ENGINEERING METROLOGY AND

ASSEMBLY

Autorské dílo v rámci projektu Rozvoj jazykových kompetencí

pracovníků VŠB-TUO: InterDV, registrační číslo projektu:

CZ.1.07/2.2.00/15.0132

autoři:

Ing. Lenka PETŘKOVSKÁ, Ph.D.

Ing. et Ing., Mgr. Jana PETRŮ, Ph.D.

Ostrava 2012

1

INTRODUCTION

In this study text we also used views and experiences of foreign participants The XXXIII.

International Conference on Aerospace, Mechanical, Automotive and Materials Engineering aims.

The aim of this conference was to bring together leading academic scientists, researchers and

scholars to exchange and share their experiences and research results about all aspects of

Aerospace, Mechanical, Automotive and Materials Engineering. Next aim is discuss the practical

challenges encountered and the solutions adopted. On the conference was presented paper with the

aim Analysis of Assembly Process of Semiautomatic Elevated Platforms for Loading and Unloading

Disabled Passengers. During the conference was discussed with experts from various country about

specific issues that can be in the planned scripts.

The project “Development of language skills of workers VSB-TU Ostrava: InterDV,

(registration number CZ.1.07/2.2.00/15.0132 project) enabled creating this educational support as

well as the participa tion on the international conference.

In this way, the authors would like to thank Czech leading administrators of this project.

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CONTENTS

1 Metrology introduction _____________________________________________ 5

1.1 The basic metrology terms ____________________________________________ 6

2 The National metrology system and the Metrology act ____________________ 8

2.1 The National metrology system ________________________________________ 8

2.2 The Metrology act ___________________________________________________ 8

2.3 The basic SI unit system ______________________________________________ 9

2.4 Metrology traceability and etalons ____________________________________ 12

3 Errors and measurement uncertainty _________________________________ 14

3.1 Measurement errors ________________________________________________ 15

3.1.1 Random errors ___________________________________________________________ 15

3.1.2 Normal distribution _______________________________________________________ 16

3.1.3 Systematic errors _________________________________________________________ 18

3.1.4 Gross errors _____________________________________________________________ 19

3.2 Uncertainty of measurement _________________________________________ 19

3.2.1 Types of uncertainties _____________________________________________________ 19

4 Measuring devices ________________________________________________ 25

5 Measurement of lengths ___________________________________________ 27

5.1 Gauge blocks ______________________________________________________ 27

5.2 Measuring devices for absolute length measuring ________________________ 29

5.2.1 Vernier Calliper __________________________________________________________ 29

5.2.2 Micrometric measuring devices _____________________________________________ 30

5.3 Fixed and limit measuring devices (Calliper gauges) _______________________ 32

5.4 Dial indicator ______________________________________________________ 34

6 Angles measurement ______________________________________________ 36

6.1 Measuring facilities _________________________________________________ 36

6.1.1 Angle gauges ____________________________________________________________ 36

6.1.2 Angles __________________________________________________________________ 37

6.1.3 Protractors ______________________________________________________________ 38

6.1.4 Water levels _____________________________________________________________ 39

6.1.5 Sine ruler _______________________________________________________________ 40

6.1.6 Profile projector __________________________________________________________ 41

7 Surface roughness control __________________________________________ 42

7.1 Height of roughness parameters ______________________________________ 44

7.2 The measurement of surface roughness ________________________________ 45

8 Thread gauging __________________________________________________ 47

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8.1 External thread gauging _____________________________________________ 47

8.1.1 Comprehensive gauging ___________________________________________________ 47

8.1.2 The pitch P gauging _______________________________________________________ 47

8.1.3 Mean diameter of thread gauging ___________________________________________ 48

8.1.4 Vertex angle thread gauging ________________________________________________ 49

8.2 Internal thread gauging _____________________________________________ 49

9 Gear gauging ____________________________________________________ 50

9.1 Some parameters of gear gauging _____________________________________ 50

9.1.1 Thickness of the teeth gauging ______________________________________________ 50

9.1.2 Dimension over teeth gauging ______________________________________________ 51

9.1.3 Peripheral run-out gauging _________________________________________________ 51

10 Introduction to ASSEMBLY ________________________________________ 52

10.1 Assembly and Its Significance for the Engineering Industry _______________ 52

10.2 Basic Terms in the Field of Assembly _________________________________ 52

10.3 Assembly Activities and Their Division _______________________________ 53

10.3.1 Definition of Assembly Activities ___________________________________________ 54

10.3.2 Division of Assembly Activities _____________________________________________ 54

10.3.3 Classification of Assembly Activities _________________________________________ 55

11 Assembly Elements and Systems ___________________________________ 58

11.1 Characteristics of Assembly Elements ________________________________ 58

11.2 Assembly Groups and Subgroups ____________________________________ 62

12 Technological Level of Product Design with Regard to Assembly _________ 63

12.1 The Term Construction Technological Solution _________________________ 63

12.2 Technological Level of Product Design with Regard to Assembly __________ 63

12.3 Influence of the Product Design and Technological Concept on the Assembly

Process 63

12.4 Indicators of Design Technology Level with Regard to Assembly ___________ 66

13 Analysis of Dimensional chains ____________________________________ 68

13.1 Dimensional Chain Basic Terminology ________________________________ 68

13.2 Division of Dimensional Chains _____________________________________ 68

13.3 Dimensional Chain Members _______________________________________ 69

13.4 Selected Types of Dimensional Chains ________________________________ 70

14 Calculation of dimensional chains __________________________________ 72

14.1 Basic Formulas for Calculating the Straight Liner Dimensional Networks ____ 72

14.1.1 Parallellinear Dimensional Networks ________________________________________ 72

14.1.2 Serial Linear Dimensional Networks _________________________________________ 73

14.1.3 Combined Linear Dimensional Networks _____________________________________ 73

4

14.2 Basic formulas for calculating plane dimensional networks _______________ 78

15 Assembly Methods ______________________________________________ 82

15.1 Division of Assembly Methods ______________________________________ 82

15.1.1 Method of Total Interchangeability of Parts __________________________________ 82

15.1.2 Method of Partial Interchangeability of Parts _________________________________ 84

16 Assembly Organization ___________________________________________ 90

16.1 Division of Assembly Types from the Organizational Point of View _________ 90

16.1.1 Concentrated Assembly __________________________________________________ 91

16.1.2 Divided Assembly________________________________________________________ 92

16.1.3 Stream Assembly ________________________________________________________ 93

16.1.4 Subject Assembly ________________________________________________________ 93

16.1.5 Line Assembly __________________________________________________________ 94

17 Assembly Lines _________________________________________________ 95

17.1 Types of Assembly Lines ___________________________________________ 95

17.2 Examples of Assembly Line Arrangements ____________________________ 96

18 Raviomalization of assembly _____________________________________ 101

18.1 Rationalization of Assembly Process ________________________________ 101

18.2 Accuracy of Manufacturing and Its Effect on Assembly Costs ____________ 102

References _________________________________________________________ 103

5

1 Metrology introduction

Metrology is a scientific and technical discipline that deals with all information and activities

regarding the measuring. It is also a core of uniform and precise measurement in all science areas

such as industry, government, defence, health and environmental protection activities. A uniform

and precise measurement is an assumption of mutual trust while tools exchange, but it is also one of

necessary requirements for effective production. The current trends lead to industry globalization

and required worldwide scale cooperation. Many parts and subparts are produced all over the world

and they have to be assembled into one functional unit in which all parts puzzle together. To meet

this requirement there is a necessity to use a uniform and precise measurement, global

communication systems, scientific and environment observation, etc. Respecting these facts there is

a high industry importance of measuring devices and metrology.

The product exchanges have always required common measurement units since civilization

beginning, as well as in the middle age, when there was a requisite to meet appropriate accuracy by

big constructions and land measurements and astrological phenomena observations. This has lead to

more or less consistent measurement units and to the measurement results unification.

Consequently it was evident that one area or state restriction was not enough and it was essential to

get a global solution. After the great French revolution the decimal measurement system was

enforced and the international agreement called Metric convention was signed (until now 51

countries have signed this agreement with the first member link by year 1875).

By a metric convention was accepted the unit system and its etalons, that is adjusted

according the knowledge development, but principally it helps to meet long-term and global

measurement unification with all positive accepts on international business and production process

cooperation scale. The organization structure for governments was set by the convention. It offers

the basement for measuring meetings that solve the currant issues. The international bureau of

weights and measures (BIPM) was established in Sevres close to Paris and it stores the international

unit etalons and operates as the international metrology centum. The Convention holds the general

conference on weights and measures (CGPM) and in between it leads the international committee

(CIPM). It accepted the modern measurement system form in 1960 by the Convention – The

international system of units SI.

Nowadays (apart from scientific and technical progress) the metrological system mainspring

are the political and industry changes based on economic opening towards the global market,

product movement release and business technical barriers elimination. A historical milestone was

the unification of the Metric convention and the sign process of mutual state etalons and certificates

recognition issued by metrological institutes (CIPM MRA) – apart from the Metric convention

member states, this advantage is also usedby 23 associated states and economies with the respect to

CGPM.

The General Conference on Weights and Measures (Conférence Générale des Poids et

Mesures, CGPM) is made up of delegates of the governments of the Member States which are

Argentina, Australia, Austria, Belgium, Brazil, Bulgaria, Cameroon, Canada, Chile, China, Croatia,

Czech Republic, Denmark, Dominican Republic, Egypt, Finland, France, Germany, Greece, Hungary,

India, Indonesia, Islamic, Republic of Iran, Ireland, Israel, Italy, Japan, Kazakhstan, Kenya, DPR of

Korea, Republic of Korea, Malaysia, Mexico, Netherlands, New Zealand, Norway, Pakistan, Poland,

6

Portugal, Romania, Russian Federation, Saudi Arabia, Serbia, Singapore, Slovakia, South Africa, Spain,

Sweden, Switzerland, Thailand, Turkey, United Kingdom, United States of America, Uruguay, Bolivia,

Rep. of Venezuela.

Observers from the Associates of the CGPM are Albania, Bangladesh, Belarus, Plurinat. State

of Bolivia, Bosnia and Herzegovina, CARICOM, Chinese Taipei, Costa Rica, Cuba, Ecuador, Estonia,

Georgia, Ghana, Hong Kong (China), Jamaica, Latvia, Lithuania, The FYR of Macedonia, Malta,

Mauritius, Rep. of Moldova, Montenegro, Panama, Paraguay, Peru, Philippines, Seychelles, Slovenia,

Sri Lanka, Tunisia, Ukraine, Viet Nam, Zambia, Zimbabwe.

There is an agreement by each more difficult technology, what is not possible to change, is

also not possible to produce (Lord Kelvin) and without more and more demanding experiments is not

possible to get new scientific knowledge. Obviously the technical demand on measurement

equipment inserts it to the most advanced technologies. The present metrology are not common the

officers with stamps and cabinets full of historical devices, but top measuring devices and

metrologists cooperate with universities and are orientated towards the modern industry demand.

The metrology has a cross cutting character and is used in all areas. The preset metrology is

characterised by:

dynamic technical measuring devices development, close relationship with progress in

physics and the utilization of metrology in new areas such as chemistry and biology,

application of electronics, computer technology and metrology and the increased level

in data communication (e-distance calibration),

implementation of basic unit standards based on the quantum phenomena and

universal physical constants (apart from units such as kg, ampere, mol and kelvin

realized by all basic units of SI)

there is a measuring results acceptance trend and tests on the international scale

recogisation trend that also utilizes, apart from the technical competence, mutual trust

strengths based on quality systems, accreditation, certification (one – stop testing).

1.1 The basic metrology terms

All term and symbols in the whole metrology are normalized by the Technical Normalized

Information TNI 01 0115 - the international metrology dictionary – The Basic and Additional terms

(VIM). The dictionary is effective since February 2009 and was introduced with the respect of ISO/IEC

directive. This norm is the terms dictionary that includes specific discipline identification and

definition.

Some terms can be defined:

Metrology (VIM 2.2) – science of measurement and its application

when in the note of this definition is introduced that the metrology includes all theoretical

and practical aspects of measurement, whatever the measurement uncertainty and field of

application.

7

Measurement (VIM 2.1) – process of experimentally obtaining one or more quantity values

that can reasonably be attributed to a quantity.

Measurand (VIM 2.3) – quantity intended to be measured.

Measurement principle (principle of measurement) (VIM 2.4) – phenomenon serving as the

basis of measurement.

Measurement method (method of measurement) (VIM 2.5) – generic description of a logical

organization of operations used in measurement.

Measurement procedure (VIM 2.6) – detail description f a measurement according to more

or more measurement principles and to a given measurement method, based on a measurement

model and including any calculation to obtain a measurement result.

Measurement result (result of measurement) (VIM 2.9) – set of quantity values being

attributed to a measurand together with any other available relevant information.

when in the note of this definition is introduced that it is the total data of data test including

information about uncertainty measurement.

Quantity (VIM 1.1) – property of a phenomenon, body, or substance, where the property has

a magnitude that can be expressed as a number and a reference.

Base quantity (VIM 1.4) – quantity in a conventionally chosen subset of a given system of

quantities, where no subset quantity can be expressed in terms of the others.

Derived quantity (VIM 1.5) – quantity, in a system of quantities, defined in terms of base

quantities of that system.

Quantity value (value of quantity, value) (VIM 1.19) – number and reference together

expressing magnitude of a quantity. For example, length of a given rod 5.34 m or 534 cm.

Conventional quantity value (conventional value of a quantity, conventional value) (VIM

2.12) – quantity value attributed by agreement to a quantity for a give purpose. For example,

standard acceleration of free fall gn = 9.806 65 m·s-2.

Measurement unit (unit of measurement, unit) (VIM 1.10) – real scalar quantity, defined and

adopted by convention, with which any other quantity of the same kind can be compared to express

the ratio of the two quantities as a number.

Derived unit (VIM 1.11) – measurement unit for a derived quantity.

Only basic terms dealing with common metrology are specified. The next notions, concerning

concrete problems can be mentioned in other chapters.

8

2 The National metrology system and the Metrology act

2.1 The National metrology system

The national metrology system is a scheme of technical tools, devices and technical

employees, legal and technical regulations that define the status and government subject inter-

relationships and the legal persons in charge of all different activities while assuring the

measurement tools and process of unification in the Czech Republic. It has a significant importance

and it is mainly:

domestic industry support (according to the statistics, cca. 50% of GDP depends on

advanced technologies development where the metrology has the primary role.

business technical barriers elimination and this is mainly on the international

measurement result and tests acceptance.

The system could be divided into the following areas:

fundamental metrology (FM), it deals with the system of units and physical constants,

state etalon storage and development, units exchange on lower levels and with science

and metrology research,

Industrial metrology (PM), it helps to assure the measurement unification as well as

accuracy in order to meet the production and services quality in a broad area range

(unregulated metrology area),

Legal metrology (LM), it helps to assure the measurement unification and accuracy in

the regulated area according to effective legal treatments.

2.2 The Metrology act

The basic building stone of the Czech metrology legislative is the Metrology act and its

implemented regulations. The metrology act is the act S.F. No. 505/1990.

The purpose of the Metrology act is the rights and obligation adjustment for individuals, who

are entrepreneurs, legal entities, governmental and other authorities and that all to the extent of

unification, correct tools and measurement assurance.

Gauges partition according to the Metrology act. Gauges help to determine a certain

measured value. Together with other auxiliary measuring devices for the purpose of this Act they are

devided into:

Etalons – these are measuring devices that help to implement and secure these units or

scales and they help their transmission on the measuring devices with the lower level of

accuracy.

Set working gauges (more often is used the term set gauges) – these are measuring

devices that the Ministry of Industry and Trade appointed to the obligatory verification

with respect to their importance.

9

Non set working gauges (more often is used the term working gauges) – theye are

devices that are not etalons nor set gauges.

Certified reference materials and others reference materials – these are materials or

substances with a well- defined composition or characteristics used mainly for

measuring tools verification or calibration, measuring methods evaluation and

quantitative material characters definition.

2.3 The basic SI unit system

Among common metrologist’s skills should belong cross-sectional knowledge such as the

basic SI unit system realization and their relationship to the fundamental nature constants. The SI

unit system (Système International d‘Unités) was officially accepted on the 11th General conference

of weights and measures in 1960. This system consists of 7 basic units and derived SI units and it

represents a coherent system. It means the units are linked together by multiplication and derivation

operations with a numerical factor that equals 1. Out of these 2 groups of units are derived their

multiples and semi parts by appropriate prefixes. The only exception is the mass unit (kg). A dot (in

English) or a comma (in French, Czech) is used as the decimal symbol with respect to national

language. The measurement points are by our state set by the Metrology act. These are the basic SI

units and other units out of the SI system. It is necessary to mention that all countries with imperial

system of units (all Anglo – Saxon countries) at the moment are under the transformation on the SI

unit system (the Great Britain has an exception until 2010).

Table 1– SI base units

Measure Name Symbol

Length meter m

Mass kilogram kg

Time second s

Electric current ampere A

Thermodynamic temperature kelvin K

Amount of substance mole mol

Luminous intensity candela cd

Further the SI system includes the SI derived units, among which (two) additional units – the

unit of angle and solid angle - belong

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Table 2 – SI derivated units

Derived unit

Derived SI unit

Name Symbol Expression by SI base and

derived units

Angle radian rad 1 rad = 1 m/m = 1

Solid angle steradian sr 1 sr = 1 m2/m2 = 1

Frequency hertz Hz 1 Hz = 1 s-1

Force, Weight newton N 1 N = 1 kg.m/s2

Pressure, Stress pascal Pa 1 Pa = 1 N/m2

Energy, work, heat joule J 1 J = 1 N.m

Voltage, electrical potencial diffrence, electromotive force

volt V 1 v = 1 W/A

Electric capacitance farad F 1 F = 1 C/V

Electric resistance ohm Ω 1 Ω = 1 V/A

Electric conductance siemens S 1 S = 1 Ω-1

Magnetic flux weber Wb 1 Wb = 1 V.s

Magnetic field strength tesla T 1 T = 1 W/m2

Inductance henry H 1 H = 1 Wb/A

Degree Celsius Degree Celsius 1) °C 1 °C = 1 K

Luminous flux lumen lm 1 lm = 1 cd.sr

Luminance lux lx 1 lx = 1 lm/m2

Radioactivity becquerel Bq 1 Bq = 1 s-1

Adsorbed dose of ionizing radiation gray Gy 1 Gy = 1 J/kg

Equivalent dose of ionizing radiation sievert Sv 1 Sv = 1 J/kg

1) Celsius Degree is a special term for the unit Kelvin with indication Celsius degree.

In general, there has already been some trend defining the basic SI units in metrology

according to basic natural constants during some decades – this has an influence on their time and

place realization. Each unit has an additional, more detailed fact dedicated:

Time - second

A second is the duration of 9 192 631 770 periods of the radiation corresponding to the

transition between the two hyperfine levels of the ground state of the cesium-133 atom (13th CGPM

(1967). The caesium is supposed to by the absolute zero temperature and completely without outer

influence.

Length - meter

1 meter (m) is defined as the length of the path travelled by light in vacuum in 1/299792458

of a second (1983).

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Weight – kilogram

1 kilogram (kg) equals to the international prototype of kilogram (1889).

The international prototype of kilogram is produced by platinum – iridium alloy and is stored

in exactly set conditions in Sèvres in Paris.

Electric current – ampere

1 ampere (A) this force is used in the formal definition of the ampere, which states that it is

the constant current that will produce an attractive force of 2 × 10–7 Newton per meter of length

between two straight, parallel conductors of an infinite length and a negligible circular cross section

placed one meter apart in vacuum.

Thermodynamic temperature - Kelvin

1 Kelvin (K) equals to 1/273.16 of thermodynamic temperature of triple point of water

(1967).

Amount of substance – mole

1 mole (mol) is the amount of substance that consists of many elementary entities that are in

0.012 carbon 126C.

Luminous intensity - candela

1 candela (cd) is the luminous intensity, in a given direction of a source that emits monochromatic radiation of frequency 540.1012 hertz and that has a radian intensity in that direction of 1/683 watt per steradian.

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Table 3 – Names and symbols of multiple and partial SI prefixes

10n Prefix

Name Symbol Name base

1024 yotta Y

1021 zetta Z

1018 exa E

1015 peta P

1012 tera T teras (Greek) – heavenly sign

109 giga G gigas (Greek) – giant

106 mega M megas (Greek) – big

103 kilo k chilios (Greek) – thousand

102 hecto h hekat o (Greek) – hundred

10 deca da dekas (Greek) – ten

10-1 deci d decem (Latin) – ten

10-2 centi c centum (Latin) – hundred

10-3 mili m mille (Latin) – thousand

10-6 micro μ mikros (Greek) – small

10-9 nano n nan o (Italian) – dwarf

10-12 pico p piccol o (Italian) – very small

10-15 femto f femton (Swedish) – fifteen

10-18 atto a atton (Swedish) – eighteen

10-21 zepto z

10-24 yocto y

2.4 Metrology traceability and etalons

Traceability

It is a feature of a measurement result or etalon value that could define the relationship to

appropriate references, mainly the state or international etalon true the uninterrupted chain of

comparisons (the follow up chain) that values are known.

Calibration

The basic mean by the follow-up measurement assurance is the low-end etalon calibration

and devices. This calibration shows the metrological features of calibrated machine.

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By etalon

is understood the material degree, machine, device or system that is appointed to define,

realize, conserve or reproduce units or one or many values and it serves as a model, reference (e.g.

1kg prototype, etalon resistor, caesium clock)

Example: 1 meter (m) is defined as the length of the path travelled by light in vacuum in

1/299792458 of a second (1983). One meter is realized on the primary level of wavelength using

helium – neon – iodine stabilized laser. On the lower level were used material extents, such as basic

gauges and the follow up process is assured by the optical interferometrie for the basic gauges length

set up in connection with the above-mentioned wavelength laser light.

Primary etalon

is an etalon understood as the one with the highest level of metrology quality and its value is

accepted without any other continuity to other etalons the same variable.

International etalon

by an agreement, this is an etalon that on the international scale helps to set the values of

other etalons with appropriate variables (e.g. BIMP etalons). At the moment there is only one such

etalon – the weight unit etalon. This etalon serves more states and can be stored in one of them –

subsequently the traceability agreement is agreed.

State etalon

is an etalon, agreed to be the basic of value set up for other etalons of appropriate variables

in the state.

Note: State etalons, mainly on the highest technical level in the country, are the resource of

metrological traceability for most of the measurements for suitable variable.

BIPM (International Bureau of Weights and Measures) Primary laboratories (National metrology institutions in the most countries) Accredited laboratories Companies Bottom end users

Figure 1 – Traceability chain

Unit definition, international etalons

Foreign state etalons Domestic state etalons

Reference etalons

Company etalons

Gauges

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3 Errors and measurement uncertainty

Measurement error (error of measurement, error) (VIM 2.16) – measured quantity value

minus a reference quantity value.

Systematic measurement error (systematic error of measurement, systematic error) (VIM

2.17) – component of a measurement error that in replicate measurements remains constant or

varies in a predictable manner.

Random measurement error (random error of measurement, random error) (VIM 2.19) –

component of measurement that in replicate measurement varies is an unpredictable manner.

Repeatability condition of measurement (repeatability condition) (VIM 2.20) – condition of

measurement, out of a set of conditions that includes the same measurement procedure, same

operators, same measuring system, same operating conditions and same location, and replicate

measurements on the same or similar objects over a short period of time.

Measurement repeatability (repeatability) (VIM 2.21) – measurement precision under a set

of repeatability conditions of measurement.

Reproducibility condition of measurement (reproducibility condition) (VIM 2.24) – condition

of measurement out of a set of conditions that includes different locations, operators, measuring

systems, and replicate measurements on the same or similar objects.

Measurement reproducibility (reproducibility) (VIM 2.25) – measurement precision under

reproducibility conditions of measurement.

Measurement uncertainty (uncertainty of measurement, uncertainty) (VIM 2.26) – non-

negative parameter characterizing dispersion of the quantity values being attributed to a measurand,

based on the information used

when in the note of this definition is introduced, that herewith parameter can be for example

a standard deviation (or its given multiple).

Type A evaluation of measurement uncertainty (Type A evaluation) (VIM 2.28) – evaluation

of a component of measurement uncertainty by a statistical analysis of the measured quantity values

obtained under defined measurement conditions.

Type B evaluation of measurement uncertainty (Type B evaluation) (VIM 2.29) – evaluation

of a component of measurement uncertainty determined by means other than a Type A evaluation

of measurement uncertainty.

Standard measurement uncertainty (standard uncertainty of measurement, standard

uncertainty) (VIM 2.30) – measurement uncertainty expressed as a standard deviation.

Combined standard measurement uncertainty (combined standard uncertainty) (VIM 2.31)

– standard measurement uncertainty that is obtained using the individual standard measurement

uncertainties associated with the input quantities in the measurement model.

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Measurement accuracy (accuracy of measurement, accuracy) (VIM 2.13) – closeness of

agreement between the measured quantity value and the true quantity value of the measurand.

3.1 Measurement errors

Measurement methods and our senses imperfection, limited accuracy of measurement

devices, variable measurement circumstances and other influences cause that the measurement can

not set the real value of physical variable x. The difference between the real and measured value is

called the absolute error of measurement. This error has two components – systematic and random.

According to the error core, errors are divided into three groups.

Systematic errors are caused by the use of inappropriate or less suitable measurement

method, imprecise gauge or measuring machine or by observer’s error. These errors distort the

numerical measurement result in completely regular manners. The result is increased or decreased

by certain circumstances, regardless the number of repeated measurements. Errors are difficult to

detect and often are detected by the result comparison with other device output data.

Examples: By the weight measurement in the air according to the Archimedes law the object

weight is always lower than the real object weight and density is lower than the density of weight.

A systematic error has occurred by neglecting the air buoyancy and it can be removed by

appropriate correction (vacuum correction). While measuring voltage by voltmeter, the voltage is

always lower than the real one due to the fact that the voltmeter has got unlimited large internal

resistance. A systematic error has its origin in equipment construction and it can be removed by the

use of device with higher internal resistance.

Due to the fact we are able to define systematic error roots, we can estimate their value and

character and by their effect evaluation on measurement results and thus we can eliminate them (by

the help of a suitable correction).

Random errors that balance accidentally in values and signs by repeated measurements are

influenced by a big number of random non-predictable effects.

Random errors are described in a certain probability distribution.

Systematic errors influence the accuracy and randomly the result precision.

Gross errors (marked as outliers or out of tolerance values) are caused by an exceptional

cause, incorrect typing result, sudden measuring machine break down or inaccuracy, wrong

measuring condition set up, etc. The measured value varies by repeated measuring activities. This

measurement result should be eliminated out of the values evaluation in order to prevent the final

result distortion.

3.1.1 Random errors

Unlike the gross and systematic errors that can be corrected by the right measurement

method, precise devices and measuring process, random errors are present during each measuring

activity and they cannot be influenced. The measuring accuracy depends on the measurement

circumstances.

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Uncontrolled factors that are changed by repeated measurement randomly and

independently of the controlled factors influence are the cause of random errors. The measurement

result is the variable xi that differs to the real variable x. Their difference is the measurement error εi

εi = xi - x0 .

The error εi can never be defined, it can be only estimated.

The error determinated as the difference of measured and actual variables is called absolute

error. It is expressed in the units of variables. The relative error, defined by the relation

ir ,i

0x

is the dimensionless quantity, often expressed in %.

Random errors that balance accidentally in values and signs by repeated measurements are

influenced by a big number of random non-predictable effects. Random errors act as random

variables and are controlled by mathematical rules of probability. By an extended amount of

repeated measurements, the statistical rules can be used in order to estimate random errors on

measurement accuracy.

3.1.2 Normal distribution

Suppose a corrected influence on systematic errors.

Taking into account the frequency of the measured value and the fact that their influence of

frequency and variable value is recorded into the graph, then it is found out that by the big number

of measurement n → ∞ (the basic set), the curve will be smooth and the distribution of the

measured values will be perfectly symmetrical. The real value of quantity x0 refers to the maximum

curve (Figure 3.1). The normal (called Gaussian) distribution is based on the following assumption:

a) The final error of each measurement is the result of a big amount of very small a not

linked together errors,

b) There is a same probability of positive and negative deviation.

The function of normal distribution is mainly states in formula

2

0

2

( x x )1( x ) exp

22

where: σ 2 - variance,

σ – standard deviation (mean measured deviation value x out of the real value of

quantity x0),

x – value of quantity out of undefined series of conducted measurements,

(x) – the probability density of variable values x.

17

Figure 2 – Gaussian distribution

By the help of function (x), it is possible to define the probability according to the fact that

the measured value of quantity was in the defined interval (Figure 3.2). Provided:

0

0

x 2

0

2

x

( x x )1exp dx 0,683,

22

then the probability that the measured value of quantity is in the interval x0−σ, x0+σ, is

68.3 %. In the interval x0 ± 2σ it is 95 %, out of the interval x0 ± 3σ it will be only 3 ppt of values of

quantity.

Figure 3 – Probability intervals

In the set with the defined number of measurements (selected set), only the most probable

values of quantity can be discussed, this values will be very close to the real values of quantity. For

the most accurate estimation of the real value of quantity x0 is used the arithmetical mean x out of

n measured values x1, x2,….. xn.

n

i

i 1

1x x

n

18

where: n – number of measurement,

xi – value of the measured values of quantity (i = 1,2,…….n).

If the number of measurements is increased, the value of arithmetical mean approximates

the real value (Figure 3.3). Despite this, it is not possible to reach an arbitrarily accurate result.

The measure of variance in the population is the standard deviation σ. The values variance of

selected characters is defined by the selected standard deviation out of one measurement:

n 2

i

í 1

x x

sn 1

.

The accuracy of measurement is increased by an increasing number of n measurements.

Therefore the repeated measurement is used the standard deviation of arithmetical (selected) mean

s and it is influenced by the difference of x0 and x (see Figure 4).

Figure 4 – The number of measurements

The full line curve shows the x values distribution around the real value x0, while the dashed

curves show the distribution of the measured values around the arithmetical mean. On the figure 3.3

is shown that with the increasing n the value of arithmetical mean is getting closer to the real value

x0. The selected standard deviation of the arithmetical mean could be calculated from the

relationship

n 2

i

i 1

x x

sn n 1

.

3.1.3 Systematic errors

By the repeated systematic errors measurement under the same measuring circumstances,

the value of the measured variable always distorts in the same way. In order to eliminate it, more

accurate measuring machines should be used or there is a need to use corrections. It is not possible

to reach this by laboratory practice. Therefore systematic errors estimation will be provided so that

19

the maximal error would always be higher or equal to the accepted error. Errors involved on

systematic error are caused by the measuring device limited accuracy, measuring method error and

measuring device operator error.

3.1.4 Gross errors

The cause of gross errors is a wrongly conducted measurement, incorrect data reading,

incorrect data processing method, device error, incorrect device manipulation, etc. The measuring

result influence of the gross error cannot be calculated. It has to be excluded out of the measured

database and the measurement cannot be conducted until the gross error source is not defined and

executed. In many cases this can be provided only after some suspected measured data testing and

there is a possibility that our decision about exclusion (non-exclusion) of the suspected data out of

the database can be incorrect.

The suspected values testing (values influenced by gross error) is possible to conduct under

the circumstances of normal probability of density distribution.

3.2 Uncertainty of measurement

The basic parameter of a measuring result is the uncertainty of measurement. Uncertainty of

measurement is based on the quantity of all measurement errors (random and systematic) that can

significantly change and essentially define the measurement result and the interval is supposed to be

the one for measuring results. Uncertainty of measurement is a fundamental part of each

measurement result and it is very important in the cases when the measurement results are related

to some limited value. The precise measurement uncertainty evaluation is a necessary part of correct

laboratory practice and it provides very valuable information about the quality and measurement

reliability or laboratory testing quality and their customers.

The measurement result expression is complete by containing its own measurement value of

a variable as well as uncertainty of measurement belonging to this value. The results of

measurement include uncertainty of measurement and are defined as:

UyY ,

where Y is the measured value, y is the measured value estimation and U is the extended

uncertainty of measurement defined in the same units.

3.2.1 Types of uncertainties

Four types of basic uncertainties are present in the practice:

Standard uncertainty - Type A: among the components of uncertainty type A belong the

components based on the statistical evaluation of repeated measurements. The components of

uncertainty - type A are characterized as a variance estimation and standard deviation based on

repeated measurement.

Standard uncertainty - Type B: among the components of uncertainty - type B belong the

components evaluated by other ways except the statistical methods. The component of uncertainty -

20

type B is determinate by the probability density function attributed to the measured value based on

the previous skills and available information.

Combined standard uncertainty: it is very rare in practice to use only type A or B evaluation

of standard uncertainty. Then it is necessary to set the resulting effect of combined uncertainties by

both types A and also B. Final combined uncertainty is determined according to the spread

uncertainty rules.

Extended uncertainty: defines the interval around the measurement result where the

measurement result will be found under certain required level of confidence. Extended uncertainty is

composed of combined standard uncertainty and multiplied according to the coefficient of coverage

that depends on requested coefficient of confidence and effective number of output value

measurement of freedom.

Uncertainty sources

All phenomena that can influence the uncertainty of clear measurement result could be set

as the uncertainty sources. They cause that the measured value differs from the real value.

Uncertainties are influenced by measuring machine choice (analogue or digital), type of filter used,

samples and other tools on the whole signal transport chain. Interference environment significantly

influences the measurement uncertainties.

The most often uncertainty sources cover:

imperfect or incomplete definition of the measured variables or their realization,

inappropriate measuring device choice (distinguish ability etc.),

inappropriate (no representative) choice of the measured samples,

incorrect measuring method,

rounding contrast out of the assumed values,

linearization, approximations, interpolation or evaluating extrapolation,

unknown or uncompensated environment influences,

measuring condition variability,

subjective operative influences,

measurement standards and reference materials inaccuracy.

Some of the uncertainty sources are shown exclusively or significantly by uncertainties - type

A and other are more significant by uncertainties - type B. However many sources may be the causes

of both types of uncertainties and this is the biggest danger in order not to fail one of the

component. It can cause very considerable distorting effect.

21

Uncertainty determination procedures

A general methodical procedure of uncertainty measurement determination can be

explained in some steps and its scheme is visible in the following Figure. This procedure is possible to

adjust according to the specifics of the concrete solved issue.

Figure 5 – The scheme of measurement uncertainty

Standard uncertainty - Type A:

By the repeated variable measurement X, we can get n data of x1, x2, …, xn. In this case,

uncertainty – type A is determined as the selected standard deviation of the specified mean:

n

i

ixAx xxnn

su1

2)()1(

1,

where the specified mean x is:

n

i

ixn

x1

1.

Standard uncertainty - Type B

1. Possible sources of uncertainty - type B Z1, Z2, …, Zm are determined:

Imperfections are uncertainty sources in a measurement process:

usage of already used tools, mainly ranges, measuring instruments and transducers,

measurement method usage,

22

measurement circumstances, mainly the values that influence the variables,

used constants for evaluation,

used relationships for evaluation.

2. Standard uncertainty - type B determination for each source:

Uncertainty values can be overtaken from technical documentation: certificates,

calibration sheets, technical standards, measuring machine technical data, technical

tables and physical constant tables.

By estimating the standard uncertainty estimation: the possible range of deviation is set

firstly zmax out of the nominal value of the uncertainty source whoseexceeding is

hardly possible. In the same interval, the possibility of deviation probability is evaluated

and the most suitable approximation is determined. Type B standard uncertainty is

linked to this source and it can be defined as:

maxz

uBz ,

where the value of quantity is overtaken from the chosen approximation table. Values of

quantity correspond to the ratio

maxjz, where 2 is variance of the distribution.

Figure 6 – Types of distribution by the determination of standard uncertainty - Type B

23

3. Conversion of uncertainties uzj on the corresponding components of the measured

value of uncertainty.

4. Merging the individual uncertainties into the resulting standard uncertainty – Type B

Combined uncertainty

Both types of uncertainty result can be merged into a combined uncertainty:

22

ByAyCy uuu ,

where uAy is the uncertainty - Type A and uBy is the uncertainty - Type B

Extended uncertainty

The extended uncertainty is defined as follows:

CyukU ,

where k is the coefficient of expansion and uCy je standard measurement uncertainty.

In the examples, where the normal (Gaussian) distribution of the measured value of quantity

is taken into the account and where the standard estimated uncertainty y is defined by sufficient

reliability, it is necessary to use the coefficient of expansion k = 2. This extended uncertainty refers to

the probability of approx. 95%.

Solved examples

Measurement of a cylinder diameter by a Vernier calliper

The task is to determine uncertainties of the measurement of a cylinder diameter by a

Vernier calliper. The measurement is conducted 10 times under the same conditions.

Measured values:

No. of measurement

1 2 3 4 5 6 7 8 9 10

di [mm]

80.1 80.2 80.1 79.9 80.0 80.2 80.1 79.9 80.0 80.1

Type A standard uncertainty:

First, the value extermination is determinated and it equals to the same selected diameter of

the measured values of quantity.

10

1

180.06

10i

i

d d mm

.

The type A standard uncertainty equals the selected standard deviation of the selected

diameter:

24

10

2

1

1( ) ( ) 0.034

10 (10 1)A i

i

u d d d mm

.

Type B standard uncertainty:

The type B standard uncertainty is determined by two components: measurement device

error and operational error. Both errors presume a uniform rectangular distribution.

An uncertainty caused by a Vernier calliper:

In the certificate is stated that the Vernier calliper in the measured interval (0 to 150 mm) is

defined by an error of 0.05 mm.

An uncertainty for type B of this source is:

max1

0.05( ) 0.029

3 3B

zu d mm .

Uncertainty caused by operator:

This uncertainty includes the imperfection of perpendicular setting of the gauge towards the

axis of the cylinder, etc.All is then added into the final error with the value of 0.1 mm.

The type B uncertainty of this source is:

max2

0.1( ) 0.058

3 3B

zu d mm .

The final type B standard deviation is:

For the final uncertainty - type B standard the Gaussian rule of uncertainty propagation is

taken into the account:

2 2 2 2

1 2( ) ( ) ( ) 0.029 0.058 0.065B B Bu d u d u d mm .

Combined uncertainty:

It is gained by merging type A and type B uncertainty:

2 2 2 2( ) ( ) ( ) 0.034 0.065 0.073C A Bu d u d u d mm

Extended uncertainty:

The extended uncertainty is defined by the formula:

( ) ( ) 2 0.073 0.146CU d k u d mm .

The result of repeated measurement of cylinder diameter:

(80.06 0.15)d mm

25

4 Measuring devices

In general, among the measuring devices belong:

measuring devices,

transducers,

auxiliary measuring devices,

reference materials.

According to TNI 0115 (The international dictionary of basic and general metrology terms),

the whole scale of terms can be defined.

Measuring device (measuring machine) (VIM 3.1) – this device is used for a separate

measurement or in connection to another one or more additional measurements.

Measuring system (VIM 3.2) – it is a set of one or more measuring devices and very often

other devices, including activators and resources, that is assembled and adjusted for information

gethering. This is used to generatethe measured values of quantity in the specified interval for the

specific type value.

Embodied rate (VIM 3.6) – is the measuring device that is reproduced or permanently

provides values of one or more types and each of them has an assigned value of a variable.

Measuring transducer (VIM 3.7) – it the device used for measurement that provides the

output variable and has a specified relation to the input variable.

Measuring chain (VIM 3.10) – are many elements of the measuring system that define the

only way for the measuring signal from the receiver to the output.

A measuring device might be defined in many ways:

1. According its usage:

operating – it a devices used for laboratory, production and other measurements

etalons – are used for unit realization, storage and reproduction. It might be material

rates, measuring devices or measuring systems (e.g. Etalon – weight 1 kg, ammeter

etalon, etalon hydrogen electrode).

2. According to the data recording form:

With a displayed function – e.g. Pointer ammeter, Vernier calliper,

With a recording function - recording spectrometer, seismograph.

3. According to the character of recorded data:

analogue – the measured data is a continuous function of the measured variable,

26

digital (analogue) – the measured data is in the value form (numbers).

4. According to the measured values of quantity with special terms:

with a variable name and suffix metr – tachometer,

with a unit name and suffix metr – ammeter,

with a variable name and suffix – tachometr,

with a measured environment name and suffix – gasmetr,

others – stopwatch, scale

5. According to the measured contact surface:

contact – there is a direct contact with the measured surface.

non-contact – there is not a direct contact with the measured surface.

Etalons have a specific place among all measuring devices. It might be an embodied rate,

measuring device, reference material or measuring system. Its task is to define, realize and

reproduce or store units of physical values or multiple the values of these variable less accurate

gauges. Only a measuring device that meets specific measuring requirements can be called an etalon.

The main etalons cover:

physical phenomenon or features for a certain variable reproduction, it must be well

known and supported by a precise processing theory and an experiment,

time stability,

low dependence on the affecting values of quantity,

transformation technique on other measuring devices must be physically realized.

The measuring device with an etalon function must meet the following requirements:

it used mainly for the value of quantity reproduction, storage and transformation on

the measuring devices,

it has a required technical and metrological features and it is equipped with an

appropriate documentation,

it is calibrated, tested and checked by a particular metrology organization based on the

provided tests and measurements. This calibration, testing and checking is repeated in

certain intervals,

it is registered as an etalon,

it used by a certain way and appointed persons,

it is stored under certain conditions on a predefined place.

27

5 Measurement of lengths

Length measurement is one of the most frequently used operations in metrological

engineering companies, where they represent up to 70% of all measurements.

5.1 Gauge blocks

Gauge blocks were introduced in measurement for the first time by a Swedish scientist C. E.

Johanson. Gauge blocks realize concrete length as distance of grinding and lapping end surface. The

most often used gauge blocks in the shape of a cross 9 x 30 mm to nominal length 10.5 mm and 9 x

35 mm over 10.5 mm of length. The measurement surfaces are machined with a very high

dimensional precision, surface roughness and flatness.

Gauge blocks (Figure 8) are produced in sets with graded sizes. To assemble an exact

proportion, gauge blocks assemble together and create a block of gauge blocks. Gauge block are

connected by insertion of one cleansed gauge block on the second and adhesive force retains thaem

together thanks to high flatness and low roughness. This process is called suction. Two connected

gauge blocks can withstand power up to 1000 N. When assembling gauge blocks, there is a rule that

to compile the dimension we use the minimum of gauge blocks. Assembled gauge blocks should be

composed only for the time necessary. They are disassembled similarly to assembly – moving

towards each other. They must not be unstuck because the surface of gauge blocks can be degraded.

Figure 9 shows description of a gauge block.

Figure 7– Set of gauge blocks

28

Figure 8 – Progress at suction of gauge blocks

Figure 9 – Description of a gauge block

where: MA – mark,

MF – measuring surface,

SF – side face.

The material of gauge blocks must be highly wear resistant, very hard, immune against rust,

(metal gauge blocks must be conserved), with a small factor of expansion and with good adherence.

The most used material is steel 19 422, wolfram carbide or oxide ceramic.

Gauge blocks are the most often used etalons of length. Accuracy of gauge block is divided

into 4 categories:

K – calibration,

0 – etalon,

1 – used as a working gauge or etalon,

2 – workshop, for checking of callipers, micrometers or comparative measurement.

29

5.2 Measuring devices for absolute length measuring

Measuring devices and gauges for length measuring is possible to divide into three groups:

one coordinate measuring technique,

more coordinate measuring technique,

multidimensional measurement technique.

5.2.1 Vernier Calliper

Vernier Calliper (Figure 10) is a simple hand measuring device for determination of length

proportions of components. It is possible to measure internal and external dimensions, depth or

recess. A standard calliper works on the principle of vernier.

Tenth (1/10) vernier is a scale long 9 mm or 19 mm divided into 10 equal pieces. A vernier

calliper with tenth vernier has the value of the smallest interval 0.1 mm. Vernier calliper with

twentieth vernier has accuracy 0.05 mm and fiftieth vernier has accuracy 0.02 mm. Since the eighties

of the 20th century, digital vernier calliper (Figure 11) started to be used.

Figure 10 - Universal Vernier Calliper

where: 1 - measuring surface for external

dimensions,

2 - blade surfaces for internal

dimensions,

3 - measurement surface to measure

depth,

4 - major scale,

5 – vernier,

6 - solid measuring arm,

7 - movable measuring arm,

8 - sliding part,

9 - square,

10 - stick to measure depth ,

11 – securing against movement.

30

Figure 11 – Digital calliper

Figure 12 – Measurement of external and internal dimension and depth

5.2.2 Micrometric measuring devices

Micrometric measuring devices are of more kinds. They are about one order more accurate

than vernier callipers. The basic part of all micrometric measuring devices is a micrometrical bolt with

lead 0.5 mm or 1 mm and length 25 mm. Longer bolts are not made for production reasons

(compliance with accurate lead) and from practical reasons (time required for measurement).

Micrometric measuring devices range: 0 up to 25, 25 up to 50, 50 up to 75 (mm), etc.

A classic example is the U-micrometric measuring devices.

Figure 13 – U – micrometric measuring device

The value of 1 run scale is 0.01 mm. A scale with vernier and with accuracy of 0.001 mm is

produced exceptionally.

31

Micrometric measuring devices with digital scale have accuracy 0.001 mm. These

micrometric measuring devices are also possible to be connected to a net of data collection for

further processing.

Figure 14 – Digital micrometric measuring device

Figure 15 – Micrometric measuring device for internal dimensions

U-micrometric measuring devices with an accurate drift indicator (micro indicating snap

gauge) can be used for setting the nominal value as a comparative gauge.

Figure 16 – Micro indicating snap measuring device

A micrometric depth measuring device is a modification of a U-micrometric measuring

device. It works on the same principle – micrometrical bolt. This gauge is intended for special depth

measuring of holes and grooves. They are delivered in a set with exchangeable extensions to make

measurement of big depths and diameter possible too. They can also be digital.

32

Figure 17 – Micrometric depth measuring device

Figure 18 – Micrometric length measuring device with extensions

Three-touch precision micrometric measuring devices for holes measurement can have a

classical micrometric head or digital head. Measurement anvils are expaned by a cone which pushes

the micrometric bolt.

Figure 19 – Three-touch micrometric measuring device

5.3 Fixed and limit measuring devices (Calliper gauges)

Fixed measuring devices, calliper gauges, measuring templates and other special measuring

device are used in series manufacturing. Using them, we don’t determine real size or deviation from

the nominal value. Controlled pieces are sorted only on the good, repairable and irrecoverable

(scrap).

33

Calliper gauges division:

non-tolerance – they have only one shape, which is compared with the controlled item,

tolerance – they have a good side and a rejected side and they are used to check the

upper (lower) limit size of shafts (holes). The controlled dimension lies in the tolerance

field if the good side passes and the rejected fails.

In practice 3 types of calliper gauges:

workshop – used in production,

take over – used when the customer takes over the products,

comparison – to check workshop and take over calliper gauges.

The standard requires a clear distinction of the rejected side from the good side, for example:

color mark rejected side,

by contraction of the rejected side,

by shortening of the measuring surface of the rejected side,

number or word mark.

Figure 20 – Calliper gauge for outside dimensions

34

Figure 21 – Example of measurement by calliper gauge

where: 1 – good side of calliper gauge

2 – good component,

3 – rejected side of calliper gauge,

4 – good component.

Figure 22 – Calliper gauge for checking a hole

5.4 Dial indicator

Dial indicators (Figure 23) are simple measuring gauges for precise measurement of a very

small distance. Dial indicators can be single-turn, multi-turn or less than single-turn.

Dial indicators have only a small stroke. That is why they are mainly used for comparative

measurements in conjunction with a hard measuring stand. Zero on the scale is scrollable so it is

possible to set the relative zero in any position of the contact.

35

Figure 23 – Dial indicators (analog on the left side, digital on the right side)

36

6 Angles measurement

Plane angle is the ratio of the length cut arc of a circle to its radius. It is most labeled by

Greek letters, eg etc. Plane angle is an angle between the two rays led from the same

point. The unit of plane angle is radian (rad).

The radian is a unit that cannot be exactly realized and in technical practice it has no use. In

practice is used a degree unit. It is an angular measure of plane angle and it is marked as „°“. It is

divided to angle minutes of arc (mark ') and angular seconds (mark "). These units cover the following

relations:

1′ = (1 / 60)°

1″ = (1 / 60) ′ = (1 / 3600)°.

Figure 24 –Definition of unit for plane angle

where: r – size of plane angle in radians,

– size of plane angle in degrees.

6.1 Measuring facilities

6.1.1 Angle gauges

Angle gauges are not used as gauge blocks, but as etalons of angles. To create an accurate

angle, we use connection of gauge blocks and a sinus ruler. Where it is not possible to use this

method, there is a use of angle gauges. These are flat prism with a thickness of 2 mm and with one or

more exactly defined angles.

37

Figure 25 – Example of angle gauges

Angle gauges are grinded, gently lapped and their surface reaches a mirror glaze. They can be

used as a reflective surface in optical measurements. They can be assembled to other values of

angles using special brackets.

6.1.2 Angles

Angles are simple angle gauges for measuring an angle. Most are with 90° angle, but threre

are also angles of 30°, 45°, 120°, etc. These are not adjustable measuring devices. They are in various

sizes and copies. They are used to control perpendicularity and marking-off. Their accuracy is defined

as a diversion of an arm of a certain length from the absolute size of the nominal angle. They are

produced in 4 classes of accuracy. In technical practice are used different shapes of angles. According

to the construction, they are divided to:

flat,

trying,

knife,

control,

with clamping grooves,

others.

Figure 26 – Angles - a) flat, b) trying, c) knife.

38

6.1.3 Protractors

Mechanical protractors belong to basic equipment of a locksmith workshop. Because they

are universal they are used for adjusting angles and marking-off angles. The scale division mostly

used is 30'. It is possible to include here mechanical and digital protractors.

A universal protractor (Figure 27 a) has an ability to measure any angle with an uncertainty of

less than 5'. It consists of fixed and movable rulers. Fixed and movable arms are closely attached to

the measured surface and their correct position is checked by daylight. The measured angle is

checked on the scale (vernier), which is a part of the moving disk. An example of measurement using

a universal protractor is in Figure 28.

A digital protractor (Figure 27 b) has an electronic sensing unit attached on the fixed arm. It

can measure angles in degrees and minutes. The advantage of a digital protractor is resetting it in

any turning or it is possible to fix it in a stand and thereby to complete the arm depending on

whether it is the measurement of small or large angles or internal angles.

a) b)

Figure 27 – Protractors a) universal, b) digital

Figure 28 – Example of measurement by a universal protractor

An optical protractor (Figure 29) is similar to a mechanical protractor in its construction, but

reading proceeds by an ocular. When turning the protractor to the light, the ocular displays a scale

and it is possible to read the angle in tens of minutes.

39

Figure 29 – Optical protractor

6.1.4 Water levels

The easiest way to measure horizontal surfaces is using a water level. In engineering practice

is used either a curved container (called U tube) or more often water levels. There are many types of

water levels for different applications with different accuracy.

Figure 24 – Engineering water level

Water levels are mostly made with one tube with liquid with an air bubble. To measure a

horizontal surface in two perpendicular directions, weuse cross water levels, which have two

perpendicular tubes. A frame water level is made from a rectangular frame and it allows to measure

not only a horizontal position, but also a vertical position.

Marks on the tube help with reading values of gradient to the horizontal position.

Figure 31 – Frame water level and cross water level

There are also optical water levels, which read the value in an ocular or digital water level.

40

Figure 32 – Digital water level

6.1.5 Sine ruler

A sSine ruler is a precisely made component which is used for accurate setting of the upper

plane angle ruler. it is mainly used for indirect measuring of slope on components as wedges or

cones and to set the accuracy tilt of the manufacturing material. They can also be used for accuracy

control of protractors (Figure 33).

Figure 33 – Sine ruler

When measuring, a sine ruler of known length is placed with one roller on the surface of the

control desk. The required angle is created by assembling of gauge blocksto the second roller. Thanks

of high accuracy of shape and dimensions of the functional surface, it is possible to obtain accurate

angle setting. For the calculation are used basic trigonometrical terms to solve a right-angled

triangle. It can measure components of various sizes. Therefore the distance of the rollers is spaced

and it uses lengths of 100, 200 and 300 mm, in special cases even longer.

Figure 34 – Example of measurement with sine ruler

41

6.1.6 Profile projector

A profile projector is an optical device, which projects the picture of a real component on the

focusing screen or projection screen. It is often used for control of shape-complicated components or

control of small dimensions. Zoom in of projectors can be 10x up to 100x, in special causes up to

1000x. Control components can be observed in reflected or passing light, or in combination of both.

An error of zooming should not exceed 1.5 %. It is possible to measure or check the zoom in picture

with a nominal shape template and tolerance limits, which are used for comparison. This method of

control is very quick. The condition of a good image is that the observed component must reflect the

light well.

Figure 35 – Profile pojector

42

7 Surface roughness control

Control of the surface structure is generally very difficult and accuracy of the results depends

on fulfillment of a number of assumptions.

Any technological method used in the implementation of technical area surface leaves

roughness which has cardinal importance at function of these surfaces. Roughness on the surfaces

represents spatial formation which is difficult to assess. The problem of roughness assessment is

solved by reduction to the section plane perpendicular to the surface. At the plane of cut is obtained

a profile which is the basic source of information to assess the texture surface.

Figure 36 – Real surface

Figure 37 – Perpendicular cut

43

Figure 38 – Real profile

Surface structure is divided to individual components according of inequality pitches. It is a

component with the smallest pitch forming surface roughness, with the component called waviness

and component with the greatest pitch inequalities of the specified basic profile. The standard ISO

4287 defines these geometric parameters:

R for surface roughness,

W for surface waviness,

P for basic profile.

Figure 39 – Basic length and evaluated length

where: l – basic length – chosen so that it did not influence waniness and deviations form,

ln – evaluated length.

According to the prevailing direction of inequality, roughness is assessed in a transverse or

longitudinal direction. Roughness parameters are evaluated on the actual profiles which are obtained

as the intersection of the perpendicular.

44

Basic characteristics of surface roughness:

height Ra, Rm, Rz,

width Sm, S,

shape tp.

Figure 40 – Determination of the profile center line

7.1 Height of roughness parameters

Figure 41 – Profiles image of surface roughness using of filters

The highest height of profile Rz – the sum of the highest height protrusions of profile Rp and

the depth maximum of profile Rv in the range of basic length (Figure 42).

45

Figure 42 – The highest height of profile Rz

The arithmetical mean roughness Ra – the arithmetical absolute value of the profile in the

range of basic length.

Figure 43 – The arithmetical mean of surface Ra

7.2 The measurement of surface roughness

Among the most used methods of this time belongs measuring of surface roughness using a

contact method by surface meters. Surface meters measure surface roughness on the set length of

the surface. The measurement point is motor-moved and picks up its transverse movement. The

measurement should be similar to standard definition of surface roughness as close as possible.

Therefore it is possible to switch the pitch of systematic inequalities, which the surface meter can

filter out and switch the evaluation way of the transverse movement of the pitch.

The workshop surface meters have a pocket dimension and they are easily mobile. They are

usually used without a tripod so that they are hand-pressed to the measured surface. Scanning pitch

has a greater apex angle and greater radius of rounding.

Besides this contact method, there also exists a method of non-contact (optical)

measurement of surface roughness as well asa method using light interference.

46

Figure 44 – A workshop surface meter

47

8 Thread gauging

A thread is a spatial surface which is geometrically created by winding one or more forming

profiles in helix to a roll or cone surface.

8.1 External thread gauging

Kinds of gauging:

comprehensive gauging (thread is controlled as a whole – it is not possible to evaluate

the actual thread dimension, but only observe the prescribed tolerance),

partial gauging ( individual parameters of thread are controlled).

8.1.1 Comprehensive gauging

Figure 45 – Groin and jingle bells gauges

control is performed using fixed threaded rings or using U-threaded gauges (groin and

jingle bells gauges, see Figure 45),

good threaded ring should be easily screwed (good side of U-threaded gauge should

easily move their own weight) by the controlled thread,

it guarantees that the mean thread diameter d2 does not exceed the upper limit of the

size and that pitch errors P and flank angle is offset by the appropriate reduction d2.

8.1.2 The pitch P gauging

Thread pitch P – axial movement of the middle part of the thread side related to the thread

axis corresponding to one full thread turning of this point.

Thread comb templates gauging – quick check of informative character.

Figure 46 – Thread comb templates gauging

48

Workshop microscope gauging

It is a non-contact method for monitoring the thread pitch by microscopea crosshairs is set

on one side of the thread, the value is read and then it is set on the other side of thread. Both values

are subtracted from each other and thus the pitch of thread is calculated.

Figure 47 – Pitch gauging by microscope

8.1.3 Mean diameter of thread gauging

Gauging of three-wire method

for a given pitch P is selected a wire with diameter dd (according to table),

by a longitudinal measuring device (micrometer measuring device, length measuring

device) is measured the dimension over the wires Md2,

if we know Md2, it is possible to determine the mean diameter of thread d2.

2xMdd 22

where: value 2x calculated from the geometric relationships is tabulated.

Figure 46 – Three-wire method

This gauging can be also done by a workshop microscope and using a micrometer with

exchangeable touches.

49

Figure 47 – Workshop microscope Figure 48 – Micrometer measuring devices with

gauging exchangeable touches gauging

8.1.4 Vertex angle thread gauging

Vertex angle thread gauging by a workshop (universal) microscope.

It is a non-contact method of monitoring the vertex angle thread by a microscope, in which

the shadow image of the thread profile is set and by crosshairs is deducted the value of the vertex

angle thread.

Figure 49 – Vertex angle thread gauging

8.2 Internal thread gauging

Options of internal thread parameters controls are limited and impossible with at small

thread (up to 10 mm). Often, the only possibility of internal thread gauging is using thread calliper

gauges.

At the internal thread, it is possible to measure the thread pitch on a microscope with a head

which forms an angle of 90° in optical axis. The vertex angle on a microscope with a special head or

it is possible to use the impression method. The mean thread diameter is possible to measure by a

micrometer pin measuring device or on a universal length measuring device with special arms.

50

9 Gear gauging

Gears are important components of engineering. In practice are used more and more high-

performance and high-speed machines that must have high efficiency and quiet operation. This is the

reason of high requirements on gears. In terms of correct and quiet shot and torque transmission,

gears must fultfill this condition:

precision and accuracy of the thickness of the teeth. It must be the same for all the

teeth (shocks),

precision and accuracy of the shape of teeth and quality of teeth surface,

minimum of radial and face run-out.

9.1 Some parameters of gear gauging

9.1.1 Thickness of the teeth gauging

The measurement shows if the established clearance on axial distance will be observed at

tooth gearing.

Tooth thickness is measured:

on the arc of the pitch circle,

on the secant,

in the constant thickness and height of the tooth.

The measurement is done by a tooth measuring device.

Measurement of tooth thickness sk at constant height hk

Constant height and thickness of the teeth is dependent on the basic pitch. It means it is not

dependent on the number of teeth but only on the module and angle of gearing.

The basic scheme is in Figure 50.

The advantage of the measurement is in simple measuring devices. The disadvantage of the

measurement is its depence on accuracy of the head diameter circle.

¨

Figure 50 – Measurememt of tooth thickness

51

9.1.2 Dimension over teeth gauging

This is the most common method for direct determination of the side clearance. A nominal

dimension over the teeth M is the distance between parallel planes in the tangent touching two

opposite teeth sides while it is assumed gearing without a side clearance. The measurement is

carried out over as many teeth z´ so that the touch of the measurement surfaces of the measuring

devices was approximately on the pitch circle.

Figure 51 – Dimension over teeth M gauging

Advantages of measurement:

simple gauge,

it can be measured directly on the machine in production,

it is easy to detect any displacement of the basic profile from the measured dimension.

9.1.3 Peripheral run-out gauging

Peripheral run-out operates as a periodic change in the dimension and direction of the axial

distance gearing.

Peripheral run-out gouging can be done:

by a dial indicator (accuracy tongue, ball touch and control tha all the gaps),

by measurement devices for control of two-side rolling.

52

10 Introduction to ASSEMBLY

10.1 Assembly and Its Significance for the Engineering Industry

The manufacturing process often ends with assembly, during which decisive conditions

concerning reliability and quality of a product are completed. Practically all engineering devices

consist of individual components. A characteristic feature of assembly processes is connecting two or

more components to assembly subgroups, groups and higher units. Technologies that are used to

connect these components are usually ones that provide direct connections without additional parts

or materials. Beyond connecting, assembly usually consists of other activities, like inspecting,

washing, breaking in, conservation, transport to assembly workplace and others.

Assembly can be characterized as a set of activities performed by people, devices and

machines, with a final product made from individual parts and assembly groups by performing of

these activities in certain order during certain time. Assembly is usually the final phase of

manufacturing process within engineering industry.

The significance of assembly in the engineering industry comes from the share of assembly

in engineering product work expenditure that makes 30 to 40 % on the average and also about 30 to

50 % of the total number of manufacturing workers is engaged by assembly. In large series

manufacture, the assembly work expenditure goes down, mainly due to more sophisticated design

and higher level of mechanization and automation of the manufacturing process.

Assembly quality requirements are equal to the requirements of assembled equipment.

Poor quality assembly can degrade quality and accurately made components. On the contrary quality

assembly can appreciate the manufacturing technology of parts, and simple technological

interventions can eliminate possible errors occurring during the part manufacturing.

For these reasons we need to actively pursue the questions of assembly processes and

search for possibilities how to decrease the costs related to it, e.g. by a suitable construction design

and its division to individual assembly groups and subgroups, by selection of simpler connecting

methods, by selection of fittings that do not need exact mating, by employing design elements with a

certain degree of freedom, by using standardized and unified elements, and by other methods.

10.2 Basic Terms in the Field of Assembly

Similarly to the theory of manufacturing technologies we encounter terms like process,

operation, section, task, movement and advance in the field of assembly work. The mentioned terms

can be characterized as follows:

Assembly Process - a subsystem of the manufacturing process, whose target is assembly of

products. The assembly process can be analyzed from the point of view of its integration into the

manufacturing process, its function and regulation properties.

Assembly Operation - a finished part of the assembly process that is performed during

assembly of a complex or a product by one or more workers at one workplace, usually without

rebuilding the assembly equipment (e.g. welding, riveting, checking of dimensions). Assembly

operations are fundamental structural units of the assembly process.

53

Assembly operations are undoubtedly very labor-intensive and costly. In practice they often

take up to 50 % of costs. The ratio of equipment assembly costs largely depends on technical and

organizational level of assembly in the company. The technical and organizational level of assembly

in an engineering company is especially influenced by:

Design Solution – construction and designed complexity of individual parts, function groups and whole products. The design solution influences labor-intensity of assembly, methods of assembly exchange, possibilities to employ mechanical and automatic assembly elements, organizational arrangements of assembly operations, etc. by more than 50 %.

Technology and Organization – from the point of view of used assembly activities, work and mechanization means, organization and progress of assembly etc.

Quality of Work Force – qualification prerequisites of workers, their skills, experience of assemblymen, etc.

Working Conditions and Environment – the sum of work environment influences, such as temperature, noise, humidity, lighting, dust, etc.

The effectivity of assembly processes can be influenced especially by quality design

preparation, together with technical and technological preparation.

Assembly Section – a part of operation that is performed on one connection by one tool

under approximately the same technological conditions (e.g. rough modification of dimensions and

fine modification of dimension on location).

Assembly Task – a complete simple work activity of a worker in an assembly process or

preparation of a product for assembly within a section (for example clamping a part to an assembly

jig, switching on a machine, etc.).

Assembly Movement – the smallest part of the work activity in an assembly process. These

are described in detail especially in mass production (e.g. grab a wrench, put the wrench on, turn the

wrench etc.).

Assembly Technological Procedure – a sum of operations related to connecting finished

parts, subassemblies and assemblies into a product using jigs, equipment and tools that correspond

to drawing requirements and technical conditions.

Assembly Base – a set of surfaces and part elements that determine the part position with

regard to other already assembled parts or basic surfaces.

Assembly Work Position – a part of operation performed with the same position of a jig or

assembly element.

10.3 Assembly Activities and Their Division

Assembly cannot be understood as mere putting together (e.g. setting up, adjusting position,

balancing etc.) and connecting of parts to an assembled unit, but as a set of a number of assembly

activities.

54

10.3.1 Definition of Assembly Activities

By Assembly Activities we understand individual activities that are done during assembly

(e.g. cleaning, arranging, driving screws, adjusting, measuring, packaging, shipping, and others).

There is a number of assembly activities that can be divided to six basic groups, namely:

preparation activities;

adjustment activities;

handling activities;

connecting activities;

inspection activities;

other activities.

These activities and their mutual relationships must fully respect requirements on quality,

reliability and durability of products, and also meet technological, organizational and economy

requirements of the assembly process itself.

10.3.2 Division of Assembly Activities

Figure 52 shows the division of assembly activities with practical examples of individual

activity group.

Preparation

Adjustment

Handling

Connecting

Inspection

Others

- Clearig

- Preparation

of Work Place,

Tools, Devices

Assembly Activities

- Sorting, Marking

- Selecting

- Balancing

- Modified

Surfaces

- Modified

Shapes

- Modified

Dimension

- Inserting

- Removing

- Moving

- Establishing

- Tilting

- Loading

- Screwing

- Riveting

- Soldering

- Forming

- Stamping

- Welding

- Bonding

- Adjustment

- Measurement

- Testing

- Controling

- Conservation

- Package

- Transport

- Disassembly

- Final Finish

Figure 52 - Division of assembly activities.

Relative shares of individual assembly activities differ depending on type of production,

degree of batch and repeated manufacturing, complexity of assembly units, available technological

degree of construction, implemented degree of assembly process mechanization and automation,

etc.

In piecemeal to small series manufacturing, especially preparation activities are critical, and

as far as the assembly itself is concerned checking and adjustments including disassembly work have

fundamental significance. These activities comprise about 80 % of the assembly work expenditure in

total.

55

Výzkumný ústav pro mechanizaci a automatizaci VUMA (The Research Institute for

Mechanization and Automation) in Nove Mesto nad Vahom, Slovakia has created “The Classification

System of Work Performed During Assembly in the Area of Engineering Industry Manufacturing” or

the analysis of internal assembly in the area of piecemeal and small series production.

The share of connecting and handling assembly activities (e.g. insertion, slipping over etc.)

increases in serial and mass production.

10.3.3 Classification of Assembly Activities

Classification of assembly activities is used to detect activities more easily. One of the

possible systems of marking activities is illustrated for example viz Figure 53.

ACTIVITY GROUP

SUBGROUP

ACTIVITY TYPE

ACTIVITY

ADDITIONAL MARK

Figure 53 – Example of the assembly activities.

Classification of activity groups (0 – 9):

0 – assembly preparation work (cleaning, workplace preparation)

1 – adaptation work (component finishing and shape modifications, sorting and marking)

2 – manual assembly work

3 – automatic assembly work

4 – connecting work

5 – work during adjusting and break in

6 – inspection work

7 – conserving and packaging work

8 – work during disassembly

9 – handling work

0 – other work

X

X

X

X

X

56

Classification of the connection work subgroup (0 – 9):

0 – folding and bending

1 – driving screws

2 – other connections possible to disassemble

3 – pressing

4 – riveting

5 – welding

6 – soldering

7 – gluing

8 – shaping

9 – rolling

0 – other connections not possible to disassemble

Classification of technological pressing method activity types (0 – 9):

0 – pressing by static axial force

1 – impulse-vibration pressing

2 – explosive pressing

3 – pneumatic pressing

4 – other methods of axial pressing

5 – by shrinking (cooling)

6 – by expansion (heating)

7 – combined (heating and cooling)

8 – by pressure oil

9 – by change of internal tension

0 – other methods of transversal pressing

A practical example of assembly activity marking in an industrial company is 431XX, which

means:

4 – connecting work

3 – pressing

57

1 – impulse-vibration pressing

X – characteristic marks of activity;

X – additional technological mark.

The efficiency of assembly process requires to focus on the selection of suitable assembly

activities, especially from the point of view of:

decreasing manual work share;

decreasing assembly work intensity;

increasing work productivity and quality;

increasing mechanization and automation level;

increasing equipment standardization level of assembly workplaces.

58

11 Assembly Elements and Systems

This chapter deals with categorization of assembly elements and systems, which helps in the

understanding of overall complexity of the unit assembly problems. An integral part of this chapter is

characteristics of the terms part, subgroup, group and assembly complex, including practical

examples.

11.1 Characteristics of Assembly Elements

From the point of view of assembly, each more complicated industrial product divides itself

into so-called assembly elements, i.e. machine groups and parts that can be assembled separately

and independently from other parts of the product. The division of products to smaller units is

usually in agreement with the unit construction documentation.

Figure 54 shows basic division of the product from the point of view of individual phases of a

manufacturing process. The schematic expresses the division of device assembly to basic assembly

elements.

Figure 54 – Schema of assembly elements.

The basic elements of an assembly process are:

Part – an element impossible to disassemble (initial assembly element), a product part that is usually made from one piece of material;

Unit – a unit created by joining two or more parts; while the way how they are joined does not matter; it is a primary assembly element;

Subgroup (Subsystem) – represents a unit created by joining two or more parts without regard to the way of joining; subgroups can be divided into a number of orders, e.g., subgroups of the 1st order can be directly assembled to groups, subgroups of the 2nd order can be assembled to the 1st order subgroups, etc.;

Group (System) – the highest assembly element that is created by joining one or more subgroups and other parts;

59

Product – mostly it is the final tangible assembly product intended for a market that is functionally and constructively finished; it is created from parts, subgroups and groups joined by dismountable or non-dismountable way;

Piece of Equipment – this is a system of industrial products that should perform given operational and technological tasks.

In case of more complicated parts, a technological assembly schematic is created that

illustrates the order of assembly of individual parts to subgroups and groups and then to the final

product or piece of equipment.

Figure 55 shows a simple technological assembly schematic consisting of the main axis with a

basic part on its beginning and a finished product at its end.

Major Part Product

Par

t

Par

t

Par

t

Par

t

Ass

embly

Gro

up

Ass

embly

Subgro

up

Par

t

Figure 55 – A simple technological assembly schema.

Assembly Schematic is the initial document for assembly technological procedure. The

assembly schematic gives you an overview of mutual part connections. Further the assembly

schematic should show what parts and in what order they should be connected together and

placement of parts for the correct organization of assembly.

Basic Part means the main part that starts a product or group assembly, for example a frame

of a milling machine. An assembly can also start with a basic group or system, e.g. an automobile

chassis. Parts that are directly placed in a product are placed on the main axis of the schematic in a

set order. Systems and subsystems are also entered under the schematic axis in a set order. If

needed, this way it is possible to present schematics of individual product systems and subsystems.

Figure 56 shows an assembly technological schematic branching into individual order

subgroups.

60

Major Part ProductP

art

Par

t

Gro

up

Par

t

Bas

ic P

art

Part

Part

Subgroup

1st OrderBasic Part

Par

t

Figure 56 – Branching out assembly technological schema.

Project documentation already uses classification during the solution of technological

equipment investment. This classification starts with an operational complex, progresses through

operational system and units to basic units and parts. The following terms are used for classification

of elements during assembly of investment units:

Operational Complex (OC) – all machinery equipment that performs complete technological process. It usually includes more basic production operational systems and a number of auxiliary production operational systems.

Operational System (OS) – machinery equipment that performs individual technological process within Operational Complex or a complete process of one type of auxiliary production.

Operational Unit (OU – machinery equipment that performs a complete part of a partial basic production technological process within Operational System or complete process of an auxiliary production. The operational unit is often a system consisting of connected basic units.

61

Basic Unit (BU) – is a machine or machinery equipment including accessories performing a certain basic function, either main or auxiliary, in individual parts of a technological process.

Operational

Complex

OC

Operational

System

OS

Operational

Unit

OU

Basic

Unit

BU

Figure 57 – Classification elements in the assembly investment complex.

Example: Design assembly schema of a pneumatic cylinder.

Figure 58 shows a simple assembly schema of a pneumatic cylinder.

Assembly of

Pneumatic

Cylinder

Cylinder

Head

Cylinder

Bottom

Cylinder

Piston

Direct Bush

Wiper Ring

Sealing Gasket

Sealing Ring

Back Cover

Sealing Gasket

Sealing Ring

Piston

Piston Rod

Damping Piston

Figure 58 – A simple assembly schema of ta pneumatic cylinder.

62

11.2 Assembly Groups and Subgroups

Division of a product into assembly groups is the initial document for an assembly

technological procedure. The proposed integration into the assembly system should also take also

into account possible disassemblyand not perform the division into assembly groups within the

system from the construction point of view only.

Assembly Group is a functionally and constructionally enclosed part of a product that is

created by joining of parts and subgroups by dismountable or non-dismountable union.

Assembly Subgroup is a basic part of a product that is functionally and constructionally open

consisting of dismountable or non-dismountable unions of two or more parts.

The current trend is to present more demanding assembly groups and subgroups using

videos and animations. For example the product 3DVIA Composer was created to ease creation of

assembly manuals and interactive 3D animations and videos. Such created videos and animations

eliminate misunderstanding, help to speed up work, and reduce transfer costs. Figure 59 shows 3D

vizualization of a lifting platform in program INVENTOR.

Figure 59 – 3D vizualization of a lifting platform.

63

12 Technological Level of Product Design with Regard to Assembly

12.1 The Term Construction Technological Solution

By the term Construction Technological Solution we understand such a construction of parts

or products that guarantees optimum production, while meeting all of its functions (productivity,

economy, efficiency, durability, etc.), production requirements (low weight, low production costs for

a given production extent and selection of suitable materials) and requirements for their use

(reliability, easy maintenance, possibly being maintenance-free, reparability, control and others).

The four following partial tasks are the most important during product design:

ensuring proper function of product mechanisms;

solution of the most suitable shapes of details and product construction groups;

selection of suitable materials and shapes of semi-products;

determination of economical way of production and assembly of parts, groups and complexes.

12.2 Technological Level of Product Design with Regard to Assembly

Similarly to the production technological procedure, an increased attention must be paid to

assembly technological requirements. The term Product Design Technological Level from the point

of view of assembly includes such modifications of dimensions, shapes, materials and other

parameters that equal the lowest assembly labor intensity and finishing of a product, possibly

improvement of its actual functions within given production possibilities.

A designer strives for a minimal number of parts of the whole and a modular product

arrangement from the point of view of assembly. A suitably chosen part design allows simplification

of assembly processes, elimination of manual workplaces and implementation of mechanization and

automation. With the trend that increases the assembly automation level requirements for better

product design technological level and accuracy of construction also increase. Assembly costs can

significantly increase production costs of a finished product due to unsuitable design of parts.

12.3 Influence of the Product Design and Technological Concept on the

Assembly Process

During the product design and technological concept proposal we also need to consider

available production and assembly technology, together with financial and operational requirements.

Cooperation of a designer and production technologist is necessary.

Generally to facilitate production and assembly, it is good to:

utilize the forms of design and technological standardization (simplification, standardization and unification), i.e. to accept only the necessary design and technological variety, both for parts and products, and utilize the benefits of modular solution;

increase the level of construction technology in relation to the assembly.

64

The example of utilization of design and technological standardization in practice is

implementation of constructionally unified series of milling machines. The target of this unification is

to lower the number of individual construction assembly groups to a certain number of unified ones.

This solution leads to increase production series of parts and thus assembly groups, decrease

material and energy production demands, decrease labor intensity and production and assembly

costs.

A number of construction, technology and operational tasks is being solved during product

design from the point of view of technological assembly level. The following solutions can be

considered possibilities to increase design technological level:

a) from the design point of view:

simplicity of product, construction group and system concepts;

minimization of part numbers for product assembly systems;

simplicity of part shapes and their modifications making assembly easier;

product modular character enabling independent assembly of groups;

division of assembly complex or product in the location of the simplest joint;

such product division that assembling or connecting does not require adjustment of assembly activities (additional fitting during assembly on site, regulation), primarily using the correct choice of dimensional chains and their tolerances for individual parts;

product division in such a way that it is possible to perform the assembly in one, two or three perpendicular directions;

selection of part shapes modified to their orientation during assembly, if possible without visual or touch control, i.e. creation of suitable areas for setting, grasping and clamping of parts, in other words selecting distinctly symmetrical or asymmetrical shapes with marked orientation marks;

designing of parts, whose centers of mass would ensure stability during handling, orientation, transport etc.;

practicality during selection of bases, dimensions, tolerances and surface roughness parameters of products;

limitation of dimensional chain closing member adjustments during assembly, securing of full replaceability and unification of parts if possible;

b) from the technological point of view:

including mechanization and automation to the assembly process;

increasing of specialization level and integration of assembly workplaces and processes;

creation of conditions for the improvement of time and dimensional structures of assembly, for example conditions for time parallelization in individual assembly phases that would primarily shorten actual assembly time;

shortening of production preparation section and also the time needed for its implementation;

allowing the use of highly productive assembly methods and assembly automation;

65

lowering of consumption of materials and energies;

securing production accuracy and accuracy of the follow-up assembly;

c) from the operational point of view:

analysis of operational reliability and durability of products;

increasing of maintenance and machine repair simplicity.

Figure 60 and Figure 61 show facilitation assembly modification shapes of parts.

Figure 60 – Facilitation of assembly modification shapes of parts.

Figure 61 – Facilitation of assembly modification shapes of parts.

66

12.4 Indicators of Design Technology Level with Regard to Assembly

Significance of the product design technology level is evaluated according to various relative

criteria that compare the design of a new type of industrial product with one or more product types

from the actual production program.

The following indicators can be used to evaluate the design technology level:

Indicator of Unit Production Costs 𝐔𝐍𝐜𝐣:

𝑈𝑁𝑐𝑗 =𝑁𝑐𝑗𝑛

𝑁𝑐𝑗𝑠 [-],

where Ncjn – unit production costs for 1 unit of a new type of product [CZK∙unit-1];

Ncjs – unit production costs for 1 unit of the current type of product [CZK∙unit-1];

Indicator of Labor Intensity 𝐔𝐏:

𝑈𝑃 =𝑃𝑛

𝑃𝑠 [-],

where Pn – labour intensity for 1 unit of a new type of product [h∙unit-1];

Ps – labourintensity for 1 unit of a current type of product [h∙unit-1];

Indicator of Material Consumption 𝐔𝐐𝐦:

𝑈𝑄𝑚 =𝑄𝑚𝑛

𝑄𝑚𝑠 [-],

where Qmn – weight of constructionfor 1 unit of a new type of product [kg∙unit-1];

Qms – weight of construction for 1 unit of a current type of product [kg∙unit-1];

Indicator of Material Utilization 𝐔𝐏𝐦:

𝑈𝑃𝑚 =𝑃𝑚𝑛

𝑄𝑚𝑛 [-],

where Pmn – weight of used components for production of 1 unit of a new type of product [kg∙unit-

1];

Qmn – weight of construction for 1 unit of a new type of product [kg∙unit-1];

67

Indicator of Costs per Product Weight Unit 𝐔𝐍𝐜𝐣𝐐:

𝑈𝑁𝑐𝑗𝑄 =𝑁𝑐𝑗𝑛

𝑄𝑚𝑛 [CZK·kg-1],

where Ncjn – unit production costs for 1 unit of a new type of product [CZK∙unit-1];

Qmn – weight of construction for 1 unit of a new type of product [kg∙unit-1];

Indicator of Standardized UNS, Unified UUS and taken over parts UPS:

𝑈𝑁𝑆 =𝑄𝑁𝑆

𝑄𝑆 [-], 𝑈𝑈𝑆 =

𝑄𝑈𝑆

𝑄𝑆 [-], 𝑈𝑃𝑆 =

𝑄𝑃𝑆

𝑄𝑆 [-],

where QS – total number of components of product [psc];

QNS – total number of standard components of product[psc];

QUS – total number of unified components of product [psc];

QPS – total number of taken components of product [psc].

Relative indicators can also be expressed from the point of view of used industrial materials,

for example, alloy steels, non-ferrous metals, plastics etc.

A criterion for evaluation of a product technical level that cannot be ignored is the indicator

of new product exploitation level that can be judged in relation to weight, selling price or costs of a

product:

Exploitation Indicator Related to Weight 𝑼𝑬𝑸𝒎:

𝑈𝐸𝑄𝑚 =𝑄𝑚

𝐸 [kg·unit-1·W-1, kg·unit-1·mm-3],

whereQm – weight of construction for 1 unit of a new type of product [kg∙unit-1];

E –exploitation characteristics of product (power,volumeetc.) [W, mm3];

Exploitation Indicator Related to Unit Production Costs 𝑼𝑬𝑵𝒄𝒋:

𝑈𝐸𝑁𝑐𝑗 =𝑁𝑐𝑗

𝐸 [CZK· unit-1·W-1, CZK· unit-1·mm-3],

where Ncj – unit production costs for 1 unit of a new type of product [CZK∙unit-1];

E – exploitation characteristics of product (power, volume etc.) [W, mm3].

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13 Analysis of Dimensional chains

13.1 Dimensional Chain Basic Terminology

Parts entering an assembly process are made with various accuracy and tolerances. During

the assembly of parts their mutual arrangement must be set within the limits of prescribed accuracy.

Fitting of certain areas must provide the prescribed clearance; on the other hand connection of

others should provide necessary overlap. The correct size of part dimension deviations depending on

required connection or mechanism accuracy can be determined according to the theory of solution

of dimensional networks.

Dimensional Network is an interrelated set of dimensions applicable to two or possibly

several product functional areas or several dimensional chains. A dimensional network consists of

one or more dimensional chains.

Dimensional Chain is an enclosed chain of interrelated dimensions that are arranged in

certain order that is crucial for mutual position of areas or axes of one or more parts. Individual part

dimensions are members of a dimensional chain, i.e. dimensions where the sum of all dimensional

chain members either gives the total required dimension or differs from the required total dimension

by a clearance or an overlap.

Dimensional Chain Schematic is a graphical display of the dimensional chain and it always is

an enclosed curve.

The target of a dimensional chain solution is to determine limit dimensions or limit deviations

from nominal values of partial dimensions according to production or construction requirements, or

to change tolerances to meet technical and assembly documentation requirements.

13.2 Division of Dimensional Chains

The basic types of dimensional chains are:

Basic Dimensional Chain – all its members have own functions during solution of a given task;

Derived Dimensional Chain – this is the dimensional chain whose initial member is one of the system members of a basic dimensional chain and so it is tied to the basic chain;

Construction Dimensional Chain – a dimensional chain that serves for solution of the task to ensure accuracy during product construction;

Technological Dimensional Chain – a dimensional chain that serves for solution of the task to ensure accuracy during product production;

Control Dimensional Chain – a dimensional chain that serves for solution of the task to find (measure) values characterizing product accuracy;

Linear Dimensional Chain – a chain whose members are length dimensions;

Plane Dimensional Chain – a chain whose members are located in one or several parallel planes;

69

3D Dimensional Chain – a dimensional chain whose members are located in non-parallel planes;

Angular Dimensional Chain – a chain whose members are angle dimensions;

Dimensional Chain Tied in Parallel – dimensional chains that have at least one common member;

Dimensional Chain Tied in Series – dimensional chains that have at least one common base;

Combined Tied Dimensional Chain – dimensional chains that have mutual members and bases;

Selected types of dimensional chains will be defined and schematically displayed further in

the text.

13.3 Dimensional Chain Members

Dimensional chain members can be divided to initial, closing and connecting ones. In the

cases of initial and closing members, accuracy of dimensions is determined by accuracy deviations of

all other dimensional chain members. If a chain starts with this member, then this member is called

the initial member, if it ends with it, then it is the closing member. Dimensional chain members are

marked by Latin alphabet capitals (A, B, C…).

Dimensional networks consist of members that can be:

- Partial Set Members Ai, where i = 1, 2, 3 … n – they are members of a dimensional network that have a functional relationship to the closing member. The connecting members are all other chain members with the exception of the closing and compensation members. The accuracy of dimensions of partial set members influences a change in dimension accuracy of the closing member.

The partial connecting members can be subdivided to:

i. Enlarging Partial MembersAi – members of a dimensional network, when

they are increased (decreased), the closing dimensional network member 𝐴𝑈increases (decreases);

ii. Decreasing Partial MembersAi – members of a dimensional network, when

they are increased (decreased), the closing dimensional network member 𝐴𝑈 decreases (increases);

- Closing Member 𝑨𝑼 – a member of a dimensional network whose value is the initial quantity during submitting of the task that is being solved or it is the final searched quantity during the solution of a dimensional network. The closing member is the resulting member that is marked as informative on a drawing, or it is the assembly resulting dimension that is calculated as the sum of individual parts, or it is a clearance, overlap, geometrical tolerance, etc.

- Compensation Member AKi – a member of a dimensional network whose changing results in achieving the required accuracy of the closing member 𝐴𝑈;

- Common Member Ai,Bi, where i = 1, 2, 3 … n – is marked by the chain symbols to which it belongs to. It is a member that acts in several dimensional chains at the same time.

70

- Independent Member Ai, where i = 1, 2, 3 … n – is a set member whose value does not depend on the value of another set member.

- Dependent Member Ai, where i = 1, 2, 3 … n – is a set member whose value functionally or correlatively depends on the value of another set member.

Each dimensional network has at least two partial connecting members and one closing

member AU. Closing members AUare usually non-quoted dimensions within part construction

drawings, or they are specified as informative dimensions in round parentheses.

Figure 62 shows linear dimensional chain and its members.

Figure 62 – Linear dimensional chain.

13.4 Selected Types of Dimensional Chains

We distinguish types of dimensional chains in terms of relative position, direction and size of

members. In practice we mostly encounter calculations of linear and planar chains.

To clarify, different types of chains can be characterized as follows:

Straight Linear Chain (see Figure 63, all members of the chain are parallel);

Figure 63 – Schema of Straight Linear Chain.

Planar Chain (see Figure 64, some or all chain members are not in parallel directions, but in one or more parallel planes);

71

Figure 64 – Schema of Planar Chain.

Spatial Chain(see Figure 65, some or all chain members are situated in divergent directions and divergent planes);

Figure 65 – Schema of Spatial Chain.

Angular Chain (see Figure 66, some or all chain members are the angular extent with the joint top). It can be planar or spatial type.

Figure 66 – Schema of Angular Chain.

72

14 Calculation of dimensional chains

This chapter deals with the solution of dimensional chains by basic formulas for calculating

the straight line and plane dimensional chains. The chapter lists a number of solved examples of the

dimensional chain.

To solve dimensional network dimension limits or nominal dimensions and tolerances are

calculated according to members of the functional, assembly and manufacturing requirements. In

general, two types of problems are solved, namely:

Control Tasks – determine the dimensions and tolerances of the closing member based on the known dimensions and tolerances of partial members in these tasks to verify the correctness of the construction solution;

Construction Tasks – determine the dimensions and tolerances of partial members based on the known dimensions and tolerances of the closing member. These are tasks that are solved in engineering and design assemblies.

Depending on the method, assembling of the product are used in the formulas for

calculations. Methods of installation will be provided with examples solved in the next chapter.

For calculations dimensional circuits will be described below:

• Basic Formulas for Calculating Straight Liner Dimensional Networks;

• Basic Formulas for Calculating the Plane Dimensional Networks.

14.1 Basic Formulas for Calculating the Straight Liner Dimensional

Networks

Assembly complex may consist of several straight liner dimensional chains. Individual

members of the various straight liner chains denote with different capital letters alphabet such as A,

B, C, etc. The individual chains can be a connecting member, with which given the relationship

between the chains.

We distinguish between linear dimensional chains by mutual relationship to:

Parallel linear dimensional networks;

Serial linear dimensional networks;

Combined linear dimensional networks.

14.1.1 Parallellinear Dimensional Networks

Figure 67 showsinteractions among the chains and the connecting members (in the figure are

members connecting members A2 = B1, B2 = C1, C2 = D1, D2 = E1).

73

A1AU

A2 = B1BU

CU

DU

B2 = C1

C2 = D1

EU

E2

D2 = E1

Figure 67 – Parallel linear dimensional networks.

14.1.2 Serial Linear Dimensional Networks

Figure 68 shows creation of mutual relations between the common base 1 and the common

base 2.

Figure 68 – Serial linear dimensional networks.

14.1.3 Combined Linear Dimensional Networks

Figure 69 shows a dimensional network with combined relation, where we can find parallel

and serial relations.

74

Figure 69– Combined linear dimensional networks.

We can use these formulas to solve a straight liner chain with a complete substitutability

method:

Nominal Dimension of the Closing Member 𝑨𝑼 – for calculation of nominal dimension of the

closing member 𝐴𝑈of linear chainis in general used:

AU = ∑ Ai

m

i=1

− ∑ Ai

n

i=m+1

where m is the number of all enlarging partial members and nis the number of all partial

members. We consider the closing member as an independent member. Then ∑ Ai mi=1 is a sum of

nominal dimensions of the enlarging partial members and∑ Aini=m+1 is a sum of nominal dimensions of

the decreasing partial members.

Tolerance of the Closing Member𝑻𝑨𝑼 – is calculated by the sum of tolerances of all

members of a linear chain:

TAU = ∑𝑇𝐴𝑖

n

i=1

The above equation shows that while the number of members in the dimension chains

increases, the number of terms of addition𝑇𝐴𝑖 increases as well resulting in either reduction of

tolerance of individual members of the chain so that the tolerance of closing member remained

constant, or while maintaining the values of tolerance of individual members to increasing the

tolerance of the closing member. It must be kept in mind that all tolerances are always positive

numbers.

75

For calculation of linear, chains, it is true that the closing member TAUis the difference

maximum AUmaxand minimum𝐴𝑈𝑚𝑖𝑛

value of the closing member and tolerances equal to the sum

of all members of the chain:

TAU =𝐴𝑈𝑚𝑎𝑥− 𝐴𝑈𝑚𝑖𝑛

= ∑𝑇𝐴𝑖

n

i=1

Tolerance of the Partial Member𝑻𝑨𝒊 –to calculate the tolerance of partial member can be

used the equation:

T𝐴𝑖 =TAU − ∑𝑇𝐴𝑖

n-1

i=1

Bottom Limit Dimension of the Closing Member 𝑨𝑼𝒎𝒊𝒏- minimum value of the closing

member can be calculated by substituting the minimum dimensions of all enlarging members and

subtracting the maximum dimensions of decreasing members:

𝐴𝑈𝑚𝑖𝑛 = ∑ Ai

min

m

i=1

− ∑ Ai max

n

i=m+1

UpperLimit Dimension of the Closing Member 𝑨𝑼𝒎𝒂𝒙- maximum value of the closing

member can be calculated by substituting the maximum dimensions of all enlarging members and

subtracting the minimum dimensions of decreasing members:

𝐴𝑈𝑚𝑎𝑥 = ∑ Ai

max

m

i=1

− ∑ Aimin

n

i=m+1

Bottom Deviation of the Closing Member 𝑰𝑨𝑼:

IAU = ∑ IAi

m

i=1

− ∑ SAi

n

i=m+1

Upper Deviation of the Closing Member𝑺𝑨𝑼:

SAU = ∑ SAi

m

i=1

− ∑ IAi

n

i=m+1

Using the formulas for calculating, IAU and SAU substitute all the values of deviations partial

members with their respective marks, i.e. positive or negative.

76

The calculated dimensions of the closing member and the partial member can be

represented schematically, shown in Figure 70 and Figure 71.

Figure 70 – Parameters of the closing member.

Figure 71 – Parameters of the partial member.

From the described analysis of the dimensional chain stems the rule of the shortest line. A

sub-task solution of the accuracy of the relative positions surfaces and axes of the individual parts

must be dealt with, similarly to their machining, by application of dimensional chains with

a minimum number of members.

77

Example:

Determine the dimension of clearance, its tolerance and deviations for the shaft and

bearing. The clearance is considered as the closing member 𝐴𝑈 of the dimensional chain.

FJigure 72 shows a schema of a linear dimensional chain. Diameter of bearing hole

𝐴1 = 50+0.16+0.28 mm, diameter of shaft 𝐴2 = 50−0.15

−0.05 mm.

A1

A2

AU

Figure 72 – Example of linear dimensional chain.

Nominal Dimension of the Closing Member 𝑨𝑼:

AU = ∑ Ai

m

i=1

− ∑ Ai

n

i=m+1

AU = 𝐴1 − 𝐴2 = 50 − 50 = 0 𝑚𝑚

Tolerances of the Members𝑻𝑨𝒊:

Tolerance of the Hole 𝑇𝐴1= 0.28 − 0.16 = 0.12 𝑚𝑚

Tolerance of the Shaft 𝑇𝐴2= −0.05 − (−0.15) = 0.10 𝑚𝑚

Tolerance of the Closing Member𝑻𝑨𝑼:

TAU = ∑𝑇𝐴𝑖

n

i=1

TAU =0.12 + 0.10 = 0.22mm

Bottom Limit Dimension of the Closing Member 𝑨𝑼𝒎𝒊𝒏:

𝐴𝑈𝑚𝑖𝑛 = ∑ Ai

min

m

i=1

− ∑ Ai max

n

i=m+1

𝐴𝑈𝑚𝑖𝑛 = 50.16 − (49.95) = 0.21 mm

78

Upper Limit Dimension of the Closing Member𝑨𝑼𝒎𝒂𝒙:

𝐴𝑈𝑚𝑎𝑥 = ∑ Ai

max

m

i=1

− ∑ Aimin

n

i=m+1

𝐴𝑈𝑚𝑎𝑥= 50.28 − (49.85) = 0.43 mm

Bottom Limit Deviation of the Closing Member 𝑰𝑨𝑼:

IAU = 𝐴𝑈𝑚𝑖𝑛− 𝐴𝑈

IAU = 0.21 − 0 = 0.21 𝑚𝑚

or you can use the formula:

IAU = ∑ IAi

m

i=1

− ∑ SAi

n

i=m+1

IAU=0.16 − (−0.05) = 0.21 mm

Upper Limit Deviation of the Closing Member𝑺𝑨𝑼:

SAU = 𝐴𝑈𝑚𝑎𝑥− 𝐴𝑈

SAU = 0.43 − 0 = 0.43 𝑚𝑚

or you can use the formula:

SAU = ∑ SAi

m

i=1

− ∑ IAi

n

i=m+1

SAU=0.28 − (−0.15) = 0.43 mm

Dimension of Clearance May Be Written in the Form:

AU = 0+0.21+0.43 𝑚𝑚

14.2 Basic formulas for calculating plane dimensional networks

The above-mentioned formulas are determined for linear dimensional chains. For calculation

of plane dimensional networks, it is possible to use these formulas:

Tolerance of the Closing Member𝑻𝑨𝑼 – it is calculated as the sum of tolerances of all

members of plane chain:

TAU = ∑|𝜕𝐴𝑈

𝜕𝐴𝑖| 𝑇𝐴𝑖

n

i=1

79

Decreasing the values of partial derivatives 𝜕𝐴𝑈 𝜕𝐴𝑖⁄ is reflected in the decreasing tolerance

of the closing member TAU, because the value of tolerance depends on the closing element of the

plane dimensional chains as weel as on the values of partial derivatives.

Bottom Limit Dimension of the Closing Member 𝑰𝑨𝑼:

IAU =∑|∂AU

∂Ai

| IAi

m

i=1

− ∑ |∂AU

∂Ai

| SAi

n

i=m+1

Upper Limit Dimension of the Closing Member𝑺𝑨𝑼:

SAU = ∑|∂AU

∂Ai

| SAi

m

i=1

− ∑ |∂AU

∂Ai

| IAi

n

i=m+1

The equations for calculation of bottom deviations of closing member 𝐼𝐴𝑈 and upper

deviations of closing member 𝑆𝐴𝑈 substitute the values of deviations of partial members with their

respective marks, respectively positive or negative.

The calculated numerical value of the partial derivative for enlarging member is positive, ie

∂AU ∂Ai >0⁄ .

The calculated numerical value of the partial derivative for decreasing member is negative, ie

∂AU ∂Ai <0⁄ .

According to the calculated numerical values of partial derivatives is decided whether a

member of the plane dimensional network is increasing or decreasing.

Example:

Determine the nominal dimensional and limit deviations of distance holes in direction

of axis x. Figure 73 shows a schema of a plane dimensional network, the required distance of

holes is a member of the closing member𝐴𝑈 of dimensional network. The distance of holes in

axis y is 𝐴2 = 90 ± 0.025 mm and the axis distance is 𝐴1 = 120 ± 0.030 mm.

80

AU

A2 A

1

AU

x

y

A1

A2

Figure 73 – Example of plane dimensional network.

Nominal Dimension of the Closing Member 𝑨𝑼 – to calculate of the value of nominal

dimension closing members 𝐴𝑈, it is possible to use:

AU = √𝐴12 − 𝐴2

2

AU = √1202 − 902 = 79.37 = 79 𝑚𝑚

Calculation of the Values of Partial Derivatives 𝝏𝑨𝑼 𝝏𝑨𝒊⁄ – continue with the calculation of

partial derivatives 𝜕𝐴𝑈 𝜕𝐴𝑖⁄ , for the value of closing member 𝐴𝑈 according to partial members of

dimensional chain:

𝜕𝐴𝑈

𝜕𝐴1 =

1

2

1

√𝐴12 − 𝐴2

2

2𝐴1 =𝐴1

√𝐴12 − 𝐴2

2

= 𝐴1

𝐴𝑈

𝜕𝐴𝑈

𝜕𝐴1 =

𝐴1

𝐴𝑈=

120

79> 0

The calculation shows that partial member 𝐴1 is a member of the solved enlarging member of

the plane dimensional network because the partial derivative 𝜕𝐴𝑈 𝜕𝐴1⁄ has a positive value.

𝜕𝐴𝑈

𝜕𝐴2 =

1

2

1

√𝐴12 − 𝐴2

2

(−2𝐴2) = −𝐴2

√𝐴12 − 𝐴2

2

= − 𝐴2

𝐴𝑈

𝜕𝐴𝑈

𝜕𝐴2 = −

𝐴2

𝐴𝑈 = −

90

79< 0

The calculation shows that partial member 𝐴2 is a member of the solved decreasing member

of the plane dimensional network because the partial derivative 𝜕𝐴𝑈 𝜕𝐴2⁄ has a negative value.

81

Bottom Limit Dimension of the Closing Member 𝑰𝑨𝑼:

IAU =∑|∂AU

∂Ai

| IAi

m

i=1

− ∑ |∂AU

∂Ai

| SAi

n

i=m+1

IAU = 120

79∙ (−0.030) + (−

90

79∙ 0.025) = −0.074 𝑚𝑚

Upper Limit Dimension of the Closing Member𝑺𝑨𝑼:

SAU = ∑|∂AU

∂Ai

| SAi

m

i=1

− ∑ |∂AU

∂Ai

| IAi

n

i=m+1

SAU = 120

79∙ 0.030 + [−

90

79∙ (−0.025)] = +0.074 𝑚𝑚

Dimension of the closing member may be written in the form:

AU = 75 ± 0.074 𝑚𝑚

82

15 Assembly Methods

This chapter deals with the assembly methods and their influence on the expected

manufacturing accuracy. Manufacturing accuracy is a significant contributor to total costs of

production. For every described assembly methods, there is an example from the practice.

15.1 Division of Assembly Methods

The prescribed accuracy when assembling the parts can be found out by the assembly

methods below. The selection of methods for solving dimensional chains is determined by structural

peculiarities of the parts and type of production.

Ways to deal with these chains that affect the assembly methods are as follows:

total interchangeability of parts;

partial interchangeability of parts;

selection of components;

compensation (fixed member);

regulation (movable member);

alignment (adjustment).

15.1.1 Method of Total Interchangeability of Parts

This method allows assembly of all components that make up individual members of the

dimensional chain made in the prescribed dimensions and tolerances without selection or adaptation

and ensures accuracy of the closing member.

Using this method, the individual components entering the assembly process must be made

with such accuracy to obtain the desired accuracy of the assembled group in any random selection.

This requires the production of parts in narrow tolerances. Assembly based on the complete

interchangeability of parts could be organized in mass and large production.

The advantages of this assembly method are simple technological preparation assembly

(structure, mechanized assembly work, standardization of work), simple and economical assembly

(without selection and adaptation, lower qualification of the workers, stable assembly time),

simplification of mechanization and automation of assembly, possibility of co-operation of

production, simple maintenance and repair of the product based on interchangeable parts,

equipment, spare parts.

The disadvantage of this method is in contrast to the increasing demands on accurate

manufacturing methods, jigs and measuring instruments, a longer production times and associated

increase in costs of production of components with the required accuracy. Dimensional networks in

the application of the method total interchangeability is solved in the example below.

83

Example:

Determine the distance between the front part and external left flat of plate AU. Figure 74

shows a schema of the solved assembly group.

Figure 74 – Schema of assembly parts.

Nominal Dimension of the Closing Member 𝑨𝑼:

AU = ∑ Ai

m

i=1

− ∑ Ai

n

i=m+1

AU = 𝐴1 − (𝐴2 + 𝐴3 + 𝐴4) = 20 − (1.8 + 16 + 2) = 0.2 𝑚𝑚

Tolerances of Members 𝑻𝑨𝒊:

Tolerance Member 𝑇𝐴1= 0 − (−0.2) = 0.2 𝑚𝑚

Tolerance Member 𝑇𝐴2= 0.1 − 0 = 0.1 𝑚𝑚

Tolerance Member 𝑇𝐴3= 0 − (−0.2) = 0.2 𝑚𝑚

Tolerance Member𝑇𝐴4= 0 − (−0.1) = 0.1 𝑚𝑚

84

Tolerance of Closing Member 𝑻𝑨𝑼:

TAU = ∑𝑇𝐴𝑖

n

i=1

TAU = TA1 + TA2+ TA3+ TA4=0.2 + 0.1 + 0.2 + 0.1 = 0.6mm

Half Tolerance of Closing Member 𝜹TAU :

𝛿TAU = TAU

2

𝛿TAU = 0.6

2= 0.3 𝑚𝑚

Dimension of Closing Member AU:

AU = 0.2 ± 0.3 𝑚𝑚

Maximal Dimension of Closing Member 𝑨𝑼𝒎𝒂𝒙:

𝐴𝑈𝑚𝑎𝑥= 0.2 + 0.3 = 0.5 𝑚𝑚

Minimal Dimension of Closing Member 𝑨𝑼𝒎𝒊𝒏:

𝐴𝑈𝑚𝑖𝑛= 0.2 − 0.3 = −0.1 𝑚𝑚

15.1.2 Method of Partial Interchangeability of Parts

This method allows assembly of all components that make up individual members of the

dimensional chain made in the prescribed dimensions and tolerances without selection or adaptation

and ensures accuracy of the closing member.

The method of partial interchangeability of parts is based on the consideration that the

actual dimensions of each member of the dimensional chain are, due to random mistakes,

distributed across the tolerance field, but with different frequency, i.e. extreme values are less

numerous than in the middle.

Using the knowledge of probability theory, it can be extended tolerance of all members of

the dimensional chain is not changed even though the specified tolerance of the closing member, ie

unchanged accuracy of assembly. This method of solving dimensional chains allows extending the

tolerance of individual members, especially where the specified tolerances proves uneconomical.

The advantage of method of partial interchangeability is a choice of wider tolerances of parts

(reducing costs of manufacturing), simple and economical assembly. It is suitable to be equipped

with automatic assembly machine devices for measuring deviations and a locking device for removal

of non-compliant components.

85

The application of probability theory to engineering manufacturing results in the fact that

dimensional accuracy of machine parts is characterized by the the location of the given dimension

and type of normal distribution curve of each member of the chain, which allows:

middle arithmetic deviation 𝑋𝐴𝑖from the nominal dimension J;

middle quadratic deviation 𝜎𝐴𝑖;

width of variance 𝑣𝐴𝑖.

The coordinates of the center real curve differ by the value:

XAi= aAi

+ αAi∙ δAi

where aAi – coordinates of the center tolerance field of the i-th member of the chain due to the

nominal dimension;

αAi – asymmetry coefficient of variance curve of the i-th member of the chain;

δAi – a half of tolerance field of the i-th member of the chain;

Figure 75 – Normal distribution curve.

Coordinates of the Center Tolerance Field of the Closing Member from the Nominal

Dimension:

𝑎𝐴𝑈= ∑𝑎𝐴𝑖

m

i=1

− ∑ 𝑎𝐴𝑖

n

i=m+1

86

Middle Quadratic Deviation of the Closing Member 𝝈𝑨𝑼:

𝜎𝐴𝑈= √∑𝜎2

𝐴𝑖

𝑛

𝑖=1

Half of the Tolerance Field of the Closing Member 𝜹𝑨𝑼:

𝛿𝐴𝑈= √∑𝛿2

𝐴𝑖∙ 𝑘2

𝐴𝑖

𝑛

𝑖=1

Coefficient Variance of Relative Dimensions of Individual Members of the Dimensional

Chain 𝒌𝑨𝒊:

𝑘𝐴𝑖=

6 ∙ 𝜎𝐴𝑖

𝑣𝐴𝑖

Upper Limit Deviation of the Closing Member𝑨𝑼𝒎𝒂𝒙:

𝐴𝑈𝑚𝑎𝑥= 𝑎𝐴𝑈

+ 𝛿𝐴𝑈

Bottom Limit Deviation of the Closing Member 𝑨𝑼𝒎𝒊𝒏:

𝐀𝐔𝐦𝐢𝐧= 𝐚𝐀𝐔

− 𝛅𝐀𝐔

Tolerance of the Closing Member 𝑻𝑨𝑼:

𝑇𝐴𝑈= 2𝛿𝐴𝑈

Dimensionof the Closing Member 𝑨𝑼:

𝐴𝑈 = 𝑎𝐴𝑈± 𝛿𝐴𝑈

Example:

Determine the tolerance N of a stroke piston pump, which according to technical

requirements should not exceed the tolerance field of 3 mm. Figure 76 shows the scheme of the

piston.

87

Figure 76 – Scheme of a piston.

Determine coordinates of a center tolerance field of the partial members from the nominal

dimension in case of𝛼𝐴𝑖= 0and𝑘𝐴𝑖

= 1.

𝑎𝐴1= +0.31 𝑚𝑚

𝑎𝐴2= +0.215 𝑚𝑚

𝑎𝐴3= −0.31 𝑚𝑚

𝑏𝐴4= +0.31 𝑚𝑚

𝑏𝐴5= −0.26 𝑚𝑚

Determine the halves of the tolerance fields of partial:

𝛿𝐴1= 0.31 𝑚𝑚

𝛿𝐴2= 0.215 𝑚𝑚

𝛿𝐴3= 0.31 𝑚𝑚

𝛿𝐵4= 0.31 𝑚𝑚

𝛿𝐵5= 0.26 𝑚𝑚

88

Limit Deviations of the Closing Member𝑨𝑼:

𝐴𝑈= [(𝑎𝐴2+ 𝛼𝐴2

∙ 𝛿𝐴2) + (𝑎𝐴3

+ 𝛼𝐴3∙ 𝛿𝐴3

) − (𝑎𝐴1+ 𝛼𝐴1

∙ 𝛿𝐴1)] ±

±√𝛿2𝐴2

∙ 𝑘2𝐴2

+ 𝛿2𝐴3

∙ 𝑘2𝐴3

+ 𝛿2𝐴1

∙ 𝑘2𝐴1

𝐴𝑈= [0.215 − 0.31 + 0.31] ± √0.233425 = 0.215 ± 0.448 𝑚𝑚

Limit Deviations of the Closing Member𝑵:

𝑁 = [(𝑎𝐴𝑈+ 𝛼𝐴𝑈

∙ 𝛿𝐴𝑈) + (𝑏𝐵4

+ 𝛼𝐵4∙ 𝛿𝐵4

) − (𝑏𝐵5+ 𝛼𝐵5

∙ 𝛿𝐵5)] ±

±√𝛿2𝐴𝑈

∙ 𝑘2𝐴𝑈

+ 𝛿2𝐵4

∙ 𝑘2𝐵4

+ 𝛿2𝐵5

∙ 𝑘2𝐵5

𝑁 = [0.215 + 0.31 − 0.26] ± √0.4482 + 0.312 + 0.262 = 0.265 ± 0.634 𝑚𝑚

Total Upper Limit Dimension𝑵𝒎𝒂𝒙:

𝑁𝑚𝑎𝑥 = 13 + 0.265 + 0.634 = 13.899 𝑚𝑚

Total Bottom Limit Dimension 𝑵𝒎𝒊𝒏:

𝑁𝑚𝑖𝑛 = 13 + 0.265 − 0.634 = 12.631 𝑚𝑚

For comparison, the calculation will be performed by a simple sum of the tolerance of

members:

Nominal Dimension of the Closing Member𝑨𝑼:

AU = ∑ Ai

m

i=1

− ∑ Ai

n

i=m+1

AU = (𝐴2 + 𝐴3) − 𝐴1 = 11 + 40 − 50 = 1 𝑚𝑚

Bottom Limit Dimension of the Closing Member 𝑨𝑼𝒎𝒊𝒏:

𝐴𝑈𝑚𝑖𝑛 = ∑ Ai

min

m

i=1

− ∑ Aimax

n

i=m+1

𝐴𝑈𝑚𝑖𝑛=11 + 39.38 − 50 = 0.38𝑚𝑚

89

Upper Limit Dimension of the Closing Member 𝑨𝑼𝒎𝒂𝒙:

𝐴𝑈𝑚𝑎𝑥 = ∑ Ai

max

m

i=1

− ∑ Aimin

n

i=m+1

𝐴𝑈𝑚𝑎𝑥 = 11.43 + 40 − (49.38) = 2.05 𝑚𝑚

Nominal Dimension of the MemberN:

N = ∑ Ai

m

i=1

− ∑ Ai

n

i=m+1

N = (𝐴𝑈 + 𝐵4) − 𝐵5 = 1 + 31 − 19 = 13 𝑚𝑚

Bottom Limit Dimension of the Member𝑵𝒎𝒊𝒏:

𝑁𝑚𝑖𝑛 = ∑ Ai min

m

i=1

− ∑ Aimax

n

i=m+1

𝑁𝑚𝑖𝑛=0.38 + 31 − 19 = 12.38𝑚𝑚

Upper Limit Dimension of the Member 𝑵𝒎𝒂𝒙:

𝑁𝑚𝑎𝑥 = ∑ Ai max

m

i=1

− ∑ Aimin

n

i=m+1

𝑁𝑚𝑎𝑥 = 2.05 + 31.26 − 18.48 = 15.19 𝑚𝑚

Tolerance of Stroke Piston Pump𝑻𝑵:

𝑇𝑁 = 𝑁𝑚𝑎𝑥 − 𝑁𝑚𝑖𝑛 = 15.19 − 12.38 = 2.81 < 3 𝑚𝑚

The calculation shows that the tolerance field of piston stroke is less than 3 mm and thus it

corresponds to the technical requirements.

90

16 Assembly Organization

This chapter talks about individual assembly types from the point of view of their

organization. Assembly organization depends on the type and extent of production, labor

expenditure of the assembly itself and other factors. The reader will get familiarized with two main

ways of assembly - stationary and non-stationary.

16.1 Division of Assembly Types from the Organizational Point of View

Planning of the assembly organization depends on dimensions, weight, and shape complexity

of parts and products. The size of assembly series is not insignificant either. The basic division of

assembly can be made according to the degree that people and machinery participate in the

assembly process:

Manual assembly

Semi-automatic (mechanized) assembly and

Automatic assembly

Manual assembly is advantageous for its great flexibility to assembly conditions and low

assembly resource investment requirements. However, it employs more workers with low work

productivity. There may be ergonomic problems; work premises are not efficiently used, etc. Solution

and implementation of manual assembly automation require great costs, significant effort for

preparation starting with design modifications of parts and products.

Automated or semi-automated assembly makes sense for certain serialization of production

and depends on economic conditions of the production. Automated assembly is performed by

machines made for the purpose or custom designed equipment. According to assembly complexity,

assembly cells, special assembly machines or even assembly lines or robots are used.

In the case of smaller series, a higher technological and organizational assembly form can be

selected using technological standardization, and assembly units with the same or similar assembly

processes can be joined together according to so-called design and technological similarity to create

“type” assembly workplaces.

Implementation of automation into assembly enables:

Smooth assembly progress without intermediate storage of parts at workplaces

Lower amount of work-in-progress during assembly

Shortening of continuous assembly time

Implementation of specialized workplaces and workers

Better overview of the movement of assembly units at workplace

Joining assembly units with the same or similar assembly processes together

91

Joining of the same assembly character activities to create “type” assembly

workplaces

Increasing of time and performance utilization of work resources

Achieving of high work productivity with lower demands on worker qualification, etc.

The method and organization of assembly primarily depend on the production type and

extent, assembly labor expenditure, supply methods, etc. We recognize two basic assembly forms:

Internal assembly and

External assembly.

Internal assembly is performed within a given production plant and the product leaves the

production process usually ready for direct use (e.g. automobiles and consumer goods). On the other

hand, external assembly is executed outside of a production plant, when individual parts of

equipment that were previously internally assembled in production plants are assembled in a

prescribed sequence (e.g. during assembly of sizeable machines and equipment, bridges and

constructions, air conditioning, pipes and armatures). This is usually a stationary assembly.

We recognize two organizational forms of internal assembly according to the movement of

parts during assembly, degree of complexity and characteristic features of assembled product:

Immovable or stationary assembly (assembly work is supposed to be concentrated in a permanent working place):

- Concentrated

- Divided

- Stream

Mobile or non-stationary assembly (takes place in several concurrent assembly operations or groups performed by workers):

- Subject

- Line

The stationary assembly is typical for piecemeal or small series production. The non-

stationary assembly is suitable for small series, large series and mass production, where moving of

assembly workers around the product is minimal.

16.1.1 Concentrated Assembly

Concentrated assembly is done by connecting individual parts in one stationary workplace

and is usually done by one group of workers. It is used during assembly of heavy or large parts that

are assembled according to general assembly procedures without detailed time analysis of individual

activities.

92

Preparation,

Storage

Figure 77 – Concentrated assembly.

Practical use: Piecemeal or small series production (e.g. assembly of large machines -

excavators, stackers, etc.).

Disadvantages of the concentrated assembly are high demands on qualification of workers,

assembly areas, long continuous assembly periods, irregular progress of assembly, approximate

determination of time standards, etc.

16.1.2 Divided Assembly

Divided assembly progresses according to the work division principle. A product is assembled

in several stationary assembly workplaces at the same time. The assumption for this type of

internal assembly is the possibility to divide the product into individual pieces, subsets and systems

according to the assembly schematic and with regard to work volume within given assembly

operations. A time standard is created for assembly groups.

Preparation, Storage

Preparation, Storage

Figure 78 – Divided assembly.

93

The advantage in using divided assembly is the possibility of concurrent pre-assembly of

individual groups, for example when more products (like machine tools) are assembled in one

assembly hall, groups of workers move from one assembly group to another and the assembly takes

place in individual phases. The whole assembly is then represented by connecting parts, subsets and

systems into a finished product. It is used in small series production.

Practical use: small series production (e.g. milling machines, lathes, etc.).

16.1.3 Stream Assembly

The stream assembly is done in stationary assembly workplaces, where specialized groups of

workers perform certain parts of the assembly. These specialized groups of workers perform only

limited parts of work and move from one workplace to another. The scheme of stream assembly is

represented by Figure. Assembly work is divided all the way to operations or tasks. This type of

assembly is suitable for automation of assembly process due to its firm synchronized cycles of parts

transport.

Figure 79 – Stream assembly.

Practical use: large series production (e.g. production of roller bearings, measuring devices,

engines, transmissions, electric switches, etc.).

The advantage of this assembly organization is synchronization of individual workplaces from

the point of view of assembly activity volume.

16.1.4 Subject Assembly

Subject assembly is characterized by a free movement of the assembled object that moves

through individual workplaces. Workers perform only certain repeated operations with free cycles of

moving parts among stationary workplaces. The assembler workplaces are always appropriately

equipped for the assembly. This type of assembly is intended for small to large series production.

94

Figure 80 – Subject assembly.

Practical use: small to large series production (e.g. machine tools, construction machines,

locomotives, electrical motors, etc.).

16.1.5 Line Assembly

Line assembly is characteristic by a forced movement of the assembled object that is given

by a speed of the assembly line, while the sequence of assembly operations must be followed.

Sometimes it is called a continuous assembly (assembly organized on a line can be synchronized or

non-synchronized, depending on the way the product is output).

Practical use: large series production, e.g. pumps

The mobile assembly can be performed with periodic cycle and continuous movement. The

assembly cycle is a time interval between the assemblies of two finished products. This cycle is

regulated by the speed of the conveyor and kept using audio and light signalization.

The assembly cycle Tm in minutes can be calculated according to the following formula:

skm

60FT =

N, where

Fsk is the real hourly assembly time fund,

N is the yearly production of assembled products in units.

95

17 Assembly Lines

This chapter deals with classification of individual types of assembly lines according to

different criteria. When designing an assembly line, we do not need to consider economical criteria

only, but also ergonomy of human work as well as disadvantages of human participation in the

assembly.

17.1 Types of Assembly Lines

An assembly line can be characterized as a sum of workplaces arranged according to a

technological procedure that are connected by interoperation transport and intended for

performance of set operations during assembly of the whole product or its part. Assembly lines are

mostly divided according to the following criteria:

use of mechanization and participation of people in assembly:

o Manual lines

o Semi-automatic (mechanized) lines and

o Automated lines

way of movement of assembled product:

o Stationary lines

o Lines with moving product

Product moves only after an operation is finished

Product moves continuously

way of assembly work performance:

o Directly on a conveyor

o Outside of a conveyor

way of spatial arrangement:

o Simple lines

o Branched lines

degree of synchronization:

o Synchronized lines (continuous)

o Non-synchronized lines (interrupted)

assembly cycle:

o Lines with firm (fixed) assembly cycle

o Lines with free assembly cycle

number of assembled objects on the line:

o One object lines (uniform)

o Multi product lines (alternating)

At the beginning of the 20th century, the manipulation during assembly and the assembly

itself were exclusively manual. Later large series and mass production originated and production

96

lines with it, especially in the automotive industry. First lines were manual: workers performed

monotonously repetitive simple assembly tasks, later partially mechanized. The productivity was

good, but work conditions were unsatisfactory due to monotonous work. Substitution of this

monotony by simple assembly automats was revolutionary.

The first moving assembly line was used in 1913 in a car factory in Detroit. It was created by

Henry Ford. The line was used to connect individual assembly workplaces and workers were divided

along the line and performed assembly work in a set sequence. High qualification was not necessary.

Production character was:

Simplification;

Specialization;

Standardization.

Figure 81 – Assembly line in Ford factory.

17.2 Examples of Assembly Line Arrangements

The basic spatial arrangement of simple and branched assembly lines can be augmented by

further division. From the spatial arrangement point of view assembly lines can be further divided

according to:

occupation of assembly line sides:

o One-side lines

o Two-side lines

direction of line movement:

o One-way lines

o Two-way lines

97

placement of workplaces on the line:

o Side placement

o Front placement

Assembly lines with side workplace arrangement offer better possibility to use assembly

machines and large jigs during assembly. In this arrangement each worker’s workplace is defined and

well-arranged.

Assembly lines with frontal arrangement usually have smaller demands for work area and

allow manipulating with an object by both hands. The disadvantage is that only small jigs and

manually operated work tools can be used in these workplaces.

The two-side arrangement significantly saves space; on the contrary with the simple (one-

side) arrangement the demands for space are higher and often satisfied by long conveyor routes.

Figure 82 – Schema of one-side arrangement assembly line.

Figure 83 – Schema of two-side arrangement assembly line.

98

Figure 84 – Schema of two-side arrangement assembly line with one direct.

Figure 85 – Schema of two-side arrangement assembly line with both direct.

Figure 86 – Schema of two-side arrangement assembly line with head workplaces.

99

Figure 87 – Schema of two-side arrangement assembly line with side workplaces.

The arrangement of branched lines is very difficult especially for spatial reasons. These lines

are very sensitive to break-downs and are hard to adapt to production program changes. For the

above-mentioned reasons, they are implemented in mass production only, with only one type of

product and high yearly production volume. On the other hand multi-object lines are more adaptive

to sudden changes in a production program.

Preparation,

Storage

Non-

confirmation

Products

Figure 88– Schema of branched assembly line.

100

Synchronized assembly lines are characterized by a firm bond between individual

workplaces, regular rhythm of individual assembly workplace changes and the transport system. A

synchronized line cycle is derived from the assembly line with the longest assembly operation time.

Non-synchronized assembly lines are typical by their free binding among individual assembly

workplaces, so the cycle is partially free, and in the case of manual lines the work rhythm is set by the

operator. The disadvantage is that an interoperation supply of the assembled products usually needs

to be created. On the other hand these lines feature high amount of flexibility during a change of the

assembled product and relatively high performance along with a high level of work humanization.

101

18 Raviomalization of assembly

This chapter deals with rationalization of assembly process with a set of typical (sample)

assembly operations, sets of group assembly of similar products, comparison of possible solution

versions, virtual assembly and so on.

18.1 Rationalization of Assembly Process

A purposeful activity focused on increasing the technical and organizational level of the

assembly process, with maximum utilization of all existing resources, means, and material and

human elements. It is used for improvement of the existing assembly system and for design of new

assembly systems. Mostly it is focused on the assembly technological preparation and assembly

process.

Rationalization of assembly technological preparation lies in automatic creation of assembly

technological procedures using computers and:

A set of typical (sample) assembly operations - it requires necessary workplace equipment for a certain serialization of production (possibility of maximum use of unification, and standardization of individual assembly complexes), tools, time standards, appropriate qualification of workers

Sets of group assembly of similar products (groups of products) - utilization of a database with verified previous solutions, classification according to the main characteristics (shapes and dimensions)

Comparison of possible solution versions and selection of optimal solution

Virtual assembly - using appropriate software (e.g. Virtual Assembly) you can virtually try how to put individual parts together into higher assembly units of the proposed product, what sequence, procedure and tools to use using computer simulation of the process. It allows finding inaccuracies from the point of view of collision detection (one part interferes with a route of movement of another)

Assembly ability (a part cannot be inserted or taken out because it is bigger or there is not enough space for manipulation by handler, tools, etc. necessary for its assembly and attachment)

Interaction of design and assembly simulation programs enables immediate changes of

dimensions, shapes and removal of found shortcomings, then a prototype can be manufactured and

production prepared. By the use of various CAM programs, we can verify technological procedures

for various ways of assembly, after their selection we can generate optimized control programs to

control assembly lines, manipulators, robots, material transport and transport of semi-finished

products and parts.

Rationalization of assembly process includes:

Layout of the assembly workplace (dimensions of work areas, arrangement of individual objects at the workplace), working environment (lighting and air-conditioning), and equipping the workplace with tools and aids

102

Selection of a suitable kind of assembly activities (from the point of view of accuracy of assembly work, optimal division of work, optimal degree of mechanization and automation)

Organization of the assembly process (division of work among individual workplaces, specialization of workplaces, movement of workers and assembly units, etc.)

18.2 Accuracy of Manufacturing and Its Effect on Assembly Costs

A large share of labor expenditure during assembly goes to adjustment work. Its limitation, or

ideally complete exclusion, depends on the production quality and accuracy of the assembled parts.

Under accuracy, we need to imagine dimension and shape tolerances and positions of planes.

Selecting accuracy itself is a serious problem for each designer.

For assembly with complete exchangeability, parts are made with very narrow tolerances in

order to exchange them arbitrarily during assembly. Manufacture of such parts is comparatively

costly since special and exact tools, jigs and measuring devices are used for their manufacture.

Production demands for accuracy differ according to the product and production types. The

dimensional tolerance is a difference between upper and lower limit dimensions. The figure shows

the dependency between production and total costs on the proposed sizes of dimensional

tolerances.

The figure presents apparent hyperbolic increase in production costs for parts when their

dimension tolerances decrease and progressive increase in assembly costs during the increase of the

tolerances. The position of the minimum of the total cost curve (cumulative curve) depends on the

shape of both partial curves, namely the production cost and assembly cost ones. The minimum on

the cumulative curve determines the size of the optimal tolerance called economical respectively.

Figure 89 – Graph of production costs for parts for variable dimension tolerances.

103

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