E D V A R D A S S A D A U S K A S

S U M M A R Y O F D O C T O R A L D I S S E R T A T I O N

K a u n a s2 0 1 5

M U T U A L P A R T A L I G N M E N T U S I N G

E L A S T I C V I B R A T I O N S

T E C H N O L O G I C A L S C I E N C E S , M E C H A N I C A L

E N G I N E E R I N G ( 0 9 T )

KAUNO UNIVERSITY OF TECHNOLOGY

EDVARDAS SADAUSKAS

MUTUAL PART ALIGNMENT USING ELASTIC VIBRATIONS

Summary of Doctoral Dissertation

Technological Sciences, Mechanical Engineering (09T)

2015, Kaunas

The research was accomplished during the period of 2010-2014 at Kaunas

University of Technology, Faculty of Mechanical Engineering and Design,

Department of Production Engineering and Department of Mechatronics.

Research was supported by Europe Structural Funds.

Scientific supervisor:

Prof. Dr. Habil. Bronius BAKŠYS (Kaunas University of Technology,

Technological Sciences, Mechanical Engineering – 09T).

Dissertation Defense Board of Mechanical Engineering Science Field:

Dr. Habil. Algimantas BUBULIS (Kaunas University of Technology,

Technological Sciences, Mechanical Engineering – 09T) – chairman;

Assoc. Prof. Dr. Giedrius JANUŠAS (Kaunas University of Technology,

Technological Sciences, Mechanical Engineering – 09T);

Prof. Dr. Habil. Genadijus KULVIETIS (Vilnius Gediminas Technical

University, Technological Sciences, Mechanical Engineering – 09T);

Prof. Dr. Juozas PADGURSKAS (Aleksandras Stulginskis University,

Technological Sciences, Mechanical Engineering – 09T);

Prof. Dr. Habil. Arvydas PALEVIČIUS (Kaunas University of Technology,

Technological Sciences, Mechanical Engineering – 09T).

The official defense of the dissertation will be held at 10 a.m. on 30th

of June,

2015 at the Board of Mechanical Engineering Science Field public meeting in

the Dissertation Defense Hall at the Central Building of Kaunas University of

Technology.

Address: K. Donelaičio st. 73 – 403, LT-44029, Kaunas, Lithuania,

Phone nr. (+370) 37 300042, Fax. (+370) 37 324144, e-mail:

doktorantura@ktu.lt

The summary of dissertation was sent on 29th

of May, 2015.

The dissertation is available on internet (http://ktu.edu) and at the library of

Kaunas University of Technology (K. Donelaičio st. 20, LT-44239, Kaunas,

Lithuania).

KAUNO TECHNOLOGIJOS UNIVERSITETAS

EDVARDAS SADAUSKAS

DETALIŲ TARPUSAVIO CENTRAVIMAS NAUDOJANT

TAMPRIUOSIUS VIRPESIUS

Daktaro disertacijos santrauka

Technologijos mokslai, mechanikos inžinerija (09T)

2015, Kaunas

Disertacija rengta 2010-2014 metais Kauno technologijos universitete,

Mechanikos ir dizaino fakultete, Gamybos inžinerijos ir Mechatronikos

katedrose. Moksliniai tyrimai finansuoti Europos struktūrinių fondų lėšomis.

Mokslinis vadovas:

Prof. habil. dr. Bronius BAKŠYS (Kauno technologijos universitetas,

technologijos mokslai, mechanikos inžinerija – 09T).

Mechanikos inžinerijos mokslo krypties daktaro disertacijos gynimo

taryba:

Habil. dr. Algimantas BUBULIS (Kauno technologijos universitetas,

technologijos mokslai, mechanikos inžinerija – 09T) – pirmininkas;

Doc. dr. Giedrius JANUŠAS (Kauno technologijos universitetas, technologijos

mokslai, mechanikos inžinerija – 09T);

Prof. habil. dr. Genadijus KULVIETIS (Vilniaus Gedimino technikos

universitetas, technologijos mokslai, mechanikos inžinerija – 09T);

Prof. dr. Juozas PADGURSKAS (Aleksandro Stulginskio universitetas,

technologijos mokslai, mechanikos inžinerija – 09T);

Prof. habil. dr. Arvydas PALEVIČIUS (Kauno technologijos universitetas,

technologijos mokslai, mechanikos inžinerija – 09T).

Disertacija bus ginama viešame Mechanikos inžinerijos mokslo krypties

tarybos posėdyje, kuris įvyks 2015 m. birželio 30 d. 10 val., Kauno technologijos

universitete, Centrinių rūmų disertacijų gynimo salėje.

Adresas: K. Donelaičio g. 73 – 403, LT-44029, Kaunas, Lietuva

Tel. (8 - 37) 300042, faksas (8 - 37) 321444, e. paštas doktorantura@ktu.lt

Daktaro disertacijos santrauka išsiųsta 2015 m. gegužės 29 d.

Disertaciją galima peržiūrėti internete (http://ktu.edu) ir Kauno technologijos

universiteto bibliotekoje (K. Donelaičio g. 20, LT-44239, Kaunas, Lietuva).

5

Introduction

Automatic assembly systems plays vital role in automating production

process. They directly affect production efficiency and quality of the goods.

According to the statistical analysis, 30-60% of the tasks in most of the

industries branches are assembly operations. Part assembling time takes 35-40%

of all manufacture time. Around 33% of all assembly operations are peg to bush

assembly operations. Because of that, assembly operation has big potential in

reducing manufacture time by improving assembly methods and installing

automatic part assembly systems and devices.

Assembly processes considering the level of automation sorted to several

categories. First is manual assembly when a worker uses tools, worktable,

grippers, conveyors etc. to perform traditional assembly operations. Second is

mechanised assembly, when workers use variety of power tools (impact wrench,

press etc.). In the third category a specialized automatic devices designated only

for the particularly assembly operation are used. Devices can be readjusted to

produce several types of products. This type of assembly used in making

different products in a big series. Assembly type of the fourth category

incorporates PLC (programmable logical controller) to control processes of

separate assembly line modules. Fifth - is an adaptive assembly system. The

process control system uses feedback signal to operate assembly equipment at

the different stages of the part assembly.

The main progress of automatic assembly is a robotic system, which

accommodates programmable assembly devices, robots and manipulators.

Because of geometrical tolerances of the parts, inappropriate basing of the parts,

tolerances of the robot/manipulator positioning, linear and angular mismatch of

the assembled parts may occur. To compensate those inaccuracies manufactures

uses passive or active part alignment methods.

This work investigates a new approach of passive vibratory part alignment

method using elastic vibrations. In this method bush placed on the assembly

plane and is free to move in a narrow space. Another component (peg) fixed in a

gripper, which has piezoelectric vibrator in it. Vibrator presses upper end of the

peg. Peg and a bush also pressed to each other with a predetermined force.

Piezoelectric vibrator generates high frequency harmonic excitation to the peg

and creates elastic vibrations of the peg in longitudinal and lateral directions. The

lower end of the peg moves in elliptical shape trajectory. Because of the friction

force between the components, bush moves to the part alignment direction. Parts

successfully assembled after the alignment occurs. This passive alignment

method allows assembling parts with circular and rectangular cross-section with

no chamfers and at their axial misalignment of few millimetres, or makes it

possible to use low accuracy robots with repeatability value of ±1-2 mm. A

vibratory part alignment device that uses elastic vibrations is more simple

6

technologically since it does not use feedback signals or sophisticated control

algorithms. Such alignment system with proper chosen excitation signal

parameters provides reliable, more efficient and cost effective part assembly

comparing to active alignment systems.

Aims and objectives of scientific research. Research objective –

theoretically and experimentally investigate vibratory part alignment in

automatic assembly when using elastic vibrations of the peg. Determine

excitation parameters for the stable and reliable part alignment. To achieve those

objectives following tasks has to be fulfilled.

Analyse scientific papers about widely used part alignment methods in a

now days industry.

Carry out experimental research of the peg’s tip vibration while he is in

a contact with bush. Determine nature of the peg’s vibrations, their relationship

to the excitation signal amplitude and bush-to-peg pressing force.

Perform part alignment experiments with circular and rectangular cross-

section pegs using their elastic vibrations. Determine influence of excitation and

mechanical system parameters to the alignment efficiency and reliability.

Compose mathematical model of circular part alignment when the peg

excited in axial and transversal direction. Determine excitation signal and

mechanical system parameters for the stable and reliable part alignment at

impact and non-impact modes.

Methods of research. Numerical and experimental methods used in this

work. Peg and bush movement expressed by the system of second order

differential equations and solved by Runge-Kuta method in Matlab. Movement

of the movable component (bush) is modelled. Obtained results represented in a

form of graphs and shows influence of excitation and dynamic system

parameters to the bush motion. Special experimental set-up designed for the

vibratory part alignment. Experiments performed with parts of circular and

rectangular cross-section and made from steel and aluminium. The peg fixed in a

gripper and vibratory excitation done to the upper end of the peg in a

longitudinal direction by mean of piezoelectric vibrator. Low frequency

generator Г3-56/1 provides excitation signal to the circular shape piezo ceramic

CTS-19. Bush and a peg alignment performed at different excitation signal

parameters, peg-to-bush pressing force and misalignment distance between the

parts. Oscilloscope PicoScope 4424 and computer Compaq nc6000 measures

alignment duration. Laser dopler vibrometer OFV512/OFV5000 used in peg’s

lateral and longitudinal vibration measurements.

Scientific novelty. The new scientific data revealed during preparation of

the thesis:

7

1. Technologically easier way for vibratory part alignment using peg’s

elastic vibrations was proposed. Piezoelectric vibrator presses upper end of the

peg and excites it in longitudinal direction.

2. As peg excitation done on one end, the other end performs elliptical

shape motion. Friction forces arise by pressing peg and bush to each other and

ensure linear and rotational motion for the bush.

3. Vibratory part alignment using elastic peg’s vibrations allows of

centring circular and square cross-section parts at non-impact and impact modes

when there is mechanical contact between them.

4. Was done peg-bush alignment simulations at non-impact and impact

modes and alignment duration dependencies on excitation frequency, amplitude,

initial pressing force were determined.

Practical value. Part alignment using peg’s elastic vibrations allows

centring circular and rectangular cross-section parts with chamfers and without it

and at axial misalignment error of several mm between the components. The

proposed method expands technological capabilities of automatic assembly. Data

collected in theoretical and practical research are useful in design and

development of vibratory devices and systems.

Scope and structure of the dissertation. Dissertation consists of

introduction, three chapters, conclusions, references and the author’s publication

list. The text of dissertation comprises 90 pages, 61 figures and two tables.

Propositions to be defended:

1. New vibratory part alignment method is technologically easier method

since it does not require feedback signal.

2. Peg’s end tip moves in elliptical shape trajectory and friction forces that

rises in a contact point between the parts provides linear and rotational motion to

the bush.

3. Nature of the bush motion and alignment duration depends on

frequency and amplitude of peg’s vibrations, phase shift between longitudinal

and lateral components, initial peg-to-bush pressing force, and axial

misalignment between the parts.

4. Mathematical models sufficiently good describe real vibratory part

alignment system and theoretical trend of part alignment duration dependencies

correlates with experimental ones.

1. Literature review

More and more companies use robotic or automated assembly lines to

ensure product quality and reduce production costs. For the parts (peg and bush)

to be assembled their connection surfaces has to match.

8

The principal problem in automated assembly is the uncertainties between

mating parts due to various errors. These are systematic positioning errors of

robot or other manipulating device as well as errors due to insufficient control

system resolution or due to vibrations in assembly area, etc. Uncertainties also

result from random dimensions of assembled parts, their basing and fixing on a

worktable or manipulation device. All those errors lead to inaccuracies between

relative position of mating parts and prevents them from joining with each other.

Alignment is the most important stage in automated assembly process, which

compensates position offset between mating parts. Mutual part alignment carried

out in many ways: by interaction between part and a chamfer or other guiding

element, using auto search method or active compliance control devices.

Using auto search methods mating parts makes translational and rotational

motion in a plane perpendicular to the joining axis until their connecting surfaces

matches. Auto search categorized into three types: non-directional search

(without feedback), directional search with feedback, directional search without

feedback (but with vibration assistance). Alignment devices with feedback

signals classify as active alignment methods. Passive alignment methods do not

use feedback signal. There is no need of feedback signal if directional motion of

movable based part done by mean of vibrational excitation.

Principal of vibrational part alignment lies in vibrational displacement effect

inherent to the nonlinear asymmetric mechanical systems. Structural, kinematic,

force etc. asymmetry might emerge in nonlinear mechanical systems. Force and

kinematic asymmetry is most common in vibratory alignment. Force asymmetry

originates from the rotational motion of the part as the components rest to each

other. Vibrational excitation of the turned part causes kinematic asymmetry.

Vibrational non-impact displacement of a mobile-based body on an inclined

plane analysed by researches B. Baksys and N. Puodziuniene [1, 2] from

Kaunas University of Technology.

Peg - hole alignment under axial peg vibrational excitation investigated by B.

Baksys and J. Baskutiene [3]. It was determined that alignment duration

depends on excitation frequency, amplitude and system stiffness parameters.

Immovable based bush also might be excited in axial direction or in two

perpendicular directions in a vertical plane.

B. Baksys and K. Ramanauskyte [4] investigated mutual part alignment

using vibratory auto search method. Horizontal vibrating plane provides search

motion for the part on it. Plane excited in a two perpendicular directions can

generate circular, elliptical or interwinding helix search paths. Motion of the

unconstrained and elastically and damping constrained part pressed with constant

and varying pressing force was investigated.

Mutual part alignment by directional vibrational displacement performed in

the same way as linear and rotational motion of the output link in the ultrasonic

motors. M. E. Archangelskyj [5] investigated oscillations of the cascade steel

9

vibrator. It was found that generating high frequency (17.7 KHz) axial vibrations

at the end of the vibrator, other end oscillates in axial and transverse directions.

Because of the phase shift between vibration components tip of the vibrator

makes elliptical motion. The brass disk starts to rotate around its axes when it

touches rear or end surfaces of the vibrator, thus indicates about periodically

intermittent mechanical contact between disk and vibrator. Authors N. Mohri

and N. Saito [6] investigated effect of lateral and longitudinal vibrations to the

part insertion. It was determined that high frequency lateral vibrations reduces

dry friction coefficient between rear surfaces of the mating parts and facilitates

part insertion. If the peg excited in longitudinal and lateral directions a motive

force generated for the parts mating.

To expand technological capabilities of automated assembly, to simplify and

reduce cost of assembly equipment a new vibratory alignment method presented.

The novelty of this method is the use of elastic high frequency vibrations. A

friction force that rises in a contact point between components provides linear

and rotational motion to the bush and directs it to the alignment direction.

Vibrational displacement makes possible alignment and joining of the parts

without chamfers and with axial part misalignment of few millimetres. Method is

suitable for the parts with circular and rectangular cross-section. A

comprehensive theoretical and experimental research presented in this work as

there is no scientific papers on this alignment method.

2. Experiments

To investigate peg’s vibrations while it is in a contact with bushing the

following experimental equipment has been used (Fig. 2.1). Peg 4 fixed in a

middle cross-section in a gripper 1. Piezoelectric vibrator 2 pressed to the upper

end of the peg with pressing force F2 and excites peg in axial direction.

Excitation signal to the vibrator provided by signal generator 3. The lower end of

the peg is pressed to the bushing 5 with initial pressing force F1 while axis

misalignment Δ.

One axis laser dopler vibrometer (LDV) used to register peg’s vibrations. The

interferometer head OFV512 measures vibrations and controller OFV5000

coverts signal from interferometer to the voltage signal corresponded to vibration

amplitude. Further signal captured with oscilloscope PicoScope 4424 and

displayed on a computer screen.

10

+Δ −Δ

Peg

Bush

Y X

Fig. 2.3 Bush placement in respect to the peg

a) b)

Fig. 2.1 Experimental setup: a – measurement scheme: 1 – gripper; 2 – piezoelectric

vibrator; 3 – excitation signal generator Г3-56/1; 4 – peg; 5 – bushing; 6 – fiber

interferometer OFV512; 7 – vibrometer controller OFV5000; 8 – oscilloscope

PicoScope4424; 9 – personal computer PC; zi – longitudinal vibrations; xi, yi – lateral

vibrations; b – measurement equipment

Vibrometer measurements were taken in a three directions X, Y, Z. Where X,

Y corresponds to lateral vibrations in a two perpendicular directions and Z are

longitudinal vibrations.

Axial misalignment of the

parts –Δ and +Δ lies on X-

axis (Fig. 2.2). Thus mutual

part alignment occurs when

bushing center coincides

with coordinate axes center.

Peg’s tip vibration

magnitude was investigated

under different pressing

forces F2 (vibrator to the

peg) and F1 (peg to the

bushing) when excitation

frequency f varies from 6523 to 6723 Hz, and tip movement trajectory was

defined in relation with misalignment position Δ.

In order to find movement trajectory of the peg’s end tip, measurements of

two perpendicular axes (X-Y, Z-Y, Z-X) were taken. Synchronization signal

related to the excitation signal synchronizes measurement process. As long as

vibrations are periodic and steady, vibrations magnitude (xi, yi, zi) of each axis

OFV5000 OFV512

Г3-56/1

PicoScope4424

PK

1

2 3

4

5 6 7

8

9

Z

Y

X

zi

yi

xi

π/2

F2

F1

+Δ

O

11

defined at the same periodic

time τi according to the

synchronization signal (Fig.

2.3). Plotting those values in

a Cartesian coordinate

system peg’s path in all three

planes found. Time interval

between two vibration

signals at the same

instantaneous phase gives us

phase difference ε between

those signals. Excitation

parameters and objects of

experiments presented in

Table 2.1.

Table 2.1, Characteristics of excitation signal and aligned parts

Influence of forces F1 and F2 to vibration amplitude was investigated on peg

No. III. Pressing force of piezoelectric vibrator to the peg was gradually

increased every 14 N and corresponding measurements of vibration magnitudes

on all three axes were taken. The results presented on figure 2.3 and 2.4.

Force F2 and excitation frequency has no impact on vibration magnitude

along axis X. Vibration amplitude in Y-axis direction gradually increases when

pressing force reaches 49 N and later stabilizes at 115 N. Meanwhile overall

vibration magnitude decreases as excitation frequency increases. Amplitude of

longitudinal vibrations increases more rapidly after F2 exceeds 90 N until that

growth relatively small. Such character of amplitude increment related with

contact area changes between peg and piezoelectric vibrator. More force is

applied bigger micro deformations between peg and vibrator thus bigger contact

area and more excitation energy transferred to the peg. Since peg excited with

vibrations of high frequency and small amplitudes, contact area between peg and

No. I II III

Peg Steel S235JR

Diameter, mm 10 10 10

Length, mm 59.8 79.65 99.75

Chamfers No

Bush Steel S235JR

Hole diameter, mm 10.1 10.1 10.1

Excitation signal parameters

Frequency, Hz 8475 6711 6623

Amplitude, V 132

0 τi 0.05 0.1 0.15 τi,

ms

0.2

xi

0.1

yi

0

zi

-0.1

-0.2

Vib

rati

on

am

pli

tud

e, µ

m 1

2

3

4

εzy

εxy

εzx

1

0

-1

Sy

nch

ron

izat

ion s

ign

al,

V

Fig. 2.3 Vibration signals and phase

difference ε: 1 – synchronization signal; 2 – Y

vibrations; 3 – X vibrations; 4 – Z vibrations

12

vibrator plays vital role. This could be seen from graph 1 and 2, as excitation

frequency increases pressing force F1 also has to be increased to keep same

longitudinal vibration amplitude. It was also experimentally set that mutual part

alignment starts when force F2 exceeds 90 N, until that process of part alignment

is not stable or it does not work at all.

Fig. 2.3 Vibration amplitude versus force

F2: longitudinal Z vibrations: 1 – f=6523

Hz; 2 – f=6623 Hz; lateral Y vibrations: 3 –

f=6523 Hz; 4 – f=6623 Hz; 5 – f=6723 Hz;

lateral X vibrations: 6 – f=6523 Hz; 7 –

f=6623 Hz; 8 – f=6723 Hz;

Fig. 2.4 X vibration amplitude versus

force F1

If force F2 had no impact on vibrations in X-axis, totally different impact had

force F1. As the peg is pressed to the bushing with axis misalignment Δ=+1.5

mm, vibration amplitude gradually increases as force F1 increases. The same

tendency retains even if excitation frequency changes in range from 6523 to

6723 Hz (Fig. 2.4). In our case, part alignment is most rapid when peg is excited

at 6623 Hz frequency and vibrations in X-axis are the biggest.

Experiment results mentioned above in generally shows what influence for

the vibrations amplitude has mounting conditions of the peg, and that excited peg

vibrates in three directions perpendicular each other. However, there still no

answer why bushing is slides toward coordinate axes centre. To find out what

factors in charge of this effect, motion trajectory and direction of peg’s tip was

determined.

After excitation frequency for stable and steady part alignment was

experimentally set to all pegs (Table 2.1), motion trajectory of the tip was taken

in all three coordinate planes. Excitation frequency mainly depends from the

peg’s natural frequency, design of the gripper and force F1. Thus for the grippers

with different design or made from different material excitation frequency for

steady and stable part alignment will be different. In our case, excitation

frequency for stable and steady part alignment have lied between second and

0.4

0.2

0

Vib

rati

on

am

pli

tud

e, µ

m

29 49 69 89 109 F2, N

1 2

3

4 5

6 7 8

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02 X

am

pli

tud

e, µ

m

0 0.5 1.0 1.5 2.0 F1, N

F2=10

1 N 6623, Hz

6523, Hz

6723, Hz

13

0.1

0.05

0

-0.05

-0.1

0.1

0.05

0

-0.05

-0.1

0.2

0.1

0

-0.1

-0.2

-0.05 0 0.05 -0.05 0 0.05 -0.1 0 0.1

Peg I Peg II Peg III

Y, µm

Z,

µm

0.05

0

-0.05

0.03

0.02

0.01

0

-0.01

-0.02

-0.03

0.05

0

-0.05

Peg I Peg II Peg III

0.02 0 -0.01 0.01 0 -0.01 -0,02 0.02 0 -0.01

X, µm

Y,

µm

third natural bending mode of the peg. Figure 2.5 shows peg’s tip path while

forces F2=101 N, F1=0 N

a) b)

Fig. 2.5 Path trajectory of unloaded peg: a – in YOX plane; b – in ZOY plane.

Longitudinal vibrations are dominant in all cases and are twice as high as

transverse ones. While in YOX plane they polarized in Y direction since peg’s

vibrations in X direction are negligible.

When the peg is pressed to the bushing with the force F1=2.2 N and axis

misalignment Δ=-1.5 mm, lateral vibrations on X axis increases significant and

peg’s end moves in elliptical shape trajectory in all three coordinate planes (Fig.

2.6). Black dots on the path indicate its direction. For the different pegs,

direction of rotation is different, that depends from excitation frequency, and

natural mode gripper–peg system vibrates.

In order the alignment of the parts could occur, bushing has to slide along

positive X direction. There are two ways how bushing aligned. First is direct

alignment (Peg II and III). In this case peg’s tip moves counter-clockwise in

ZOX plane (Peg II and III, b), thus direction of the normal force in the contact

point lies on the positive X direction and bushing is directly pushed toward

coordinate axes center. Vibrations along Y-axis has little effect since their

amplitude smaller than X, and overall vibrations are more polarized along X-axis

(Peg II and III, a). Normal peg to bushing pressing force is bigger when

longitudinal vibrations amplitude is negative. Thus, propellant force is bigger

when peg vibrates along positive X-axis rather than negative.

Second way of part alignment is indirect alignment (Peg I). Here peg’s

motion is clockwise in ZOX plane (Peg I, b) and bushing is pushed from the

coordinate axes center. However, because of the peg’s tip elliptical movement in

14

0.1

0

-0.1

0.1

0

-0.1

0.1

0

-0.1

Y,

µm

Z,

µm

Z,

µm

-0.1 0 X, µm -0.1 0 X, µm -0.1 0 Y, µm

εxy=-0.02 εzx=-2.07 εzy=-2.05

εxy=0.22 εzx=-2.95

εzx=-1.11 εxy=2.91

εzy=3.11

εzy=2.26

I

II

II

I

-0.15 0 X, µm -0.08 0 X, µm -0.15 0 Y, µm

-0.1 0 X, µm -0.1 0 X, µm -0.1 0 Y, µm

0.15

0

-0.15

0.1

0

-0.1

0.08

0

-0.08

0.1

0

-0.1

0.1

0

-0.1

0.1

0

-0.1

Y,

µm

Z,

µm

Z,

µm

Z,

µm

Z,

µm

Y,

µm

YOX plane (Peg I, a), bushing is turned by the angle so the alignment trajectory

lie on the major axis of the ellipse and then pushed towards coordinate axes

center.

a b c

Fig. 2.6 Path trajectory of loaded peg when Δ=-1.5 mm: a – in YOX plane; b – in ZOX

plane; c – in ZOY plane

Results of peg’s tip trajectory while Δ=+1.5 mm presented in figure 2.7. As

contact conditions between peg and bushing has changed (contact area crescent

now faced to opposite side), phases between vibrations also changed. In this case,

for the bushing to align with the peg, bushing has to slide along negative X

direction. Peg’s I and II tip moves clockwise in a ZOX plane (Peg I and II, b)

thus direct alignment is going.

Bushing with the Peg III aligned during indirect alignment. The bush is

propelled along negative X direction, but because of rotation effect in YOX

plane (Peg III, a) bushing is turned and pointed to the coordinate axes center.

It is clear that during direct alignment peg’s motion in ZOX plane plays key

role, meanwhile during indirect alignment there is combination of peg’s

movement in ZOX and YOX planes.

15

a b c

Fig. 2.7 Path trajectory of loaded peg when Δ=+1.5 mm: a – in YOX plane; b – in ZOX

plane; c – in ZOY plane

Experimental setup designed and made to investigate part alignment when

elastic vibrations applied to the peg (Fig. 2.8). The peg fixed in a gripper 8.

Gripper can move in vertical direction in order to insert peg into the bush when

alignment occurs. Spring 7 works as gravity force compensator for the gripper

and helps to capture the moment as the peg falls into the bush hole. Vertically,

moving table 6 adjusts pressing force of mating parts. The table moved

horizontally in order to change axis misalignment, which is measured with

indicator 9. Low frequency signal generator 3 provides signal to the piezoelectric

vibrator. The amplitude and frequency of the signal are measured by multimeter

1. Switch 5, oscilloscope 4 and personal computer 2 are used for alignment event

triggering and alignment duration measurement respectively.

0.1

0

-0.1

0.1

0

-0.1

0.1

0

-0.1

Y,

µm

Z,

µm

Z,

µm

-0.1 0 X, µm -0.1 0 X, µm -0.1 0 Y, µm

εxy=1.53 εzx=1.28 εzy=-0.25

εxy=2.55

εzx=0.59

εzx=3.09 εxy=2.60

εzy=-1.97

εzy=0.49

III

I

I

I

-0.08 0 X, µm -0.08 0 Y, µm -0.06 0 X, µm

-0.15 0 X, µm -0.2 0 X, µm -0.2 0 Y, µm

0.06

0

-0.06

0.2

0

-0.2

0.08

0

-0.08

0.2

0

-0.2

0.08

0

-0.08

0.15

0

-0.15

Y,

µm

Z,

µm

Z,

µm

Z,

µm

Z,

µm

Y,

µm

16

3

2

1

4

5

6

13 8

Δ 12 10 F1

F2

7

11

9

Fig. 2.9 Measurement circuit: 1 – Peg; 2 –

piezoelectric vibrator; 3 – housing; 4 – bush;

5 – plate; 6 – force sensor; 7 – 9 V power

supply; 8 – switch; 9 – light-emitting diode

(LED); 10 – oscilloscope; 11 – signal

generator; 12 – computer; 13 table

Fig. 2.8 Experimental setup: 1 – multimeter FLUKE 110; 2 – computer Compaq nc6000;

3 – signal generator Г3 – 56/1; 4 – oscilloscope PicoScope 4424; 5 – switch; 6 – table; 7 –

spring; 8 - gripper; 9 – indicator BDS Technics

The peg 1 is hold in a middle

cross-section by the clamps of

the gripper (Fig. 2.9).

Piezoelectric vibrator 2 is

implemented in a housing 3.

Threaded end of the housing can

freely rotate in a gripper at the

same time performing linear

motion towards the peg.

As the piezoelectric vibrator

lean to the peg, further torque

increment sets pressing force for

the piezoelectric vibrator to the

peg. Bush 4 is mobile based on

the electrically conductive plate

5 while the latter is located on

the force sensor 6. The bush,

plate, and force sensor fixed to

the table 13 and moves together.

The following electrical

circuit was designed to measure

the alignment time. Anode of the power supply 7 connected to the electrically

1 2 3 4 5 6 7 8 9 10

17

conductive plate. Cathode first connected to the switch 8 and LED 9 and later to

the gripper. Oscilloscope 10 is measuring voltage signal on the LED. When the

switch closes electrical circuit, the voltage jump on the LED occurs. At the same

time, excitation signal from generator 11 connected to the piezoelectric vibrator

6. As the bush slides to the peg’s center, contact resistance alternating and

electrical signal has unstable manner. When alignment between peg and the bush

occurs, there is no mechanical contact between them and the voltage jump on the

LED is the lowest. Measured signal transferred to the computer 12 and by mean

of the software alignment time is calculated.

During investigation, the peg is excited in axial direction by mean of

cylindrical shape piezoelectric vibrator with 30 mm in diameter and 13 mm in

height. Pressing force vibrator-to-peg is set to 101 N and kept constant

throughout the experiments. Harmonic excitation signal generated by low

frequency generator. Each time experiment repeated four times and a mean value

of four trials is taking as a result. Influence of axis misalignment Δ, excitation

frequency f, excitation signal amplitude U and initial peg-to-bush pressing force

F1 to the alignment duration Δt is investigated. Experiments were carried out

with steel and aluminium pegs with circular (C) and rectangular (R) cross-

sections and their counterparts steel and aluminium bushings. The alignment of

rectangular parts was done along short side of the peg. The parts were both type

with chamfers and with no chamfers. Measurements of the parts used in

experiments are given in Table 2.2

Table 2.2, Material and geometrical data on specimens

The dependencies of alignment duration Δt on axis misalignment Δ is

presented in figure 2.10. Steel peg I excited under different excitation frequency

No. Peg Bush Cross-

section

Chamfers

Steel S235JR

Diameter, mm Lengh, mm Diameter, mm No

I 10 99.75 10.1 C

II 10 79.65 10.1 C

III 10 59.8 10.1 C

IV 7.95 99.85 8.05 C

V 5.95 99.6 6 C

Aliuminium SAPA6082-T6

VI 10 99.95 10.05 C No

VII 10 99.95 10.05 C 0.55x43º

Steel S235JR

Lengh x Widh x Heigh, mm Lengh x Widh, mm

VIII 10.1x5.3x99.3 10.4x5.45 R Nėra

IX 10.05x5.1x99.5 10.4x5.45 R 0.33x49º

18

and initial pressing force F1. Excitation signal amplitude U=142 V is same to all

investigated pegs.

It is determined that alignment duration increases as axis misalignment

increases. The character of a graph is linear and do not depend on excitation

frequency. We can also see that alignment duration depends on misalignment

direction. In a direct alignment case the alignment duration is shorter (Fig 2.10,

a). However, excitation frequency has significant influence to the alignment

duration (Fig. 2.11).

a) b)

Fig. 2.10 Alignment duration dependencies on axis misalignment Δ: a) bush placement

+Δ, b) bush placement –Δ; 1 – f=7000 Hz; 2 – f=7050 Hz; 3 – f=7100 Hz; 4 – f=7150 Hz;

5 – f=7200 Hz

a) b)

Fig. 2.11 Alignment duration dependencies on excitation frequency f: a) bush placement

+Δ, b) bush placement –Δ; 1 – Δ=0,4 mm; 2 – Δ=0,6 mm; 3 – Δ=0,8 mm; 4 – Δ=1,0 mm;

5 – Δ=1,5 mm; 6 – Δ=2,0 mm; 7 – Δ=2,5 mm; 8 – Δ=3,0 mm; 9 – Δ=3,5 mm

0,0

1,7

3,3

5,0

7000 7050 7100 7150 7200

Δt, s

f, Hz

F1=2.2 N

7

6

5

4

3

2

1

9

8

5.0

3.3

1.7

0.0 0,0

0,5

1,0

1,5

7000 7050 7100 7150 7200

Δt, s

f, Hz

F1=2.2 N

5 6 7 8 9

1 2 3 4

1.5

1.0

0.5

0.0

0,0

0,5

1,0

1,5

0,4 1,4 2,5 3,5

Δt, s

Δ, mm

F1=2.2 N 1

4 2

5

3

1.5

1.0

0.5

0.0 0.4 1.4 2.5

0,0

1,7

3,3

5,0

0,4 1,4 2,5 3,5

Δt, s

Δ, mm

F1=2.2 N

1

2

3

4

5.0

3.3

1.7

0.0 0.4 1.4 2.5

19

The alignment of the parts is most rapid when excitation frequency is

between 7050-7100 Hz and this trend visible under different axis misalignment.

As frequency changes from these values, alignment duration increases. It was

also determined that for a small axis misalignment (up to 1 mm) the influence of

excitation frequency is negligible.

Excitation frequency at which alignment of the parts is most rapid increases

as geometrical dimensions of the peg decreases. However, size of the peg is not

the only reason of frequency changes. Contact quality between peg and

piezoelectric vibrator plays significant role in an excitation frequency. More is

the area the end surface of the peg touches vibrator, more acoustic energy

transferred to it, as well as excitation frequency is lower. During experiments

was noticed that end surface of smaller diameter peg was harder to make parallel

to the end surface of the vibrator. That circumstance should be taken in

consideration making any conclusions on excitation frequency using smaller

diameter pegs.

a) b)

Fig. 2.12 Alignment duration dependencies on force F1: a) bush placement +Δ, b) bush

placement –Δ; 1 – Δ=0,4 mm; 2 – Δ=0,6 mm; 3 – Δ=0,8 mm; 4 – Δ=1,0 mm; 5 – Δ=1,5

mm; 6 – Δ=2,0 mm; 7 – Δ=2,5 mm; 8 – Δ=3,0 mm; 9 – Δ=3,5

Figure 2.12 represents dependencies of alignment duration Δt on initial

pressing force F1 under different axis misalignment. Influence of initial pressing

force on alignment duration is relatively small when axis misalignment is up to 1

mm. In a case when Δ>1 mm alignment duration decreases as force F1 increases.

3. Numerical simulation of part alignment at non-impact and impact

modes

During vibratory alignment, two solid bodies like peg and a bush interact

with each other. Peg presses bush with predetermined force and its elastic

vibrations are excited. Friction forces that rise during interaction of those two

0,0

0,5

1,0

1,5

2,0

2,5

1,5 2,0 2,5

Δt, s

F1, N

f=7050 Hz

1 2 3 4

5

6

7

1.5 2.0 2.5

2.5

2.0

1.5

1.0

0.5

0.0 0,0

0,3

0,7

1,0

1,5 2,0 2,5

Δt, s

F1, N

f=7050 Hz

1 2 3

4

5

6

7

8

9

1.5 2.0 2.5

1.0

0.7

0.3

0.0

20

bodies guide bushing to the axis alignment direction. It is necessary to make

systems consisting of two interactive bodies dynamical modelling in order to

examine alignment process further.

In general case peg is fixed in a specially designed gripper while bush is

based on a plane. Piezoelectric vibrator presses the top end of the peg and excites

its elastic vibrations. Experimental research has showed that longitudinal and

lateral vibrations of the peg are created. There is a phase shift between them thus,

peg’s tip moves in elliptical trajectory on a vertical plane. Alignment process is

modelled by two-mass dynamical system in a reference frame XOY (Fig. 3.13).

A mass m1 depicts peg

that oscillates in two

perpendicular directions

while mass m2 is a bush that

has to be aligned to the peg.

Alignment process is possible

only when peg press bush

with predetermined force and

oscillation amplitude is on

the proper level. Vibration

amplitude depends on

excitation frequency and is

the biggest when system

oscillates close to their

natural mode. Natural mode

itself depends on the

geometrical characteristics of

piezoelectric vibrator and a

peg, the way in which peg fixed in a gripper, peg-to-bush pressing force

magnitude. Typical excitation frequency is in a range of kilohertz and amplitude

of few micrometres.

Since vibration amplitude is at the same measurable level as roughness of the

surfaces, rheological properties of the bodies should be taken into account. At

the contact point, the surface texture deforms in a normal and tangential

directions. During high frequency elastic vibrations not only elastic but also

elasto-plastic deformations may occur. To evaluate deformations of this kind we

use rheological Kelvin–Voigt model. Thus, the surface of the mass m2 in normal

and tangential directions constructed by stiffness (K2X, K2Y) and damping (H2X,

H2Y) elements connected in parallel. Surface deformations induce reaction forces

Rx, RY:

.

,

120122

1221222

YHYYKR

XXHXXKR

YYY

XXX (3.1)

K2y H2y K2x m2

X2

Bsin(ωt+ε)

K1y H1y Asinω

t

H1x

K1x

H2x Y0

m1

Y1

X

Y

0

0

Δ

N X

1

Fig. 3.13 Dynamical model

21

where Y0 – deformation of the contact surface because of the initial pressing

force, (X′2-X′1) – relative deformation speed in X direction, Y′1 – deformation

speed in Y direction.

Mass’s m1 tip longitudinal tA sin and lateral tBsin vibration

amplitudes vary according to the law of the sinus. Stiffness and damping forces

restricts mass movement in X and Y directions.

.

,

11111

11111

YHYKR

XHXKR

YYY

XXX (3.2)

When two bodies are in contact, friction forces arise in their contact zone. Its

magnitude expressed using dry friction model. Force F1fr affects mass m1. Force

F2fr that is sum of friction forces peg-bush and bush-base acts on mass m2

.2111 XXsignNF fr (3.3)

.221212 XsignNXXsignNF fr (3.4)

where μ1 – coefficient of friction between mass m1 and m2, μ2 – coefficient of

friction between m2 and base, N – normal pressing force.

All friction forces formulated taking into account that they are not affected by

relative speed. We get equation of motions for mass m1 and m2 by projecting all

acting forces in X and Y axes:

.0

,sin

,sin

2212112212222

1012112111

1211111111

XsignNXXsignNXXKXXHXm

tAKYYKKYHHYm

tBKXXsignNXKXHXm

XX

YYYYY

XXX

where 22 /;/ dtdXdtdX

We are using following dimensionless parameters to have generalized results

of simulation:

.1

;;;;;;;;

;;;;;;

;;;;;;;;

100

02

22

1

212

1

222

1

11

2

1121

1

22

1

11

2

22

1

11

11

22

11

2

2

ml

yyknll

Yy

lpm

Nn

pl

Aa

l

Bb

m

m

kkkpm

Kk

pm

Kk

K

Kkhhh

pm

Hh

pm

Hh

pm

Hh

pm

Hh

l

Yy

l

Xx

l

Xx

m

Kppt

y

yyyY

yY

y

X

Xxyyy

Yy

Yy

Xx

Xx

x

(3.5)

22

Then motion equations written in a dimensionless form:

.0

,sin

,sin

22121121222

01111

121111111

xsignnxxsignnxxxxhx

ykakykyhy

bkxxsignnxkxhx

x

yyyy

xxx

(3.6)

where 22 /;/ ddxdtdx

We used a program code written in MATLAB environment to obtain

numerical simulation results. Solver ode15s was used in calculating stiff

differential equation. Since our dynamic system is described by second order

differential equations, we had to rewrite them to a pair of simultaneous first order

differential equations (Esfandiari, 2013) to obtain the solution. The following

initial values of the parameters of the dynamic system were used ,3b ,2a

,51 xk ,3,01 yk ,3,02 yk ,7,02211 yxyx hhhh ,3,1 ,2,01

,1,02 ,32,0 ,1000 ,2,0 .30 y Initial conditions alignment

conditions ,01 x ,01 x ,01 y ,01 y ,02 x 02 x . As τ=0 normal

reaction force is equal to initial pressing force 0ykn y . After the excitation

signal is applayed, force n alternates and it‘s value depends on mass‘s m1

coordinate y1, thus 1ykn y . Alignment occurs when 2x . During

simulation, we analysed the effect of each parameter on the part alignment

process by adjusting only one parameter and keeping all the others constant.

Alignment duration dependency on excitation frequency at different phase

shifts between longitudinal and lateral vibrations represented in figure 3.14.

Alignment duration is lower at the lower frequency values if phase shift is

between 0 and π/2. As excitation frequency increases, alignment duration also

increases. When phase shift is grater then π/2, alignment duration decreases as

excitation frequency increases since ellipsis of peg movement trajectory have

changed inclination angle and short axis of the ellipsis have shortened. There is

also a peak in alignment duration at the excitation frequency ν=1 no matter the

phase shift between vibration components. This is because excitation frequency

became equal to the natural frequency of the bush along X-axis.

Figure 3.15 shows alignment duration dependency on the phase shift

between vibrations at different excitation frequencies. Dependencies have

parabolic character, thus, there exist phase shift at which alignment process is the

most rapid. All curves have intersection points in the region between π/4 to

5π/12 and it means that excitation frequency has little effect on alignment

duration in this region.

23

ε=0 ε=π/6 ε=π/3 ε=π/2

a a a a

-20 -10 0 10 b

-20 -10 0 10 b

-20 -10 0 10 b

-15 -10 -5 0 5 b

0

-1

-2

-3

-4

-5

0

-2

-4

-6

0

-2

-4

-6

0

-2

-4

-6

ε=2π/3 ε=5π/6 ε=π

a a a

-10 -5 0 5 10 b

-10 -5 0 5 10 b

-15 -10 -5 0 5 10 b

ε=7π/6

a

-20 -10 0 10 b

0

-1

-2

-3

-4

-5

0

-1

-2

-3

-4

-5

0

-1

-2

-3

-4

-5

0

-1

-2

-3

-4

-5

Fig. 3.14 Alignment duration dependencies

on excitation frequency ν; 1 – ε=0; 2 –

ε=0.79; 3 – ε=1.57; 4 – ε=2.36; 5 – ε=3.14

Fig. 3.15 Alignment duration dependency

on phase shiftε; 1 – ν=0.1; 2 – ν=0.3; 3 –

ν=0.5; 6 – ν=0.7

a) b) c) d)

e) f) g) h)

Fig. 3.16 Peg’s tip motion trajectory on phase ε

Motion trajectory of the peg’s end tip depends on the phase shift between

lateral and longitudinal vibrations (Fig. 3.16). In our case for the bush to be

aligned with the peg, necessary that peg’s end tip moves counter clockwise

direction (Fig. 3.16, b-f). Phase shift is between π/6 and 5π/6 radians in this case.

Peg’s end tip moves in a clockwise direction when the phase shift is grater then π

(Fig. 3.16, h). The bush does not align with the peg anymore, but rather moves

away from it. In case when phase shift is 0 or π radians (Fig. 3.16, a, g)

0,00 1,05 2,09 3,14ε

1

2

3

4

τ∙105

1.2

0.9

0.5

0.2

0,1 0,6 1,1 1,5 2,0ν 0.1 0.6 1.1 1.5 2.0

1

2

3

4

5

τ∙105

1.7

1.2

0.6

0 0.00 1.05 2.09 3.14

24

alignment process has unstable manner and depending on excitation frequency

alignment of the parts may occur or not.

Fig. 3.17 Alignment duration dependencies

on lateral vibration amplitude b; 1 – a=1; 2

– a=2; 3 – a=4; 4 – a=6; 5 – a=8

Fig. 3.18 Alignment duration dependencies

on longitudinal vibration amplitude a; 1 –

b=1; 2 – b=3; 3 – b=9

Another important parameter that has direct influence on the alignment time

is amplitude of lateral and longitudinal vibrations of the peg figure 3.17, 3.18

respectively. When lateral vibrations reach certain limit (in our case b=2) their

influence on the alignment duration becomes negligible. Much bigger influence

on the process time has longitudinal vibrations. As amplitude increases,

alignment duration constantly decreases.

Fig. 3.19 Alignment duration dependencies

on initial deformation kyy0; 1 – b=1; 2 –

b=2; 3 – b=3; 4 – b=4; 5 – b=5

Fig. 3.20 Alignment duration dependencies

on dry friction coefficient μ1; a=6: 1 –

μ2=0,06; 2 – μ2=0,08; 3 – μ2=0,1; 4 –

μ2=0,12; 5 – μ2=0,16; 6 – μ2=0,18

Because repulsive force created due to the friction between a peg and a bush,

initial pressing force between those parts plays key role in the part alignment

(Fig. 3.19). For the process to be stable and reliable initial pressing force has to

be at a certain limit, but not < 1.5. If it is less, the system runs into the impact

1 4 7 10b

1

2 3

4 5

τ∙105

1.4

0.9

0.5

0 1 4 7 10 1 3 5 7 9a

1

2 3

τ∙105

1.4

0.9

0.5

0 1 3 5 7 9

1,5 2,9 4,3 5,6 7,0kyy0

1 2 3

4

5

τ∙104

0.7

0.5

0.4

0.2 0.9 1.7 2.6 3.4 0,1 0,3 0,5 0,7μ1

τ∙104

3.4

2.2

1.1

0

2

3

4 5

6 1

0.1 0.3 0.5 0.7

25

mode and our model cease to be valid. We have to increase initial pressing force

to rule out system from the impact mode. However to obtain the shortest

alignment duration we have to keep it as low as possible but avoiding system to

fall into the impact mode. As initial pressing force increases, alignment duration

also increases. In case when lateral vibration is lower, alignment duration

increases more rapidly in comparison to the cases when vibration is higher. Two

friction forces acts on the bush during part alignment. One is the friction force

between peg and the bush that moves bush to the alignment direction. Second

one is friction force between bush and the base which causes bush movement to

slow down. Friction forces directly proportional to the coefficient of dry friction

between acting surfaces (Fig. 3.20). Alignment duration keeps stable as

coefficient μ1 increases, there is only small decrease of alignment duration at

μ1=0.3. There is rapid increase in the alignment duration when coefficient of

friction is more than 0.5. Dry friction coefficient between bush and the base has

to be as small as possible. When μ2>0.12 alignment process starts at μ1>0.2.

When μ2>0.19 alignment process stops.

a) b)

Fig. 3.21 Alignment duration dependencies a) on axis misalignment δ; 1 – ν=0.1; 2 –

ν=0.3; 3 – ν=0.6; 4 – ν=0.8; 5 – ν=1.0, b) on excitation frequency ν; 1 – δ=400; 2 –

δ=800; 3 – δ=1500; 4 – δ=2500; 5 – δ=3500

Alignment take place at different axis misalignment between the parts and at

different excitation frequency (Fig. 3.21, a, b). The alignment duration increases

as axis misalignment increases. Dependencies have linear character no matter the

excitation frequency. We can see that excitation frequency has low impact on the

alignment duration when axis misalignment is small (Δ<800). Only when

misalignment increases the influence of excitation parameter becomes apparent.

Work pieces align most rapidly when mass m1 vibrates at resonant frequency. As

excitation frequency rangers from resonant, alignment duration constantly

increases until process becomes impossible. Qualitatively dependencies have a

400 1175 1950 2725 3500 δaa

1 5

4

3

2

τ∙104

12.0

6.0

0

0,1 0,4 0,7 1ν

1 2

3

4

5 τ∙105

1.5

1.0

0.5

0

0.1 0.4 0.7

26

good match with experimental results thus confirms validity of our mathematical

model.

During experimental alignment of the parts was observed part alignment at

impact mode when contact disappears between bush and the peg tip. At this

moment peg breaks away from the bush and later hits it at certain speed level.

Such alignment regime forms when longitudinal oscillations has bigger

amplitude or pressing bush-to-peg

force is not sufficient. During

impact part alignment a recurrent

interaction between peg and a bush

is going. Peg breaks from the bush

when normal component of

excitation force higher then peg-

to-bush pressing force. To simulate

impact part alignment it is

necessary to form equations

describing motion of the impact

body before the impact with the

bush and the impact interaction.

Peg rendering mass oscillates in

normal and tangential directions.

Resultant body motion trajectory

depends from excitation amplitude

components and their phase. The motion trajectory in the vertical plane could be

circular, elliptical or linear inclined at the certain angle to the horizontal axis (Fig.

3.22).

Motion of the mass m1 when it breaks from the bush m2 defines equations:

.sin

,sin

10111111

1111111

tAKYYKYHYm

tBKXKXHXm

YYY

XXX (3.7)

Diagonal impacts of mass m1 causes motion of the mass m2 defines by

equation:

02222 XsignNXm (3.8)

We use dimensionless parameters to get generalised version of motion

equation:

Bsin(ωt+ε)

K1y H1y Asinω

H1x

K1x m1

Δ

X

I1X I2

Y

I1Y

Y0

m2

N

Fig. 3.22 Model of the contact interaction

27

.1;;;;;;

;;;;;

;;;;;;;

00222

2

2

1

2

1

11

2

11

1

11

1

11

11

22

11

1

1

mlll

Yy

lp

gdgmN

lpm

Nn

p

l

Aa

l

Bb

m

m

pm

Kk

K

Kk

pm

Hh

pm

Hh

l

Yy

l

Xx

l

Xx

m

Kppt

Yy

X

Xx

Yy

Xx

x

Motion of the bouncing mass m1 in dimensionless form:

.sin

,sin

01111

11111

ykakykyhy

bkxxhx

yyyy

xx

(3.9)

Dimensionless equation of motion of mass m2:

.0222 xsigndx (3.10)

Interaction of the bodies at the moment of the diagonal impact defines

impact equations. Describing diagonal impact, we assume that velocity

components of normal impact vary according to the linear impact law and do not

depend on tangential velocity component. When impact is linear, normal

velocity of the mass m1 after the impact defined by equation:

.11

yRy (3.11)

where

1y – mass m1 velocity before the impact, R – impact restitution coefficient.

To define impact interaction we use hypothesis of dry friction that

determines link between normal and tangential impact impulses:

.11 yx II (3.12)

where I1x, I1y – impact impulses, μ – coefficient of dry friction.

Studying diagonal impact, we assume that slipping velocity between the

bodies in the impact interval is always positive. Such impact called a sliding

impact. Normal velocity restitution equation valid for the sliding impact only.

There are two phases of the sliding impact. First is a load phase. It starts from the

moment of the contact between bodies and continues until reaches maximum

surface deformation. Second is load reduction phase. It starts at the moment of

the deformation end until break of the contact between the bodies. When m1 hits

m2 during load phase in accordance with impulse hypothesis, we can write:

.11

1

1

ymI y (3.13)

28

.101

1

1

xxmI x (3.14)

where ,1

x

1y – body m1 velocity before impact, 0x – absolute sliding velocity

at the end moment of the first impact phase.

By inserting (3.13) and (3.14) to (3.12) we get:

.110

yxx (3.15)

Direction of the tangential velocity cannot be changed at the impact

moment, thus 00 x . From (3.15) we get:

./ 11 yx

That is a self-stop condition for the body m1. Tangential displacement of

body m1 stops at the first stage of the impact and it bounces from the body m2 in

a normal direction if this condition not fulfilled. Because we investigate case of

the sliding impact, the body m1 does not bounce at the end of the first impact

phase and impact process continues. Thus at the end of the second impact mode,

we can write:

.11

2

1

ymI y (3.16)

.011

2

1 xxmI x (3.17)

where ,1

x

1y – body m1 velocities after the impact.

Expressions (3.16) and (3.17) linking with (3.12) and taking in to account

(3.11) and (3.15), we can calculate m1 tangential velocity after the impact:

.1111 Ryxx (3.18)

Tangential impulses of the body m1 make body m2 to slide towards axis

misalignment direction. Bodies m1 and m2 impact impulses according to the

impulses hypothesis is:

.

,

2222

1111

xxmI

xxmI

x

x

(3.19)

Impact impulse to the body m2 transferred by the dry friction thus we can

write:

.12 xx II (3.20)

Composing impulse expressions (3.19) to the (3.20) and taking in to the

account (3.18), we get equation for calculating velocity of the body m2 after the

sliding impact:

29

.11

1

2

22

yRxx (3.21)

Expressions (3.11), (3.18) and (3.21) used to calculate bodies m1 and m2

velocities after the impact.

During numerical simulation we used the following constant values:

,3b ,2a ,51 xk ,3,01 yk ,7,011 yx hh ,57,1 ,2,01 ,1,02

,32,0 ,1000 ,4,1 ,10 y ,1n 7,0R . Initial conditions:

,01 x ,01 y ,01 y ,10002 x 02 x . Part alignment condition: 02 x .

Excited peg oscillates in longitudinal and lateral directions and hits the

bush. Restitution coefficient R valuates deformation of the bush. At the impact

moment, the impact energy transferred to the bush and it slides to the axis

misalignment direction. Peg bounces from the bush after the energy transferred

meanwhile a bush keep sliding because of inertia until the next impact.

Proper settings of the excitation and mechanical system parameters must

be chosen to have alignment process stable and reliable. The peg excited in the

frequency range from 1 to 1.7 to have alignment of the bush reliable (Fig. 3.23).

in this frequency range alignment duration do not depend on the phase shift

between vibration components and is easily predictable. As excitation frequency

increases, alignment duration decreases and reaches minimal value at 1.4.

Subsequent increase of the excitation frequency makes alignment duration to

increase. When ν<1 or ν>1.4 alignment duration is hardly predictable and

changes rapidly if small excitation

frequency changes applied.

As axis misalignment

increases, alignment duration also

increases. Dependencies has linear

character and do not depend on

excitation frequency (Fig. 3.24, b).

Influence of excitation frequency

to the alignment duration is

minimal when δ>800 (Fig. 3.24, a).

Only when excitation frequency

increases we can observe

frequency range at which part

alignment is the fastest. If

excitation frequency is more than

1.7 alignment process stops. As

ν≥1.9 alignment process recurs

again, but alignment duration rapidly decreases as excitation frequency increases.

0,1 0,7 1,4 2,0ν

τ∙103

0,8

0,6

0,4

0,2

1

2

3 4

5

6

7

Fig. 3.23 Alignment duration

dependencies on excitation frequency ν:

1 - ε=0; 2 - ε=0.26; 3 - ε=0.52; 4 -

ε=0.79; 5 - ε=1.05; 6 - ε=1.31; 7 -

ε=1.57

30

a) b)

Fig. 3.24 Alignment duration dependencies on: a) excitation frequency ν: 1 - δ=400; 2 -

δ=600; 3 - δ=800; 4 - δ=1000; 5 - δ=1500; 6 - δ=2000; 7 - δ=2500; 8 - δ=3000; 9 -

δ=3500; b) axis misalignment δ: 1 - ν=1; 2 - ν=1.2; 3 - ν=1.5; 4 - ν=1.7; 5 - ν=1.9; 6 - ν=2;

As longitudinal vibration amplitude increases, alignment duration

decreases exponentially (Fig. 3.25, a). Lateral vibration amplitude if it is not

equal to zero, has no influence to alignment duration at all (Fig. 3.25, b).

a) b)

Fig. 3.25 Alignment duration dependencies on: a) longitudinal vibration amplitude a: 1 -

b=1; 2 - b=2; 3 - b=3; 4 - b=4; 5 - b=5; b) lateral vibration amplitude b: 1 - a=1; 2 - a=2; 3

- a=4; 4 - a=5;

Friction forces between bush and peg and between bush and base also

have influence to the process duration. Their influence evaluates dry friction

coefficients μ1 and μ2. As friction force between bush and peg increases,

alignment duration decreases exponentially (Fig. 3.26, a). Meanwhile if friction

forces between bush and base increases, alignment duration increases linearly

(Fig. 3.26, b).

1,0 2,3 3,7 5,0b

τ∙103

0.6

0.4

0.2

0

1

2

3

4

1.0 3.3 3.7 5.0 1,0 2,3 3,7 5,0a

τ∙103

0.6

0.4

0.2

0

1, 2, 3, 4, 5

1.0 3.3 3.7 5.0

1,0 1,3 1,7 2,0 400 1433 2467 3500δ

τ∙103

2.4

1.7

0.9

0.2

5

1

6

2 4 3

τ∙103

2.6

1.7

0.9

0

8 9

7 6 5 4 3

2 1

1.0 1.3 1.7 2.0

31

a) b)

Fig. 3.26 Alignment duration dependencies on: a) dry friction coefficient μ1: 1 – μ2=0,1; 2

– μ2=0,2; 3 – μ2=0,3; 4 – μ2=0,4; 5 – μ2=0,5; 6 – μ2=0,6; 7 – μ2=0,7; 8 – μ2=0,8; b) dry

friction coefficient μ2: 1 – μ1=0,1; 2 – μ1=0,2; 3 – μ1=0,3; 4 – μ1=0,4; 5 – μ1=0,5; 6 –

μ1=0,6;

Initial deformation y0 and longitudinal peg vibration amplitude has

influence to the alignment duration in the close relation to each other. When

deformation y0 increases alignment duration slightly decreases, but process stops

if longitudinal vibration amplitude becomes insufficient and peg no longer hits

the bush (Fig. 3.27, a). As longitudinal vibration amplitude increases, alignment

duration decreases exponentially. However alignment process possible only

when amplitude is bigger than initial deformation (Fig. 3.27, b).

a) b)

Fig. 3.27 Alignment duration dependencies on: a) initial pressing deformation y0: 1 - a=1;

2 - a=2; 3 - a=3; 4 - a=4; 5 - a=5; 6 - a=6; b) peg‘s longitudinal vibrations a: 1 - y0=1; 2 -

y0=2; 3 - y0=3; 4 - y0=4, 5 - y0=5

When impact restitution coefficient R increases, alignment duration

constantly decreases because increases amount of energy bush receives during

0

175

350

525

700

1,0 3,7 6,3 9,0y0

τ∙103

0.52

0.35

0.17

0

1

2 3

4

5 6

1.0 3.7 6.3 9.0 30

197,5

365

532,5

700

1,0 2,7 4,3 6,0a

τ∙103

0.53

0.36

0.20

0.03

1

2

3

4

5

1.0 2.7 4.3 6.0

0

2000

4000

6000

0,1 0,3 0,6 0,8μ1

τ∙103

4.0

2.0

0

8

7 6

5 4

3 2

1

0.1 0.3 0.6 0.8 0

2000

4000

6000

0,1 0,3 0,6 0,8μ2

τ∙103

4.0

2.0

0

1

2 3

4 5 6

0.1 0.3 0.6 0.8

32

impact. During pure elastic impact when R=1 alignment duration would be the

shortest (Fig. 3.28, a). Friction forces between bush and base restricts bush

motion. Thus as normal pressing force between bush and base increases,

alignment duration increases (Fig. 3.28, b). If impact force is less then friction

force, alignment process does not occur.

a) b)

Fig. 3.28 Alignment duration dependencies on: a) impact restitution coefficient R: 1 -

N=0,2; 2 - N=0,4; 3 - N =0,6; 4 - N =0,8; 5 - N =1; 6 - N =2; 7 - N =3; 8 - N =4; 9 - N =5;

b) normal pressing force N: 1 - R=0,2; 2 - R=0,3; 3 - R=0,4; 4 - R=0,5; 5 - R=0,6; 6 -

R=0,7; 7 - R=0,8

Conclusions

1. Proposed and investigated part alignment method when using elastic

vibrations of the peg. piezoelectric vibrator pressed to the upper end of the peg

provides high frequency excitation oscillations. The lower end of the peg starts

to vibrate in longitudinal and lateral directions. Part alignment occurs only when

mechanical contact ensured between mating parts. Such method compensates

axial part misalignment of 1-1.5 mm for the chamferless parts with circular and

rectangular cross-section. Vibratory part alignment when using elastic vibrations

of the peg enhances productivity and reliability of automatic part assembly

operations like: insertion of the shaft to the bearing, tooth wheel, electric motor

rotor etc.

2. Experiments have proved that lower end of the peg moves in elliptical

shape trajectory in all three coordinate planes while excitation done in

longitudinal direction to the upper end. When mating parts pressed to each other,

friction force propels bush to the part alignment direction. During part alignment

bush makes not only linear motion, but also rotates about contact point to the peg.

3. In all vibratory part alignment experiments, alignment process fastest

when peg’s oscillation frequency is closest to the third natural bending mode. As

excitation frequency ranges from it, alignment duration increases until process

100

400

700

1000

1300

0,2 0,4 0,6 0,8R

τ∙103

1.0

0.7

0.4

0.1

1 2 3 4 5

6

7

8 9

0.2 0.4 0.6 0.8 100

400

700

1000

1300

0,0 1,7 3,3 5,0N

τ 103

1.0

0.7

0.4

0.1

1

2

3 4

5 6

7

0.0 1.7 3.3 5.0

33

stops. As longitudinal vibration amplitude increases, alignment duration

decreases. The shortest alignment duration is at excitation signal level of 142 V.

Alignment regime depends on longitudinal peg vibration and part-to-part

pressing force. Impact alignment regime starts when longitudinal peg’s

vibrations are at high level and part-to-part pressing force is not sufficient.

However, for the practical usage non-impact regime is more suitable since it is

more stable especially during indirect part alignment. If pressing force higher

then 2.9 N lateral vibrations are supressed and alignment stops. If pressing force

is less then 1.5 N, alignment process falls in to the impact mode.

4. The mathematical models of part alignment at impact and non-impact

regimes were constructed. Computer simulations revealed that alignment

duration and reliability mostly depends on part-to-part pressing force, excitation

frequency and amplitude, phase shift between vibration components.

Mathematical models and simulation results were verified by the experimental

alignment of the cylindrical and rectangular cross-section parts.

Alignment duration increases as excitation frequency increases if phase

shift between vibration components is 0-π/2 at non-impact alignment regime.

When phase shift is more than π/2, alignment duration decreases as excitation

frequency increases. Alignment duration increases rapidly or alignment process

stops at all when excitation frequency equal to bush’s natural frequency. Stable

part alignment at impact mode goes when excitation frequency is from 1 to 1.7.

Alignment duration, depending on a phase shift between vibration components

could differ 7 %.

Literature

1. Baksys, B.; Puodziuniene, N. Modeling of Vibrational Non-Impact Motion of

Mobile-Based Body. International Journal of Non-Linear Mechanics, 2005,

40(6), p. 861-873.

2. Baksys, B.; Puodziuniene, N. Modelling of Vibrational Impact Motion of

Mobile-Based Body. International Journal of Non-Linear Mechanics, 2007,

42, p. 1092-1101.

3. Baksys, B.; Baskutiene, J. The Directional Motion of the Compliant Body

Under Vibratory Excitation. International Journal of Non-Linear Mechanics,

2012, 47, p. 129-136.

4. Baksys, B.; Ramanauskyte, K. Motion of a Part on a Horizontally Vibrating

Plane. Mechanika, 2005, 55(5), p.20-26.

5. Архангелский, М. Е. О Превращение ультравуковых колебаний

поверхности во вращательное и поступательное движение тела.

Акустический журнал, 1963, 9(3), p. 275-278.

34

6. Mohri, N.; Saito, N. Some Effects of Ultrasonic Vibration on the Inserting

Operation. The International Journal of Advanced Manufacturing

Technology, 1994, 9(4), p. 225-230.

List of author‘s publications

Articles in publications from the master Journal List of the Institute

for Scientific Information (ISI)

1. Sadauskas, Edvardas; Bakšys, Bronius. Alignment of the parts using high

frequency vibrations // Mechanika / Kauno technologijos universitetas,

Lietuvos mokslų akademija, Vilniaus Gedimino technikos universitetas.

Kaunas : KTU. ISSN 1392-1207. 2013, Vol. 19, no. 2, p. 184-190. DOI:

org/10.5755/j01.mech.19.2.4164. [Science Citation Index Expanded (Web of

Science); INSPEC; Compendex; Academic Search Complete; FLUIDEX;

Scopus]. [0,500]. [IF (E): 0,336 (2013)]

2. Sadauskas, Edvardas; Bakšys, Bronius; Jūrėnas, Vytautas. Elastic vibrations

of the peg during part alignment // Mechanika / Kauno technologijos

universitetas, Lietuvos mokslų akademija, Vilniaus Gedimino technikos

universitetas. Kaunas : KTU. ISSN 1392-1207. 2013, Vol. 19, no. 6, p. 676-

680. DOI: 10.5755/j01.mech.19.6.6014. [Science Citation Index Expanded

(Web of Science); INSPEC; Compendex; Academic Search Complete;

FLUIDEX; Scopus]. [0,333]. [IF (E): 0,336 (2013)]

3. Sadauskas, Edvardas; Bakšys, Bronius. Peg-bush alignment under elastic

vibrations // Assembly Automation. Bradford : Emerald. ISSN 0144-5154.

2014, Vol 34, no. 4, p. 349-356. DOI: 10.1108/AA-05-2014-031. [Science

Citation Index Expanded (Web of Science); EMERALD; Compendex].

[0,500]. [IF (E): 0,711 (2013)]

Articles in other referred publications from list of the Institute for

Science Information (ISI proceedings)

1. Sadauskas, Edvardas; Bakšys, Bronius. Alignment of cylindrical parts using

elastic vibrations // Mechanika 2012 : proceedings of the 17th international

conference, 12, 13 April 2012, Kaunas University of Technology, Lithuania /

Kaunas University of Technology, Lithuanian Academy of Science,

IFTOMM National Committee of Lithuania, Baltic Association of

Mechanical Engineering. Kaunas : Technologija. ISSN 1822-2951. 2012, p.

267-270. [Conference Proceedings Citation Index]. [0,500]

35

Information about author of the dissertation

Name, Surname: Edvardas Sadauskas

Date and place of birth: 13 October 1980, Kaunas, Lithuania.

E-mail: edvardas_sadauskas@hotmail.com

Education and training

2010-09 – 2014-08 Doctoral student at Kaunas University of Technology

in the field of Mechanical Engineering Sciences.

2004-09 – 2006-07 Kaunas University of Technology, Master of Sciences

in Mechanical engineering, Mechanical engineering.

2000-09 – 2004-07 Kaunas University of Technology, Bachelor of

Sciences in Mechanical engineering, Mechanical

engineering.

Reziumė

Detalių tarpusavio centravimas yra vienas svarbiausių automatinio rinkimo etapų,

kurio metu kompensuojamos renkamų detalių tarpusavio padėties paklaidos. Tik

diegiant efektyvius automatizuotus rinkimo metodus, galima sumažinti rinkimo

darbų sąnaudas, užtikrinti stabilią renkamų gaminių kokybę, palengvinti darbo

sąlygas, likviduoti varginančius monotoniškus rankinio rinkimo veiksmus.

Rinkimo darbų automatizavimo pažanga daugiausiai priklauso nuo naujų

rinkimo technologinių procesų, pagrįstų efektyviais komponentų centravimo

metodais sukūrimo. Vienas perspektyvių, iki šiol mažai nagrinėtų yra vibracinis

centravimo metodas, pagrįstas strypo tampriaisiais virpesiais. Vieno iš

komponentų (įvorės) kryptingas poslinkis ir posūkis užtikrinamas frikcine

sąveika su virpančiu strypo laisvuoju galu.

Eksperimentiškai ištirtas cilindrinių ir stačiakampio skerspjūvio detalių be

nuožulnų centravimas, žadinant strypą. Sudarytos centravimo trukmės

priklausomybės nuo detalių pradinio prispaudimo jėgos, ašių nesutapimo,

žadinimo dažnio ir amplitudės.

Šiame darbe nagrinėjamas vibracinis detalių tarpusavio centravimas besmūgiais

ir smūginiais režimais. Sudaryti detalių tarpusavio centravimo įtaisų dinaminis

bei matematinis modeliai, įvertinantys detalių sąveiką viso centravimo proceso

metu. Ištirtas detalių tarpusavio centravimas kuomet velenas kinematiškai

žadinamas sujungimo ašies kryptimi, o įvorė bazuojama paslankiai. Teoriniai

tyrimai patvirtinti atliktais eksperimentais.

Pateiktos tarpusavio centravimo trukmės priklausomybės nuo dinaminės

sistemos bei žadinimo parametrų. Nustatyta, kad didžiausią įtaką centravimo

besmūgiais ir smūginiais režimais trukmei turi detalių pradinio prispaudimo jėga

36

bei žadinimo dažnis, išilginių veleno virpesių amplitudė, trinties jėga tarp strypo

ir įvorės.

Atlikti teoriniai ir eksperimentiniai vibracinio centravimo tyrimai patvirtino, kad

vibraciniu metodu galima centruoti automatiškai renkamas cilindrinio ir

stačiakampio skerspjūvio detales, nenaudojant jutiklių, vykdymo įtaisų, specialių

valdymo algoritmų. Gauti teorinių bei eksperimentinių tyrimų rezultatai gali būti

pritaikyti rinkimo įrenginių projektavimui bei automatinio rinkimo technologijų

tobulinimui.

Darbo struktūra ir apimtis

Disertaciją sudaro įvadas, trys skyriai, išvados, autoriaus publikacijų

disertacijos tema ir naudotos literatūros sąrašai bei priedai. Disertacijos apimtis

90 puslapių, 61 paveikslas ir 2 lentelės. Literatūros sąrašą sudaro 78 šaltiniai.

Pirmame skyriuje, remiantis moksline literatūra išanalizuoti automatiškai

renkamų detalių centravimo metodai, jų privalumai ir trūkumai. Pateikta su

disertacijos tema susijusių tyrimų apžvalga. Suformuluoti pagrindiniai tyrimų

uždaviniai.

Antrame skyriuje pateikti eksperimentiniai virpančio strypo galo tyrimai,

kai jis liečiasi su įvore. Nustatytos strypo judesio trajektorijos bei strypo–įvorės

prispaudimo jėgos ir žadinimo signalo dažnio įtaka išilginių ir lenkimo virpesių

amplitudėms. Atlikti apvalaus ir keturkampio skerspjūvio strypo ir įvorės

centravimo tyrimai, kai strypas žadinamas sujungimo ašies kryptimi. Sudarytos

centravimo trukmės priklausomybės nuo detalių pripaudimo jėgos, ašių

nesutapimo, strypo žadinimo parametrų.

Trečiame skyriuje pateikti vibracinio centravimo, naudojant tampriuosius

strypo virpesius, dinaminiai modeliai, esant nesmūginiam ir smūginiam

centravimo režimui. Pateiktos įvorės sąveikaujančios su dviem statmenomis

kryptimis judančiu strypo galu, judesio lygtys bei centravimo proceso

skaitmeninio modeliavimo rezultatai. Išaiškinta dinaminės sistemos ir žadinimo

parametrų įtaka centravimo procesui, Sudarytos parametrų derinių sritys, kai

centravimas būna sėkmingas.

Mokslinių tyrimų tikslas ir uždaviniai

Tyrimų tikslas – teoriškai ir eksperimentiškai ištirti renkamų komponentų

vibracinio centravimo, naudojant tampriuosius strypo galo virpesius, procesą.

Nustatyti žadinimo parametrų ir mechaninės sistemos įtaką centravimo

efektyvumui. Siekiant įgyvendinti šį tikslą, reikia išspręsti šiuos uždavinius:

Atlikti mokslinės literatūros apžvalgą apie šiuo metu plačiai pramonėje

naudojamus automatiškai renkamų komponentų centravimo metodus.

37

Atlikti apvalaus skerspjūvio strypo galo virpesių eksperimentinius tyrimus,

kai strypas liečiasi su įvore. Nustatyti virpesių pobūdį, jų priklausomybę nuo

žadinimo signalo amplitudės bei nuo įvorės ir strypo prispaudimo jėgos.

Atlikti apvalaus ir stačiakampio skerspjūvio detalių centravimo tyrimus,

naudojant tampriuosius strypo virpesius, išsiaiškinti žadinimo bei mechaninės

sistemos parametrų įtaką centravimo efektyvumui ir patikimumui

Sudaryti cilindrinių komponentų centravimo matematinį modelį, kai

naudojami tamprieji strypo galo virpesiai, kuomet strypas žadinamas

sujungimo ašies kryptimi, esant nesmūginiam ir smūginiam centravimo

režimui. Atlikti nesmūginio ir smūginio centravimo proceso modeliavimą,

išsiaiškinti žadinimo bei mechaninės sistemos parametrų įtaką centravimo

efektyvumui ir patikimumui.

Mokslinis naujumas

Rengiant disertaciją buvo gauti šie mechanikos inžinerijos mokslui nauji

rezultatai:

1. Pasiūlytas naujas technologiškai paprastesnis detalių centravimo metodas,

panaudojant vienos iš centruojamų detalių (strypo) tampriuosius virpesius,

kai virpesių žadinimas vyksta sujungimo ašies kryptimi iš galo prispaustu

pjezokeraminiu vibratoriumi.

2. Iš galo išilgine kryptimi žadinamo strypo laisvasis galas juda elipsine

trajektorija erdvėje. Tokiu dėsniu virpantį strypo galą prispaudus prie įvorės

atsiradusi trinties jėga užtikrina įvorei poslinkį ir posūkį

3. Pasiūlytu metodu galima centruoti nesmūginiu ir smūginiu režimu apvalaus ir

stačiakampio skerspjūvio strypines detales su įvorės tipo detalėmis

nepriklausomai nuo jų tarpusavio padėties, esant mechaniniam kontaktui tarp

jų.

4. Sudaryti strypo ir įvorės centravimo nesmūginiu ir smūginiu režimu

matematiniai modeliai bei nustatytos centravimo trukmės priklausomybės

nuo žadinimo dažnio ir amplitudės, komponentų tarpusavio prispaudimo

jėgos.

Darbo rezultatų praktinė vertė

Taikant tampriuosius strypo virpesius, galima centruoti įvorės tipo detales

turinčias apvalaus ir stačiakampio profilio skyles su atitinkamo profilio strypais,

kai komponentai yra su nuožulnomis ir be jų, o komponentų tarpusavio padėties

paklaida siekia kelis milimetrus, taip praplečiant automatizuoto rinkimo

technologines galimybes. Tyrimo rezultatai leidžia nustatyti ir parinkti žadinimo

ir įtaiso parametrus ir juos suderinti, kad centravimas būtų sėkmingas, o jo

trukmė mažiausia.

38

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