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
4.3.3. Analysis of Geometric Feature 25 ............................................................................... 47
4.3.4. Analysis of Geometric Feature 30 ............................................................................... 49
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
Figure 4.12 Graphical description of process variation as observed for machine room A for feature 24
.............................................................................................................................................................. 46
Figure 4.13 Indicating the feature number 25 ....................................................................................... 47
Figure 4.14 Graphical description of process variation as observed for machine room A for feature 25
.............................................................................................................................................................. 48
Figure 4.15 Graphical description of process variation as observed for machine room B for feature 25
.............................................................................................................................................................. 49
Figure 4.16 Indicating the feature number 30. ...................................................................................... 50
Figure 4.17 Graphical description of process variation as observed for machine room A for feature 30
.............................................................................................................................................................. 51
Figure 4.18 Graphical description of process variation as observed for machine room B for feature 30
.............................................................................................................................................................. 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.
18
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
alu
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
alu
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.
Table 4.3 Measured values of the feature number 24 for a sample size of 30.
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
Figure 4.12 Graphical description of process variation as observed for machine room A
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
alu
es
Sample No.
Process Variation-24A
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.
Table 4.4 Measured values of the feature number 25 for a sample size of 30.
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
It is to be noted that, along with the above values, the feature 25 has the following data with it and this
is same for both the machine rooms:
Tolearnce-0.05
Target Value-0.0
USL-0.05, LSL-0.0
Figure 4.14 Graphical description of process variation as observed for machine room A
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
alu
e
Sample No.
Process Variation-25A
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.
Figure 4.15 Graphical description of process variation as observed for machine room B
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
alu
es
Sample No.
Process Variation-25B
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
It is to be noted that, along with the above values, the feature 30 has the following data with it and this
is same for both the machine rooms:
Tolearnce-0.031
Target Value-164.0
USL-164.031, LSL-163.969
Figure 4.17 Graphical description of process variation as observed for machine room A
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.
Process Variation-30A
LSL
USL
52
Figure 4.18 Graphical description of process variation as observed for machine room B
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
alu
es
Sample No.
Process Variation-30B
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.
4.4.2. Errors Affecting the Accuracy of the Geometric
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]
4.4.3. Errors Affecting the Accuracy of the Geometric
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
nominal point of 29.5mm. This motion is with respect to the movement of tool in Y axis, in steps of
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).
The green line in the graph is the nominal or desired line after machining, but due to the presence of the
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
nominal point of 29.5mm. This motion is with respect to the movement of tool in Y axis, in steps of
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).
The green line in the graph is the nominal or desired line after machining, but due to the presence of the
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
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