Tolerance Treatments of Pairs and Repeatability in Assembly

VACLAV STEFAN, KAMENSZKA ADRIANA, MACHAC TOMAS

Institute of Production Technologies

Slovak University of Technology in Bratislava, Faculty of Materials Science and Technology in

Trnava

Jana Bottu 2781/25, 917 24 Trnava

SLOVAKIA

Abstract: In practical applications, statically predetermined pairs are frequently encountered. Designers use them

to improve product stiffness. This paper describes a method to reduce the effects of problematic assembly in

sliding pairs, which are often statically predetermined. Additionally, it addresses the crucial assembly problem

of repetitive accuracy, particularly in assembly line design.

Key-Words: Assembly, Kinematic Couples, Tolerance, Repeated Precision

Received: July 24, 2022. Revised: November 5, 2023. Accepted: December 7, 2023. Published: December 31, 2023.

1 Introduction

Assembly and assembly procedures are essential

elements of production. In many cases, the

production process is completed by the assembly,

which establishes the critical preconditions for the

reliability and quality of the product. Almost every

piece of engineering equipment consists of individual

components. A characteristic element of assembly

processes is the joining of two or more components

into sub-assemblies, groups and larger assemblies. A

variety of technologies are typically used to join

components, including those that provide a direct

connection without the need for additional

components or materials. In addition to the actual

joining, the assembly usually includes other activities

such as inspection, commissioning, conservation,

transportation of components to the assembly site,

and others [6,7].

The importance of assembly in the mechanical

engineering industry can be seen from the share of

assembly in the labour intensity of mechanical

engineering products, which averages 30 to 40 %. Of

the total number of employees in manufacturing,

about 30 to 50 % are employed in assembly. In high-

volume production, the share of assembly labour

decreases, which is mainly influenced by the

sophistication of the design, a higher degree of

mechanisation and automation of the assembly

process.

Therefore, it is necessary to actively address the

issue of assembly processes and look for ways to

reduce the associated costs, e.g. appropriate

structural design of the equipment and its division

into individual assembly groups and subgroups,

selection of simpler connection methods, selection of

such beddings that do not require fitting, use of

structural elements with a certain degree of freedom,

use of standardised and unified components and

others [9].

The paper is a contribution to the improvement of

the technological construction of product design

methodologies in terms of assembly, i.e. the

methodologies for the field of DFA (Design for

Assembly) [4]. The main objective of improving the

assembly process is mostly to reduce the unit cost per

product.

A systemic approach can be applied to achieve

continual improvement of all components of the

assembly system. This paper is aimed at improving

the components of assembled product and assembly

machines. Researchers all over the world focus

mainly on improving the elements of an assembled

product in the assembly system. Reduction of e.g. the

number of components may lead to a dramatic

decrease of the assembly laboriousness’ and

consequently also the assembly unit cost. The savings

can be thus achieved exclusively by brainpower

activities while incurring only a minor investment.

Such a methodology is necessary since the well-

known methods in this field are characterised by

excessive subjectivity of evaluators or by relativeness

of the results reflecting the current economic

situation.

DESIGN, CONSTRUCTION, MAINTENANCE

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Vaclav Stefan, Kamenszka Adriana, Machac Tomas

E-ISSN: 2732-9984

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Volume 3, 2023

This paper is aimed at developing an objective

method to increase the assembly product quality by

using the indicators calculated on the basis of

generally accepted laws of geometry, statics,

kinematics and dynamics, where assembly quality of

construction is assessed by objective indicators such

as a number of needed rotators and translators, the

necessary volume of rotations and translations,

power consumption, optimum dimension and

tolerance treatment, as well as other objective

indicators permanently associated with the

construction of the product, independently from

either the evaluators’ opinions or the current

economic situation in the country.

The objective of this paper is to use the basic

sciences (Mathematics, Mechanics, etc.) to

generalise the findings from practice through the

theoretical examination of the assembly process from

manufacturing the parts through their assembly,

testing and shipping.

2 Statics in Assembly

The structures are based on pairs, i.e. connections

of two bodies to each other. In Fig. 1 we can see a

basic overview of the most used engineering pairs,

which are: ball joint, rotary sliding pair, rotary pair,

sliding pair A, sliding pair B and screw pair.

This problem is also dealt with by prof. Whitney

who uses real shapes of solids to represent pairs, but

these are not compatible with the theory of statics

[13].

Doc. Valentovič recommends the use of spherical

models, which are compatible with statics, instead of

photographic images and drawings for the illustration

of engineering pairs and structures. Such an approach

is more convenient because it is clearer and in

accordance with statics terminology [11,12].

On (Fig. 1), the diagrams of pairs in the shape of

ball models and their brands, which we use when

drawing the whole structure, are listed, so that we do

not have to create it from ball models in a laborious

and complex way.

According to the principle of the spherical model,

bodies are considered to be perfectly rigid according

to statics, so they meet at single points instead of

surfaces.

Fig. 1 illustrates that the ball joint placed in the

spherical seat and shown by a spherical model is a

three-point joint (it touches the other object at three

points); the rotary sliding pair is a four-point pair; the

rotary pair is a pair that is prevented from moving

(sliding), so it is a five-point pair; sliding pairs A and

B are five-point pairs and also screw pair is five-point

pair. Next to the spherical models is a suggestion for

the use of markers.

Fig. 1 Basic space movable non-singular couples

with proposal for standardisation of signs

Such pairs are correct and can be assembled

without problems if they are formed according to the

spherical models given above. However, it is

important to note that these pairs can also be

incorrect. In particular, gear pairs require special

attention as they are generally considered to be

rolling.

However, this is not true because if a tooth of one

wheel engages in the gap of the other wheel at a given

axial distance according to the diagram (Fig. 2d), the

tooth will only make contact at one point in the gap.

Contact is made at two points only when the

wheel is relaxed and pushed into the other wheel, as

shown in Fig. 2e.

In particular, it is necessary to be careful about

this phenomenon, as the notion is generally used that

the gears always form a one-point pair, i.e. rolling.

In the case of rolling couples, it is important to

distinguish whether a ball rolls on a plane (Figure 2a),

a prism rolls on a plane (Figure 2b) or balls roll in a

V-groove (Figure 2c). It is obvious that in each case

there is a different number of contact points.

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Fig. 2 Movable rolling couples (a, b, c) and

tooth couples (d, e)

In practice, the cases of statically predetermined

couples are often used by designers to increase

product stiffness.

A ball bearing is a classic example of a

mechanism that should, theoretically, have a

maximum of three balls from a static point of view.

However, in reality, more balls are added to increase

the stiffness of the bearings.

Fig. 3 shows a column press which is statically

predetermined from a structural point of view.

When, as in this case, the support moves on two

longitudinal rollers, this pair is statically

predetermined. According to Grübler –

Valentovič´s equation [11, 12], the number of

degrees of freedom is minus two.

Fig. 3 Toleration treatment of statically

redetermined movable couples

Other conditions must also be considered for

trouble-free assembly, taking into account the fact

that these pairs are not parallel, but divergent or even

non-intersecting. In such cases, it is necessary to

leave enough space between the pairs so that they can

be assembled without problems, even if they are not

parallel. However, in this case, we lose some of the

stiffness.

Often it is necessary to connect two solids in the

assembly process, where it is obvious that it is a

statically predetermined kinematic pair.

There are similar cases with flange connections,

e.g. connection of a white flange with a black flange

that has four pins (Fig. 4a). We must quote both parts

so that this assembly is possible under all

circumstances.

Fig. 4 Assemblability of statically predetermined

unmovable couples: a, b - dimension method,

c - degrees of freedom number,

d - assemblability check

This implies the condition that the pins must be in

these holes at all times (Fig. 4d).

This can be achieved by quoting the pins (Fig. 4,

pos. 2), and also the holes (Fig. 4, pos. 1), separately

from the same base. We shall proceed in the same

way with the joints (Fig. 4b).

The advantage of this quotation method is that

manufacturing tolerances do not accumulate.

The presented pairs represent very strongly

statically predetermined structures, where the

expected number of degrees of freedom is one, due to

the stability of the displacement.

In fact, according to the formula given in Fig. 4c,

we calculate the number of degrees of freedom "is =

-10", which implies that the above system is statically

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Vaclav Stefan, Kamenszka Adriana, Machac Tomas

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Volume 3, 2023

9 times predetermined. However, with proper

quotation, a trouble-free assembly can be achieved.

After such processing, it is possible to produce the

above components in series with the assurance of

trouble-free assembly.

Fig. 5 presents an alternative method for

reducing the effects of problematic assembly in

very common sliding pairs, which are usually

statically predetermined. The sliding pair shown

in Fig. 5a is obviously statically predetermined

and we can reduce the consequences by first

assembling the sliding table with the rods and

then attaching them to the base (Fig. 5b). The

example in Fig. 5c demonstrates trouble-free

assembly with two rotary-sliding pairs (four-

point) on one shaft and a single-point pair

(touching from the left or right side) on the

other shaft. This structure is not statically

predetermined but determined, which as we

already know will ensure a completely trouble-

free assembly.

Fig. 5 Assemblability of movable couples [10]

a – statically predetermined structure, b – gradual

assembly of statically predetermined structure, c –

statically determined structure

If we do not need high precision on our product

there is the possibility of using so-called clearance

limiters. The principle is shown in Fig. 6.

Fig. 6 Assemblability of movable couples.

1 – frame, 2 – screws, 3 – extenders, 4 – bench

3 Statics Repeated Precision of

Assembly

Repetitive accuracy is a crucial aspect of

assembly, especially in the design of assembly lines.

The inaccuracy of the robot inserting the pin into the

hole and the hole's position in the drift may cause the

pin to fail to insert into the hole. This can happen for

two reasons.

The first cause of assembly failure is that, for

example, the robot that removes the pins from the

pallet has a certain repetitive positioning accuracy

(inaccuracy), i.e. when inserting the pins, their spikes

are not pointed or positioned at one and the same

point (position), but they fill a circle with eccentricity

"Ek" (Fig. 7a).

Fig. 7 Repeated accuracy of the robot and clamp

increases the probability of the insertion of the pin

into the hole [10]

a – robot and clamp, b – conditions of insertion, c –

chamfers improve the probability of insertion, d –

vibration improves the probability of insertion

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Vaclav Stefan, Kamenszka Adriana, Machac Tomas

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Volume 3, 2023

The second cause of assembly failure is that the

centre of the hole is not always in the same place

relative to the machine frame, but may be different in

each clamp. This is caused by the inaccuracy of the

carrier and the inaccuracy of the clamp, but also by

the inaccuracy of the component shape in the clamp

(Fig. 7a).

The combination of these causes can lead to

inaccuracies that accumulate. It is easy to

demonstrate, as shown in Fig. 7b, that if a pin with a

maximum radius of Rkmax is to fit into a hole, the

radius of the hole must be:

Rdmin = Ed + Ek + Rkmax (1)

where:

Rdmin – minimum radius of opening [mm],

Ed – eccentricity of opening [mm],

Ek – eccentricity of pin [mm],

Rkmax – maximum radius of pin [mm].

It is generally known from practice that with this

method of assembly, the clearance between the pin

and the hole must be disproportionately large. This

issue can be avoided by using the necking of both

components (Fig. 7c) and trying to use flexibility to

get the pin into the hole.

The second option is to place the pin against the

hole and make oscillating movements until it catches

the hole under slight pressure (Fig. 7d). However,

this method is more complicated.

4 Conclusion

We have just proved that the more accurate the

assembly line, and therefore the "better quality", the

lower the values of Ek and Ed will be. As shown in

Figure 7b, the assembly will be trouble-free under

this condition:

Rdmin = Ed + Ek + Rkmax [mm] (2)

or for the diameters:

(∅dmin)/2 = Ed + Ek + (∅kmax)/2

Ødmin = 2Ed + 2Ek + Økmax [mm] (3)

After supplying particular values, e.g. Økmax = 40.1

mm, Ek = 0.2 mm, Ed = 0.1mm: Ødmin = 0.2 + 0.4 +

40.1 = 40.7 [mm].

The clearance (0.7mm) is unacceptable, so we

will reduce the hole diameter. In practice, it is often

the clearance between the pin and the hole that has to

be unreasonably large for this type of assembly.

This issue can be avoided by using the necking of

both components (Fig. 7c) and trying to use

flexibility to get the pin into the hole.

Another option, but more complicated, is to place

the pin against the hole and move it in an oscillating

movement until the hole is caught in the pin with a

little pressure (Fig. 7d).

If these tasks are to be performed not only by

humans but also by machines, these devices must

have "artificial sight and feel", but these systems are

then very complex, which leads to an increase in the

cost of the assembly process.

The paper is a contribution to the improvement of

the assembly methods in the field of technological

construction of product design in terms of assembly

[5, 8] or in the area of methodologies known as DFA

(Design for Assembly) [3].

The general objective of improving the assembly

process is mostly a reduction of the unit cost per

product.

Reduction of the number of components may lead

to a dramatic decrease of the assembly laboriousness’

and consequently also of the assembly unit cost. The

savings can be thus achieved exclusively by

brainpower activities while incurring only a minor

investment.

The well-known methods in this field are however

characterised by excessive subjectivity of evaluators,

or by relativity of the results related to the current

economic situation.

This paper was therefore aimed at developing an

objective methodology to increase the assembly

product quality by using the indicators calculated on

the basis of generally accepted laws of geometry,

statics, kinematics and dynamics, where assembly

quality of construction is assessed by objective

indicators such as a number of required rotators and

translators, the necessary volume of rotations and

translations, power consumption, optimum

dimension and tolerance treatment, as well as other

objective indicators permanently associated with the

construction of the product, independently from

either the evaluators’ opinions or the current

economic situation in the country.

The methodology is not only a tool for evaluation;

it also reveals the causes of so-called “troublesome

assembly”, indicates the ways of problem elimination

and reduces the overall complexity and laboriousness

of assembly work.

This does not however mean that the known

methodologies [1, 2] should be ignored.

Further research will aim to make the effort to

improve the methods known under the abbreviation

DESIGN, CONSTRUCTION, MAINTENANCE

DOI: 10.37394/232022.2023.3.30

Vaclav Stefan, Kamenszka Adriana, Machac Tomas

E-ISSN: 2732-9984

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Volume 3, 2023

of the DFA (Design for Assembly) and verification

of the methodology with Artificial Intelligence.

Acknowledgement:

This paper was written in the framework of project

001STU-4/2022 entitled “Support of the distance

form of education in the form of online access for

selected subjects of computer aided study programs”.

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[13] WHITNEY, D. E., Mechanical assemblies, New

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors Stefan Vaclav, Adriana Kamenszka and

Tomas Machac prepared both the theoretical and

practical parts of the article together.

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Vaclav Stefan, Kamenszka Adriana, Machac Tomas

E-ISSN: 2732-9984

306

Volume 3, 2023

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

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