

Worst Case 0.193434944

Statistical 0.111683618

Simulation 0.065344216


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Figure 5-8: Tolerance Chart of ABS part

Figures 5-9 shows the dimension histograms of the concluding link (L8) and the contributing

links (L1, L4 and L7). Number of iterations required to achieve certain accuracy can be

predicted from figure 5-10. This figure shows means and standard deviations values predicted

using the simulation in terms of number of iterations (x axis). Clearly, 4000 iterations seem to

have very close results to the 100,000 iterations. Therefore, 4000 iterations can be considered

as proper choice for the sample size virtually machined parts (iterations.)


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Figure 5-9: Dimensions histogram using simulation

Figure 5-10: Progress of results with sample size increase

Suppose that we need to maintain 0.066 mm for dimension shown in line 8 in the tolerance

chart (figure 5-8). Then, we will need to allocate proper tolerances for the concluding links

(dimensions shown in lines: 1, 4 and 7 in the same figure). According to our simulation,

assigning 0.060 mm for each contributing link will be good enough to meet what we need.

However, if we need to make the allocation using the worst case and statistical methods,

0.022 mm and 0.038 mm will be needed for each contributing link.


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If we choose processes that are capable to achieve our simulation requirements, then we will

be satisfied with 2,700 parts per million (PPM) rejects when having the process capability

index equals to 1. However, if we use thesame process considering the traditional methods,

much more rejects per million will be expected (refer to table 5-5 and figure 5-11). This

shows the importance of having less conservative method for tolerance allocation.

Figure 5-11: Rejection areas comparison when allocating concluding links tolerance using worst case,

statistical and simulation methods

Table 5-5: Part per million (PPM) rejections comparison when allocating tolerance using worst case,

statistical and simulation methods

Worst Case Statistical Simulation

Tolerance 2.02 3.5 6

Cp 0.337 0.583 1

PPM rejects 307,728 76,727 2,700


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6. Simulation is a proper choice for this problem because of its complexity. Furthermore,

simulation is not restricted to normal error distributions only; rather, it can take any

probability distribution function (Normal, Uniform, Weibull, Triangular, etc) depending on

the actual error distribution. It is precious to mention that even though statistical tolerances

are assumed to be normally distributed; a lot of evidences in the real world defy this

assumption [Lin, Wang and Zhang (1997)].

7. Monte Carlo simulation is believed to be a powerful tool to solve problems that include

stochastic variables. However, the main critique to this method is the need for a quite large

number of iterations to converge to accurate enough results. 10,000 iterations are considered

as large enough by most Monte Carlo practitioners [Cvetko, Chase and Magleby (1998)]. In

our work here, we benchmark the results at large iterations size (like 100,000 or 1 million)

and consider these results as absolutely accurate ones. Afterwards, we calculate the errors by

increasing the sample size. We can choose a sample size that has close results to the

benchmarked ones.

6.2 Recommendations for Future Works

1. Simulation validation and verification increase users confidence of the results accuracy.

Verification can be defined as the assessment of how close simulation results are to the

conceptual model. In other words, it is the task of ensuring that the simulation was built

accurately as the modeled indeed wanted.


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Validation is used to ensure that the simulation model matches accurately enough with the

real world behavior. In our problem here, some experiments are recommended to be

conducted for validation. Some experiments must be done to determine simulation input in

case they are not already known. These inputs are the considered manufacturing errors

included in the simulation. Afterwards, a real machining must be done for a large enough

number of parts according to the same setup plan adopted in the simulation to evaluate the

tolerances of the dimensions. The output of the simulation experiment will match the results

of the experiment if the simulation model is valid. Typically, statistical inference tests can be

good to test the closeness between simulation results and the real world behavior.

2. Monte Carlo simulation is known as a computationally extensive tool of calculation.

Therefore, developing more efficient methods in terms of calculation time could be a

valuable future work.


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