Relation between Process Capability Indices and Geometric...

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School of Industrial Engineering and Management

Department of Production Engineering

Relation between Process Capability Indices and Geometric

Errors of Machine Tool

Harikishan Veluru Ramanaiah

M.Sc. Thesis

KTH Royal Institute of Technology

Stockholm

November 2016

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SAMMANFATTNING

Högsta kvalitet, har blivit det viktigaste kravet från kunder oavsett segment och kravet på att ha högsta

kvalitet har ökat oerhört inom tillverkningssektorn. För att hålla jämna steg med de ständigt ökande

kraven från kvalitetsstandarder, måste industrier använda olika tekniker och metoder som stöd för att

producera den tillverkade delen med högsta precision. Detta beror på flera faktorer såsom

maskinverktyg, skicklighet och kunskap hos operatören, skärande bearbetning och parametrar,

noggrannhet och precision hos mätutrustning.

Trots att ingenjörer är väldigt noggranna med att säkerställa att den tillverkade delen är av bästa kvalitet

med högsta precision, kommer det alltid att finnas slumpmässiga faktorer som kommer att resultera i

en viss avvikelse i artikel dimensionerna vilket påverkar den slutliga produkten vid montering. För att

övervinna detta, har industrier valt att tillämpa kapacitets index för att möjliggöra regelbundna

kontroller av hur väl en process kan producera delarna. Studie av duglighets faktorer är kända för att

vara mycket effektiv.

I kombination med detta, övervakar industrier noga eventuella fel som uppstår antingen från

maskinverktyg, process eller arbetsmiljö, detta för att kunna studera dessa fel och deras orsakande

faktorer, som elimineras och minimeras för att uppnå högsta möjliga noggrannhet hos produkterna. Det

har skett en omfattande forskning kring fel som påverkar produkt noggrannhet och olika metoder för

kompensering har utformats för att minimera effekterna av dessa fel. Diskussioner kring dessa två

ämnen ledde till frågeställningen, "finns det någon koppling mellan kapacitets index och

maskinverktygs fel" och "om det finns ett samband, vad är det och hur kan det bidra till att uppnå en

bättre noggrannhet.

För att bedöma genomförbarheten av denna frågeställning, har denna forskning bedrivits. Kärnan i detta

examensarbete är att studera realtidsdata av kapacitetsindex och kontrollera om det finns låga index

värden för någon process. Sedan associera teoretiska överväganden om eventuella fel som orsakar ett

lågt värde av kapacitetsindex. Vilket i sin tur kommer att bidra till identifieringen av relationen mellan

kapacitetsindex och fel i maskinverktyg.

Detta teoretiska övervägande kommer att valideras via simulationstester i MATLAB. Detta kommer att

genomföras med stöd från företaget Leax, Falun. Kapacitets data som testerna baseras på kommer att

förses från Leax, och avser maksinverktyget Mazak VMC.

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ABSTRACT

Appropriate quality, has become the most important requirement of a customer from any segment and

the demand to have the highest quality has tremendously increased for a manufacturing sector. To keep

up with the ever-rising demands of quality standards, industries must employ various techniques and

methodologies which assist them in producing the manufactured part with the highest accuracy. This

depends on several factors such as machine tool, skill and knowledge of the operator, cutting process

and parameters, accuracy and precision of measuring equipments.

Although the engineers take at most care to make sure the manufactured part is of the best quality with

highest part accuracy, there will always be some random factors which will add some amount of the

deviation in the part dimensions and this might affect the final product during assembly. To overcome

this, industries have known to follow the application of capability indices in order to have regular check

on how well a process can produce the parts. Study of the capabilities have known to be very effective.

Along with this, industries closely monitor for the possible errors arising either from the machine tool,

process or working environment, to study these errors and their causes, which will be eliminated and

minimized to have the highest part accuracy. There has been an extensive research done on the errors

affecting the part accuracy and various compensation methods have been devised to minimize the

impact of these errors. Discussions about these two topics led to the thought, ‘is there any link between

the capability indices and the machine tool errors’ and ‘if there is a link, what is it and how can it help

in achieving a better accuracy’.

To assess the feasibility of this thought, this research has been carried out. The core of this thesis

research is to study the real-time data of capability indices and check for the presence of any low

capability indices for any process. Then, associate the theoretical considerations of possible errors

causing a low value of capability indices. Which in turn will help in identification of relation between

capability index and the errors of machine tool.

This theoretical consideration will be validated by carrying out simulation runs in MATLAB. This

research will be carried out with the support from industry Leax, Falun and the data related to capability

study is also collected from the same industry. The data of capability study that has been obtained is

recorded for the machine tool Mazak VMC.

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ACKNOWLEDGEMENT

This thesis research would not have been possible without the able support and guidance of my

supervisor Dr. Andreas Archenti and I extend my sincere thanks and gratitude to him. Also, this thesis

would not have shaped up in a good way without the help of Ph.D. student, Theodoros Laspas. He has

always been my reliable support and was always available to guide me during the entire thesis phase

and it was Theodoros Laspas, who was responsible to help me setup the LDBB testing equipment and

carrying out the tests in the industry.

I would like to extend my gratitude to Mr. Björn Johansson, production engineer at Leax, Falun, for

giving me the opportunity to visit the Leax industry and for helping me in understanding about their

current methods employed to carry out capability study and for providing me all the relevant data

required regarding the capability study. I, humbly thank all the faculty of Leax, Falun, for helping me

carry out the testing in one of their machine tool.

Lastly, I would like to thank my parents and friends who have always been of high motivational support

all through the thesis research.

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Nomenclature and Abbreviations

Cp Process capability

Cpk Adjusted process capability

Cpm Process capability, when target is of essence

Cm Machine Capability

Cmk Corrected machine capability

USL Upper specification limit

LSL Lower specification limit

μ Process mean

σ Standard deviation

T Target value

n Sample number

N Sample number

N-1 Bessel’s correction

S Standard deviation for sample

�̅� Mean

xi Value in the population

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Table of Contents 1. Introduction .............................................................................................................................. 11

1.1. Research Background ....................................................................................................... 11

1.2. Research Objective ............................................................................................................ 11

1.3. Research Scope ................................................................................................................. 11

1.4. Research Motivation .......................................................................................................... 11

2. Literature Research ............................................................................................................... 12

2.1. Capability ............................................................................................................................. 12

2.1.1. Capability Definition ....................................................................................................... 12

2.2. Literature on Capability index ........................................................................................... 12

2.2.1. History of Capability Index ............................................................................................ 12

2.2.2. Process Capability ......................................................................................................... 13

2.3. Quality Tools Associated with Cp .................................................................................... 15

2.3.1. Control Charts ................................................................................................................. 15

2.3.1.1. Control Limit Choice ................................................................................................... 16

2.3.2. Histograms ...................................................................................................................... 16

2.4. Capability index for Varying Distributions ....................................................................... 17

2.5. Calculation of Standard Deviation ................................................................................... 18

2.6. Machine Capability ............................................................................................................. 19

2.7. Applications of Capability Indices .................................................................................... 21

2.8. Limitations of Capability Indices ....................................................................................... 21

2.9. Process Capability Study .................................................................................................. 22

2.9.1. The Steps of a Capability Study ................................................................................... 22

2.10. Recommendations of Capability Indices..................................................................... 23

2.11. Summary of Capability Index ........................................................................................ 24

3. Introduction to Machining System ........................................................................................ 25

3.1. Description of a Machining System ................................................................................. 25

3.2. Accuracy Definition ............................................................................................................ 26

3.3. Accuracy in a Machining System ..................................................................................... 27

3.4. Errors in a Machine Tool ................................................................................................... 32

3.4.1. Thermally Induced Errors .............................................................................................. 32

3.4.2. Load Induced Errors ...................................................................................................... 33

3.4.3. Errors due to Geometric Inaccuracies ........................................................................ 33

4. Methodology ............................................................................................................................ 35

4.1. Capability in Leax ............................................................................................................... 36

4.2. Data Collection ................................................................................................................... 36

4.3. Data Analysis ...................................................................................................................... 38

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4.3.1. Analysis of Geometric Feature 22 ............................................................................... 42

4.3.2. Analysis of Feature Geometric 24 A ............................................................................ 45



4.4. Identification of Errors causing Low Cpk Index (Theoretical) ...................................... 52

4.4.1. Errors Affecting the Accuracy of the Geometric Feature 22 .................................... 52

4.4.2. Errors Affecting the Accuracy of the Geometric Feature 24 and Geometric

Feature 25 ...................................................................................................................................... 54

4.4.3. Errors Affecting the Accuracy of the Geometric Feature 30 .................................... 56

4.5. Validation of Theoretical Considerations of Errors ........................................................ 57

4.5.1 Simulation using MATLAB ............................................................................................ 57

4.5.2 Simulation Method ................................................................................................................ 58

4.5.3 Assumptions Considered for Simulation ..................................................................... 60

4.5.4 Simulation of Feature-Symmetry ................................................................................. 60

4.5.4.1 Tool Path of Symmetricity for Simulation ................................................................ 61

4.5.5 Simulation of Feature-Dimension....................................................................................... 62

4.5.5.1 Tool Path of Dimension for Simulation .......................................................................... 62

5. Results ......................................................................................................................................... 64

5.1 Simulation of Feature-Symmetry ..................................................................................... 64

5.1.1 Assignment of Error Values .......................................................................................... 64

5.1.2 Symmetry Calculation .................................................................................................... 65

5.1.3 Results of Symmetry-Simulation with Tool Length of 1mm ..................................... 66

5.1.4 Results of Symmetry-Simulation with Tool Length of 100mm ................................. 69

5.2 Simulation of Feature-Dimension .................................................................................... 70

5.2.1 Assignment of Error Values .......................................................................................... 70

5.2.2 Dimension Calculation ................................................................................................... 72

5.2.3 Results of Dimension-Simulation with Tool Length of 1mm .................................... 72

5.2.4 Results of Dimension-Simulation with Tool Length of 100mm ................................ 74

6. Discussion and Conclusion ................................................................................................... 76

7. Future Scope .............................................................................................................................. 78

8. References .................................................................................................................................. 79

9. Internet References ................................................................................................................... 81

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List of Figures

Figure 2.1 Illustration of centering of Cp. ............................................................................................. 14 (a)-Cp is large and well centered. Process will produce parts within the specification limits. (b)-Cp is

large but off-centered. Process will produce parts with deviation from the specification limits. (c)-Cp

is small but well centered with a large spread. Process will have parts with large deviation from the

specification limits. ............................................................................................................................... 14

Figure 2.3 Illustration of a Control Chart. ............................................................................................ 16

Figure 2.4 Normally distributed curve against the plot of histogram. [B] ............................................ 17

Figure 2.5 Illustration of transformation of non-normal distribution to normal distribution using

central limit theorem; (A) - Parent distribution which is non-normal. (B) - Transformation from parent

distribution after considering sample size of 3. (C) -Transformation to nearly normal distribution, after

iteration of 30 samples. ......................................................................................................................... 18

Figure2.6 Flowchart of the machine capability methodology. ............................................................. 20

Figure 2.7 Illustration of steps of capability. ........................................................................................ 22

Figure 3.1 Illustration of a system and its entities. ............................................................................... 25

Figure 3.2 Illustration of the entities of a machining system responsible for machining accuracy. ..... 26

Figure 3.3 Illustration of the term accuracy. ......................................................................................... 27

Figure 3.4 Illustration of errors of linear axis, where; EBX-Angular error around axis-B, ECX- Angular

error around axis-C, EAX- Angular error around axis-A, EXX- Linear positioning error, EYX-

Straightness error in Y direction, EZX- Straightness error in Z direction. [12] ..................................... 29

Figure 3.5 Illustration of errors in rotary axis. where; EXC-Radial error of C in X direction, EYC-

Radial error of C in Y direction, EZC- Axial error of C, EAC-Tilt error of C around X, EBC- Tilt error of

C around Y, Ecc- Angular positioning error. [12] ................................................................................. 29

Figure 3.6 Illustration of the kinematic structures. Where; (A)- Cantilever type, (B)- Portal type, (C)-

Bridge type and (D)- Joint arm type. [12] ............................................................................................. 31

Figure 3.7, Illustration of dynamic characters, k-stiffness, m-mass, d-damper .................................... 32

Figure 4.1 Thesis methodology shown in a flow chart. ........................................................................ 35

Figure 4.2 Top view of the End yoke and Side view of the end yoke .................................................. 36

Figure 4.3 Bottom view of the End yoke .............................................................................................. 37

Figure 4.4 a special fixture to house the workpiece .............................................................................. 37

Figure 4.5 Compilation of all the data collected in Leax, this data comprises of only the critical

features .................................................................................................................................................. 39

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Figure 4.6 Drawing of the critical features along with their geometrical tolerance .............................. 40

Figure 4.7 Drawing of the critical features along with their geometrical tolerance .............................. 41

Figure 4.8 Indicating the feature number 22 ......................................................................................... 42

Figure 4.9 Graphical description of process variation as observed for machine room A for feature 22

.............................................................................................................................................................. 44

Figure 4.10 Graphical description of process variation as observed for machine room B for feature 22

.............................................................................................................................................................. 44

Figure 4.11 Indicating the feature number 24 A ................................................................................... 45


.............................................................................................................................................................. 46

Figure 4.13 Indicating the feature number 25 ....................................................................................... 47


.............................................................................................................................................................. 48


.............................................................................................................................................................. 49

Figure 4.16 Indicating the feature number 30. ...................................................................................... 50


.............................................................................................................................................................. 51


.............................................................................................................................................................. 52

Figure 4.19 Illustration of a perpendicular control tolerance. [C] ........................................................ 53

Figure 4.20 Illustrating how a straightness error might affect the perpendicularity. [D]...................... 54

Figure 4.21 Illustration of symmetric control tolerance. [E] ................................................................ 55

Figure 4.22 Illustration of Squareness error. [F] ................................................................................... 55

Figure 4.23 Illustration of the roll error. [G] ......................................................................................... 56

Figure 4.24 Illustration of the elements causing cyclical error. [H] ..................................................... 57

Figure 4.25 Illustration of distribution of median points derived for side 1 and 2 ............................... 61

Figure 4.26 Illustration of Tool Path for simulation of symmetry. ....................................................... 62

Figure 4.27 Illustration of Tool Path for simulation of dimension. ...................................................... 63

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Figure 5.1 Graph of error difference of ECX1 (er=0.001um). .............................................................. 67

Figure 5.2, Graph of error difference of EBX1 (er=0.001). .................................................................. 68

Figure 5.3 Graph of error difference of EXX1 (er=8um). .................................................................... 69

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1. Introduction

1.1. Research Background High quality, long life products in an economical price range is what the customers demand and due to

the enormous number of suppliers for similar products is resulting in highly competitive world. In this

era of competition each manufacturing firm is focusing on producing the product with the highest

possible quality. Thus, the manufacturing firms are constantly thriving to have better accuracy with

minimal maintenance of the machine tool yet having the highest productivity with minimum lead time.

This demands the manufacturers to thoroughly understand their machining system, as machining system

poses lots of challenges and can be considered as the house of numerous errors. Upon understanding

the machining system, it will be easier to identify the error and it sources, which helps to achieve better

productivity with greater accuracy and minimum rejections/loss.

Tool wear, machine wear, thermal changes and reactions, vibrations, spindle errors and many more

such errors are the factors contributing to poor part accuracy. Through the history of development of

manufacturing systems and process, it can be witnessed that many propositions have been done on how

to overcome these errors or how to minimize them. One such key element is Capability of the Machining

System.

1.2. Research Objective From the literature review it has been understood that, a machine tool can be subjected to several forms

of error and these errors are known to have a high level of impact on the machining process and on the

overall quality of the part being machined. While some errors exhibit a minimum effect, and are often

hard to avoid them. Some will have a major effect and cannot be neglected. Thus, this thesis mainly

focusses on the geometric errors of the machine tool.

The main objective of this research is to identify the impact of geometric errors on the accuracy of the

part machined. Thereby, making use of the Cp and Cpk data obtained from the industry to investigate if

this can be co-related with the geometric errors. This in turn will be assessed to determine if any link

can be established between process capability indices and the geometric errors of the machine tool.

1.3. Research Scope Capability index is a statistical tool to measure and know how accurately a part can be produced. This

thesis focuses on using the data of Cp obtained from the industry and an attempt is made to see if the Cp

data can be considered to check what error is responsible for low capability index and if yes, what is

the source of the error.

1.4. Research Motivation After carrying out the relevant literature study required to perform this research, it has been identified

that, right from early 1900 until today, a lot has been researched about the capability indices and its

importance along with that a lot of research has been done regarding machine tool errors and how they

can be minimized. But, very less or almost nothing has been discussed or done with respect to how the

capability indices can be linked to machine tool errors. If it can be linked what are the advantages of

establishing this link has not been studied, this literature gap serves the motivation to carry out this

research with objective mentioned in the section, 1.2

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2. Literature Research

This chapter comprises of a detailed explanation of the concepts regarding capability and process

capability indices, its applications, limitations and capability of machine.

2.1. Capability Each part to be machined will have a set of design specifications, which explains about the tolerance

limits, positioning of geometric features. It becomes highly important to meet these requirements and

usually as the machine tool ages the capacity to produce parts as per specification will start to decline.

Also, sometimes even if the machine tool is quite new there might be process related errors which

produces the part with deviations from the specifications. Thus, to overcome this and predict if the

process is scaling down to not meet the design specifications capability indices were defined and

introduced.

Consequently, capability measure is one of the methods to visualize the ability of the process being

carried out to produce the required product. Usually, capability measure is a good method of application

in the industries, but this has some of its limitations as well and this will be discussed later in the report.

2.1.1. Capability Definition Though there is no standard way to define Capability, the most general and widely used definition is:

Capability is the ability of the process to produce units with dimensions within the tolerance limits (with

respect to the characteristics of interest). [1]

It is very important to understand that the above definition of capability holds good in case of univariate

process capability.

In industry two indices of capability are used, namely: Process Capability Index (Cp) and Machine

Capability Index. (Cm)

2.2. Literature on Capability index Upon detail study of numerous papers on the topic Capability indices, several concepts and

methodologies related to Cp was understood and these have been explained in the further sections.

2.2.1. History of Capability Index It has been well understood that a process never exhibits a normal distribution curve, there is always

some deviation, this deviation can either be natural and controlled or uncontrollable which might be

because of numerous factors emerging during the process. This was first studied by Walter A Shewhart,

a renowned statistician born in United States. It is learnt from the book, “Economic Control of Quality

of Manufactured Product, 1931” which documents all the work done by Shewhart. Shewhart prepared

theoretical models and framed the problems as “Assignable Cause Variation” and “Chance Cause

Variation”.

Shewhart’s main observation was that, production process must be in statistical control and only with

chance cause variation in order to predict the future output. The theoretical model prepared by Shewhart

was the “Control Chart” and its application is based on the experience mainly and on probability.

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After Shewhart, many other researchers such as, Kane, Juran etc. have further developed the statistical

tool and have derived indices for capability measure. Process capability analysis has always been used

as the technique to ensure the product produced is of right quality and this is applied in various segments

of the product cycle. The first to derive the Capability index was Joseph.M.Juran in 1974, this was to

relate the actual process spread to the allowable process spread. The Cp [2] is termed as follows:

𝐶𝑝 =𝑈𝑆𝐿 − 𝐿𝑆𝐿

6𝜎 (1)

In 1986, Victor E. Kane derived the new index of capability as a measure of process performance, this

was denoted by Cpk [3]. Index Cpk is related to Cp, but unlike Cp it utilizes the process mean. Cpk is

estimated in two formulations, each formulation being assessed as unilateral tolerance against the mean.

This is expressed as follows:

𝐶𝑝𝑢 =𝑈𝑆𝐿 − 𝜇

3𝜎 (2)

𝐶𝑝𝑙 =𝜇 − 𝐿𝑆𝐿

3𝜎 (3)

Cpk = Minimum (CPL, CPU) (4)

An alternative to Cpk is the notation Cpm [1], which was derived in 1985 by Hsiang and Taguchi and by

Chan, Sheng, and Spiring in 1988 independently. Cpm is used when the process mean is no to deviate

from the target value. Cpm is expressed as follows:

𝐶𝑝𝑚 =𝑈𝑆𝐿 − 𝐿𝑆𝐿

6√σ2 + (μ − T)2 (5)

2.2.2. Process Capability Along with understanding the definition well, it is essential to know that, computing of Cp is based on

two assumptions [1] and they are:

▪ Quality characteristics under consideration follows normal distribution. This assumption is

made to have computational advantages.

▪ Process being studied is under statistical control. This assumption is made as the absence of

stability in the process makes it unpredictable and this results in failure of Cp to reflect the

actual capability level of the process.

The quality characteristics are of three types:

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▪ Nominal the best, which means the process has both USL and LSL.

▪ Smaller the better, process with only USL.

▪ Larger the better, process with only LSL.

One of the main factor to be considered during the computation of Cp is that, a process must display

large value of Cp and the process must be well centered. In case of a non-centered process, even if the

Cp is large enough, there will be some parts produced beyond the specification limits. Similarly, if the

centering is good but the Cp is small, the process will result in producing parts beyond the specification

limits [1]. This also explained in the following figures.

LSL (a) USL LSL (b) USL LSL (c) USL

Figure 2.1 Illustration of centering of C p.

(a)-Cp is large and well centered. Process will produce parts within the specification

limits. (b)-Cp is large but off-centered. Process will produce parts with deviation from

the specification limits. (c)-Cp is small but well centered with a large spread. Process

will have parts with large deviation from the specification limits.

Process is considered capable when the value of Cp is large and this value is 1.33, if the value is less

than 1.0, then the process is not capable. The value 1.33 is obtained based on the standard deviation and

this is considered as 3.99σ from the mean (μ) on either side. Since the value of standard deviation is

what determines the drift in process, +/- 3σ is considered. As higher the standard deviation the

possibility of dispersion and centering will be affected [4].

Also, the standard deviation under +/- 3σ is not considered good as this leads to identifiable area under

the curve, which is an indication of the fact that the probability of conformity is less and the process

will not be capable. In case of +/-3σ the area under the curve is almost nil. If the deviation is the specified

limits of tolerance, then the Cp can be estimated as:

𝐶𝑝 =𝑈𝑆𝐿 − 𝐿𝑆𝐿

6𝜎=

3𝜎 − (−3𝜎)

6𝜎= 1.0 (6)

The following table and the figure indicates the probability value of different standard deviations.

Table 2.1 Indication of different standard deviation against the respective conformity of parts in

probability and percentage. Also, indicating the possible value of Cp.

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Figure 2.2 Illustration of Standard deviation against the percentage of conformity and Cpvalue. [A]

2.3. Quality Tools Associated with Cp Of the several available quality tools two of them generally associated with Cp are Control Charts and

Histograms. These both are explained in the following sections.

2.3.1. Control Charts Very often control charts are used in industry as a support to Cp in order keep track of variation in the

process. Shewhart first introduced control charts with the aim to find the possible assignable causes of

variation and thus to make the process more predictable. These charts also served as a great tool to

graphically depict the output of the process with respect to time. In manufacturing industry, the data

collected, which is based on number of produced units and then is weighted to a standard deviation and

plotted in the chart [1].

Control charts serves two main purposes [1];

▪ First, to identify assignable causes of variation in order to maintain the process stable. (This

also helps in maintaining the assumption for Cp computation as mentioned in the Section 2.2.2)

▪ Second, to be able to detect whenever a change is occurring in the stable process, which

generally results in mean variation.

Standard

deviation

Probability of

conformity

Conformity

in Percentage

Cp

+/-1σ 0.6828 68.28% 0.33

+/-2σ 0.9546 95.46% 0.66

+/-3σ 0.9973 99.73% 1.0

+/-4σ 0.9999 99.99% 1.33

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Control charts is a process quality indicator, where the process will have to stay between the calculated

limits, called as the control limits. As long as the process is within these limits, the process is said to be

under statistical control. It is important to note here that there is a considerable difference between

control limits and tolerance limits. They both are defined as following [1]:

▪ Control Limits: Calculated limits used in control charts, as an indicator of stability of the

process. The distance between the limits is often set as six times the standard deviation.

▪ Tolerance Limits: Usually design based specification limits seeking a dimension to be

produced for the part manufactured. Hence, these limits are based for a singular unit produced.

Figure 2.3 Illustration of a Control Chart .

2.3.1.1. Control Limit Choice Principle to be followed during calculation of the control limit is to ensure that the false alarms are

almost nil [1]. According to this principle, the control limits are often:

𝜇 ± (3𝜎/√𝑛), where μ is the central line and n is number of samples.

This expression explains that for every 0.3% of the cases the process will be stopped for deviation from

μ. And this risk is generally considered okay. [Shewhart, 1925]. Assuming that, 𝑥 is seen as observation

from a normal distribution, this risk is calculated.

2.3.2. Histograms One of the important things to do during data analysis is the right manner of illustration of the data

collected. Generally, the amount data collected and to be analyzed is huge and one of the best ways to

analyze them is by plotting of histograms. Histogram is one such tool which also enables in identifying

if the process is normally distributed [1]. Some of the characteristics of a normally distributed curve

such as, Symmetricity, unimodal can be easily visualized in a histogram.

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Figure 2.4 Normally distributed curve against the plot of histogram. [B]

2.4. Capability index for Varying Distributions The concepts and equations expressed in the previous sections are for estimating capability of the

process, only under the condition of normal distribution. But reality, there will be many other cases

where the distribution need not necessarily be normal. Even if the distribution is normal, it might be

non-uniform. Several researchers such as, Clements (1989), Chan et al. (1991), Karl et al. (1994),

Mukherjee and Singh (1994), Veevers (1995,1998), Boyles (1996), Perakis and Xekalaki (2002), Maiti

et al (2010) and many more have proposed different forms of equations best suitable for a certain type

of distribution [5].

Since, this research does not focus about the different types of distributions, much will not be explained

about this, although for further understanding of these topics, the papers published by the authors

mentioned above can be of good reference. But it in case of a non-normal distribution, a theorem called,

Central Limit Theorem will be used to convert it into a normal distribution.

Central limit theorem states that:

“Distribution of the average or the sum of a large number of independent-identically distributed

variables will be approximately normal, regardless of the underlying distribution.” [3]

Central limit theorem uses iteration of several samples taken from the parent distribution and computing

the averages, usually statisticians have suggested this sample size to be a minimum of 30 and also

depending on the type of parent distribution, the required sample size might get larger. The following

figures explain the transformation of non-uniform normal distribution to nearly uniform normal

distribution.

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Figure 2.5 Illustration of transformation of non-normal distribution to normal

distribution using central limit theorem; (A) - Parent distribution which is non-normal.

(B) - Transformation from parent distribution after considering sample size of 3. (C) -

Transformation to nearly normal distribution, after iteration of 30 samples .

2.5. Calculation of Standard Deviation All through the different indices that have been developed for different process distributions, standard

deviation is one of the most important factors under consideration. Thus, it is essential to know what

standard deviation is and how it is calculated. Standard deviation is defined as follows:

“Standard deviation is the measure to quantify the amount of variation in a set of data value.” [1]. A

low standard deviation indicates that the data values are close to mean, whereas high standard deviation

indicates that the data value is more dispersed and far from the mean value. Usually standard deviation

The distribution in this figure

is Non-Normal, let us

consider this as the parent

distribution.

fo1

i

0 Xi

1

(A)

As per Central Limit

Theorem, by taking three

samples from the parent

distribution and computing

the averages, produces the

distribution similar to this.

0 1

(B)

fo3i

Continuing the application of

Central Limit Theorem,

iteration of up to 30 samples

and computation of the

averages, results in a

distribution very near to

normal distribution.

0 1

(C)

f32i

19

is calculated under two conditions, one during estimation of sample and the other during the estimation

of the whole population. Estimation of standard deviation for population [6] is expressed as:

√(1

𝑁) ∑ (𝑥𝑖 − 𝜇)2𝑁

𝑖=1

While for the estimation of sample [6], it is expressed as:

√(1

𝑁 − 1) ∑(𝑥𝑖 − �̅�)2

𝑁

𝑖=1

Here, N – 1 is used instead of N, which is called as the correction factor also called as Bessel’s

Correction. 𝑥 is the sample mean; whereas μ is the mean of the population. “s” is denoted as standard

deviation for sample, whereas σ is the standard deviation for population [6].

2.6. Machine Capability Capability of machine is usually defined as the, “capability of the machine to carry out the set process

efficiently and effectively to produce parts as per the specification limits [1].” Usually, the factors

affecting the machining capability are its own inherent properties such as, feed, speed, tools, coolant

flow rate etc. The expression for machine capability is same as the process capability. But it is denoted

by Cm [1] and the corrected machine capability is denoted by Cmk [1].

Cmk = Minimum (CPL, CPU) (9)

𝐶𝑝𝑢 =𝑈𝑆𝐿 − 𝜇

3𝜎 (10)

𝐶𝑝𝑙 =𝜇 − 𝐿𝑆𝐿

3𝜎 (11)

Study of capability of machine is considered as short-term unlike process capability which is considered

as long-term. The reason for capability of machine to be considered short-term is because, in industry

this study is carried out only 2-3 times during the machine’s life; this is first carried out at the time of

purchase of a new machine, then once upon installation of the machine in the shop floor and once before

the expiry of the warranty period.

Study of capability of machine is well planned and organized, as this is not repeated often and this test

is carried out in controlled temperature, with a single tool and by a skilled operator, as a result of this,

capability of machine is not considered accurate and also due to the fact that only the first 30 or 50

parts produced in a row are used as samples. No random samples are selected. This study is carried

out in a certain methodology and it is explained in the following flow chart.

(7)

(8)

20

Figure2.6 Flowchart of the machine capability methodology.

START

Preparation of machine tool, i.e...Pre-production

run/setup, in order to check and prepare for stability of the

process and thereby having the measured values in the

middle of the tolerance zones (as much as possible)

This depends on the type of tolerance.

Manufacturing of a representative number of parts in an

un-interrupted production. (Theoretical requirement is 100

parts/ can be minimized to 50 in case of economic needs.)

Measurement of part characteristics and documentation

of results and the deviations observed/measured. This is to

be done in production sequence.

Statistical Evaluation:

1. Qualitative evaluation.

2. Study of the distribution.

3. Calculation of capability indices.

Assessment of

results.

Requirement met?

Machine is Capable.

Problem Analysis.

Make improvements.

YE

S

NO

21

2.7. Applications of Capability Indices The intention of devising various types of indices was to serve the purpose of having a good summary

of the process and to know it’s good and bad in a language which can be easily understood by all

involved from the shop floor to management. Some of the applications as mentioned by Viktor E. Kane

[2] are:

▪ Prevention of Non-conforming product: In most of the industries, a particular part is

produced in bulk continuously and in such cases, it is recommended to have benchmark set as

measure. This resulted in proposing standard Cpk as 1.33 and this can be used in any sort of

industry, this ensures that the percentage of non-conforming products are at minimum.

▪ Continuous Improvement: Since there can be numerous factors affecting the process, it is

hard to maintain a process same throughout production. So, monitoring capability indices will

help in analyzing when the process is tending to decline in terms of quality and even if a process

is exhibiting a good capability throughout, it can always be made better.

▪ Communication: Use of capability indices will result in dimensionless study of the data, thus

making it easy to understand and communicate between the departments of the same

organization. This also aids in development of better design and better manufacturing

techniques, enabling a more economically stable process.

▪ Prioritization: As a result of above mentioned advantages, a simple summary might help in

prioritizing various elements in the shop floor. Some such elements can be prioritizing the

process improvement’s or process changes or tool changes.

▪ Quality Audit: Various auditing tools are used in a company in order to have the right kind of

quality. Capability study is one such which aids in having the desired quality right at the time

of manufacturing, rather than having to assess it post production.

2.8. Limitations of Capability Indices It has been learnt that until today, there are no potential drawbacks due to the use of capability indices.

Instead problems have aroused due to lack of knowledge in handling statistical data or misinterpreting

it. Some of the limitations as mentioned by Viktor E. Kane [2] are:

▪ Statistical control: Capability study is most effective if the process is in control and there can

be numerous factors affecting the process; presence of special causes or unknown random

causes, makes it hard to retain the process in statistical control. Also, as mentioned earlier, lack

of good tools to read the statistical data can lead to un-foreseen problems.

▪ Sampling Plan: According to Deming, “he could make process appear in statistical control by

merely spacing out the samples within a subgroup over time”. This statement makes us

understand that, depending on the sampling plan, value of σ can be estimated and by changing

the width of the control limits, it is more likely to have a process in statistical control, but by

doing this, the capability of the process reduces, thus it is important to have a right kind of

sampling plan.

22

▪ Computation: In case of a person with less experience, it might get complex to compute the

indices and might end up in wrong estimations. Training the people and having workshops can

be solution to this.

▪ Tool Wear: The performance of a process and its capability is dependent on how often the tool

is changed and under what circumstances the tool has been changed.

▪ Non-normality: it is well known by now that, estimation of Cp and Cpk are based on the

assumption that, the process distribution is normal and it is also known that, not at all times a

process exhibits normal distribution, this might be a problem for computing the Cp and Cpk.

Although there exists new set of modified of indices and methods to transform, these are still

considered to be complex for implementation and also these are not well understood yet and

lack a more generalized form of estimation during the non-normal distribution.

2.9. Process Capability Study In modern day, carrying out capability study is done in most of the industries and this is happening as

result of higher customer awareness and the demand from their side. In 1998, Deleryd defined the

process capability as follows:

“Process capability study is an improvement methodology where a product characteristic is measured

and analyzed in order to determine the ability of the process to meet the specification for the

characteristic studied” [1]

2.9.1. The Steps of a Capability Study It is learnt that the steps involved in process capability study are similar to the improvement cycle and

these steps are shown in figure [1] below:

Figure 2.7 Illustration of steps of capability .

23

▪ Identification of important characteristics and plan the study: Before starting with the

study, it is essential to plan the study and to plan the study well, some of the questions need to

be answered and these can be; What is to be measured? How to measure? Why to measure?

Along with this a good brainstorming is necessary. After this, it is important to identify the

character, property or a feature to measure.

▪ Establish statistical control and gather data: The most important thing to be done to compute

and study capability is to bring the process under statistical control or stable. If the process is

not in control, then the capability value measured cannot be reliable. Like explained in the

section 2.3.1, it is essential to present a basis for the final capability evaluation using control

charts and before doing this the data has to be collected in a right manner and analyzed well.

▪ Asses the capability of the process: Upon the computed value of the capability, certain steps

must be taken in order to improve the process, in order to do this, it is very important to assess

the capability. Assessing the capability can be done by using any of the suitable tools of

improvement such as, histograms, fish bone diagram, measuring values on a probability

plotting paper.

▪ Initiate Improvement Efforts: This is the stage where; certain actions will have to be taken

after a thorough analysis and study of the process capability. Some of these actions can be

changing the parameters of process, improving the process, prioritizing the improvement of a

process.

2.10. Recommendations of Capability Indices Since there have been quite a few modifications of the indices and also with the motive to help the

industries implementing the capability study for the first time, few researchers such as Viktor E. Kane,

K. Palmer, K.L.Tsui and others have given recommendations [7] concerning the use of a particular type

of capability index based on the specified limits These are;

▪ If the specification limits are one sided, use Cpu or Cpl, as appropriate.

▪ If the specification limits are two sided, if the conformance is less than 95%, Cp, Cpk and k are

recommended.

▪ If the specification limits are two sided, if the conformance is between 95% and 99%, Cp, and

Cpm are recommended.

▪ If the specification limits are two sided, if the conformance is between 95% and 99%, Cp, and

Cpm are recommended.

▪ In case where the process mean is deviated from the midpoint of the specification limits, index

k is recommended.

▪ Index k is denoted by: 2|𝑚−�̅�|

𝑈𝑆𝐿−𝐿𝑆𝐿 m is the midpoint.

24

2.11. Summary of Capability Index With the availability of various forms of indices, each has its importance for the right kind of parameters

and factors involved in the process. Overall, the collective use of these indices serves as great measure

of process, its capability to produce parts as per the desired quality. Following is a table summarizing

the index Cpk and the actions need to be done based in its value.

Table 2.2 Summary of the Index Cpk and its value.

Index Capability Value Quality level Action

Cpk >1.0 Inadequate Immediate action is a must to avoid large

uncontrollable variations.

Cpk 1.0-1.33 Adequate Process needs regular monitoring and check on

statistical control.

Cpk 1.33-1.50 Good No serious action needed, but there is scope to make

it better.

Cpk <1.50 Excellent No actions are required, though it is advised to check

if this value is attained in an economic manner.

25

3. Introduction to Machining System

This chapter discusses in detail about what a machining system is, what machining accuracy is, what

the different types of machine tool errors are and how the errors of a machine tool impact the accuracy.

3.1. Description of a Machining System Machining system can be referred to, as a system which is linking the elastic structure of the machine

tool and the machining process. This said, a system is defined as;

“A set of interacting or independent component parts forming a complex/intricate whole” [8].

These set of interacting components which are either dependent or independent with each other work

for a common purpose. Usually this purpose is the behavior of the system. Every system is portrayed

by a spatial boundary which is surrounded and influenced by an environment.

Figure 3.1 Illustration of a system and its entities .

Similarly, a machining system is a system with machining technology, machine tool elastic structure

and cutting process parameters as its entities or elements. The accuracy of the machining system thus

depends on these three elements and these are usually interdependent, which means that variation in

one of these will trigger a new effect in one of the other two thus affecting the accuracy. This makes it

essential to understand these three entities well to understand, how machining accuracy might vary and

why it varies.

The following gives a brief understanding about each of these entities:

▪ Cutting Process and Parameters: During a production of a part, cutting process is selected

based on the geometry and the material of the work piece and the surface finish required. Some

26

of the general of mostly employed cutting processes are; turning, drilling, grinding, milling.

During these cutting processes, the parameters opted to have the best product also affect the

accuracy of the machining, some such parameters are; depth of cut, feed rate, speed of cut and

spindle speed, tool geometry, etc.

▪ Machining Technology: This entity is mostly based on the type of machine tool used, it can

be anything from a simple turning machine to most advanced CNC machine tools.

▪ Elastic Structure of Machine Tool: This can be expressed as combination of the tool, tool

holder, work piece, clamping device, etc. apart from these there exists two other subsystems of

a machine tool which also contribute to machining accuracy and they are; the drives and the

control system.

Figure 3.2 Illustration of the entities of a machining system responsible for machining

accuracy.

3.2. Accuracy Definition According to the publication of ISO-5725, accuracy is defined as:

“The closeness of a measurement to the true value and it involves a component of a random error and

a component of systematic error” [9]. It is also referred to as error.

27

Figure 3.3 Illustration of the term accuracy.

3.3. Accuracy in a Machining System The range of deviation between the cutting tool and the work piece is considered to estimate the

accuracy of a machined part in the machining system. As mentioned in the section 3.0, a machining

system consists of different entities and a large number of variations amongst these entities individually

will contribute majorly to the accuracy of a part being machined. Some of the factors which affect the

accuracy [10] are as follows:

Elastic structure of machine tool

▪ Guiding elements

▪ Servo motors

▪ Geometry

▪ Linear and rotary encoders

▪ Fixturing

▪ Thermal effects

▪ Work piece and its residual stress

▪ Material and geometry of work piece

▪ Tool

Cutting process

▪ Chatter

▪ Thermal influence

▪ Cutting forces

▪ Programming

Machining technology

▪ Complexity of the machine tool

▪ Sensitivity

▪ Dynamic forces

28

The complexity in this data will always increase in number and in order to simplify this, the factors

affecting the accuracy were classified into three categories, namely; Systematic errors, Random errors

and a combination of these two. Due to possibility of such high number of variations from each of the

entities of the machining system, in the year 1984, Weck mentioned that there are mainly four factors

[11] affecting accuracy and these are:

▪ Temperature Influence

▪ Geometric and Kinematic errors

▪ Static stiffness

▪ Dynamic stiffness

Before understanding these factors, it is necessary to understand what a structural loop of a machine

tool is, as most of these arise within the structural loop. Structural loop is defined as, an assembly of a

set of mechanical components which maintain a relative position between specified objects [12]. In a

machine tool, spindle shaft, bearings, guideways, frame, housing, fixtures form the structural loop. In

case of any change in the geometry of this structural loop components the actual end effector position

and orientation relative to work piece vary, which in turn affect the accuracy.

It is required to understand the four factors affecting the accuracy and interaction between these errors

plays a significant role in the behavior of the overall system. These are explained in the following

manner:

▪ Temperature Influence: During the entire period of operation of a machine tool, it is

subjected to presence of heat and this heat is usually found varying in a certain range. Source

of this heat can either be internal or external and the difference in co-efficient of expansion

among the parts of the machine. Some of the parts of which exhibit variation of temperature

are servo motors, lead screw, spindle as well as changes in the air temperature of the working

environment.

This results in changes in the geometry and dimensions of the machine tool and bring about

the cumulative thermal distortion in the entire structural loop of the machine tool which affects

the interface of tool and work piece resulting in poor accuracy. It is important to consider the

differences in co-efficient of expansion as they often lead to thermal stresses. [12]

▪ Geometric and Kinematic Errors: Sometimes, there is a considerable deviation existing

between the workpiece and the cutting tool, usually this results in positioning error and also

axis imperfections, such errors are known to be geometric characteristics of the machine tool

[11]. These are further classified into component errors and location errors. Component errors

are position dependent, errors of axis and usually caused by imperfections in the drive system.

Location errors are not dependent on the position, but the axes orientation and position play a

significant role.

As component errors are related to axis imperfections, it must be noted that every linear axis

and rotary axis have six possible geometric errors [13] [14]. These errors are pictorially

depicted in the following figures:

29

Figure 3.4 Illustration of errors of linear axis, where; E BX-Angular error around axis-B,

ECX- Angular error around axis-C, EAX- Angular error around axis-A, EXX- Linear

positioning error, EYX- Straightness error in Y direction, EZX- Straightness error in Z

direction. [12]

Figure 3.5 Illustration of errors in rotary axis. where; E XC-Radial error of C in X

direction, EYC- Radial error of C in Y direction, E ZC- Axial error of C, EAC-Tilt error of

C around X, EBC- Tilt error of C around Y, E cc- Angular positioning error. [12]

One of the kinematic character of the machine tool is, the components of the machine tool having a

relative motion between the tool and the workpiece. According to Weck, it is important for the machine

components to have a good co-ordination between them, for example, the feed rate. Kinematic errors

can be as errors rising out due to errors in angle and errors in length. The kinematic structure of the

machine tool is defined by the layout of its components and their axes [12]. Generally, the kinematic

30

structures are of the types; Cantilever, Portal/Gantry, Bridge and Joint arm, these are pictorially

described as in figure3.6.

(A)

(B)

31

(D)

Figure 3.6 Illustration of the kinematic structures. Where; (A) - Cantilever type, (B)-

Portal type, (C)- Bridge type and (D)- Joint arm type. [12]

▪ Static stiffness: From the kinematic structure, it is known that, each component is connected

by a joint and these joints can be rigid and affect its smooth functioning and most of the spring

effect arises from these joints [12]. Some of the factors affecting these joints are, gravity,

acceleration, dead weight, load exerted by workpiece and fixtures, these affect the geometry

of the machine tool, which there affects the accuracy of the system.

Static stiffness is calculated based on deflection measurement result and is generally non-linear

for wider load ranges. Feed rate does not directly affect the machine tool’s static stiffness.

(C)

32

▪ Dynamic Stiffness: Forces of a machine tool which are usually varying are considered as the

dynamic forces, some of these forces are; machining force, measuring force [Schwenke].

These varying forces will generate vibrations affecting poor surface finish, accelerated tool

wear. Deformations caused due to vibrations are hard to compensate due to the fact that the

amplitude of these vibrations are unknown [12]. The characters of dynamic forces in a machine

tool are; stiffness, mass and damping. Sources of these dynamic force can be due to, cutting

forces, tool break, and play in machine tool joints.

Figure 3.7 , Illustration of dynamic characters, k -stiffness, m-mass, d-damper.

3.4. Errors in a Machine Tool From the previous section, it is learnt that there are four main characteristics affecting the accuracy of

the machining system, similarly those four main characteristics very often exhibit errors originating

from them which reduces the accuracy in a significant magnitude, if not rectified. Thus, it gets important

to learn about these errors. The accuracy of the machine tool itself is sometimes seen as the limiting

factor to produce the part with highest accuracy and quality [15].

There are three main error sources in a machine tool which determine the accuracy of the machine tool,

which in turn has effect on the accuracy and quality of the part being manufactured [15]. These three

errors are:

▪ Thermally induced errors

▪ Load induced errors

▪ Errors due to geometric inaccuracies

3.4.1. Thermally Induced Errors There are many different types of thermal errors which have been identified to affect the machine tool.

The most basic form of heat transfer is well known and they are; Conduction, Convection and Radiation.

d

33

The various forms heat induced in and around machine tool will be transferred in these forms alone.

Various forms of heat sources affecting the machine tool are:

▪ Room temperature

▪ People

▪ Heat added/removed by coolant systems

▪ Heat generated by the machine

▪ Heat generated by the cutting process

Since it is learnt that, thermal gradients are the worst kind to occur as they result in thermal warping,

here are some of the causes of thermal gradients [17]:

▪ The bed may be exposed to excess amounts of thermally controlled fluids.

▪ Evaporative cooling.

▪ Overhead lights can sometimes result in gradients at sensitive structures.

▪ Internal heat sources (motors, spindle, and process).

3.4.2. Load Induced Errors There are three main types of forces present during the machining [A.C.Okafor, Y.M.Ertekin] and these

are:

▪ Workpiece weight

▪ Forces from cutting process

▪ Gravity forces because of mass displacement of the machine tool components

It has been learnt that, these errors can be significant in comparison to kinematic errors, for instance,

due to the weight of moving slide, the guideways might bend resulting in vertical straightness and pitch

error motion [12].

3.4.3. Errors due to Geometric Inaccuracies It has been well established by J.Bryan (1990) [16] and M.Weck et al. (1995) [18] that errors arising

due to geometric inaccuracies may exceed 50% of the total machining error. This makes it highly

essential to understand these errors, enabling to avoid some of these errors completely and minimize

the some of these errors by compensatory measure. Neglecting these errors will result in poor accurate

parts and with a high level of quality distortion.

Some of the geometric errors are known to arise during the cold start conditions, while majority of them

are due to imperfections in the machine tool structure and due to misalignment in the drive systems

such as guide ways. These errors amplify as the components of machine tool start to wear either due to

improper usage or due to regular aging of the machine tool. Geometric inaccuracies usually result in

squareness and parallelism errors in the moving elements of the machine and position and orientation

errors of the cutting tool [15].

In the year 1997, Soons JA has identified that, there are 21 geometric errors for 3 axis machine tool, 52

geometric errors for 5 axis machine tool [19]. The 21 geometric errors which have been identified for a

three-axis machine tool is described in the table 3.1.

34

Table 3.1 Description of 21 geometric errors in a three-axis machine tool.

Geometric errors Number of error components

Linear positioning error (scale error) 3

Straightness error 6

Angular error 9

Orthogonal/squareness error 3

Total 21

Linear positioning errors: Linear errors used to be one of the largest error until the advancements that

have happened in the design of the machine tools. This has resulted in more accurate lead screw and

the linear scale, further the machine tools have been provided with the controllers which ca compensate

for the repeatable position errors [20].

Straightness error: Straightness error is the error which is in the direction perpendicular to the direction

of motion. It is caused by the misalignment of in guideways and also this may happen during the

installation of the machine tool or shipping [20].

Angular error: Errors observed in pitch, yaw and roll are the angular errors.in most of the machine

tools, angular errors are related to straightness error as, in most of the times, straightness error

measurement already includes the errors caused due to angular errors [20].

Squareness error: Presence of non-perpendicularity between the axes is considered as squareness

error. Squareness error is defined as,

“the difference between the inclination of the reference straight line of the trajectory of the functional

point of a linear moving component with respect to its corresponding principal axis of linear motion

and the inclination of the reference straight line of the trajectory of the functional point of another

linear moving component with respect to its corresponding principal axis of linear motion’. [21].

This error usually arises during the installation or shipping of the machine tool, also machine levelling

during the new installations will cause some squareness error [20].

35

4. Methodology

This chapter consists of a detailed explanation of the methods employed to meet the research objective.

Methodology employed for this thesis starts with an extensive literature research followed by visit to

the industry Leax in Falun and collection of data required for carrying out this research from the

industry, this is followed by data analysis with respect to the literature research done. Based on the data

analysis and literature research, the problems are identified and the cause for these problems are

theoretically identified, cause identified will be validated by carrying out simulation of a mathematical

model using MATLAB.

Figure 4.1 Thesis methodology shown in a flow chart.

START

Literature Survey

Visit to Leax and Data collection

Data Analysis

Identify the Problems

Identification of causes of low Cpk

(Theoretical)

Simulation of mathematical model using

MatLab.

Results

Discussion and scope for future research.

CONCLUSION

36

4.1. Capability in Leax

Leax is an industry which manufactures several components required for the automotive industry. Upon

interaction with the production managers, it is learnt that LEAX carries out its capability measure in the

following manner:

▪ Capability of the machine, Cm is carried out twice, once before the purchase of the machine

tool and once in the plant after the machine tool has been installed. Cm index is determined by

the using sample size as 50.

▪ Process capability, Cpk is regularly performed and the data collection from sample is random,

they have distinguished the features machined as critical features and not-so critical features.

Based on this, they measure the critical features for every 15th machined part whereas the non-

critical features are measured once in every shift. Apart from this, the most critical features are

measured using Go, No-Go instruments. The sampling size is 30 and the desired Cpk value is

chosen as 1.33.

4.2. Data Collection Data regarding the machine tool used, the part manufactured and historical data of the capability study

carried out with respect to the part manufactured were obtained.

Part Data: The part which is manufactured and the process capability to produce this part accurately

studied in this thesis is End yoke. End yoke also called as pinion yoke is a part which is used as a joint

between rear axle drive shaft end joints between the final sprung final drive and unsprung rear wheel

stub axle. This part has several features to be machined, out of which eleven features are identified as

critical. The reason to choose these selected features as critical will be explained in the further sections.

(a) (b)

Figure 4.2 Top view of the End yoke and Side view of the end yoke

37

Figure 4.3 Bottom view of the End yoke

Machine tool Data: This end yoke is machined in a series of two machines. First the part is processed

in a turning center for machining the cylindrical spline and this can be seen in the side view of the end

yoke from the Figure 4.2. Next, this part is machined in a three axis Mazak VMC, for machining of the

other features and this can be seen in the top view of the end yoke from the figure 4.2. Furthermore,

this part is processed in a broaching machine to create the cylindrical splines, as seen in the bottom

view of the end yoke from the figure 4.3.

This thesis focusses only on the Mazak VMC, thus the features machined only in this machine tool will

be considered. The Mazak VMC is a three-axis machine and the work bed is divided into two stations

and which are named as Machine room A and Machine room B, the intention of dividing the work bed

into stations is to produce two parts at a faster rate. The loading and unloading of the part into and out

of the machine is done by a robot. A special hydraulic fixture has been used to house the work piece

firmly.

Figure 4.4 a special fixture to house the workpiece

38

Capability Study Data: Leax provided the data of the capability study performed for the process

carried out in the Mazak VMC to produce the required features, this data is only for the critical features

and not all the features machined and this includes the capability for both the machine rooms. This data

provides with the sample number, tolerance type, target value, UTL, LTL, average, Cpk requirement

and calculated value of Cp and Cpk. Also, this data is supported with control charts, which indicates the

trend of the process performance.

4.3. Data Analysis Since, the data collected was quite big, it is first compiled into one file. All the data has been sorted and

organized according to the feature number and the machine number, also it must be noted that, the data

contains information of only the critical features. According to Leax, of all the features machined, only

the critical features are included in the data, since, these are the most highly interacting features when

assembled and made to work with other components. To mention few, the flatness of the feature number

23, the dimensional width between the towers etc.

39

Figure 4.5 Compilation of all the data collected in Leax, this data comprises of o nly the

critical features

Also, the drawing showing these features has also been obtained as a part of data collection. The

dimensions and the perspective drawing of other features have been erased due to confidentiality.

40

Figure 4.6 Drawing of the crit ical features along with their geometrical tolerance

41

Figure 4.7 Drawing of the critical features along with their geometrical tolerance

From the data compiled, it is clear, the capability of the process carried out in the Mazak VMC is very

good for most of the features machined. The Cpk demand for all the features is 1.33, irrespective of the

type of the tolerance. For some of the features such as feature 26, the Cpk computed is well over 10, in

fact it is 21.66, 17.63 and similarly most of the Cpk values are very high in comparison to the required

Cpk. Refer to Figure #, for all the values in detail.

The Cpk values computed for the features as 22, 24 in machine room A, 25 and 30 are mentioned in the

following table:

42

Table 4.1 Details of the features with low Cpk

This thesis will be focusing on these features only, as these are exhibiting low Cpk and most of these

features are showing low Cpk index in both the machine rooms. Hence, during machining of these

features, there are some machine tool errors which will are affecting these operations. So, it is necessary

to analyze the data of these features further. In this attempt, the values of each of these feature is

tabulated and a graphical chart is plotted to see how the process variation is occurring.

4.3.1. Analysis of Geometric Feature 22 The four towers which can be seen in the figure 4.8, is the feature number 22 and the design specification

for this is, the towers must be perpendicular to the spline.

Figure 4.8 Indicating the feature number 22

The capability of the process to machine this feature is bad in both the machine rooms and upon plotting

a control chart for the measured value of sample size 30, it is seen that the process produces all the 30

parts within the specified limits, but the process variation is high, which means that the process does

not produce parts with similar dimensions and the variation in dimensions produced is high. This is

depicted graphically, refer to figure 4.9 and 4.10.

Feature

No.

Machine

Room

Cpk-

Computed

22 A 1.07

22 B 0.94

24 A 1.29

25 A 0.87

25 B 0.58

30 A 0.98

30 B 1.31

Datum A, Spline

of cylindrical axis

Feature 22

Feature 22

43

Table 4.2 Measured values of the feature number 22 for a sample size of 30.

Sample No. Measured Value-A Measured Value-B

1 0.07 0.08

2 0.048 0.082

3 0.034 0.048

4 0.036 0.043

5 0.055 0.049

6 0.056 0.074

7 0.05 0.061

8 0.045 0.052

9 0.063 0.087

10 0.092 0.055

11 0.062 0.078

12 0.068 0.078

13 0.068 0.063

14 0.037 0.049

15 0.045 0.077

16 0.054 0.071

17 0.051 0.065

18 0.071 0.051

19 0.039 0.052

20 0.073 0.072

21 0.04 0.035

22 0.037 0.053

23 0.028 0.055

24 0.053 0.066

25 0.039 0.068

26 0.054 0.038

27 0.033 0.049

28 0.029 0.052

29 0.061 0.049

30 0.05 0.042

It is to be noted that, along with the above values, the feature 22 has the following data with it and this

is same for both the machine rooms:

Tolearnce-0.1

Target Value-0.0

USL-0.1, LSL-0.0

44

Figure 4.9 Graphical description of process variation as observed for machine room A

for feature 22

Figure 4.10 Graphical description of process variation as observed for machine room B

for feature 22

From the above graphs, it is evident that both the machine rooms exhibit different range of process

variation even though they are machined in the same machine tool. Also, most of the parts produced

are systematically located between the mid-point of the range and the USL rather than target value.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Mea

sure

d V

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es

Sample No.

Process Variation-22A

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Mea

sure

d V

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es

Sample No.

Process Variation-22B

USL

LSL

USL

LSL

45

4.3.2. Analysis of Feature Geometric 24 A The inner sides of the towers must be symmetric with respect to datum-A, spline. This is feature number

24 and is shown in the figure 4.11. It is important to know that, the Cpk index is low only for the

machine room A and machine room B shows a good Cpk index value of 1.84.

Figure 4.11 Indicating the feature number 24 A

The capability of the process to machine this feature is bad only in machine room A and upon plotting

a control chart for the measured value of sample size 30, it is seen that the process produces all the 30

parts within the specified limits, but the process variation is high, which means that the process does

not produce parts with similar dimensions and the variation in dimensions produced is high. Though

the process produces few parts close to the target value, most of the parts are produced close to the

average value. This is depicted graphically, refer to figure 4.12.


Sample

No.

Measured

Value-A

1 0.039

2 0.025

3 0.012

4 0.046

5 0.035

6 0.04

7 0.038

8 0.028

9 0.043

10 0.004

11 0.036

12 0.039

13 0.037

14 0.009

15 0.029

Datum A, Spline

of cylindrical axis

Feature 24

Feature 24

46

16 0.003

17 0.045

18 0.045

19 0.021

20 0.042

21 0.028

22 0.029

23 0.027

24 0.025

25 0.019

26 0.008

27 0.007

28 0.035

29 0.007

30 0.017

Also, the other values are:

Tolearnce-0.08

Target Value-0.0

USL-0.08, LSL-0.0


for feature 24

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Mea

sure

d V

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es

Sample No.


LSL

USL

47

4.3.3. Analysis of Geometric Feature 25 The inner sides of the towers have an angle feature machined at higher position than feature number 24

and these must be symmetric with respect to datum-A, spline. This is feature number 25 and is shown

in the figure#. In this case, both the machine rooms exhibit a low Cpk index and most importantly the

Cpk index of machine room B is very low with a value of 0.58 against the required Cpk of 1.33.

Figure 4.13 Indicating the feature number 25

Upon plotting a control chart for the measured value of sample size 30, it is seen that the process

produces all the 30 parts within the specified limits, but the process variation is high, which means that

the process does not produce parts with similar dimensions and the variation in dimensions produced is

high. This is depicted graphically, refer to figure 4.14 and 4.15.


Sample

No.

Measured

Value-A

Measured

Value-B

1 0.025 0.047

2 0.006 0.047

3 0.015 0

4 0.004 0.003

5 0.028 0.015

6 0.024 0.013

7 0.026 0.004

8 0.018 0.015

9 0.014 0.018

10 0.027 0.01

11 0.014 0.007

12 0.011 0.012

13 0.007 0.006

14 0.031 0.001

15 0.002 0.045

Feature 25

Datum A, Spline

of cylindrical axis

48

16 0.027 0.042

17 0.007 0.045

18 0.041 0.033

19 0.006 0.029

20 0.046 0.046

21 0.017 0.015

22 0.017 0.026

23 0.017 0.03

24 0.021 0.035

25 0.009 0.045

26 0.042 0.023

27 0.013 0.021

28 0.009 0.025

29 0.03 0.028

30 0.03 0.02



Tolearnce-0.05

Target Value-0.0

USL-0.05, LSL-0.0


for feature 25

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Mea

sure

d V

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e

Sample No.


LSL

USL

49

From this graph, it can be seen that, though there are some parts produced close to the target value,

there are sudden spikes in the process, especially with the sample number 18, 19 and 20, where it can

be seen there is sudden dip for the 19th sample and again a high spike for the 20th sample.


for feature 25

Like machine room A, from this graph it can be seen that, though there are many parts produced close

to the target value, there are sudden spikes in the process, especially with the sample number 2 and 3,

where it can be seen there is sudden dip for the 3rd sample where it produces the part with the exact

dimension as the target value and again a high spike for the 15th sample from 14th sample. Due to such

high variation in the process, the index Cpk is very low.

4.3.4. Analysis of Geometric Feature 30 The width between the towers is the feature number 30, this is basic dimensional tolerance and is shown

in Figure 4.16. For this feature, both the machine rooms exhibit a low Cpk index and unlike the other

features analyzed, this feature has a two-sided tolerance, which is it has both USL and LSL specified

and the range is 164+/-0.031.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Mea

sure

d V

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es

Sample No.


LSL

USL

50

Figure 4.16 Indicating the feature number 30.

Just like the other features, control chart is plotted for this feature too, in order to have a better

understanding of the process. The chart is plotted for the measured value of sample size 30, it is seen

that the process produces all the 30 parts within the specified limits, but the process variation is high,

which means that the process does not produce parts with similar dimensions. Unlike the other features,

an interesting and strange observation for this feature is that, the process does not exhibit a very high

variability and is still within the specified limits. But almost all of the parts are systematically produced

close to USL. This is depicted graphically, refer to figure 4.17 and 4.18.

Table 4.5 Measured values of the feature number 30 for a sample size of 30

Sample

No.

Measured

Value-A

Measured

Value-B

1 164.019 164.017

2 164.019 164.016

3 164.017 164.018

4 164.018 164.016

5 164.029 164.017

6 164.028 164.024

7 164.029 164.026

8 164.022 164.026

9 164.026 164.021

10 164.02 164.021

11 164.025 164.024

12 164.023 164.022

13 164.024 164.022

14 164.023 164.021

15 164.024 164.018

16 164.019 164.017

17 164.021 164.02

18 164.02 164.018

Feature 30

51

19 164.021 164.02

20 164.02 164.018

21 164.021 164.019

22 164.02 164.019

23 164.021 164.018

24 164.02 164.017

25 164.021 164.017

26 164.018 164.014

27 164.02 164.015

28 164.018 164.018

29 164.019 164.016

30 164.019 164.017



Tolearnce-0.031

Target Value-164.0

USL-164.031, LSL-163.969


for feature 30

163.93

163.95

163.97

163.99

164.01

164.03

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Mea

sure

Val

ues

Smaple No.


LSL

USL

52


for feature 30

Both the graphs of the machine room A and B, are indicating a similar process variation. Like mentioned

earlier, this feature seems more interesting as it does not exhibit high range of variation, but still has a

low Cpk index.

4.4. Identification of Errors causing Low Cpk Index

(Theoretical) Based on the literature research and the data analyzed, it is evident that there are some possible machine

tool errors affecting the process which in turn is resulting in a low capability index of the features

mentioned in the previous section. In this section, the geometric tolerance control of each of this feature

is considered and possible errors affecting the Cpk will be identified on theoretical assumption. The

theoretically assumed errors will be further validated with the help of certain testing tools, which will

be explained in the next section.

4.4.1. Errors Affecting the Accuracy of the Geometric

Feature 22 According to design specification, the geometric control for this feature is surface perpendicularity.

Perpendicularity is an orientation control, which means it controls the orientation of the feature with

respect to a datum plane/axis. Perpendicularity is to indicate how much a surface can deviate from a

900 plane. Perfect perpendicularity is known to occur when a surface is exactly at 900 to the datum. In

real time manufacturing due to presence of some type of errors which cannot be controlled, perfect

perpendicularity is hard to achieve, thus perpendicularity tolerance zone is specified, this is the volume

between two parallel planes that are perpendicular to the datum plane and the surface being controlled

will be machined within this volume, as defined by the tolerance zone.

163.93

163.95

163.97

163.99

164.01

164.03

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Mea

sure

d V

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Sample No.


LSL

USL

53

Figure 4.19 Illustration of a perpendicular control tolerance . [C]

In this thesis, the feature 22 is a plane surface which is to be perpendicular to the datum A, refer to

Figure 4.8 in section 4.3.1. Some of the possible errors affecting the perpendicularity of the geometric

feature are:

▪ Straightness error: A problem in guideways, ball screw can cause straightness error, this

means that if the drive system is subjected to different loads at different points along the length

of the drive system, then it can exhibit somewhat a torsional movement or a jumpy movement.

That is the guideway might show minor bending or drift in different axes of the machine tool.

The following figure helps in having a better understanding of this, if the guideways is meant

to travel in the X direction in a straight manner and between the red dotted lines, but due to the

presence of this kind of error, the guideways will be subjected to move in a wave form and this

is what could be causing the perpendicularity error. This is also indicated in the figure.

54

Figure 4.20 Illustrating how a straightness error might affect the perpendicularity.

[D]

▪ Linear motion error: As explained in the section 3.3, a linear axis might experience motion

error in six forms, and a slight problem in any of this one form will affect the perpendicularity.

▪ Rolling element error: Similar to linear axis, a rotary element also will have six elements of

errors, refer to section 3.3. Just as in the linear motion error, any slight problem in the rotary

element might lead to perpendicularity error.

▪ Unequal clamping forces: Though this is not a type of error arising from the machine tool, it

is worth mentioning that, if the forces applied to the workpiece when clamped is not equally

distributed, it might cause the workpiece to lose its alignment, thus this can lead to

misalignment and in turn affecting the perpendicularity.


Feature 24 and Geometric Feature 25 The geometric control for both these features is symmetry with respect to the datum spline-A, symmetry

is also orientation control, which means it controls the orientation of the feature with respect to a datum

plane/axis. Symmetry is a three-dimensional tolerance, where the features are symmetrically disposed

about the plane of another feature, this another feature is the datum plane. A symmetry tolerance zone

is the width of the tolerance within which the feature is meant to be present.

55

Figure 4.21 Illustration of symmetric control tolerance . [E]

The red line in the figure is the median line, around which the measured values will lie, theoretically

this is meant to be a straight line, but in real time manufacturing, there will be a scatter of points, but as

long as these are within the tolerance zone, the part will still be accepted. Refer to section 4.3.2 and

4.3.3, to know the geometric control of the part being studied in this thesis.

Some of the errors affecting the symmetricity are:

▪ Squareness Error: An error relating to the right angle is squareness error. If the angle between

the two axes of the machine tool exceeds 90degree or if it is less than 90degree, then a

squareness error is seen. Usually squareness error leads to production of a part with differed

angles, although it might not affect the dimension, it certainly affects the positioning, this leads

to dispersion several points around the median line and causes a poor symmetry with respect to

tolerance.

Figure 4.22 Illustration of Squareness error. [F]

56

This figure expresses the angle of deviation in both the axes. Such deviations could be one of

the reason for the geometric feature 24 and 25 to have low Cpk index.

▪ Lateral play: Sometimes the guideways might exhibit a play or slop in its motion. Suppose a

guideway is made to traverse in x-direction in a to and fro manner and just in one way. This to

and fro movement must be smooth but sometimes when the guideways reach the end point and

it displaces from that point in lateral direction before returning to the initial position, this

displacement is referred to as play.

▪ Roll error: This could be one of the other possible errors affecting the symmetricity. If roll

error is present then, the axis undergoing roll error will show drift in its motion. This drift will

cause high distortion which in turn will have effect on the symmetric tolerance of the feature.

Figure 4.23 Illustration of the roll error . [G]


Feature 30 This feature is a measure of regular dimension and it is not of any specific geometric control. This

feature checks if the width between the towers is as per specification, refer to figure# in section 4.3.4.

Some of the errors which could be possible affecting the dimension are:

▪ Cyclical error: This type of error is generally causes the part to have dimensional errors.

Cyclical error is caused by a faulty ballscrew which makes the axis to move in a wayward

direction. Even a poor counterbalance system of the machine can cause this error. Also,

eccentricities in axis transducers or encoders will cause cyclic errors.

57

Figure 4.24 Illustration of the elements causing cyclical error . [H]

▪ Scaling mismatch: this type of error is another main cause of dimensional error. Scaling error

occurs when one of the axis is travelling extra or less in comparison to the other axis. Even a

faulty ball screw can cause this error, also a damaged guideway can cause this error. Axis tape

maybe subjected to over tension, causing scaling error. Due to these reasons, when one axis is

known to travel 1mm extra or 1mm less, then we can directly see the deviation from the required

dimension.

4.5. Validation of Theoretical Considerations of

Errors The previous section gives a description regarding some of the machine tool errors that could potentially

affect the processes which are indicated by low Cpk index. It is necessary to conduct an experiment

which can stand as a proof whether these considerations were right or if they differ and if they differ,

new considerations can be carried out. The test method chosen to validate the theoretical considerations

is by carrying out simulation of a mathematical model using MATLAB.

4.5.1 Simulation using MATLAB

A mathematical model has been designed using the homogeneous transformation matrix, which

describes the relative position and orientation of the machine axis with respect to each other and in

addition it can describe the errors of the machine tool [28]. This matrix has been evaluated using

MATLAB, the purpose of this simulation is to validate the theoretical considerations made in the section

4.4. To run the simulation, there a few factors which need to be defined and they are:

▪ Geometric error definition: As mentioned in the section 3.4.3, a three-axis machine tool has

21 geometric errors. These errors must be named and these are:

58

Table 4.6 Translational Errors of the three-axis machine tool

X-Axis Y-Axis Z-Axis

EXX: Positioning error in X-

axis

EXY: Straightness error of X in

Y axis

EXZ: Straightness error of X in

Z axis

EYX: Straightness error of Y

in X axis

EYY: Positioning error in Y-axis

EYZ: Straightness error of Y in

Z axis

EZX: Straightness error of Z in

X axis

EZY: Straightness error of Z in Y

axis

EZZ: Positioning error in Z-axis

Table 4.7 Rotational Errors of the three-axis machine tool

X-Axis Y-Axis Z-Axis

EAX: Roll in X-axis EAY: Pitch in Y-axis

EAZ: Pitch in Z-axis

EBX: Pitch in X-axis

EBY: Roll in Y-axis

EBZ: Yaw in Z-axis

ECX: Yaw in X-axis

ECY: Yaw in Y-axis ECZ: Roll in Z axis

Location and Orientation Errors:

C0X; B0Z; A0Z.

▪ Homogeneous Transformation Matrix (HTM): HTM is used for expression of 4 by 4 co-

ordinate transformation matrix, transforming a coordinate system or a position vector in the

body coordinate system (CS) to the base/preceding CS. This matrix is defined as following:

𝑇 = [

𝑜1𝑥 𝑜2𝑥𝑜3𝑥 𝑝𝑥

𝑜1𝑦 𝑜2𝑦𝑜3𝑦 𝑝𝑦

𝑜1𝑧

0𝑜2𝑧

0𝑜3𝑧

0𝑝𝑧

1

] (12)

where o1, o2 and o3 describe the orientation (direction of the coordinate vectors) of a coordinate frame

and vector p the position of the origin or position of the vector

4.5.2 Simulation Method The errors have been defined in the previous section 4.5.3, each of these errors will be assigned a value

(theoretical value) and the simulation will be run. At the end of the run, results will indicate the

magnitude of the impact of that error on the tool trajectory that will realize the geometry of

the part being machined. Each of the 21 errors will be run individually, meaning that when one error is

assigned with a value then the other errors will remain to be zero. In this manner, the magnitude of

impact of each error individually will be assessed.

59

Also, the value of each error will be increased by unity for a second run. This will provide information

as to how each error will impact the geometry as its magnitude increases. Using all of the input data,

Tactual (tool trajectory under the effect of the examined error) and Tnominal (nominal tool trajectory) will

be estimated. Difference between these two will give us the error.

The geometric features studied in the simulation run are:

▪ Dimension-referred to as Feature 1 in MATLAB

▪ Symmetricity- referred to as Feature 2 in MATLAB

To carry out the simulation run, certain variables and matrices will have to be defined and these are:

▪ Tool Position Matrix: This matrix is used to denote the tool position with respect to the

different axes. In this case, the position of the tool with respect to x and y axis is zero. It only

has a position in Z axis, indicating the depth of cut. This matrix is denoted by Tool;

𝑇𝑜𝑜𝑙 = [

1 0 0 𝑡𝑥0 1 0 𝑡𝑦00

00

10

𝑡𝑧1

] (13)

where tx, ty and tz is the position of the tool with respect to the axes x, y and z.

▪ Axis Location and Orientation Matrix: This matrix is used to denote the relative position and

orientation between two axes. A matrix will be computed for the axes, XY-YZ-XZ. This

denoted as:

𝑋𝑜𝑟𝑖𝑒𝑛𝑡 = [

1 −𝐶0𝑋 𝐵0𝑍 0 𝐶0𝑋 1 0 0

−𝐵0𝑍 0 1 0 0 0 0 1

] (14)

𝑌𝑜𝑟𝑖𝑒𝑛𝑡 = [

1 0 0 0 0 1 −𝐴𝑂𝑍 0 0 𝐴𝑂𝑍 1 0 0 0 0 1

] (15)

𝑍𝑜𝑟𝑖𝑒𝑛𝑡 = [

1 0 0 00 1 0 000

00

10

01

] (16)

▪ Position Matrix of Axis: Position of each axis of the machine tool will have to be defined

and these are denoted as:

𝑋𝑝𝑜𝑠 = [

1 0 0 𝑥 0 1 0 0 0 0 1 0 0 0 0 1

] (17)

Ypos and Zpos matrix are formulated in a similar way to Xpos matrix

60

▪ Homogeneous Transformation of the errors associated with an axis: Error function of each

of the axis must be transformed using a function named HTM

T= HTM (EC, EB, EA, EX, EY, EZ, axis)

Where; EC, EB, EA: the angular or tilt errors of the axis

EX, EY, EZ: the positioning errors of the axis

𝑇 = [

1 −𝐸𝐶 𝐸𝐵𝐸𝐶 1 −𝐸𝐴

−𝐸𝐵 𝐸𝐴 1

𝐸𝑋𝐸𝑌𝐸𝑍

0 0 0 1

] (18)

These errors matrices are denoted as follows for each axis:

Ex = HTM (ECX, EBX, EAX, EXX, EYX, EZX, 'x')

Ey = HTM (ECY, EBY, EAY, EXY, EYY, EZY, 'y')

Ez = HTM (ECZ, EBZ, EAZ, EXZ, EYZ, EZZ, 'z')

So, for each axis the matrix that describes the actual motion under the effect of the errors can be

expressed as follows:

Tx = Xorient*Xpos*Ex; Ty = Yorient*Ypos*Ey; Tz = Zorient*Zpos*Ez;

4.5.3 Assumptions Considered for Simulation Due to the unavailability of the specifications of the tool used, few assumptions have been considered

to carry out the simulation and these are:

The diameter of the tool is of zero unit, denoted by tx and ty (tx=0, ty=0).

The length of the tool is of unit length, denoted by tz. (tz=1)

All the errors as well as axis positions are considered in mm (displacement errors) and μm/mm (angular

errors) respectively.

To run the simulation, a workpiece coordinate system was defined. Due to lack of detailed information

about the machining process and the toolpath realization, the assignment was done in a way that would

facilitate the simulation. The workpiece CS was selected so as the Z axis will be collinear with datum

A (or the spline axis). The nominal plane defined by feature 22 was used to define the origin position.

The other axis (X and Y) are specified according to the right-hand rule. All simulations and calculations

further on are performed with respect to this coordinate system.

It was necessary to make these assumptions because, during machining, the presence of tool deflections

will contribute to the other errors impacting the machining. Thus, the above-mentioned assumptions

have been considered which will indicate that the tool deflections is almost zero and does not have

impact on the other errors.

4.5.4 Simulation of Feature-Symmetry Definition of the symmetry feature has been explained in the section 4.4.2. It is important to understand

how this measured. The following key points will summarize the measurement method:

61

▪ The two planes whose symmetry needs to be measured are chosen.

▪ The median points of these planes are calculated and then are distributed along the specified

tolerance zone with respect to the datum, refer to figure 4.26. Median points are usually derived

using CMM.

▪ The distribution or scatter pattern of these points will indicate the quality of symmetricity.

Figure 4.25 Illustration of distribution of median points derived for side 1 and 2. [I]

4.5.4.1 Tool Path of Symmetricity for Simulation

A defined tool path is essential to estimate the symmetry in MATLAB. This tool path is as follows;

Y2 = 40.5:0.5:86;

Y22 = 86:-0.5:40.5;

Y = [Y2 Y22];

X2 = ones (1, length(Y2)) * 29.5;

X22 = ones (1, length(Y2)) * (-29.5);

X = [X2 X22];

Z2 = ones (1, length(Y2)) * (-15);

Z22 = ones (1, length(Y2)) * (-14);

Z = [Z2 Z22];

62

Figure 4.26 Illustration of Tool Path for simulation of symmetry.

This tool path indicates the following:

▪ The tool moves along Y axis in both positive and negative directions. As the tool moves in

positive direction (Y2), the tool has traversed 29.5 mm in X direction(X2) from the origin (0,0)

▪ In Y2, the tool position starts at 40.5 mm and moves up to 80.6mm.

▪ Similarly, the tool moves a distance of -29.5mm in X direction(X22) and starts at 80.6mm in Y

and moves down to 40.6mm (Y22-direction).

▪ As the tool moves in the specified direction and at specified points, the impact of each error is

simulated at the step of 0.5mm in both Y2 and Y22 direction.

▪ Z2 and Z22 is the position of the tool in the Z axis (depth of cut).

4.5.5 Simulation of Feature-Dimension This feature is a simulation of regular dimension and even the measurement of this feature is very direct

and simple, unlike the measurement of feature symmetry, which is complicated. Basics of this feature

has been explained in the section, 4.3.4 and 4.4.3.

4.5.5.1 Tool Path of Dimension for Simulation

The defined tool path for simulation of the symmetry is as shown in the image below:

% Toolpath 1

% X positions array

X = [5: +0.5: -5 -5:0.5:5]; % position of X axis in mm

% Y position array

Y = [ 74*ones (1, length(X)/2) – 74*ones (1, length(X)/2)];

% Z axis

Z = -20*ones (1, length(Y));

Y2 = 40.5 – -86 mm

Y22 = 86 – -40.5 mm

X2 = 29.5 mm

X22 = -29.5 mm

Datum A

63

This tool path indicates the following:

▪ The tool moves along X axis in both positive and negative directions. The start point of the tool

is +5 and moves to -5 (negative direction of X axis) and then moves back to the point +5.

▪ Similarly, in Y direction the position of tool is at 74mm in both positive Y and negative Y axis.

▪ As the tool moves in the specified direction and at specified points, the impact of each error is

simulated at the step of 0.5mm in both positive and negative direction of X-axis

▪ Z is the position of the tool in the Z axis (depth of cut).

Figure 4.27 Illustration of Tool Path for simulation of dimension.

X = +5.0 – -5.0 mm

Y = +74.0 mm

X = -5.0 – +5.0 mm

Y = -74.0 mm

Datum, A

64

5. Results

Simulation has been run for with values assigned for all 21 errors and the results obtained are an

indication of impact of magnitude of each error individually.

5.1 Simulation of Feature-Symmetry

To run the simulation, a well described program is essential, this program includes several equations,

as per our need.

5.1.1 Assignment of Error Values

As mentioned in the section, 4.5.3, every error is individually assigned two values, the second value of

error is a unit incremental of the first value (The two error values assigned are denoted by er1 and er2).

The value assigned to each of these errors will indicate its impact on the geometric feature symmetry.

Then the actual tool path (Tact) and nominal tool path (Tnom) is estimated for both the error values

assigned.

Then, the difference between the Tact and Tnom, gives an indication of the magnitude of the error, as the

error value is increased by unit value. It is important to note here that since we are interested about a

three-axis machine only the last column (position vector) of the matrices Tact and Tnom is of interest. The

equations for achieving this, as executed from MATLAB is as follows:

“Tact = Tx*Ty*Tz*Tool; Tnom = Xpos*Ypos*Zpos*Tool;

D(:,i) = Tact(:,4) - Tnom(:,4); T(:,i) = Tact(:,4);

Err.(temp) = D

D1(:,:,er) = D; Err. (temp2).Act = T;”

The notations used in the above part of the program have been explained in the section, 4.5.3. Here,

“i” is the value of the errors assigned er1 and er2.

The initial error values that were assigned to all the errors (for the first and second simulation) can be

seen in Table 5.

65

Table 5.1 Error names and the values assigned for each error

Error

Name

er1( error

value)

er2( error

value)

ECX 0.001 0.002

EBX 0.001 0.002

EAX 0.001 0.002

EXX 8 9

EYX 8 9

EZX 8 9

ECY 0.001 0.002

EBY 0.001 0.002

EAY 0.001 0.002

EXY 8 9

EYY 8 9

EZY 8 9

ECZ 0.001 0.002

EBZ 0.001 0.002

EAZ 0.001 0.002

EXZ 8 9

EYZ 8 9

EZZ 8 9

COX 0.01E-03 0.011E-03

BOZ 0.01E-03 0.011E-03

AOZ 0.01E-03 0.011E-03

5.1.2 Symmetry Calculation

The symmetry for the planes chosen was calculated using the following equation:

“MedianMin =min(median([Err.(temp)(1,1:92);fliplr(Err.(temp)(1,93:184))])); MedianMax =max(median([Err.(temp)(1,1:92);fliplr(Err.(temp)(1,93:184))]));

if MedianMax<0 && MedianMin<0 SymmetryRange = 0 - MedianMin; elseif MedianMax>0 && MedianMin>0 SymmetryRange = MedianMax - 0; else SymmetryRange = MedianMax - MedianMin; end Err. (temp2).Symmertry(er) = SymmetryRange;”

This above-mentioned equation is a part of the program used in MATLAB, which runs the simulation.

The numbers 92, 184 are the number of arrays present in the matrix. Based on the magnitude of each

of the error, MATLAB runs the simulation and indicates if that particular has any impact on the

symmetrical feature and if yes, the magnitude of impact will also be shown.

66

5.1.3 Results of Symmetry-Simulation with Tool

Length of 1mm

Based on the input values of the error, its impact on the symmetry has been noted and this is tabulated

as follows:

Table 5.2 Results for the feature, symmetry with tz=1mm

Error

Name

er1(error

value, um)

Symmetry

Range(er1)

er2(error

value, um)

Symmetry

Range(er2)

ECX 0.001 8.59E-05 0.002 1.72E-04

EBX 0.001 1.35E-05 0.002 2.70E-05

EAX 0.001 0 0.002 0

EXX 8 0.008 9 0.009

EYX 8 0 9 0

EZX 8 0 9 0

ECY 0.001 0 0.002 0

EBY 0.001 1.35E-05 0.002 2.70E-05

EAY 0.001 0 0.002 0

EXY 8 0.001 9 0.002

EYY 8 0 9 0

EZY 8 0 9 0

ECZ 0.001 0 0.002 0

EBZ 0.001 1.00E-06 0.002 2.00E-06

EAZ 0.001 0 0.002 0

EXZ 8 0.008 9 0.009

EYZ 8 0 9 0

EZZ 8 0 9 0

COX 0.01E-03 0.00086 0.011E-03 0.000946

BOZ 0.01E-03 0.000135 0.011E-03 0.0001485

AOZ 0.01E-03 0 0.011E-03 0

From the above table, it can be explained that, the errors highlighted in red color will significantly

contribute to cause geometric inaccuracy when symmetry is being machined. Main errors of X axis

affecting the symmetry are; Pitch and yaw along with the positioning error in X axis. Also, roll error in

Y axis and Straightness error of X in Y axis. Yaw of Y axis and Straightness error in X of Z axis also

significantly contribute for inaccuracy in machining of asymmetry. Along with these, the squareness

error between X axis and Y axis, squareness error between X axis and Z axis are also major contributors

towards the machining inaccuracy of the feature symmetry.

It is also important to understand the magnitude of impact of each of the major errors mentioned in the

above table, when their error value increases. This is shown by the error difference matrix. Due to the

large size of arrays (184), only four columns are shown to understand the increase in error magnitude

as the error value(er) increases.

67

▪ Error Difference in ECX: from this it can be seen that presence of ECX affects only X-axis.

x -0.0405 -0.041 -0.0415 -0.042 -0.0425 -0.043

y 0 0 0 0 0 0

z 0 0 0 0 0 0

0 0 0 0 0 0

To understand the effect of this error better, a simple graph is plotted for value of the error, 0.001um.

This graph depicts how the yaw error of X axis, restricts the movement in X direction from meeting the

nominal point of 29.5mm. This motion is with respect to the movement of tool in Y axis, in steps of

0.5mm.

Figure 5.1 Graph of error difference of ECX1 (er=0.001um).

The green line in the graph is the nominal or desired line after machining, but due to the presence of the

error ECX, the length in X axis is considerably reduced as the tool moves every step-in Y axis. Blue

line indicates the deviation present due to ECX. Similarly, there will be further increase in this deviation

when the error value increase, this can be seen form the results table mentioned above.

▪ Error Difference in EBX: from this it can be seen that presence of EBX affects only X-axis.

x -0.016 -0.016 -0.016 -0.016 -0.016 -0.016

y 0 0 0 0 0 0

z 0 0 0 0 0 0

0 0 0 0 0 0

To understand the effect of this error better, a simple graph is plotted for value of the error, 0.001um.

This graph depicts how the Pitch of X axis, restricts the movement in X direction from meeting the


29

.49

98

529

.49

992

9.4

99

95

29

.52

9.5

00

05

40

41

.543

44

.546

47

.549

50

.552

53

.555

56

.558

59

.561

62

.564

65

.567

68

.570

71

.573

74

.576

77

.579

80

.582

83

.585

Po

siti

on

of

Too

l in

X a

xis

Position of Tool in Y axis

Error DIfference ECX1 (er1)

X Nominal X Actual

68

0.5mm. It must be noted that, unlike the ECX, EBX does not exhibit dependency on Y axis position but

rather it is a constant error with one value (of course under the condition that the error has a constant

value across the length of X axis).

Figure 5.2, Graph of error difference of EBX1 (er=0.001).


error EBX, the length in X axis is considerably reduced as the tool moves every step in Y axis. Blue

line indicates the deviation present due to EBX. Similarly, there will be further increase in this deviation

when the error value increases, this can be seen form the results table mentioned above.

▪ Error Difference in EXX: from this, it is seen that the presence of EXX affects only X-axis.

x 1 1 1 1 1 1 1

y 0 0 0 0 0 0 0

z 0 0 0 0 0 0 0

0 0 0 0 0 0 0

To understand the effect of this error better, a simple graph is plotted for value of the error, 8um. This

graph depicts how the positioning in X axis, affects the movement in X direction from meeting the


0.5mm. It must be noted that, EXX does not exhibit varying error, rather it is a constant error with one

value.

69

Figure 5.3 Graph of error difference of EXX1 (er=8um).


error EXX, the length in X axis is considerably over travels as the tool moves every step, in Y axis.

Blue line indicates the deviation present due to EBX. Similarly, there will be further increase in this

deviation when the error value increases, this can be seen form the results table mentioned above.

Similarly, to the example plots of some errors indicated above, based on the magnitude of error, each

error will impact the accuracy of the symmetry feature.

5.1.4 Results of Symmetry-Simulation with Tool

Length of 100mm

Initially, tool length (tz) was assumed to be 1mm, (refer section 4.5.3). Another consideration was made

and the tool length is now assumed to be 100mm, this is done to see how the tool length affects the

accuracy of the feature being machined. Results of this simulation run feature the effects of each error

when with new tool length.

70

Table 5.3 Results for the feature, symmetry with tz=100mm

Error

Name

er1(error

value)

Symmetry

Range(er1)

er2(error

value)

Symmetry

Range(er2)

ECX 0.001 8.60E-05 0.002 0.000172

EBX 0.001 8.55E-05 0.002 0.000171

EAX 0.001 0 0.002 0

EXX 8 0.008 9 0.009

EYX 8 0 9 0

EZX 8 0 9 0

ECY 0.001 0 0.002 0

EBY 0.001 8.55E-05 0.002 0.000171

EAY 0.001 0 0.002 0

EXY 8 0.001 9 0.002

EYY 8 0 9 0

EZY 8 0 9 0

ECZ 0.001 0 0.002 0

EBZ 0.001 1.00E-04 0.002 0.0002

EAZ 0.001 0 0.002 0

EXZ 8 0.008 9 0.009

EYZ 8 0 9 0

EZZ 8 0 9 0

COX 0.01E-03 0.00086 0.011E-

03 0.000946

BOZ 0.01E-03 0.000855 0.011E-

03 0.0009405

AOZ 0.01E-03 0 0.011E-

03 0

It is clear form this simulation run that, the length of the tool affects the magnitude of the existing error

and the deviations it will cause is not by a small margin, but rather by a big margin. When EBX is

considered the magnitude varies from 1.35E-05 um to 8.55E-05 um. Similarly, for other errors, the

magnitude has either increased or remained the same. In particular, we can see a clear effect on the

angular errors (ECX, EBY and EBZ) whereas displacement errors are unaffected by the tool length

increase.

5.2 Simulation of Feature-Dimension

To run the simulation, a well described program is essential, this program includes several equations,

as per our need.

5.2.1 Assignment of Error Values

Error values assigned to the simulation of dimension is same as that of the feature symmetry, every

error is individually assigned two values and the second value of error is a unit incremental of the first

71

value (The two error values assigned are denoted by er1 and er2). The value assigned to each of these

errors will indicate its impact on the geometric feature symmetry. Then the actual tool path (Tact) and

nominal tool path (Tnom) is estimated for both the error values assigned.

Then, the difference between the Tact and Tnom, gives an indication of the magnitude of the error, as the

error value is increased by unit value. The equations for achieving this, as executed from MATLAB is

as follows:

“Tact = Tx*Ty*Tz*Tool; Tnom = Xpos*Ypos*Zpos*Tool;

D(:,i) = Tact(:,4) - Tnom(:,4);

Err.(temp) = D

D1(:,:,er) = D;

The notations used in the above part of the program has been explained in the section, 4.5.3. Here, “i”

is the value of the errors assigned er1 and er2.

Table 5.4 Error names and the values assigned for each error

Error

Name

er1( error

value)

er2( error

value)

ECX 0.001 0.002

EBX 0.001 0.002

EAX 0.001 0.002

EXX 8 9

EYX 8 9

EZX 8 9

ECY 0.001 0.002

EBY 0.001 0.002

EAY 0.001 0.002

EXY 8 9

EYY 8 9

EZY 8 9

ECZ 0.001 0.002

EBZ 0.001 0.002

EAZ 0.001 0.002

EXZ 8 9

EYZ 8 9

EZZ 8 9

COX 0.01E-03 0.011E-03

BOZ 0.01E-03 0.011E-03

AOZ 0.01E-03 0.011E-03

72

5.2.2 Dimension Calculation

In MATLAB, the following code was used to run the simulation to study the feature, dimension.

“Err.(temp2).Diff = (D1(:,:,2)-D1(:,:,1))*1000; Err.(temp2).Norm = norm(Err.(temp2).Diff(:,1)'); Err.(temp2).MaxX11 = max(D1(1, 1:length(D1)/2 ,1)); Err.(temp2).MinX11 = min(D1(1, 1:length(D1)/2 ,1)); Err.(temp2).MaxY11 = max(D1(2, 1:length(D1)/2 ,1)); Err.(temp2).MinY11 = min(D1(2, 1:length(D1)/2 ,1)); Err.(temp2).MeanY1plus = mean(D1(2, 1:length(D1)/2 ,1)); Err.(temp2).MeanX1plus = mean(D1(1, 1:length(D1)/2 ,1)); Err.(temp2).MaxX12 = max(D1(1, length(D1)/2+1:end ,1)); Err.(temp2).MinX12 = min(D1(1, length(D1)/2+1:end ,1)); Err.(temp2).MaxY12 = max(D1(2, length(D1)/2+1:end ,1)); Err.(temp2).MinY12 = min(D1(2, length(D1)/2+1:end ,1)); Err.(temp2).MeanX1minus = mean(D1(1, length(D1)/2+1:end ,1)); Err.(temp2).MeanY1minus = mean(D1(2, length(D1)/2+1:end ,1));

Err.(temp2).YDist1=164+(((Err.(temp2).MaxY11>abs(Err.(temp2).MinY11))*Err.(

temp2).MaxY11 + (Err.(temp2).MaxY11<=abs(Err.(temp2).MinY11))*

Err.(temp2).MinY11) - ((Err.(temp2).MaxY12 >=

abs(Err.(temp2).MinY12))*Err.(temp2).MaxY12 + (Err.(temp2).MaxY12 <

abs(Err.(temp2).MinY12))* abs(Err.(temp2).MinY12)) );

Err. (temp2). YDist12 = 164 + Err. (temp2). MeanY1plus

Err.(temp2).MeanY1minus;”

This above-mentioned equation is a part of the program used in MATLAB, which runs the simulation.

Based on the magnitude of each of the error, MATLAB runs the simulation and indicates if that has any

impact on the symmetrical feature and if yes, the magnitude of impact will also be shown.

In the above code, “Err. (temp2). YDist1 and Err. (temp2). YDist12”, indicate the error

in the feature, dimension.

This code mentioned above produces the error in the dimension for only er1 values and to check the

errors for er2 values, similar coding is used, where the er1 values are replaced by er2 values.

5.2.3 Results of Dimension-Simulation with Tool

Length of 1mm

Based on the input values of the error, the impact of each error on dimension has been noted and this is

tabulated as follows:

Table 5.5 Results for the feature, dimension with tz=1mm

Error

Name

er1(error

value)

Dimension-

Ydist1

Dimension-

Ydist12

er2(error

value)

Dimension-

Ydist2

Dimension-

Ydist22

ECX 0.001 164 164 0.002 164 164

73

From the above table, it can be explained that, the errors highlighted in red color will significantly

contribute to cause geometric inaccuracy when dimension is being machined. The only error

significantly affecting the dimension is Squareness error between X axis and Y axis.

Though there are no significant changes in the value for the feature, dimension, it must be noted that

since the values of each error is very small and since the Ymax and Ymin values cancel out each other,

no change in the value of dimension is observed. But, there will be small changes caused due to the

errors, since these are very small and hence not reflected in the simulation.

This can be understood better when we look the matrix of each error. Since, the matrix has 42 arrays,

only five columns are shown here to understand the effect of error.

▪ Error matrix of EAX: It can be seen from the below matrix that, the error EAX with er value

of 0.001, introduces a difference of 0.00021 um in Y axis. This means that Y travels 0.00021um

further out in positive direction when compared to a nominal value of 74mm. But, in the table

5.5, error EAX is indicated to not have any impact on the dimension. This is due to the

EBX 0.001 164 164 0.002 164 164

EAX 0.001 164 164 0.002 164 164

EXX 8 164 164 9 164 164

EYX 8 164 164 9 164 164

EZX 8 164 164 9 164 164

ECY 0.001 164 164 0.002 164 164

EBY 0.001 164 164 0.002 164 164

EAY 0.001 164 164 0.002 164 164

EXY 8 164 164 9 164 164

EYY 8 164 164 9 164 164

EZY 8 164 164 9 164 164

ECZ 0.001 164 164 0.002 164 164

EBZ 0.001 164 164 0.002 164 164

EAZ 0.001 164 164 0.002 164 164

EXZ 8 164 164 9 164 164

EYZ 8 164 164 9 164 164

EZZ 8 164 164 9 164 164

COX 0.01E-03 163.9999 164 0.011E-03 163.9999 164

BOZ 0.01E-03 164 164 0.011E-03 164 164

AOZ 0.01E-03 164 164 0.011E-03 164 164

74

cancelling out effect of Ymax and Ymin (Ymax is the maximum deviation in y axis and Ymin

is the minimum deviation in y axis).

▪ Error matrix of EAY: Similar to the error EAX, EAY will also exhibit a small deviation in

the nominal dimension due to the presence of this error.

x 0 0 0 0 0 0 0

y 0.00021 0.00021 0.00021 0.00021 0.00021 0.00021 0.00021

z 0 0 0 0 0 0 0

0 0 0 0 0 0 0

▪ Error matrix of COX: From the table 5.5, the COX shows a deviation in the dimension value

for the, er value of 0.01*10^-3. When the error matrix of COX is observed, it can be noted that

the deviation varies at each step movement of X axis. And as a result, even though the deviation

is small, it can be seen in the result of simulation.

X -0.00074 -0.00074 -0.00074 -0.00074 -0.00074 -0.00074 -0.00074 -0.00074

Y 5.00E-05 4.50E-05 4.00E-05 3.50E-05 3.00E-05 2.50E-05 2.00E-05 1.50E-05

Z 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

5.2.4 Results of Dimension-Simulation with Tool

Length of 100mm

Initially, tool length (tz) was assumed to be 1mm (refer section 4.5.3). Another consideration was made

and the tool length is now assumed to be 100mm, this is done to see how the tool length affects the

accuracy of the feature being machined. Results of this simulation run features the effects of each error

when with new tool length.

x 0 0 0 0 0 0 0

y 0.00021 0.00021 0.00021 0.00021 0.00021 0.00021 0.00021

z 0.00074 0.00074 0.00074 0.00074 0.00074 0.00074 0.00074

0 0 0 0 0 0 0

75

Table 5.6 Results for the feature, dimension with tz = 100mm

Error

Name

er1(error

value)

Dimension-

Ydist1

Dimension-

Ydist12

er2(error

value)

Dimension-

Ydist2

Dimension-

Ydist22

ECX 0.001 164 164 0.002 164 164

EBX 0.001 164 164 0.002 164 164

EAX 0.001 163.9984 164 0.002 163.9982 164

EXX 8 164 164 9 164 164

EYX 8 164 164 9 164 164

EZX 8 164 164 9 164 164

ECY 0.001 164 164 0.002 164 164

EBY 0.001 164 164 0.002 164 164

EAY 0.001 163.9984 164 0.002 163.9982 164

EXY 8 164 164 9 164 164

EYY 8 164 164 9 164 164

EZY 8 164 164 9 164 164

ECZ 0.001 164 164 0.002 164 164

EBZ 0.001 164 164 0.002 164 164

EAZ 0.001 163.998 164 0.002 163.996 164

EXZ 8 164 164 9 164 164

EYZ 8 164 164 9 164 164

EZZ 8 164 164 9 164 164

COX 0.01E-03 163.9999 164 0.011 E-03 163.9999 164

BOZ 0.01E-03 164 164 0.011E-03 164 164

AOZ 0.01E-03 163.9984 164 0.011E-03 163.9982 164

It can be evidently seen that, change in tool length will increase the magnitude of the existing errors

and increases the magnitude of errors which were very small when the tool length was assumed to be

1mm. The deviations observed are also very large with increasing tool length. The observed dependency

of angular errors on the tool overhang is expected due to the lever effect.

76

6. Discussion and Conclusion

Sometimes, even though the dimensions of the part are within the specified limits, the process will not

be considered as capable (i.e. Cpk<1.33). This is mostly due to the occurrence of random or unexpected

events during machining. Tool break or gradual tool wear can example of such events, these are known

to bring about a significant variation in the current process being carried out. Such variations will trigger

the process to produce parts with dimensions away from the mean value and very close to the specified

limits (USL and LSL). This is evident from the process variation graphs which can be referred to from

the section 4.3.1 up to section 4.3.4.

Form the literature review conducted in this work it is known that, high process variation occurs mainly

due to the presence of machine tool errors and due to wear in the cutting tool, forces generated due to

cutting tool. This thesis mainly focuses on the geometric errors of the machine tool, thus a set of possible

errors affecting the process variation is considered as theoretical assumption (Refer to section 4.4). The

literature review forms as the basis for the theoretical considerations made and the presence of the errors

as per considerations in the section 4.4 will lead to the deviations in the process accuracy.

Furthermore, simulation carried out validates that the presence of errors did affect the process

capability, indicating the geometric accuracy of the feature machined will be affected. Though the

magnitude of impact of these errors varies based on factors such as, tool length, tool diameter, tool path.

The tool specification information was unavailable and this turns out to be one of the major setbacks

for a more realistic validation. Nevertheless, these factors were assumed before running the simulation

(Refer to section 4.5.3).

As expected, the magnitude of the errors considered in the section 4.4(Section 4.4 describes the presence

of possible errors which can have impact on the process capability) had increased with the increase in

tool length. The tool length in this thesis is an assumption. The figures 5.1, 5.2, 5.3 indicate these

changes. Interestingly, the results reveal that, few additional errors apart from the theoretical

considerations were also present affecting the process. For example, the feature, symmetry was assumed

to be affected due to the presence of squareness error, roll error. Along with these, the results from

MATLAB indicates that pitch error and straightness error of Z axis in X direction will also affect the

feature symmetry.

As the tool length of 1mm is not industrially practical, another run was made with new tool length being

100mm. with the change in length, the most interesting changes were seen for the feature-dimension,

as lot more errors were seen to be affecting. This was quite unexpected as compared with the

assumptions of section 4.4, the assumptions made in this section are the presence of possible different

errors such as roll error, squareness error etc. The likely cause of this could be, increased deflections of

the tool associated with the increased tool length.

Based on the results as seen from the simulation, it can be concluded that the different geometric errors

assumed in the section 4.4, significantly affect the features machined. Apart from the considerations

made in section 4.4, geometric errors affecting only certain features have been identified from the results

Also, the simulation results suggest that, the magnitude of the error affecting the feature is very small.

This probably is due to certain variations which are not considered during simulation such as, tool

deflections due to cutting forces, tool diameter.

The main finding of this research is, there exists a co-relation between process capability indices and

the geometric errors of the machine tool, the results which have been obtained after MATLAB

simulation support this. Also, the results of the simulation correlate well with the theoretical

considerations made. Although the identification of exact link between the two needs further research,

77

simulations and tests to be carried out with respect to different forms of errors (other than geometric

errors) and possible variants affecting the process.

78

7. Future Scope

This whole research was a theoretical effort made to study if there exists any relation between the

capability indices and the geometric errors of the machine tool. Validation of the theoretical

consideration was made using MATLAB, which is also a theoretical method. Due to limitations, such

as, lack of actual measured data and time constraints, no physical testing method was conducted.

In the future research, more practical tests such as DBB, LDBB, laser interferometer etc., can be carried

out along with simulation using MATLAB. This will give a better understanding of the error and helps

in obtaining stronger results which will help in better analysis. The results obtained in such manner can

be of great help in knowing, if these errors can be eliminated at the shop floor by engineers or if some

errors need to be considered right at the design stage of the machine tool.

Also, relatively new data of the capability indices can be used for study. Along with this, during

development of the mathematical model and simulation, it is desirable if more real time data regarding

error values, tool specifications can be used and if various parameters such as temperature, dynamic

stiffness can be considered. As these might help in understanding the problem in a deeper level. Tool

deflections will cause some errors to increase their magnitude and affect the accuracy further.

A model of tool deflection can also be studied and can be added to modeling and simulation. In this

research, very few features were studied, in the future research different and more number of features

need to be studied. It is of higher value if the same study is conducted on different machine tool and

different processes. As the number of errors affecting a five-axis machine tool is more than the three-

axis machine.

79

8. References

[1] Bergman Klefsjö, “Quality from Customer Needs to Customer Satisfaction”, Third Edition,

Studentlitteratur, Lund, 2010.

[2] Juran, J.M. (1974): “Juran’s Quality Control Handbook”, 3rd Ed. McGraw-Hill, New York, USA.

[3] Kane, V.E. (1986): “Process Capability Indices, Journal of Quality Technology”, Vol18, No 1.

[4] Doty, L.A. (1996): “Statistical Process Control”, Industrial Press In., ISBN 0-8311-3069-5.

[5] Mahendra Saha, Sudhansu S. Maiti, “Trends and Practices in Process Capability Studies”.

[6] https://www.mathsisfun.com/data/standard-deviation-formulas.html

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9. Internet References

[A] HTTP://WWW.STATIT.COM/SUPPORT/QUALITY_PRACTICE_TIPS/ESTIMATING_STD_DEV.SHTML

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[C] GD&T-PERPENDICULARITY OF A SURFACE AND CENTER PLANE ‘CIRCLE 1’

[D] HTTP://WWW.ETEL.CH/MOTION SYSTEMS/DEFINITIONS/

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[G] HTTP://WWW.WYLERAG.COM/EN/PRODUCTS/DISPLAY-DEVICES-NETWORK-

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[H] COMPARIOSN BETWEEN CNC AND NC MACHINES, 2015, QHUNT.COM

[I] HTTP://WWW.GDANDTBASICS.COM/SYMMETRY/

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