Georgios Bris & Johannes Torres
II
Abstract
In product development, it is important to understand what impact each process step has on a product. This way, the engineer has the ability to monitor the behaviour of a product throughout the manufacturing pro- cesses. By monitoring the processes, the engineer can do adjustments to the processes in order to achieve a more accurate end result. A more accurate end result is characterised by the ability to manufacture products within permissible specifications. These permissible specifications are also known as tolerances. An im- portant factor when trying to monitor the behaviour of a process is the variation observed. The variation indicates the ability to produce a product within the permissible tolerances. Big variation is translated as the inconsistency of the machine to produce standard quality. When variation is accumulated throughout the different process steps it creates a chain of variation called tolerance chain. The tolerance chain is an indis- pensable aspect of manufacturing because by managing variation and deviation it provides the final toler- ances of the finished products.
The type of workpiece examined in this work is a cemented carbide insert. Cemented carbide inserts consist mainly of cobalt and tungsten carbide. When assembled in a tool holder, they form a manufacturing tool for metal cutting.
This thesis focuses on the study of an insert’s manufacturing processes’ ability to meet the tolerance re- quirements in each process as well as in its final stage. The processes examined are grinding, edge rein- forcement (ER) and coating. A methodology based on Monte Carlo simulation for a specific insert type, acting as a demonstrator, was developed.
Measurements were conducted in order to see the variation of each process examined. Consequently, equa- tions were derived which are describing the dimensional changes occurring. The results of the derived equa- tions were compared with the results of the measured data. An assembly function comprised of the derived equations was used to perform Monte Carlo simulations.
The assembly function predicted the tolerances in each manufacturing process. The results agreed in most of the cases with the measured data. From this experiment, grinding proved to be the biggest contributor to variation. The reason is that the thickness produced had a large spread.
In conclusion, proposals for adjusting specific tolerance limits were presented. Moreover, the standard de- viations which describe each process were provided.
In the chapter Future work proposals were made concerning a different and more accurate way of measur- ing. Finally, a proposal on investigating the manufacturing processes, prior to grinding, in order to have a better view of the tolerance chain.
.
Keywords: tolerances, variation, tolerance chain, cemented carbide inserts, Monte Carlo Simulation,
III
Sammanfattning
Inom produktutveckling är det viktigt att förstå vilken inverkan varje tillverkningssteg har på en produkt.
På så sätt har ingenjören möjlighet att övervaka hur produkten förändras under hela tillverkningsprocessen.
Genom att övervaka tillverkningsprocesserna kan ingenjören göra justeringar för att uppnå ett mer ackurat slutresultat. Detta uppnås genom att tillverka produkter inom tillåtna specifikationer. Dessa tillåtna specifikationer kallas även för toleranser. En betydelsefull faktor när man övervakar förändringen hos en tillverkningsprocess, alternativt en produkt, är den observerade variationen. Variationen indikerar förmågan att producera en produkt inom de tillåtna toleranserna. När variationen ackumuleras från flera sammanlänkade processer kallas detta fenomen för toleranskedja. Toleranskedjan är en nödvändig del av tillverkningen genom att den hanterar variationen och standardavvikelsen och på så sätt förser en med de slutliga toleranserna för slutprodukten.
Det granskade arbetsstycket i detta examensarbete är vändskär av belagd hårdmetall. Dessa består huvudsakligen av kobolt och volframkarbid. När vändskäret är monterat i en verktygshållare, bildar de ett tillverkningsverktyg för skärande bearbetning.
Denna avhandling inriktas på undersökningen av ett vändskär och dess tillverkningsprocessers förmåga att uppnå toleranskraven. De undersökta processerna är slipning, edge reinforcement (ER) - förstärkning av vändskärets egg) och ytbeläggning. Som demonstration utvecklades en metod, som bygger på Monte Carlo- simulering, för en viss typ av vändskär.
Mätningarna genomfördes för att granska variationen i varje undersökt tillverkningsprocess. Därefter erhölls ekvationer som beskriver dimensionsförändringar som inträffar i varje tillverkningsprocess. Resultaten av dessa jämfördes med resultaten av avmätta data. En sammansättning av de erhållna ekvationerna användes för att utföra Monte Carlo-simuleringar.
Sammansättningen prognosticerade toleranserna i varje tillverkningsprocess. Resultaten överensstämde i de flesta fall med avmätta data. Från detta experiment, visade sig slipning vara den största bidragsgivaren till variationen. Detta på grund av är att tjockleken på vändskären varierade väldigt mycket efter slipning.
Sammanfattningsvis presenterades förslag på att justera vissa toleransgränser. Dessutom har standardavvikelser för varje enskild tillverkningsprocess blivit tillhandahållna.
I kapitlet Future work ges förslag på ett annorlunda och mer precist sätt att mäta tjockleken. Slutligen ges ett förslag på att undersöka tillverkningsprocesserna före slipning för att få en bättre bild av toleranskedjan.
Nyckelord: toleranser, variation, toleranskedja, vändskär av belagd hårdmetall, Monte Carlo simulering
IV
Acknowledgements
This master thesis project was carried out at Sandvik Coromant in Gimo. We would like to express our gratitude to our supervisor Jan-Olov Willerström, from department GHT, for his excellent supervision and his continuous encouragement when we faced problems during our project. Moreover, we would like to thank Arne Bjerkehagen, from department GHTP2, who provided us with all the necessary data in order to understand the nature of the problem and carry out our experiments and of course his inmost knowledge about the manufacturing processes. We are also grateful of the staff in department GHTB who welcomed us in the company’s environment and treated us like family and also by aiding us with valuable information.
Additional thanks to the department GHMG1 for helping us with issues about grinding.
Finally, we would like to thank our supervisor at KTH Jonny Gustafsson for his on-point supervision and help on how to tackle problems that occurred during the duration of this project.
Abbreviations
α Clearance angle, 27°14’ (in the demonstrator case) β Measuring angle, 15°36’ (in the demonstrator case)
θ Fixture angle
γ Chamfering angle, 20° (in the demonstrator case) S Thickness of the insert
ICC Inscribed Circle Circumference
PL Position accuracy PL
D The sum of ICC radius and M
M The distance between the ICC radius and the tip of the insert
WC Worst case methods
RSS Root sum square
ER Edge reinforcement
ω Translation of measuring probe Cpk Process Capability Index LSL Lower Specification Limit USL Upper Specification Limit LTL Lower Tolerance Limit UTL Upper Tolerance Limit
V
Table of Contents
Abstract ... II Sammanfattning... III Acknowledgements ... IV Abbreviations ... IV
1. Introduction ... 1
1.1. Aim and Background ... 1
1.2. Delimitations ... 1
1.3. Method ... 1
2. Frame of reference ... 2
2.1. Previous work ... 2
2.2. Insert ... 3
2.3. Manufacturing Processes ... 3
2.4. Tolerances ... 6
2.5. Statistical concepts ... 8
2.6. Tolerance chain analysis ... 11
3. Insert Geometry and measurement of interest ... 15
3.1. Insert Geometry ... 15
3.2. Measurement of interest ... 17
4. Examined operations ... 18
4.1. Grinding ... 18
4.2. Edge Reinforcement (ER) ... 19
4.3. Coating ... 21
5. Experimentation ... 22
5.1. Goodness-of-fit test ... 22
5.2. Grinding ... 24
5.3. Edge Reinforcement (ER) ... 30
5.4. Coating ... 32
5.5. S-ICC relationship ... 35
6. Monte Carlo Simulation-Assembly function ... 39
7. Discussions and conclusions ... 45
8. Future work ... 46
References ... 47
VI
1
1. Introduction
In this section, the main objective of this work alongside with the delimitations and the method description are mentioned.
1.1. Aim and Background
In product development projects, it is important to understand how parameters and tolerances affect a prod- uct or process. It is of interest to know the consequences of variation in a number of dimensions and their ability to meet the tolerance requirements in each process step in order to meet the final specification of a product.
This thesis aims to find the root cause of variation upon the dimensions of an insert caused by the processes by using the Monte-Carlo method. The model created should be able to approximate the measurement in each of the different processes by using estimated standard deviations. By doing so, one can monitor the tolerance of each process and if there is a need for adjustments in the process in order to reduce variation.
This work should therefore reveal which is the most contributing process to the tolerance chain and where improvements can be made to reduce it. Additionally, this work should provide a general instruction on how to work with tolerance chain analysis as a standardised method for all kinds of products (inserts). Pro- posals defining if there is a need for adjusting the tolerances should be included.
1.2. Delimitations
This report will only examine three process steps of the whole manufacturing process; grinding, edge rein- forcement (ER) and coating, since those are considered to be the major contributors to the final variation of the insert type chosen as the demonstrator. In addition, only one type of insert, type A, will be examined in this report.
1.3. Method
The project started with the study on how the chosen insert type was produced in the factory. Starting from metal powder to being formed into the desired shape, the insert is undergoing a variety of processes - as mentioned above – until its final form.
Data was provided to detect where the problem lies. Afterwards, several measurements were performed to understand the nature of the measurements and collect additional data. After collecting the additional data, mathematical equations describing the change in dimensions were derived. This was accomplished by ap- plying trigonometry and utilising CAD software. The derived equations were tested by input of real data and comparing the results with the measured data. After the validity of the equations was confirmed, an assembly function was formed comprised of all the derived equations.
Afterwards, Monte-Carlo simulations were performed in the assembly function, using estimated standard deviations. Comparisons between the Monte-Carlo results and the measured data were performed to test the accuracy of the model.
2
2. Frame of reference
In this chapter, the necessary theoretical background used in this report and a reference to some previous work from different researchers are included.
2.1. Previous work
Researchers have used different methods for analysing the tolerance chain, some examples are:
Gerth and Hancock [1] performed a case study which describes where tolerance analysis techniques were used to find the cause of poor product performance (a DC motor) and determines new specifications by using Monte-Carlo simulation. The authors do also a comparison between the Monte-Carlo method and the RSS and WC methods.
As a conclusion, the Monte Carlo method was successfully demonstrated on the DC motor. Its major ad- vantages are the unrestricted number of variables used. Moreover, this methodology has the ability to com- pare true system performance with intended system performance as intended by the specifications.
Cheng and Tsai [2] tested a method using the Lagrange multiplier for minimizing manufacturing cost which is subject to constraints on dimensional chains and machining capabilities. By performing this test, the re- searchers introduced the Lambert W function to assign optimum component tolerances. The reciprocal power and exponential cost-tolerance models for statistical tolerancing were investigated for employing this method. The optimization problem is solved by applying the algorithmic approach.
Bagge, et al., [3] developed a custom method, named dimension dependency chart (DDC)which analyses the in-process workpiece tolerance chain by simulating the shape variation propagation caused by system- atic and random errors. The case study examined a fictitious part named “Xshaft”.
The results show how the traditional process capability index (PCI), limits the process performance utiliza- tion. The effect of a process chain and the in-process workpiece (IPW) tolerances can be calculated to esti- mate the final part shape variation and define the allowed outcome of the manufactured process. According to the author, a well-established method to evaluate the relation between manufacturing process behaviour and a desirable tolerance is to use some kind of PCI and in this case, Cp.
Yu, et al., [4] presented a nonlinear tolerance analysis method which uses MATLAB as a tool to construct the nonlinear tolerance analysis mathematical formulation and automatically calculate the result of nonlinear tolerance analysis based on the principle of worst-case tolerance analysis.
The results of this method allowed the engineers to input some parameters such as the value and deviation of dimensions. The result of the tolerance analysis has a very high accuracy and can meet the requirement of product design and assembly analysis.
3 2.2. Insert
The insert, or the cutting tool, is any tool used to perform material removal from the work piece by means of shear deformation. Figure 1 shows an example of how an insert could look like.
There is a variety of different compositions of an insert, also known as grades. There are plain carbon steel inserts composed of carbon and manganese, high speed steels consisting of carbon manganese tungsten and chromium but the most widely utilised material today is the cemented carbide [5]. Cemented carbide is a powdery metallurgical material; a composite of tungsten carbide (WC) particles and a binder, rich in metal- lic cobalt (Co).
The materials that the insert is composed of, have a combination of hardness, toughness and wear resistance and are divided into different grades providing different properties [6]. The key characteristics of the insert are:
• Hard, to resist wear and deformation
• Tough, to resist bulk breakage
• Non-reactive with the workpiece material
• Chemically stable; to resist oxidation and diffusion
• Resistant to sudden thermal changes
2.3. Manufacturing Processes
There are five major steps to manufacture an insert. The operation sequence for the specific type that is examined is:
A. Pressing B. Sintering C. Grinding
D. Edge Reinforcement (ER) E. Coating
The first two steps (A and B) are in fact one manufacturing process, called powder metallurgy (PM). This is a forming process, where metal powders are formed into components. With powder metallurgy, the need of further machining operations can be avoided or at least greatly reduced since the manufacturing process produce components that are close to the finished shape and size.
A. Pressing
Pressing is one of the basic technological operations in powder metallurgy. The purpose of pressing is the making of semi-finished products with the dimensions and shape necessary for producing the finished prod- uct, discounting any deformation that may occur in the subsequent operations (sintering, grinding, etc.). The pressing material is usually prepared by mixing a binder or lubricant with the powder and applying pressure to the powder to increase density and shape the powder [7]. The product of pressing is called “green” com- pact. The strength of the green compacts must be sufficiently great to withstand the intervening operations (handling purposes) prior to sintering. [8].
Figure 1: Random insert (Source: Sandvik Coromant web site)
4
The shape and the size of the insert depends on the pressing characteristics of the powder. The steps when performing pressing are:
1. The cavity of the die is filled with metal powder (cemented carbide)
2. The upper and lower punch apply a specific force to press the powder into one piece The steps mentioned are illustrated in figure 2.
“Green” compacts are not to be seen as finished products because of their low strength and brittleness.
Therefore, sintering is applied to increase the mechanical properties of the green body.
B. Sintering
“Sintering is a thermal process used to bond contacting particles into a solid object” [7]. An example of sintering is pottery. Wet clay is shaped into a pot, then heat or fire is applied to increase the mechanical strength consequently creating a strong pot.
According to Randall, sintering is a thermal treatment which is crucial to the success of several engineering products; including most ceramics and cemented carbides. While most metals and their alloys can be melted and cast, forged or machined to attain the final shape, ceramic materials have a high melting point and are at the same time brittle in character. Such manufacturing processes are therefore not optimal when producing ceramic components. Thus, after the firing cycle the ceramic material becomes very strong and competitive in properties found in other manufacturing routes such as casting, machining, grinding or forging.
“For industrial components, sintering is a means to strengthen shaped particles to form useful objects such as electronic capacitors, automotive transmission gears, metal cutting tools, watch cases, heart pacemaker housings and oil-less bearings”. [7]
When sintering is completed the green body holds improved material properties and is referred to as a blank.
Figure 3 shows an example of a sintering furnace.
Figure 2: Pressing operation (Source: Sandvik Coromant)
5 C. Grinding
Grinding is a term used in modern manufacturing to describe a material removal process using abrasive methods such as high speed abrasive wheels, pads and belts [9]. Nowadays, grinding has been applied to almost all types of material providing a high material removal rate and accurate surface finish. Polishing, lapping, honing and related superfinishing processes can be applied after the grinding process to reduce surface roughness [10].
Abrasive processes are the natural choice for machining very hard materials. The general rule is that the tool used for machining should be harder than the material being removed. Grinding allows high accuracy to be achieved and close tolerances can be held for a variety of size, shape and surface texture [9].
D. Edge Reinforcement (ER)
After the grinding process, ER is performed on an insert’s cutting edges. See figure 4.
Figure 3: Sintering furnace (Source: Sandvik Coromant)
Figure 4: Insert's cutting edges
6
When the insert is undergoing edge reinforcement, mechanical force is applied on its cutting edges, resulting in a small material removal. Thus, the cutting edges are rounded which makes the insert more durable and capable of bearing larger loads. The ER value can be defined as a circle, see figure 5. Note that the dashed part of the circle’s contour is unimportant. The chequered area indicates the material being removed.
E. Coating
“Coating is the application of synthetic substances which can be applied to a carrier, in this case the insert, as uniformly as possible over the entire surface” [11].
There are two types of coatings used:
According to Sandvik Coromant “Chemical Vapour Deposition (CVD) coatings are generated by chemical reactions at temperatures of 700-1050°C. CVD coatings have high wear resistance and excellent adhesion to cemented carbide. Modern CVD coatings combine Ti(C, N), Al2 O3 and TiN layers. These components are used for improved adhesion, toughness and wear properties. CVD coated grades are the primary choice when wear resistance is important. Such applications are found in general turning and boring of steel”.
Moreover “Physical Vapour deposition (PVD) coatings are formed at temperatures of 400-600°C. The pro- cess involves the evaporation of a metal which reacts with substances leading to the creation of coating on the cutting tool surface. PVD coatings add wear resistance to grades due to their hardness. Their compressive stresses also add edge toughness and comb crack resistance. PVD coated grades are recommended for tough, yet sharp, cutting edges, as well as in smearing materials. Applications include solid end mills, threading, and milling. PVD coated grades are also used for finishing applications” [12].
2.4. Tolerances
With the continuous increased performance requirements in industry, the increase in tolerance analysis is considered a major topic. The effects of variation have a direct impact on cost performance of a product.
Inadequate performance and increased cost will affect the market strength of the product [4].
When manufacturing a part, it is impossible to reproduce it exactly according to the drawing specifications.
The different manufacturing processes produce variations which may have a critical impact on how accurate Figure 5: Chequered area - removed material
7
the finished part will be. The differences in the dimensions of the produced parts can be described as devi- ations which are due to size, location or orientation.
The produced deviations are direct results of the repeatable aspect of manufacturing such as machine per- formance, how the part is mounted on the fixture, the specific tools that are used etc. These variations are acceptable as long as they fall within the permissible limits. These permissible limits are defined as toler- ances. A critical aspect of tolerances is the size of the tolerance limits. If the tolerance limits are wide, then the possibilities of having a product within the limits are increased and the manufacturing cost is likely lower.
High manufacturing precision, results in a product with less variation and thus more stable performance.
However, the manufacturing cost is likely to be higher. Tolerances which are set improperly are a major cause leading to scrap or poor functionality [13].
From an engineering point of view, tolerance is a question of how much a geometry is allowed to deviate from the designer’s specifications. Using a common language which includes symbols, rules, nominal di- mensions called Geometric Dimensioning and Tolerances (GD&T), the designer establishes communication with the people from manufacturing [14]. As a result, the manufacturing can decide the appropriate manu- facturing processes and operation sequences. The designer uses a list of symbols to depict the requirements from the production as shown in table 1:
The purpose of these symbols is to describe the geometry of the products and how they are related with other functional parts or assemblies. The design engineer describes how the part should be manufactured with precision.
When utilizing geometrical tolerancing and dealing with component’s real geometrical surface one can cat- egorise the deviations from the nominal shape, orientation and location into either single or related to a datum feature geometric tolerance type.
Datum is a real feature of a part which is used to establish the location of a datum according to ISO 5459. The datum can be identified from the drawing of the part. The datum features are expressed on the technical drawing by a triangle attached to the datum feature [13]. The identification contained within the box is shown as a capital letter as shown in figure 6.
Table 1: GD&T symbols
8 2.5. Statistical concepts
Normal/Gaussian bell shaped curve
The normal distribution is an important probability distribution used for describing random variation. The location and scale parameters are dependent on the mean (μ) and the standard deviation (σ) [13].
±1σ indicates that 68.27% of the values lie in the range μ+σ to μ-σ.
±2σ indicates that 95.45% of the values lie in the range μ+2σ to μ-2σ.
±3σ indicates that 99.73% of the values lie in the range μ+3σ to μ-3σ.
Figure 6: Datum and GD&T symbol
Figure 7: Normal distribution
9 Weibull distribution
The Weibull distribution is used as an alternative to the normal distribution when the data is skewed and is defined as:
1
( ) f
exp
0Where θ>0 is the scale parameter and β>0 the shape parameter [15]. The mean μ and variance σ2 are:
1 1
and
2
2 2 2 1
1 1
Figure 8 shows the probability density function of Weibull distribution when θ=1 and β varies.
P-value
The P-value is the probability that the test statistic will take on a value that is at least as extreme as the observed value of the statistic when the null hypothesis (H0) is true. Thus, a P-value discloses information about the weight of evidence against the null hypothesis (H0) and so one can draw a conclusion at any specified level of significance. It is customary to call the test statistically significant when the null hypoth- esis (H0) is rejected; therefore, the P-value is the smallest level at which the data is significant [15].
Figure 8: Weibull distribution with different shape values
(1)
(2)
(3)
χ
10 2-Sample t-test
A hypothesis test for two populations means to determine if they are significantly different [16]. The hy- potheses are:
• Ho
:
1
2 0
• H1
:
1
2 0
if pvaluea ,do not reject Hoif pvaluea ,reject Ho
Where a is the significance level.
After examining the two hypotheses, the factor indicating which hypotheses stands, is the p-value. If the p- value is less than the significance level chosen, the null hypothesis is rejected.
An example of this technique is the following:
A study in order to evaluate the effectiveness of two devices for improving the efficiency of gas home- heating systems was performed. Device 1 uses electricity and device 2 uses thermal energy. The energy consumption is the measured factor and it is in British thermal unit (BTU).
For device 1 the is mean values is d1=9.91 BTU and for device 2 is d2=10.14 BTU. Their difference in means is d1-d2= -0.235 BTU. The p-value from the test result is p=0.707 which is greater than the confidence level chosen (95%) indicating the difference is not significant [16] thus the energy consumption is not af- fected by the type of device used.
2-Variances test
A hypothesis test determining whether two population variances are significantly different [16]. The hy- potheses are:
• Ho
:
12/
22 1
• H1:
12 / 22 1if pvaluea , do not reject Ho
if pvaluea,reject Ho
Where a is the significance level
If the p-value is less than the significance level chosen, the null hypothesis is rejected.
Goodness-of-fit tests
To determine the type of distribution that characterises a set of values, a procedure called “goodness-of-fit”
test determines whether a sample of n observations,x1
, ,
xn , can be considered as a sample from a given distribution [17]. A more common type of test is the normality test. These types of tests are divided into two categories [18]:1. Tests based on the empirical distribution function (EDF).
2. Tests based on regression and correlation tests.
1. Tests based on the EDF
The Anderson-Darling test and the Kolmogorov-Smirnov test are examples of EDF. These types of tests evaluate the fit of a distribution to the data or to compare different sample distributions [16]. The idea of the EDF tests in testing normality of data is to compare the empirical distribution function which is estimated based on the data with the cumulative distribution functions (CDF) of normal distribution to see if there is a good agreement between them [19].
(4)
(5)
11
This specific type of test investigates if a sample is taken from a normal distribution with parameter µ(mean) and σ2 (variance).
The assumptions are that the data is measured at least on an ordinal, meaning sorted. scale. The sample’s random variables
1, ,
n are identically, independently distributed with observations x1, ,
xn and a continuous distribution function F(x) [18].There are two hypotheses for the Anderson-Darling test for the normal distribution.
H0 : The data follows the normal distribution H1 : The data do not follow the normal distribution 2. Tests based on regression and correlation tests.
Correlation tests are based on the similarity between two or more paired datasets.
The Ryan-Joiner test for normality is similar to the Shapiro-Wilk test but is simpler to implement on a software. The Shapiro-Wilk test is using the observation that a normal probability plot that examines the fit of a sample dataset to the normal is like linear regression and by conducting an analysis of variance the quality of the fit can be examined [20].
The Shapiro Wilk normality test (in this case Ryan Joiner) is the most powerful normality test for a big number of samples (n≥50). The factors that affect the strength are the skewness and the kurtosis of the plot [19].
In normality tests the variable that indicates whether the distribution is normal or not is the P-value.
Process capability (Cp)
Cp is a statistical measure showing the ability of a process to produce output within the upper specification limit (USL) and the lower specification limit (LSL) [15].
If the process is not centred the equation is:
2.6. Tolerance chain analysis
According to Gerth and Hanckock (2000) “one analysis capable of describing the cumulative effect of var- iation is commonly known as accumulation of variation, accumulation of error, stack-up analysis or toler- ance analysis” [1]. This variation commences from the first manufacturing process and they are carried throughout the rest of the processes thus creating a chain of accumulated variation.
Tolerance chain analysis is a common term in manufacturing industry and aims to reduce the product defects but also improve the quality of a product by reducing the variation of the final product. It provides a quan- titative design tool for predicting the effects of manufacturing variation on performance and cost [21].
There are some popular techniques available for the tolerance stack up analysis [22]
1. Worst-case methods (WC):
The worst-case method determines the absolute maximum variation possible for a selected distance or gap. Various variables such as dimensions are performing a tolerance stack up and in the end, they are either added or subtracted in order to obtain the total variation of the considered distance [23].
6 USL LSL
Cp
(6)
min ,
3 3
USL LSL
Cp
(7)
12
1 n
final i
i
T T
Where Tfinal is the tolerance value Example
A component is comprised of four different parts. The overall tolerance for the assembly is 4±0.1 and each component has a tolerance of 1±0.03.
By using
1 n
final i
i
T T
the result is UTLfinal 1.03 1.03 1.03 1.03 4.12
mm and0.97 0.97 0.97 0.97 3.88
final
LTL
mm which are out of specifications therefore the individual component need tighter tolerances. This may lead to a higher manufacturing cost.Advantages:
• It is safe since it takes into account that the variables are taking their worst-case values. If the inputs meet their tolerances, then the output is assured of being within its worst-case tolerance [24].
Disadvantages:
• Is expensive because the results lead to tighter tolerances for each individual component. As a result, it requires more frequent tightening of the inputs’ tolerances in order to meet the specifications thus increasing the costs [24].
•
2. Statistical Tolerance Analysis Root Sum Square (RSS) method:
It characterises the tolerance of each part as a statistical distribution [25]. However, the statistical model proves to be unappealing to the designers because there is a slim probability of a defective assembly to occur [26].
2
1 n
final i
i
T T
Figure 9: Worst-case assembly
(8)
(9)
13 Example
The RSS method is describing each component with a mean and a standard deviation. It assumes the process is centred meaning that the component has a target value. The overall tolerance for the assembly is 4±0.06 and each component has a tolerance of 1±0.03. If all the components are added with RSS method the result will be like:
2
1 n
final i
i
T T
for UTL
0.03
2 0.03
2 0.03
2 0.03
2 0.06 4.06
and LTL
0.03
2 0.03
2 0.03
2 0.03
2 0.06 3.94
. The results satisfy the specification limits.On the other hand, if the process has drifts and shifts and the individual component has different values, e.g.
μ=0.985 σ=0.04 instead of μ=1 σ=0.03 then:
2 2 2 2
0.04 0.04 0.04 0.04 0.08 4.02
UTL
< 4.06 however2 2 2 2
0.04 0.04 0.04 0.04 0.08 3.86
LTL
< 3.94. As a result, the component will be outside the specifications, see figure 11.Figure 11: RSS method results Figure 10: RSS assembly
(10)
14 Advantages:
• RSS is less expensive than the worst-case method because it does not require tight tolerances for each individual component.
Disadvantages:
• It assumes that all the inputs are exactly at target meaning that the process is centred. As a result, various errors and process shifts can result in performance worse than the predicted [24] .
3. Taylor series, known also as non-linear propagation:
If the assembly response function is highly non-linear the RSS method can lead to serious errors. The extended Taylor series approximation can be employed. Taylor series provide a quantitative estimate on the error by using this approximation [27]. The purpose of the Taylor series is to linearise the equation using [28]:
1 1
0
n m
i
i j j
j j j j
h
dh hdx du
x u
Where: dxj is the specified tolerances of the component dimension
duj is the resultant variations in the dependent assembly dimensions h is the system of loop equations
This expression is put in vector form and for matrix [A]=
j
h x
and [B]=i j
h u
The new form is:1
[ ]{ } [ ]{ } {0}
{ } [ ][ ]{ } A dx B du du B A dx
The product of the matrices
B A1 will give the sensitivities of the dependent assembly dimensions with respect to the component dimension. So, the standard deviation can be calculated as:2
1 n
i
i j
j j
du u dx
x
Advantages:
• Estimation of errors from the RSS method.
Disadvantages:
• Issues when computing the partial derivatives.
4. Monte Carlo Simulation:
A method for iteratively evaluating a deterministic model using random numbers as inputs [29]. The random numbers are based on the type of statistical distribution which describes them. The randomly generated values are combined through an assembly function in order to determine a series of values of the assembly variable [21]. The Monte Carlo method is proven to be a useful tool in applications involv- ing uncertainty due to its low complexity and the ability to model phenomena with significant uncertainty in inputs. [29]
(11)
(12)
(13)
15
A random number generator is a procedure that produces an infinite stream of random variables independent and identically distributed (iid) according to some probability distribution [30].
1, 2, 2,.... ~
iid
Dist
The generated values are used in the assembly function ex.
y =f(X ,…,X )
1 n in order to determine a series of values of the assembly variable. This procedure is replicated N times.As result, random samples are yielded for y. It is important to state that the precision of this analysis is proportional to √𝑁 so the more numbers the better the accuracy [27].
The steps on performing a Monte Carlo simulation are the following:
1. Generation of random numbers x1, x2, …, x(n-1) with a specific mean (x) and standard deviation (σ).
2. Input the random numbers in a mathematical model f(x) 3. Compute the output
4. Analyse the results Advantages:
• Can yield more precise estimates than the three previous mentioned methods. Because of the data being generated, Monte Carlo simulation can create graphs of different outcomes. The user can see which inputs influence the possible scenarios.
Disadvantages:
• The Monte-Carlo method is completely dependent on the accuracy of the model. If the model has inconsistencies, it can provide results different than expected.
3. Insert Geometry and measurement of interest
In this section of the report an explanation of the insert’s geometry and the measurement of interest is going to be described.
3.1. Insert Geometry
Figure 12 displays an insert of type A, from a three-dimensional (3D) view. It is a trapezoidal prism with symmetrical sides but the corners are cut off and hold a steeper inclination than the chamfered sides.
In order to understand the geometry better, the 3D solid is broken down into 2D sketches which will describe the changes in geometry and also how the measuring equipment is affected by these dimensional changes.
Figure 12: Insert type A
16
The critical dimensions of the insert are the inscribed circle circumference (ICC) and the thickness (S) which are demonstrated in figure 13 and 16. The first figure illustrates the ICC and also the variables D and M which are of valuable interest and will be used in chapter 5.5. Note that the circle in figure 13 is purely there for visualisation. In reality, there is no circle marking on the insert. The ICC measurement shown in figure 13 is the distance from one side of the insert to the opposite side wherein a corresponding circle is tangent.
D is the distance from the centre of the insert to the cutting edge and M is the difference between D and the radius of the inscribed circle.
To understand how the inclination differs between the chamfered sides and the cutting edges, figure 13 shows insert A, from top view. Imagine inserting cross-sections, in this view, perpendicular to the contours of the insert. The resulting lines are either intersecting two chamfered surfaces, opposite of each other (cross- section 1), or the surfaces of the clearance angle (cross-section 2). Either way, the cross-sections display a trapezoid shape yet with different angles.
These angles, the chamfering angle γ and the clearance angle α, seen in figure 14 and 15, are important dimensions that help describing the insert. The first mentioned, γ, is the angle between the vertical and the sloping surface which is cross-section 1 in figure 13.
The clearance angle, α, is the angle between the vertical and the surface of the cutting edge, see figure 15 which is cross-section 2 in figure 13. Note that α is larger than γ.
Figure 13: Top view of insert A
Figure 14: Cross-section 1, chamfering angle γ
17 3.2. Measurement of interest
To monitor that the insert is fulfilling the correct requirements, i.e. has the right dimensions, it is desired to measure positional accuracy PL. A correct PL is the result of a correct correlation between ICC and S. Note that when measuring PL, the insert is mounted to a fixture tilted with the measuring angle β. The measuring equipment is a digital dial gauge incorporated into a fixture. It consists of a spring-loaded extensible arm (measuring probe) which measures the PL, as shown in Figure 17.
The reason for measuring in this way is to simulate when the insert is mounted into the tool holder. If the inserts’ cutting edges share the same PL-value, in spite of having different sizes in ICC and S, the workload applied will be equally distributed.
The reference measuring point is provided by a master. This master has a geometry of its own but when mounted to the fixture it creates the measuring reference point for all inserts.
Figure 17: Measured insert, front and left view Figure 15: Cross-section 2, clearance angle α
Figure 16: Front view of insert showing the thickness
PL
18
4. Examined operations
In this part of the report, an explanation on how the three operations are affecting the geometry of the insert is provided. The outcome of this chapter is the equations describing the processes.
4.1. Grinding
After pressing and sintering is finalised, the blanks are ground. The operator places the inserts in the grinding machine with the bottom facing down. Only the thickness is affected by the bottom grinding. As a result of bottom grinding the new bottom surface is larger than before. When placing the blank, with the new larger bottom surface, into the measuring equipment, the blank translates diagonally as shown in figure 18 thus touching the probe of the measuring equipment at a higher position. The reason for this is the inclined (chamfered) surfaces that are in contact with the fixture when conducting the measurements, see figure 17.
In order to calculate how much the probe is translating when the insert undergoes grinding, the following equation was derived which corresponds to figure 19.
Figure 18: Insert before and after grinding
19 4.2. Edge Reinforcement (ER)
After the grinding is completed the next operation is ER, where each of the four edges and the periphery is ER-treated. This operation removes a very small part of the insert yet it is very vital in order to achieve the right strength of the insert. Since ER is operated on a small part of the insert, CAD was utilised in order to visualise and understand how ER is affecting PL. This exercise was conducted numerous times for inserts with different ER-values. The results were still the same. Considering ER being defined as a circle, the dimensions shown in figure 20 are proportional to one another, despite different ER-values. In other words, the ratio between the dimensions is constant. However, for this particular example the ER-value given is 0,035.
The measuring probe will always touch the insert on its highest point. There are two reasons for that. Firstly, the insert’s clearance angle is larger than the measuring angle. Secondly, the measuring probe’s surface area covers all the area where material has been removed. After the insert has been ER-treated the probe will touch the insert on the highest part of the arc shown with a red circular mark in figure 20. The dashed horizontal lines represent where the probe touches the insert before and after ER is applied.
Figure 19: Translation of the probe
cos cos
tan tan
tan
tan cos cos sin
tan
tan cos sin tan cos sin
PROBE
PROBE
PROBE
PROBE
PROBE
S S
S
S S
S
S S
S
(14)20
The desired point (the red circular mark) can be located by knowing its two coordinates; x and y, where x is the horizontal coordinate and y is the vertical coordinate. The horizontal distance (in this case 0,049) is obtained when multiplying the ER-value with √2. This value (0.049) represents the length, in horizontal direction, from the vertex to where the probe touches the insert after ER is performed. To obtain the vertical distance which the probe translates is more complex.
The vertical distance (0.010) does not equal the desired length which the probe translates, yet the green line does. Both vertical lines (turquoise and green) form right triangles, together with slope of the insert and the dashed line (the measuring probe before ER is applied). Since both lines share the same angle (α-β), uni- formity can be applied to obtain a ratio between them. More specifically, to obtain an accurate value for material removal, the following steps were conducted:
0, 035 2 0, 0491
0, 0491 tan( ) 0, 0491 tan11, 63 0, 0101 0, 0108
1, 07 0, 0101
The ratio (1.07) is needed to calculate the length of the probe translation. The combination, 1.07×√2, is valid for any ER-value for this insert type. Hence the material removal equation is:
Note that the (-)-sign indicates that material is being removed.
Figure 20: Method of calculating ΔER.
[1.07 2 tan( )]
ER ER
(15)21 4.3. Coating
The last operation which is examined is coating. Depending on what grade the insert consists of, a different coating is applied providing expansion to both ICC and S as shown in figures 21 and 22. Like ER, coating values that are applied are in μm. The thickness of the insert is increased when coating is applied. The increase is two times the coating value since the coating expands both in bottom and top surface, so the equations is:
Where SC is the total thickness with coating according to figure 21 SG is the ground insert thickness
In order to calculate the changes in ICC dimension D is used. This dimension is increasing when the insert undergoes coating. Since coating is assumed to be equally distributed all over the insert the following equa- tion can be used:
Where DC corresponds to D after coating
Figure 21: Thickness expansion
2
C G
S S Coating
DC DCoating
(16)
(17)
Figure 22: D expansion
22
5. Experimentation
In this chapter, the goodness of the fits for all the variables needed are presented. Furthermore, the validity of the derived equations is tested. Additionally, a 2-sample t-test and 2-variance test were conducted, show- ing if there was a significant difference between measured and calculated data. The tests were conducted in Minitab while the validity of the equations was conducted in MATLAB. When testing the equations, some graphs may not appear the same as the distribution implies. The reason is that the number of samples generated is not large enough to form the distribution properly.
5.1. Goodness-of-fit test
Even though there is no Monte-Carlo simulation in this chapter, random numbers for specific variables were needed to be generated to complete the validity of the equations. Each of the variables are described by a specific distribution which is derived from the goodness-of-fit test and is based on the measurements. The confidence interval level chosen was 95%. All the normality figures and values are shown in figure 23 and 24 and in table 2 and 3. The graphs include a red line which is the distribution pattern. If the majority of the values are following this pattern, the p-value will be larger than the significance level thus agreeing with the distribution examined.
The p-values are shown in Table 2:
Figure 23: Normality tests for ICC, ER, Coating and S
Table 2: P-values for ICC, ER, Coating and S
23 The p-values for PL are shown in Table 3:
ICC, ER, Coating and PLER passed the normality test because their P-value is greater than the significance level (0.05). This means that they are normally distributed.
S failed to pass the normality test and this outcome is depicted also in PL values for grinding and coating where S is changing. On the other hand, because ICC is also a factor for these operations the assumption will be made that PL grinding and coating are normally distributed.
The closest distribution describing S and thus affecting the PL values for grinding and coating is the 2- parameter Weibull distribution as shown in figure 25. The reason behind this decision is that the P-value for this distribution is the closest to the significance level chosen.
Variable Samples P value
PLGR 444 0.028
PLER 444 >0.1
PLCoating 444 0.024
Table 3: P-values for PLGR, PLER and PLCoating
Figure 24: Normality test for PLGR,PLER and PLCoating
24
Figure 25: Weibull goodness-of-fit test for S
5.2. Grinding
In order to fully understand how the grinding process adds to the variation of the insert by altering its thick- ness, a test was conducted to evaluate the relationship between PL and S. Six inserts were examined with the dial indicator before and after the grinding process to acknowledge the change in PL according to the reduction in thickness. To be able to see how much the PL value changes depending on how much is ground, all six inserts and all four cutting edges of each insert were marked individually to be distinguishable. In total 24 measurements were conducted.
The main interest in this experiment was to monitor how much the mean and variation increases from one step to the other. Table 4 presents the grinding values measured after the first time and when the inserts are ground a second time. Figure 26 presents the two stages in histogram form assuming a normal distribution curve.
Operation 𝑆̅G(mm) σG 𝑃𝐿̅̅̅̅(mm) σPL
1 6.19 0.01 0.01 0.017
2 6.16 0.01 0.03 0.023
Table 4: Grinding data at stage 1 and 2
25
The results of the 2-sample- t-test and variance are shown in table 5 and indicate a significant difference between the mean and the variance in the two stages.
The next step is to test equation (14). Because the clearance angle is not measured, its mean and standard deviation had to be approximated. 24 random values were generated for α in order match the amount of measurements from S and PL.
There are two cases with two different Cp-values. The reason of selecting these Cp values was that Cp=1 shows a good process with low variation while Cp=0.5 shows the opposite. The purpose is to see how much the clearance angle affects the outcome. Additionally, the assumption is that α is normally distributed.
Variable P-value Mean 0.003 Variance 0.09
Table 5: P-values for between PL1 and PL2
Case Cp
α
σαA 1 27.24 0.17 B 0,5 27.19 0.33 Table 6: Cases A and B for different Cp
Figure 26: Grinding data at stage 1 and 2
26 Case A
Figure 27: α-values for Cp=1
After α is generated, all the variables required are put into (14) resulting in:
ω ̅(mm) σ
ω0.0175 0.0085
Table 7: ω-values for Cp=1
ω indicates the probe translation after grinding. The next step is to sum PL1 and ω. This summation will test if ω can describe the changes occurring between PL1 and PL2. Then, the outcome will be compared with the measured values (PL2) in order to see their similarity.
Figure 28: ω-values for Cp=1
27
1
PLGRC
PL
Where PLGRC is the calculated PL for grinding The results are shown both in table 8 and figure 29.
Values
PL(mm) σ
PLPL
20.029 0.023 PL
GRC0.028 0.02
Table 8: Comparison between PL2 and PLGRC with cp=1
The results of the 2-sample t-test and variance in table 9 show that (14) provided results with no significant difference to the measured.
Variable P-value
μ 0.864
σ2 0.66
Table 9: Case A P-values for significance Figure 29: Comparison between PL2 and PLGRC with cp=1
28 Case B
Figure 30 shows the generated a-values for Cp=0.5
Figure 30: α-values for Cp=0.5 Table 10 and figure 31 show the results with Cp=0.5.
ω ̅(mm) σ
ω0.0175 0.0085
Table 10: ω-values for Cp=0.5
Figure 31: ω-values for Cp=0.54
29 This leads to a PLGRC:
Values
PL(mm) σ
PLPL2 0.029 0.023 PL
GRC0.028 0.02
Table 11: Comparison between PL2 and PLGRC with Cp=0.5
Finally, the significance test shows:
After completing the experiment, it is proved that (14) can describe the changes occurring in the grinding process. Furthermore, the clearance angle is not a big contributing factor because the results for both cases are similar.
Variable P-value
μ 0.858
σ2 0.65
Table 12: Case B P-values for significance
Figure 32: Comparison between PL2 and PLGRC with Cp=0.5
30 5.3. Edge Reinforcement (ER)
The next process examined is ER. The μ and σ for PL in both operations are shown in table 13 and figure 33.
Operation 𝑃𝐿̅̅̅̅ σPL
Grinding 0.002 0.016 ER -0.007 0.017 Table 13: Measured values for PLGR and PLER
The mean value shows that there is a significant difference, the variance shows the opposite.
For this experiment the ER values generated were taken from data provided by Sandvik Coromant and they are according to the grade the insert consists of. As in the grinding experiment, random values with the same mean and standard deviation for the clearance angle were generated. These values were input in equation (15). The values obtained were:
Operation ER
ERER -0.01 0.0017 Table 15: Material removal values
Variable P-value
μ 0.007
σ2 0.56
Table 14: P-values between PLGR and PLER
Figure 33: Measured values for PLGR and PLER
31 Afterwards, ΔER is added to PLGR providing:
ERC GR
PL
PL
ER Where PLERC is the Calculated PL for ERThe results are the following:
Variables PLER(mm) σER
PLER -0.007 0.0178 PLERC -0.008 0.0172 Table 16: Comparison between PLER and PLERC
Figure 35: Comparison between PLER and PLERC Figure 34: ΔER values
32
After implementing equation (15) the mean and variance show no significant difference thus (15) is capable of calculating the dimensional changes for ER.
5.4. Coating
The last operation is coating. The following average PL and σ, for ER and coating, are shown in table 18:
Operation PL σPL ER -0.004 0.012 Coating 0.011 0.014 Table 18: Measured PLER and PLcoating values
Figure 36: Measured PLER and PLcoating values
Table 19 show significant difference both in mean and variance. After generating random values for coat- ing, the values are input in equations (16) and (17).
Variables P-value
μ 0.68
σ2 0.56
Table 17: P-values between PLER and PLERC
Variable P-value
μ 0
σ2 0.4
Table 19:P-values between PLER and PLCoating
33 For the thickness:
C G
S
S
CoatingS
G(mm) σ
sgS
c(mm) σ
sc6.20 0.061 6.21 0.062
Table 20: SG and SC values
The reason for not multiplying coating by 2 is that when the measurement is performed, only the coating on the bottom surface is affecting the translation, see figure 18. Note that an increase in thickness leads to a decrease in PL since the insert translates down the slope of the fixture.
By using equation (16) the changes due to the coating applied are:
ω ̅(mm) σ
ω-0.01 0.001
Table 21: ω-values for Coating ω is showing the produced mean and variation from the changes in S.
Figure 37: ω-values for Coating
34 For the ICC:
C is multiplied by 2 because in the measurement, both bottom and upper surfaces of the clearance angle are contributing to the rise of the probe. Because D is unknown, PLER is used as the point where the insert was in the ER operation.
(2 )
C ER
D
PL
Coating𝐷
𝑐̅̅̅(mm) σ
Dc0.02 0.013
Table 22: Dc Values
Figure 38: Dc Values
Finally, for the Coating PL the following calculation was conducted:
coatc PROBE C
PL
D Where PLcoatcis the PL calculated for coatingThe measured and the calculated PL and standard deviation for coating are compared in table 23 and in figure 39.
Type PL σPL
PLCoating 0.011 0.014 PLcoatc 0.010 0.014
Table 23: Difference between PLCoating and PLcoatc
35
Figure 39: Difference between PLCoating and PLcoatc
After combining equation (16) into (14) and then adding (17) the PLcoatc was calculated and the results show no significant difference both in mean and variance.
As a conclusion, all the three derived equations are capable of describing the dimensional changes and calculating the mean and the standard deviation of each measurement in each process. Judging by the 2- sample variance tests, the least contributing processes are coating and ER respectively.
5.5. S-ICC relationship
Even though the previous equations can describe the processes, there is still one major piece missing to create the assembly function. This point is, the state before grinding, meaning what PL value existed before grinding. Because there were no measurements prior to grinding, there is no evidence how much the insert should be ground to produce a specific PL. This creates the necessity to develop a model for the S-ICC.
Since ICC remains constant after grinding, by calculating the theoretical grinding S value and using it into (13) the PLGRcan be calculated from the start.
Figure 40 shows how these dimensions must agree on a certain level. When the values match, they should produce a PL=0. For an insert with thickness S2 the D-value (D2) must have the appropriate length so that its cutting edge reaches the measuring probe (dashed line). For a thinner insert (S1), D1 must be smaller than D2 because the workpiece is moving up the slope. For a thicker insert (S3), D3 must be bigger than D2 because the workpiece is moving down the slope.
Variable P-value
μ 0.92
σ2 0.5
Table 24: P-values between PLCoating and PLcoatc
36 .
Figure 40:Illustration of S-ICC relationship
For a model to be developed, a reference point with specific S and ICC must be considered. The reference point taken to create this model is derived from measurements, from a specific batch, where there was a full picture on how much the ICC, S and PLGR were. The reason behind this decision is that for other measure- ments performed, there was no clear picture about how much the ICC was.
The key values from the measurement of the specific batch were:
Variable Values(mm) (≈)
ICC 18.05
S 6.24
PL 0
Table 25: Observed values
From this reference point, 3 different ICC’s will be calculated for 3 different S.
The first step is to transform the reference ICC into D by using:
D r M Where: r=radius of the circle (
2
ICC)In order to calculate the M for inserts with different dimensions, the ratio between M and r must first be found. By dividing nominal radius with the nominal M-value, found in the drawing, the ratio is determined:
9, 012
3, 508 2, 569
r
M
(18)
