A R T Ū R A S T A D Ž I J E V 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

D Y N A M I C S A N D D I A G N O S T I C S O F

V E R T I C A L R O T O R S W I T H N O N L I N E A R

S U P P O R T S S T I F F N E S 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 )

KAUNAS UNIVERSITY OF TECHNOLOGY

ARTŪRAS TADŽIJEVAS

DYNAMICS AND DIAGNOSTICS OF

VERTICAL ROTORS WITH NONLINEAR

SUPPORTS STIFFNESS

Summary of Doctoral Dissertation

Technological Sciences, Mechanical Engineering (09T)

2015, Kaunas

The doctoral dissertation was prepared in 2010-2014 at Kaunas University of

Technology, Faculty of Mechanical Engineering and Design, Department of

Mechanic Engineering. Research was supported by Research Council of

Lithuania.

Scientific supervisor:

Prof. Dr. Habil. Vytautas BARZDAITIS (Kaunas University of Technology,

Technological Sciences, Mechanical Engineering – 09T).

Dissertation Defence Board of Mechanical Engineering Science Field:

Prof. Dr. Rimvydas GAIDYS (Kaunas University of Technology,

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

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

Technological Sciences, Mechanical Engineering – 09T);

Assoc. Prof. Dr. Vytautas JŪRĖNAS (Kaunas University of Technology,

Technological Sciences, Mechanical Engineering – 09T)

Prof. Dr. Habil. Rimantas KAČIANAUSKAS (Vilnius Gediminas Technical

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

Prof. Dr. Juozas PADGURSKAS (Aleksandras Stulginskis University,

Technological Sciences, Mechanical Engineering – 09T).

The official presentation of the dissertation will be held at 11 a.m. on June

19, 2015 at the public meeting of the Board of Mechanical Engineering

Science field in the Dissertation Defence Hall at the Central Building of

Kaunas University of Technology.

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

Phone: (370) 37 300042, fax. (370) 37 324144, e-mail: doktorantura@ktu.lt

The Summary of Dissertation was sent on the 19th

of May, 2015.

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

of Kaunas University of Technology

(K. Donelaičio g. 20, Kaunas).

KAUNO TECHNOLOGIJOS UNIVERSITETAS

ARTŪRAS TADŽIJEVAS

VERTIKALIŲ ROTORIŲ, SU NETIESINIO

STANDŽIO ATRAMOMIS, DINAMIKA IR

DIAGNOSTIKA

Daktaro disertacija

Technologijos mokslai, mechanikos inžinerija (09T)

2015, Kaunas

Disertacija rengta 2010 – 2014 metais Kauno technologijos universitete,

Mechanikos inžinerijos ir dizaino fakultete, Mechanikos inžinerijos

katedroje. Moksinius tyrimu rėmė Lietuvos mokslo taryba.

Mokslinis vadovas:

prof. habil. dr. Vytautas BARZDAITIS (Kauno technologijos universitetas,

technologijos mokslai, mechanikos inžinerija – 09T).

Mechanikos inžinerijos mokslo krypties taryba:

Prof. dr. Rimvydas GAIDYS (Kauno technologijos universitetas,

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

Doc, dr. Giedrius JANUŠAS (Kauno technologijos universitetas,

technologijos mokslai, mechanikos inžinerija – 09T);

Doc. dr.. Vytautas JŪRĖNAS (Kaunas University of Technology,

Technological Sciences, Mechanical Engineering – 09T);

Prof. habil. dr. Rimantas KAČIANAUSKAS (Vilniaus Gedimino technikos

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

Prof. Dr. Juozas PADGURSKAS (Aleksandro Stulginskio universitetas,

technologijos mokslai, mechanikos inžinerija – 09T).

Disertacija bus ginama viešame mechanikos inžinerijos mokslo krypties

tarybos posėdyje 2015 m. birželio 19 d. 11 val. Kauno technologijos

universiteto, centrinių rūmų, disertacijų gynimo salėje.

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

Tel. (370) 37 300042, faksas (370) 37 324144, el. paštas:

doktorantura@ktu.lt.

Disertacijos santrauka išsiųsta 2015 m. gegužės 19 d.

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

universiteto bibliotekoje

(K. Donelaičio g. 20, Kaunas).

INTRODUCTION

With the rapid development of technologies, new technological

processes are being created and old processes are being improved.

Rotating machinery plays a very important part in this. Most of the

technological processes, such as mechanical separation, grinding,

hydro energy production, are exclusively gravity-based technological

processes. Therefore, vertical rotor machinery is inevitably used in

industry, hydro power plants and maritime. When assessing the spread

of vertical vs. horizontal machinery among low to medium power

rotary equipment, horizontal rotor machinery makes up a significantly

larger proportion among the overall rotary machinery. However,

vertical rotor machinery is much more widely spread in industrial

areas, or in navigation, where technological processes are gravity-

based. Regulatory documentation, in relation to the rotor system

operation and diagnostics, mainly focuses on the horizontal rotors.

Vertical rotors in such documents are distinguished only in normative

documentation related to the large hydro turbines. In the material,

provided by the review of scientific publications, one can find

instances of scientists conducting research with vertical rotors, when

there is a need to avoid gravitation effects on the rotary system in

radial direction, for instance, when the object of study is the influence

of support anisotropy on the dynamics of the rotor system. In other

instances, research is conducted with horizontal rotor systems.

In analyzing the most common rotor system faults, according to

their frequency, one can distinguish that the most prevalent fault of

rotor systems is the imbalance, followed by the coupled rotor shaft

axis misalignment. The third most common fault is the rotor support

bearing defect. Shaft axis misalignment is found more frequently in

horizontal rotor systems, because the rotor drive unit and the work

unit may be mounted on different foundations and, eventually, the

weight of the system may deform the foundation. Vertical rotors have

common foundations and axis misalignment is a less prevalent defect

6

of the vertical rotor system, therefore, in this study, it has not been

analyzed in greater detail.

Topicality of the problem

In everyday life, people are surrounded by a number of

household appliances, many of which work based on some kind of

rotating parts. It is easy to notice that most of the household

appliances and devices used domestically, function on some part with

the help of some kind of mechanism with horizontal axis rotor.

However, even domestically, there are numerous areas, where it is

necessary that the rotor of the household appliance be vertically

oriented, otherwise, we could not be capable of performing certain

processes. These are particular technological processes that are

gravity-based. There are many household appliances that cannot

function without the help of the gravity, although, without going into

much detail, one would not say,that they would have to be particularly

with vertical rotors. Examples of such appliances would be:

separators, various household cutting equipment, grinders (household

grater), whisks, mixing equipment, juice extractors etc. Much of the

above mentioned household equipment would not operate with

horizontally oriented rotation axis, however, when using them, we do

not realize, that these processes are influenced by gravity.

In industry, maritime and energetic, as well as in households,

equipment with horizontally oriented rotation axis is more

widespread. Much of the fundamental scientific research, as well as

regulatory documentation for evaluation of technical condition and

monitoring are dedicated to this kind of machinery. Although, there

are many areas, where systems with vertically oriented rotation axis

must be used. Otherwise, technological processes, production of

energy or technological supply are impossible. Such equipment has

some certain specific features that are not fully described nor in

scientific works, nor in regulatory documentation. Devices of this kind

differ from the horizontally oriented ones in that their rotors are not

stabilized by the gravity (the weight of the rotor in radial direction).

Therefore, they are much more sensitive to radial loads in dynamic

stability sense. In addition, their foundation has significantly lower

stiffness in the upper part of the rotor, because they are not mounted

on some massive base, but rather attached to some certain prepared

construction, the stiffness of which, as compared to the stiffness of the

system mounted on the massive base, is significantly lower. These

factors complicate the assessment and monitoring of such equipment

conditions therefore, service personnel with huge experience in

technical condition assessment of such rotor machinery are required.

The major differences between the vertical and horizontal rotor

systems can be found out through examining the dynamic forces

acting on rotor supports in greater detail. If analyzing horizontal rotary

systems, a plane perpendicular to the axis of rotation of the rotor is

sufficient to define the operation of forces acting on the support, then,

assessing forces, acting on the vertical rotor supports, it is necessary to

define spatial forces, because as the rotation speed of the rotor

changes, so does the radial component of the overall dynamic force.

When speaking of support-mounted roller bearing defects, we

have to signify that the great majority of the defects in horizontal rotor

systems form at the most loaded part of the bearing. Meanwhile in

vertical rotor systems with changing operating mode, it is very

difficult to distinguish, which part of the rotor system is going to be

loaded most, i.e., whether the radial forces outweigh the gravity, and

how the load direction changes etc.

The objective of this work is to quantifiably assess and find

similarities and differences in the evaluation of technical conditions of

vertical and horizontal rotor systems and to present it in quantitative

manner, easily perceivable to the experts of this field.

Aim and tasks for the thesis

The aim of this work was to investigate and quantifiably assess

the dynamics of vertical rotor, rotating in the non-linear stiffness

supports and establish its correlation to diagnostic of defects of

vertical rotor elements.

8

These tasks were formed in order to achieve the objective of

this work:

1. To analyze scientific works published in periodic scientific

journals and international standards where the rotor dynamics are

examined in practice and theory, also to examine scientific works

published in periodic scientific journals, elaborating on the existing

and novel methods of defect diagnostic research for rotors with roller

bearings.

2. To conclude an analytical model to calculate forces, acting on

vertical rotor supports, as well as to calculate the forces acting on the

radial axial load of the vertical rotor.

3. To develop a generalized vertical rotor model, that can evaluate the

effect of the gyroscope, gravity and its spatial direction and which

could allow to theoretically determine the radial – axial forces acting

on the vertical rotor support as well as to research the rotor dynamics

phenomena.

4. To perform the comparative diagnostic research on the defects of

vertical and horizontal rotor roller bearings and quantifiably assessing

the differences in diagnostic research on defects among the two

systems.

5. To perform the diagnostic investigation of vertical rotor system

with inner rolling bearing ring race defect at different rotor imbalance

excitation force and inner ring defect angular positions.

6. To perform diagnostic research on vertical rotors with outer rolling

bearing ring race defect, while changing the tilt angle of the rotor

rotation axis from the vertical, also to determine how the dynamics of

the rotor shifts with the change of the tilt angle, to examine, how it

affects the results of the diagnostic.

Scientific novelty

Analytical model has been constructed, allowing assessing

forces acting on radial-axial supports of the vertical rotors.

A generalized theoretical model of the vertical rotor has been

designed, capable of evaluating the effect of the gyroscope, gravity

and its spatial direction, the non-linearity of the rotor supports as well

as the radial and axial gaps in the supports.

The major differences in vertical and horizontal rotor

diagnostics have been determined and quantifiably evaluated through

scientific research.

With the help of a generalized theoretical model of vertical

rotor, it has been determined, how the dynamics of the vertical rotor

shifts, when rotor sways from the vertical at an angle of up to 15˚.

Experimental studies have shown how this affects the results of the

rotor’s diagnostic research.

Practical value

The results of the research have been applied in conducting

diagnostic studies of vertical rotor equipment at „DFDS Seaways“

ferries “Vilnius Seaways“, „Optima Seaways“, „Victoria Seaways“

and „Athena Seaways“. The results of these studies have been

approbated at such industrial enterprises as PLLC “„Lifosa“, PLLC

„Nordic Sugar“, “JSC „Arvi fertis“ and others. PLLC “Lifosa” has

been advised on the renewal and the selection of new vertical pumps

for the sulfuric acid.

A universal model of the vertical rotor has been created, with

the consideration on the gravity and its spatial direction. Because of its

versatility, this rotor system model can be applied for the research on

the dynamics of both vertical and horizontal rotors.

Statements for defense

1. The analytical model of radial - axial forces acting on rotor

supports, allows for determining the magnitude and the spatial

position of overall forces F acting on radial - axial supports of the

vertical rotor.

2. In order to examine the dynamics of rotor systems, a generalized

model of the vertical rotor system, capable of evaluating the gravity

and its spatial direction, could be applied both to vertical as well as

horizontal rotor systems with nonlinear stiffness of supports.

10

3. A new created relative parameter – Defect Visibility Ratio (DVR)

allows to quantifiably assessing and compare the diagnostic research

features both for the horizontal as well as vertical rotors. It also

facilitates for the machine technical condition monitoring as well as

defect diagnostic research in situ.

The work approbation

6 (six) scientific articles were published on this topic (4 ISI

Web of Science with a citation index, and 2 other peer - reviewed

scientific journals) as well as presentations at in 9th scientific

conferences:

1. „MSM 2011“, topic of presentation: „Modeling and diagnostic of

rotary system powered by multi gear“.

2. „Mechanics 2012“, topic of presentation: „Comparison of Vertical

and Horizontal Rotor System Models and Simulation Results".

3. „Vibroengineering 2012“, topic of presentation: „Features of

Vertical Axis Rotor with Rolling Bearings Diagnostics".

4. „Scientific work on technology in western Lithuania 2012“, topic

of presentation: „Research on vertical rotor dynamics“.

5. „Mechanics 2013“, topic of presentation: "Vertical Versus

Horizontal Rotors Vibration and Diagnostics"

6. „ICOVP-2013“, Lisbon, topic of presentation: "Vertical Versus

Horizontal Rotors Dynamics and Diagnostics"

7. „Vibroengineering 2013“, topic of presentation “Influence of

imbalance phase angle to vertical and horizontal rotors bearings

diagnostics“.

8. „Mechanics 2014“, topic of presentation: „New deep groove ball

bearings high frequencies vibration testing“.

9. „Marine science and technology 2014“, topic of presentation:

„The specifics of vertical rotor machinery testing in marine

technology“

The structure of the work

Doctoral thesis consists of an introduction, four chapters,

general conclusions, references, list of author publications and

appendixes. The total scope of the dissertation of 142 pages, they

contain 107 pictures, 91 bibliographic references are used.

1 ROTOR DYNAMICS AND DIAGNOSTICS

1.1 REVIEW OF ROTOR DYNAMICS

The dynamics of rotors, as a science, started from W.J.M. Rankine

and his mathematical model of rotor that he published in 1869 year.

This model was dedicated to calculate the first critical rotational speed

of rotor. To determine the critical rotational speed of rotor, he choose

a model with two degrees of freedom, that consists of point mass that

is fixed onto stiffness element and another end of stiffness element is

fixed. All mentioned components rotate about fixed node of stiffness

element. In 1883 engineer from Sweden K.G.P. de Laval created the

first impulse turbine, which reached rotational speed of 40 000 RPM.

He derived a relationship, through which it was possible to determine

the centrifugal forces, which affect whirling motion of shaft, he also

found the self-centering phenomenon of rotors. The first

comprehensive model of rotor was created by a professor of Munich

University, August Föppl in 1885 year. A. Föppl was the first to

explain the self-centering phenomenon, investigated by de Leval, in

detail [1]. Theoretical model of A. Föppl has allowed establishing a

stable supercritical rotational speed and theoretically opened a rotor

work opportunity at supercritical rotational speed. This model has

been ignored by investigators, because in practice, it seldom allowed

to get a stable supercritical rotation speed [2]. Later, in 1919, the

Royal Society of London, commissioned an independent research of

the Irish Royal College professor Henry Jeffcott to improve a model

of August Föppl and solve the disagreements between theoretical

model and experiments. The subsequent, very important step in the

history of rotor dynamics was made in 1924 by A.Stodola. A.Stodola

12

proved in his works, that there is a second (not only one) critical speed

of rotors. During analysis of gas and steam turbines, A.Stodola

noticed, that the disc, mounted on the rotating shaft, changes the

dynamics of rotor. And that depends on the inertia moment of disk,

mounted on the rotor. This way, the damping component of

gyroscopic effect appears in mathematical expressions of rotor

dynamics. In 1924 A.Stodola published a book, in which the

phenomenon of second resonance has been described in detail. In this

book he also describes a static rotor balancing methods, approximate

calculation of critical speeds (eigenvalues) of stepped rotors and etc.

in detail. The fundamentals published in Föppl, Jeffcott and Rankine

works are also relevant today. They placed grounds of rotor dynamics

which was the basis for Campbell’s diagram of rotor critical speeds

diagram, which is often referred as the rotor critical speeds map [3].

The Vertical rotors are particularly suitable for investigations of

nonlinearity of supports. There is a couple of works where the

nonlinearity of supports investigated by analyzing a whirling motion

of rotors with different non-linearity of rotor supports. These studies

are described in works of worldwide recognized investigators of the

rotor dynamics [4]. Some authors uses the vertical rotors to create new

or to develop existing models to better describe the kinematic

movement of shaft neck in supports, but those models rarely takes a

greater theoretical or practical value, as they often are very

complicated or needs a complex mathematical ability or describe the

motion of rotor accurately only under certain conditions consolidation

[5]. During the analysis of scientific periodical publications we can

observe a one or other work that are dedicated to investigate the

dynamics of vertical rotors using finite element method. Some authors

investigated the dynamics of vertically oriented rotating discs [6].

Other authors use the vertical rotors to investigate an accuracy of

finite element solvers by comparing the results of eigenvalues of

vertical rotor to experimental results [7]. By increasing the popularity

of FEM (that are induced by development of semiconductor

technologies and FEM software development) investigations and

capabilities of complex multi stage gas turbines can be examined.

There is a few works published in this field of investigations [8]. In

periodic scientific publications, we can find authors that investigates

the multi rotational rotors of jet engines by using FEM software

ANSYS. The multi rotational speeds rotors of jet engines consists of

two gas turbines, the first and main rotor rotates fixed in supports of

motor second rotor rotates mounted using the hydrodynamic bearings

on first rotor shaft [9]. We also can find a recently defended PhD

thesis dedicated to investigate the dynamics of vertical rotors. Some

works are dedicated to parametric optimization of large rotors work

conditions to prevent an undesirable working mode of rotor [10].

Other works dedicated to theoretical investigations of rotors using

FEM [11].

1.2 REVIEW OF DIAGNOSTICS OF ROTORS

The diagnostics of rotor systems is an integral part of modern

rotor systems commissioning process. Therefore, it is necessary to

mention the possible rotor diagnostic techniques and explore some of

the reasons for their popularity and development trends in

contemporary science works. The vibration diagnostics of rotor

systems is one of several stages of technical condition assessment,

which is primarily used for, the evaluation of technical condition of

rotating machinery. The main aspects of the use of this method are

described in detail in the international standard ISO 13373-1: 2002

Condition monitoring and diagnostics of machines - Vibration

condition monitoring - Part 1: General Procedures" [12], processing

and analysis of measured vibration data are described in detail in

international standards ISO 13373-2: 2005 "Condition monitoring and

diagnostics of machines - Vibration condition monitoring - Part 2:

Processing, analysis and presentation of vibration data" [14] and ISO

13379: 2003, "Condition monitoring and diagnostics of machines -

General Guidelines on Interpretation date and diagnostics techniques

[13]. The main requirements of vibration measurement and

monitoring equipment are described in detail in the international

14

standard ISO 2954: 2012 "Mechanical vibration of rotating and

reciprocating machinery - Requirements for instruments for measuring

vibration severity" [15]. The relative vibration measurement,

evaluation of the rotor shaft neck vibrations in supports, described in

relation to an international standard, which consists of 5 - the essential

parts of this [16 – 20]. The standard describes in detail the vibration

measurements and interpretation of measurement data by measuring

the vibrations of rotor systems on non-rotating parts, consists of six

parts [21 – 26].

The traditional rotor diagnostic methods are in detail described

in practical tutorials and international standards. We should try to

identify some of them: the vibration signal spectral analysis method;

whirling motion analysis method; vibration shape factor method,

envelope method; acoustic emission method; shock pulse method and

etc. All of those methods are used in rotor vibration monitoring and

diagnostics for more then 10 - 20 years. But there is no relation

between those methods and influence of rotor dynamics to diagnostics

using mentioned traditional methods. This work is an attempt to find

trends of vertical axis rotor dynamics and it’s relation to diagnostics of

those rotors.

Among the developed new diagnostic research methods are

increasingly visible a diagnostic methods based on artificial neural

networks and the use of a hybrid, combining several diagnostic

methods together to increase the precision [27 – 28]. During the

analysis of scientific publications we can find the research works that

are dedicated to develop of a new analysis methods for spectrum or

spectrum cascades [29, 30]. The significant steps to rolling bearings

diagnostics with all kinds of defects were made by R.B. Randall with

colleagues and other well known researchers [31, 32], but all of these

works are concentrated to identify a bearing element fault indications

in different formats of data, but not related with the rotor dynamics.

We also can find some scientific publications dedicated to investigate

rolling bearings wearing problems, one of the most frequent topics -

growth of a radial fault of bearing [33, 34]. Among the new,

developed methods of rotor diagnostic, many of them based on the

new signal processing capabilities. Many of them showing good

results during investigations on testing rigs in laboratories but in

practice are hardly applicable. Many of them have a limited usage and

need of expensive research equipment.

2 THEORETICAL INVESTIGATION OF GENERALIZED

VERTICAL ROTOR MODEL

2.1 FORCES ACTING IN ROTOR SYSTEM

Mechanical vibrations are an integral environment of dynamic

systems, which, often, interpreting vibration data in a right way can

provide a lot of information about the observed system. In order to

properly assess technical monitoring of the condition of the rotor

system, it is necessary to know that some of the effects of rotor

dynamics help in stabilizing the system, but there are cases, where

summative these effects can get the opposite phenomenon -

spontaneous destabilization. In order to analyze forces acting at

supports of rotors in deep, we need to examine the main components

of summary dynamic forces in detail. There are two cases of analysis.

The first one is to analyze the acting forces in orthogonal system; a

second is to analyze acting forces in spatial system. We can see that

when we are talking about a horizontally oriented rotor, the planar

formulation of acting dynamic forces is sufficient. However, when we

talk about vertically oriented rotor system, a spatial formulation of

task is necessary, because the force of rotor weight acts in axial

direction of rotor, while the dynamic force of imbalance, which

depends from rotational speed, acts in direction that is perpendicular

to axis of rotation. It was accepted, that two acting forces exist in

orthogonal planes in vertical axis rotors: eccentric inertia force,

induced by imbalance u, acting perpendicular to rotation axis and

second – axial force as gravity force Fmg of rotor‘s mass m acting

parallel to vertical axis. In Fig.1 we see, that the total dynamic force Fs

depends on centrifugal dynamic force of imbalance and when the

16

rotational force increases, not only the magnitude of total force is

changing, the angle between rotation axis and total force also change

Fig. 1 Scheme of the forces acting in the vertical rotating rotors with

imbalance

The magnitude of total force can be calculated using mathematical

expression (2.1.).

22

cmgs FFF (2.1.)

Where: Fs – total dynamic force, N; Fmg – gravity force, N; Fc –

dynamic force of imbalance, N.

The angle between rotation axis of rotor and total dynamic force

can be calculated using mathematical expression (2.2.)

;c

mg

F

Farctg (2.2.)

We see, that if the position of rolling bearing fault is placed in

the wrong direction on the bearing ring race (the dynamic force acting

a fault indirectly), we could not see the indications of fault during the

diagnostics of such rotor. There is a lot of vertical axis equipment in

ships and diagnostics of such equipment in this environment plays an

important role. During movement, the ship makes a pitching (we

didn't analyze pitching motion types; there is six basic pitching motion

types). If the ship makes pitching, the axis of equipment with vertical

axis of rotation is tilted from vertical position; there is no data about

diagnostics of vertical rotors that are tilted from vertical. We need to

analyze how the dynamic forces affect supports (in what direction and

how it affects the magnitude), when the rotor is tilted from vertical. In

Fig.2, the forces, acting supports, when the axis of rotation of rotor is

tilted from vertical, is presented. In fig. 2 we see that the total force Fs

is not of the same direction and magnitude in different sides of rotor

support. If we have an oscillating total force, we know that it affects

and axial force.

Fig. 2 Scheme of the forces acting vertical rotors with imbalance,

when the rotation axis tilted from vertical

The total force in the side where rotor is tilted can be calculated

using (2.3.) mathematical expression. In the opposite side, it can be

calculated using (2.4.).

sin222

1 mgcmgcs FFFFF (2.3.)

sin222

2 mgcmgcs FFFFF (2.4.)

18

Then the amplitude of total force can be calculated using (2.5.).

sin

2

sin2sin2 2222

mgc

mgcmgcmgcmgc

s

FF

FFFFFFFFF

(2.5.)

And the amplitude of axial force is given in (2.6.) mathematical

expression.

cosmgmgas FFF (2.6.)

The amplitude of centrifugal force given in mathematical

expression (2.7.).

2sin)sin()sin( mgmgcmgcc FFFFFF (2.7.)

These varied axial and radial forces make rotor‘s and bearings

condition monitoring and failure diagnostic procedure complicated.

Modeling of vertical axis rotors and simulating forces acting on

bearings makes failure diagnostics procedure more adaptive for

practical usage.

2.2 FE MODEL OF VERTICAL AXIS ROTOR

Modeling and simulation of vertical axis rotor provided with

FEM and ANSYS software [35]. The designed physical model of

vertical rotor is shown in Fig. 3. and model designed with FEM in Fig.

4. The rotor total mass 2,80 kg, shaft length is 0,6 m, diameter 0,02 m,

wheel diameter 0,15 m with radius ru=0,06 m for the fixing imbalance

masses, two stiffness elements in radial direction of 1st (upper)

bearing and three stiffness elements - two in radial direction and one

in axial direction - of the 2nd

(lower) bearing.

Fig. 3 Physical model of vertical rotor elements

The FEM beam type BEAM188 10 elements used for modeling

shaft of the rotor. The beam type BEAM188 3 elements used for the

wheel with unbalance mass modeling by element MASS21. For

nonlinear bearings stiffness elements the COMBIN39 element and for

stiffness-damping linear bearing elements the COMBIN14 element

was used. The image of elements of FE model given in Fig.4.

Fig. 4 Visualized elements of FE model

The bearings nonlinear stiffness elements Kxi, Kyi and Kz2

parameters for COMBIN39 elements experimentally measured with

testing machine Zwick/Roell Z100 (Germany) in axial and radial

directions. Simulations of FEM model were provided in series: with

20

static analysis evaluating constant forces; frequency analysis for

determining resonance frequencies; harmonic analysis and transient

dynamics analysis axis symmetrical problem and axis asymmetrical

problem solving. There were used such mathematical models: Static

analysis model (2.8.).

}{}{ FuK (2.8.)

Modal analysis is using such mathematical expression for

determination of eigenvalues of rotor (2.9.).

0}]{[ 2 uMK (2.9.)

Harmonic response analysis can be implemented using (2.10.)

mathematical expression.

},,{}{}{}{ tuuFuKuCCuM gyr (2.10.)

Where the real and imaginary parts of centrifugal force can be

defined in such way (2.11., 2.12.).

tj

ccx eFtFF cos (2.11.)

tj

cccy ejFtFtFF )2/cos(sin (2.12.)

2.3 THE RESULTS OF THEORETICAL INVESTIGATIONS

In this part of this work attention is focused to the unique

features of diagnostics of vertical rotor with rolling bearings.

Therefore, the simulation is performed using the experimental tests

measured stiffness values, which measured in radial and axial

directions on rolling bearing and the experimentally measured radial

and axial gaps of rolling bearings. The simulation results presented in

Fig. 5 and Fig. 6 as vertical axis tilt angle γ influences on dynamics of

Fs and position angle φ. The simulation provided when variable

vertical axis tilt angle γ values change in discreet steps 5º, 10 º, 13,5º

degrees. The absolute value of Fs varies in wide range from 168 N up

to 203 N with increasing tilt angle γ, Fig. 5 and inflated with

increasing rotor’s imbalance. According to simulation results the

absolute value of dynamic force Fs and angular position of force φ

increases nonlinearly with increasing rotational speed n, imbalance u

and rotors axis tilting angle γ.

Fig. 5 The 2nd bearing’s dynamic forces Fs (n, γ) plots versus rotor’s

rotational speed and tilt angle γ with imbalance of 120 gmm

Fig. 6 The 2

nd bearing’s dynamic forces Fs position angles φ plots versus

rotational velocity of rotor and tilt angle γ (5º, 10 º and 13,5 º) with imbalance

of 120 gmm

It happens, that up to first resonance frequency ωR = 204 1/s the

dynamic force Fs position angle φ decreases and becomes φ = 0º

22

value, Fig. 5 and Fig. 6. Further increasing the angular velocity from

ωR=204 1/s, the Fs position angle φ increases nonlinearly. For

example, when ω = 314 1/s, imbalance uL=80,4 gmm the simulated

position angle is φ = 22º and when angular velocity is high ω = 559

1/s imbalance uL=80,4 gmm the position angle increased up to φ=77º.

At resonance at angular speed of ωR= 204 1/s kinetic orbit drastically

changes in magnitude and angle in comparison with ωL= 100 1/s and

ωH= 314 1/s angular velocities, Fig.7. These results indicated that

failures diagnostics indications of vertical axis rotors are too different

in comparison with the horizontal axis rotors. ωL= 100 1/s and ωH=

314 1/s angular velocities, Fig.7. These results indicated that failures

diagnostics indications of vertical axis rotors are too different in

comparison with the horizontal axis rotors.

a) b)

c)

Fig. 7 Kinetic orbits of vertical rotor shaft neck when rotor tilted

13,5˚ from vertical: a) n = 1000 RPM; b) nc = 1950 RPM; c) n = 2

850 RPM;

3 EXPERIMENTAL INVESTIGATIONS

3.1 COMPARATIVE INVESTIGATION BETWEEN VERTICAL AND

HORIZONTAL ROTOR DIAGNOSTICS

In this section of the work, author's research using original

testing stand is presented. This subpart is dedicated to compare the

indications of diagnostics of same rolling bearings with same faults of

vertically and horizontally oriented rotor. The objective of this section

is to identify the essential differences of the horizontal vs. vertical

rotary system fault indications, during diagnostic tests, to determine

the quantitative and qualitative differences between them. To establish

quantitative relationships as vertical rotary equipment defects

diagnostic tests and correlate with the results of rotor parameters

describing the dynamics of systems. The experimental test stand,

shown in Fig. 8 has been set up in order to investigate differences of

diagnostic features during tests with damaged 6004 single row deep

groove ball and new bearings in horizontal and vertical axis rotors.

The rotor is driven by a variable - speed AC motor controlled by

frequency inverter. Rotational speed during, measurement has been

ramped up from 100 to 3050 RPM.

Fig. 8 Rotor researches stand. 1 – AC motor; 2 – Coupling; 3 –

20 mm diameter shaft; 4 – Supports with rolling bearings.

The rotor bearing supports 1 and 2 were (Fig. 9) positioned in a

= 50 mm, a + b = 550 mm distances from flywheel disc. During a tests

24

rotation axis of the rotor has been switched from horizontal to vertical

position. At first, the brand new 2nd

ball bearing was examined, then,

it has been replaced with the faulted bearing.

Fig. 9 Orientation of rotational axis and positions of

accelerometers

Separate tests were provided: the first one with artificial defect

on the inner ring race and second - with artificial defect on the outer

ring race, as shown in Fig.10.

a) b)

Fig. 10 Defects of the deep groove ball bearing 6204: a - inner

ring race fault; b - outer ring race fault;

Tests were carried out with imbalance of 80 gmm (maximum

permissible imbalance according to ISO 1940-1 is 125 gmm) and with

determined residual imbalance of 240 gmm as found in balancing

quality grade G6.3 (ISO 1940-1). The balancing mass was attached to

the rotor flywheel disc at radius ru. The absolute vibration velocity of

bearing supports has been measured with four accelerometers 1x, 1y

and 2x, 2y mounted in two perpendicular directions at each bearing

support. Experimental data has been processed using multi-channel

vibration signal analyzer "OROS".

The order of test, during the experimental investigations, given

in Table 1.

Table 1. The vibration measurements order

No. Orientation of

rotor axis Imbalance

Rolling bearing

defects

1. Horizontal 80 g∙mm Without

Vertical

2. Horizontal 240 g∙mm Without

Vertical

3. Horizontal 80 g∙mm Inner ring race defect

Vertical

4. Horizontal 240 g∙mm Inner ring race defect

Vertical

5. Horizontal 80 g∙mm Outer ring race defect

Vertical

6. Horizontal 240 g∙mm Outer ring race defect

Vertical

Initial condition of bearing fault position, fixing in supports, given in

Fig. 11.

a) b)

Fig. 11 Initial conditions of measurements: a) position of inner

ring race defect; b) position of outer ring race defect

26

The absolute vibration data of the rotor bearing supports was

evaluated as root mean square values vRMS and vibration velocity

spectrum and cascade diagrams. The 2nd

bearing was under

investigation. The first test was performed with the brand new 2nd

bearing and another, second and third test has been carried out with

the faulty 2nd

bearing. The 1st bearing was brand new throughout the

entire experiment.

The 2nd

bearing support vibration measurement data plotted in

vibration velocity spectral cascade diagrams (Fig. 12 and Fig. 13)

shows that the 1X frequency vibration magnitudes dominated in

horizontal axis rotor at run up mode at wide rotational speed range

(1000-3050 r/min).

a) b)

Fig. 12 Vibration velocity vRMS cascade plots of 2nd

new 6004

bearing, measured with 2ya accelerometer, at run up mode of the rotor

240 gmm unbalance: a – horizontal axis rotor; b – vertical axis rotor.

The vertical axis of rotor with the new bearing generates 1X, 2X,

3X harmonics at wide rotational speed range and indicated existence

of radial gaps as nonlinearities (Fig. 12 b). Therefore, it is difficult to

diagnose imbalance in this systems. The vertical rotor with faulty

bearing generates 1X, 2X,…, 7X frequencies vibration harmonics

from 1500 RPM (Fig. 13 b). It shows us that it is difficult to diagnose

the imbalance in the rotor with significant defect in bearing. The

nonlinearities of radial gaps in the bearings dominated without acting

gravity force as shock form.

a) b)

Fig. 13 Vibration velocity vRMS cascade plots of 2nd

faulty bearing

6204 with outer ring race fault, measured with 2ya accelerometer, at

run up mode of the rotor with 240 g·mm imbalance: a – horizontal axis

rotor, b – vertical axis rotor

The physical effect stated that the rotor’s gravity force augments

vibrations velocity vRMS values in horizontally oriented rotor, although

the anisotropy of supports is significantly noticeable and stiffness of

supports in y - direction is higher than in x – direction. “Defect

visibility ratio (DVR)” parameter was designed for the quantitative

evaluation of the dynamics features of the vertical and horizontal

rotors with deep grove ball bearings.

XRMS

DEFRMS

v

vDVR

1

._ (3.1)

Where: ._ DEFRMSv – dominant defect nX harmonic vibration

velocity vRMS value, mm/s;

XRMSv1

– 1X frequency vibration velocity vRMS value,

mm/s;

As shown in table 2, the damaged bearings with defects of inner

and outer rings races, kinematic vibration frequencies were simulated.

Constant rotational speed of inner ring ni = 3050 RPM and outer ring

was fixed: ball diameter dr = 6,35 mm, number of balls 9.

The statistical data of dominating faulty bearing vibration velocity

vRMS level value divided by 1X (synchronous frequency) harmonic

vibration velocity vRMS level values was presented in Fig.14, Fig.15

DVR diagrams. The graphs shows that, in some cases, when the rotor

oriented vertically the bearing fault frequency vibration velocity vRMS

28

level is very low compared with 1X harmonic vibration velocity vRMS

level. It's very complicates such rotors bearing diagnostics.

Table 2. Kinematic vibration frequencies with stationary outer ring for the

bearing 6004

Constant rotation speed of inner ring, ni=3050 RPM=

=50,83 Hz and outer ring ne= 0 RPM

Typical

vibration

frequencies,

Hz

Rotational frequency of rolling element cage, fc 20,2

Vibration caused by radial fault of the rolling

element, with consideration to its impacts only

against the inner or only against the outer ring, fr1

[Hz]

119

The passage of rolling elements over defect in the

rotating inner ring, fip 276

The passage of rolling elements over defect in the

stationary outer ring, fep 182

Vibration caused by radial fault of the rolling

element, with consideration to its impacts against the

inner and outer rings, frp [Hz]

238

Fig. 14 Horizontally and vertically oriented rotors “Defect

Visibility Ratio” calculated form 2x accelerometer measurements data

The vibration velocity vRMS cascades, which presented in Fig. 12,

Fig.13 shows that in some cases measured vibrations of the vertical

rotor vRMS level has a lot of vibrations “noise” around the bearing

defect frequency. It very complicates such rotor bearing diagnostics,

because those vibrations can be awaking by work chain rubbing, shaft

alignment, coupling defect and etc.

Fig. 15 Horizontally and vertically oriented rotors “Defect

Visibility Ratio” calculated form 2y accelerometer measurements data

The measuring data of vibration velocities vRMS of the first

support (the second plane data) allows as only a partial determination

of a second support bearing fault. When rolling bearing with inner

ring race fault where mounted on 2nd

support, in second plane (1st

support plane) kinematic bearing fault frequency harmonics cannot be

detected. However when bearing with outer ring race fault where

mounted on 2nd

support, the accelerometers mounted on 1st support

(second plane accelerometers) captures relatively high level of the

outer ring race defect frequency 2x and 3x harmonics.

Conclusions of subchapter:

1. Vibration velocity spectrums of the vertical rotor are rich of

higher level vibrations in higher frequencies in comparison with

horizontal axis rotors. This can able to happen due to the chaotic

vertical rotor movement kinematics in radial bearing clearance.

2. Vibration intensity of horizontal axis rotor is higher in

comparison with vertical axis rotor in y-direction. Due to gravitational

influence to horizontal rotor in radial y-direction. Horizontal rotation

axis rotor is more sensitive to imbalance that generates high level 1X

30

frequency vibration amplitudes in comparison with vertical axis rotors

that is more sensitive to the values of radial gaps in the bearings.

3. The designed DVR values provide quantitative evaluation of

horizontal and vertical rotors vibration ratio levels which enables

determination how many times the defect frequency band vibration

level is less than the first harmonic vibration level. It quantifies the

complexity of the defect diagnosis.

3.2 INFLUENCE OF DYNAMIC FORCE ANGULAR POSITION TO

VERTICAL AND HORIZONTAL ROTORS ROLLING

BEARINGS FAULTS DIAGNOSTICS

Instead of using the traditional periodic planned assessment of

technical condition, monitoring systems, based on acceleration

transducers to measure and supervise mechanical vibrations often has

been used for diagnostics of modern technological machinery.

Although, sometimes it is difficult to identify rolling bearing faults,

even if constantly monitoring and analyzing the machinery vibration

acceleration or vibration velocity FFT spectra or their cascades. We

usually run into such problems when analyzing gravity based

technological processes (separators, diffusion machinery) in vertical

machinery observations. This is because the angular position of

imbalance mass varies often in this type of machinery, which causes

the inner ring race fault in the bearing. This subchapter of this work is

a comparative experimental research data between the vertical and

horizontal axis rotors. The test stand consists of the disk mounted onto

the end of the shaft; the deep groove rolling bearing 6004/C3 with

inner ring race defect is mounted behind the support. Throughout the

investigation tests, the angular position of dynamic force of imbalance

mass and inner ring race local fault is switched from 0° to 360

°, using

a 450 angular step value. The experimental studies are carried out both

with rotor axis oriented vertically and horizontally. In order to

simplify the data being analyzed and to quantifiably assess the

diagnostic experiments of vertical and horizontal rotor defects,

statistical processing parameter called “Defect Visibility Ratio”

(DVR) has being designed. Through the use of this statistical

parameter, the defect-identification capabilities of rotor with bearing

with inner ring race fault can be determined quantifiably.

Fig. 16 The experimental test stand. 1 – AC motor; 2 –

Coupling; 3 – Supports with rolling bearings; 4 – 20 mm diameter

shaft; 5 – Flywheel disc.

The experimental test stand, shown in Fig. 16, consists of rigid

frame and fixing plate on which the rotor system is mounted. The

rotor system consist of: asynchronous electric motor with current

frequency inverter 1, elastic aluminum coupling 2, 20 mm diameter

and 600 mm length shaft 4, two supports with 6004 deep groove ball

bearings 3 and flywheel disc 5, with holes, bored using 45° angular

step, for imbalance excitation mass fixing. The signal from

acceleration transducers were recorded with multi-channel vibration

signal analyzer OROS Mobi-pack OR-36. The data were analyzed

using vibration signal processing and analysis software OROS

32

NVGate V8.00. The experimental tests were carried out using deep

groove ball bearing, with inner ring race artificially designed fault as

shown in Fig. 17a. This Bearing was mounted in 2nd

support of test rig

located near the flywheel disk 5. During experimentation, the angle β,

between the bearing’s inner ring fault position and the imbalance force

vector Fu angular position, was changed from 0° to 360

° using 45

°

angular steps. The scheme of the angle ß evaluation is shown in Fig.

17 b. The experimentation was carried out using three values of

imbalance on flywheel: 72 g·mm, 120 g·mm and 156 g·mm. The

maximum allowable imbalance, according to ISO 1940-1, for

machines of such type is 125 g·mm (grade G 6.3).

a) b)

Fig. 17 Photography of the inner ring race artificial defect (a) and

explanation scheme of the angular position of imbalance force vector

Fu relative to inner ring race fault location

Dimensions of 6004/C3 rolling bearing: n = 9, number of balls;

fr = 50 Hz rotation frequency shaft with inner ring; BD = 7,8mm ball

diameter; PD = 31 mm, pitch diameter; α = 0° contact angle.

Fig. 18 The vibration velocity spectrum of vertical rotor, with 72

g·mm imbalance value and β = 0° phase angle

Synchronous frequencies 50 Hz vibration velocities dominant in

FFT spectra of vertical rotor, with 100 gmm imbalance and angle β =

0° of imbalance force vector Fu as shown in Fig. 18. Vibration

amplitudes of frequencies of 1 x BPFI, 2 x BPFI, 3 x BPFI and 4 x

BPFI are visible in the FFT vibration velocity spectrum. The 1 x =

50 Hz synchronous frequency vibration velocity amplitude is

dominant in the spectra. Higher harmonics as 2 x= 100 Hz and 3 x=

150 Hz of rotation speed vibration velocity amplitudes indicated

chaotic kinematic motion of the vertical rotor with smaller imbalance

and due to higher radial clearance in C3 class rolling bearing.

Fig. 19 The vibration velocity spectrum of vertical rotor, with 100

g·mm imbalance and β =90° phase angle

34

Comparing the vibration velocity spectra in Fig.18 and Fig. 19,

we observe that changes in angular position of imbalance force Fu

vector changed vibration velocity magnitudes of BPFI frequencies

vibration velocities amplitudes. Vibration velocity root mean square

values vRMS significantly decreased however the vRMS magnitudes of

rotor’s synchronous rotation frequency provide little changes.

In order, to simplify the processing procedure of bearing defect

diagnostics using vibration measurements experimental data and to

make data practically quantifiable and useful, we have designed the

statistical parameter "Defect visibility ratio" (DVR). The defect

visibility ratio DVR is calculated as the vibration parameter velocity

(or acceleration) of dominant bearing defect frequency value vRMS (or

aRMS) dividing by rotors synchronous rotation frequency 1x vibration

vRMS value (or aRMS) according equation (3.1).

The DVR plots versus imbalance values and positions of

imbalance force vector phase angle β of experimental tests processed

by 2x and 2y acceleration transducers signals are shown in Fig. 20 and

Fig. 21. Vertical axis rotors provide higher DVR values independent

of the imbalance values and phase angles in comparison with

horizontal axis rotors. Vertical axis rotor with damaged inner ring of

bearing is more sensitive to imbalance and has higher DVR values.

Rotor generates higher vibration velocities amplitudes of dominant

bearing inner race fault frequency vRMS values, because dynamic

stiffness of vertical rotor is less in comparison with horizontal axis

rotor.

Fig. 20 The DVR plots versus imbalance values and positions of

imbalance force vector phase angles β processed by 2x acceleration

transducer

Fig. 21 The DVR plots versus imbalance values and positions of

imbalance force vector phase angles β of experimental tests processed

by 2y acceleration transducer

Conclusions of subchapter:

1. The vertical axis rotors are more sensitive to imbalance values

in comparison with horizontal axis rotors.

36

2. The identification of bearings inner ring defect using BPFI

frequency in practical diagnostics is more informative in vertical axis

rotors.

3. The bearing’s defect identification in horizontal rotor with

acceleration transducer measuring absolute vibration in direction of

gravity force is not informative.

4. The DVR of vertical axis rotor decreases valuable by

magnifying imbalance force and bearing defect diagnostics of vertical

rotor is more applicable in practice because the DVR is greater in

comparison with horizontal axis rotors.

3.3 INFLUENCE OF MAGNITUDE AND SPATIAL DIRECTION OF

TOTAL DYNAMIC FORCE Fs AND THE BEARING FAULT

SIZE ON DYNAMICS AND DIAGNOSTICS OF VERTICAL

ROTOR

This subpart presents experimental research results of a variable

tilting of axis of vertical rotor with imbalance and artificially damaged

deep groove ball bearings 6004/C3. The imbalance has been varied in

order to determine how the magnitude of imbalance contributes to the

bearing‘s outer ring raceway spalls masking defect. The general

dynamic force acting in variable vertical rotational axis rotors

composed of rotor mass gravity force and unbalance generated force.

In order to increase rotor‘s defects diagnostic procedure accuracy and

identification of technical condition of rotor‘s with bearings housings

vibration measurement data the statistical parameter DVR (Defect

Visibility Ratio) was designed and implemented in practice.

a) b)

Fig. 22 Vertical axis rotor testing stand a) scheme of angle γ tilting

and positions of bearings b) photo of the testing stand. 1 – Tilting part

of foundation; 2 – Guiding rails; 3 – Foundation part that are fixed to

rigid wall; 4 – Rolling axis of foundation tilting mechanism; 5 – 20

mm diameter shaft; 6 – AC motor; 7 – Supports with rolling bearings;

8 - Flywheel disc.

The experimental test stand in Fig. 22a comprises of rigid frame

on which the rotor system is mounted. The system powered with

370 W asynchronous electric motor with current frequency inverter

(SSD Drives, model 650V/003/230F) control, elastic coupling, 20 mm

diameter and 600 mm length shaft, two bearing’s housings with

6004/C3 deep groove ball bearings 1, 2 and flywheel disc with holes

for imbalance masses fixation at radius ru = 60 mm. The artificially

damaged 2nd

bearing outer raceways fixed in the housing.

Each tested bearing 6004/C3 has artificially manufactured outer

ring raceway spalls: first tested has a 1,4 mm and second 3,0 mm

diameter spall, Fig. 23.

38

a) b)

Fig. 23 Artificially manufactured rolling bearings faults a) 1,4 mm

diameter outer ring spall b) 3,0 mm diameter outer ring spall

The bearings outer raceway defect in the housing has been

located in the y transducer position as shown in Fig.24. The

experiment has been carried out with increasing rotor‘s rotational

speed up to 3000 RPM, constant running at 3000 RPM and decreasing

to stand stop.

Fig. 24 Position of bearings outer raceway fault in housing

The 6004/C3 bearing’s with outer ring raceway one spall ball

pass frequency (BPFO) theoretically calculated at nominal rotational

speed 3000 RPM is fBPFO = 168,4 Hz. Imbalance has been changed

Position of outer

raceway fault

from low value uL = 80 g mm, permissible value umax_per = 125 g mm

for G6,3 balancing quality grade (ISO 1940-1), to high value uH =

156 g mm. The imbalances were variable in order to determine how

the magnitude of imbalance contributes to the bearing‘s outer raceway

failures masking effect. Angular position of vertical axis rotor has

been changed as tilting angle γ varies from 0º to 13,5 º. Vibration of

bearings housings measured using the Wilcoxon Research transducers

model 793 (sensitivity 100mV/g) and processed with signal analyzer

OROS Mobi-Pack OR-36, software OROS NVGate V8.00 and OROS

ORBIGate V4.00 and oscilloscope Scopi XII OX7104 (Metrix).

The 2nd

bearing vibration velocity (measured with 2y transducer)

spectrum in cascade format is presented in Fig.25. The vibration

measured with 2x and 2y transducers and plotted in cascade format

has limited information for vibration sources identification in bearings

failures diagnostics. The BPFO and harmonics vibration velocity

amplitudes are too low and not significant for bearing outer raceway

failure diagnostics procedure. The imbalance generated vibration

velocities amplitudes of 1x frequency synchronized with increasing

rotational speed up to n = 3000 RPM dominated in the spectrum and

BPFO frequency fBPFO = 168,4 Hz and harmonics (nX def.) vibration

velocity amplitudes values are too low.

Fig. 25 The 2

nd bearing vibration velocity spectrum in cascade format

plot during acceleration up to 3000 RPM measured by 2y transducer;

nX - synchronous rotational speeds frequencies, nX-def. BPFO

40

vibration velocity amplitudes, tilt angle γ = 13,5º, spall diameter

3,0 mm, imbalance u = 120 g mm

In order to make the procedure of bearings defect diagnostics

practically quantifiable and efficient in technical condition evaluation,

the statistical parameter "Defect Visibility Ratio" (DVR) has been

designed (described in 3.1 chapter). In the case of vertical rotor axis γ

= 0º bearing small spall diameter (1,4 mm) DVR increases with

increasing rotor’s imbalances and slightly dependent of transducers 2x

and 2y locations shown in diagram given in Fig.26. However when

the outer raceway spall diameter is more than two times larger (3,0

mm) the DVR decreases with increasing imbalances because the

bearing’s balls rotate through outer ring spall in enlarge contact area.

The spall damping effect increases and BPFO vibration velocity

values decreases Fig.26.

With increasing rotor’s vertical axis tilting angle γ the DVR

value decreases and slightly depends on rotor’s imbalance (Shown in

Fig. 27) and transducers locations 2x or 2y.

Fig. 26 The DVR versus dynamic force Fs plots

This factor indicates that DVR can be adopted to evaluate rotors

vertical angle position in the cases of installation of new machines not

only for evaluation of technical condition of vertical rotors axis

position and damaged bearings.

Fig. 27 The DVR versus axis tilt angles γ

Conclusions of subchapter

Theoretically simulated vertical axis rotor dynamic force Fs

position angle φ can be used for evaluations not only for degradations

of technical condition of bearings or bearings housing, but for control

of the vertical axis position reference to ideal vertical axis position; it

is important when angular velocities of rotors in technological

machines varies at run up and coast down including resonance

rotational speeds.

In the condition of small tilting angle (e.g. γ < 5º) the

vibration velocity values measured with x and y transducers are the

same order. However as tilting angle increased (e.g. γ > 5 º, as

characterized for sea ships machines) up to γ =13,5 º, the acting forces

on bearings are different and vibration velocities values differs about

20% when measured with x and y transducers.

In the case of outer raceway small spall (1,4 mm as the initial

defect in first phase of degrading bearing) with increasing dynamic

force Fs, the ball contact area with spall increases, and damaged

bearings indications parameter DVR increases, but in the case of large

spall (3,0 mm) the DVR values decreases, because the damping effect

during ball contact with enlarged spall increases.

The DVR parameter useful not only for vertical axis rotors

with antifriction bearing diagnostics, but for rotors with hydrodynamic

bearings technical condition monitoring, technological process control

42

in food and chemical industries (sulfuric acid pumps, sugar production

centrifuges, etc.).

4. EXPERIMENTAL VERIFICATION OF THEORETICAL MODEL

This section of work is devoted to verify a generalized

theoretical model of the vertical rotor. In second chapter of this work

the methodology of calculation of forces acting a vertical rotor

supports were provided. In second chapter of this work also presented

a generalized model of vertical rotor using finite elements. The

generalized rotor model can evaluate the gravity direction and

supports nonlinearity as well as radial gap of bearings.

The experimental research setup presented in Fig. 28, it consist

of AC motor driven by frequency inverter, labeled 1 in figure, elastic

coupling labeled 2, two supports with sliding bearings 3, dv = 20 mm

diameter shaft 4, flywheel disk 5 on which the unbalance mass can be

fixed, optical transducer for the rotational speed measurement 6. On

each of the rotor supports two displacement sensors EPRO PR6423

were mounted.

Fig. 28 Experimental investigation setup for theoretical model

verification

During experimental studies in the first support of rotor the

rolling bearing 6004/ C3 were mounted, in the second support of rotor

(located next to the disk) friction bearings with different radial gaps

were used, the material of friction bearings were PET HD 500. The

frictional bearings used in tests shown in Fig.29. The smallest used

radial gap of frictional bearing was 0,23 mm and the largest was 0,53

mm.

Fig. 29 The frictional bearings used for investigation

During the experimentation, three masses of imbalance where

used, lowest imbalance of 10 gmm, 85 gmm imbalance and 145 gmm

imbalance. In order to compare experimental results with theoretical

model calculations the radial stiffness of frictional bearings was

measured according to methodology described in second chapter. The

comparison of experimentally measured and calculated using FEM

model peak - peak values of shaft center, during first resonance of

rotor system presented in Fig. 30.

The comparison of experimentally measured and calculated

using FEM model rotational speed of first rotor resonance versus

radial gap in 2nd bearing of rotor shown in Fig. 31.

44

Fig. 30 Peak – peak values of shaft center during first resonance,

experimental and calculated results

Fig. 31 First resonance rotational speed of rotor versus radial gap in

2nd

frictional bearing, experimental and calculated results

Conclusions of subchapter

In order to assess the accuracy of the theoretical model,

additional experiments were carried out, when in the second support

of rotor radial sliding bearings with different radial gaps were

mounted. The radial stiffness curve of sliding bearings, for theoretical

model calculations, was measured experimentally.

A comparison of theoretical and experimental results of first

critical rotational speed nk values were used. It was found that at 0,39

mm radial sliding bearings gap, values between the theoretical model

and experimental data showed the highest relative error of 2,94%.

Relative error values indicate that the theoretical generalized vertical

rotor model, quite high accuracy and can be used to investigate the

behavior of the rotor.

MAIN CONCLUSIONS

1. Literature analysis of rotor dynamics fundamentals covered

stages of developments, analysis and review of modern methods and

tools applied to rotor dynamics research field. It was found that the

rotors with a vertical axis of rotation, as a research object, are usually

examined in order to analyze the influence of variable rotor properties

(mass, moment of inertia, nonlinear stiffness, variable damping, rotor

design) precession motion or in order to analyze the influence of

different non-linearity’s of supports stiffness to precession motion.

The material presented in the scientific literature has been analyzed,

focusing on the traditional, newly designed research methods for rotor

systems and their elements faults diagnostic and technical condition

monitoring. It was found that identification of technical condition of

rotor systems and diagnostics of defects are extremely relevant to the

creation of modern rotortronic systems and to renewal of long time

operated ones. Due to this fact, experimental research are being

developed continuously improving traditional diagnostic testing

methods and tools to create automatic failure prevention and forecast

methods. It was found that the diagnostic research methods, newly

developed under laboratory conditions, showed positive results of

rotor defects identification, but practical application becomes limited

when testing in Situ. Applying newly developed diagnostic methods in

practice, usually rotor dynamics is not proper taken into account. Thus

application of such methods becomes limited or difficult and expected

output is not reached. Literature analysis has showed that there is no

46

any detailed study which quantifiably aims to link vertical rotors

dynamics and defects diagnostics methods and their industrial

application.

2. Methodology for calculation of forces, acting to vertical rotors

supports, was developed. In accordance with methodology radial axial

loads of vertical rotors were calculated. It was established, that at rotor

speed of n = 3000 RPM the double increment of laboratory stand

imbalance mass (from 80 gmm to 154 gmm) results increasing by only

5 N of total force Fs, acting to rotor support next to the disk, while the

angle φ of both rotor rotation axis and the total force Fs increases by

17˚. It was found that due to the imbalance increased radial force

acting to rotor support significantly effects direction of the total rotor

support force Fs while magnitude is affected insignificantly.

Calculations were carried out not considering the rotor dynamics, gyro

effect, so this finding is theoretical.

3. In this work generalized vertical rotor model, which allows

estimating the effect of gyro and gravity, was developed. Using this

numerical model reaction forces at rotor supports were calculated,

capturing influence of imbalance. From constructed Campbell chart

critical angular velocity map ωk = 559 rad/s of the system being

analyzed was defined. It was found that even a small deviation of the

rotor rotation axis from the vertical, leads to appearance of subcritical

rotational speed. It is caused by bearing radial gap at the second

support. This gap also defines the system dynamic stiffness. It was

estimated that assessment of the effects of rotor dynamics affecting

the rotor system at rotational speed of the rotor n = 3000 RPM, double

increase of imbalance (from 80 gmm to 153 gmm) leads to increase of

the total force Fs next to the disk only slightly (only 6 N, from 31 N to

37 N). Total force Fs, angle φ, which it forms with the axis of rotation,

increases significantly by 15,5 degrees. At the critical rotational speed

ωk = 559 rad/s of the rotor the total force Fs acting to the second

support of rotor increases to 123 N, while the angle φ between the

total force and the rotor axis of rotation increases insignificantly by 6

degrees only.

4. After the comparative study of diagnostic tests of defects in

vertical and horizontal rotor rolling bearings, a new relative parameter

- Defect Visibility Ratio (DVR) was derived. DVR allows the quantity

assessment of differences among rotor defects diagnostic results.

Studies of rotor systems with differently oriented axis of rotation have

showed that defect indications of horizontal rotor system in many

cases are clearer. It allows for the researcher to identify the rolling

bearing defects in the early stages of their initiation. It was found that

defect on the rolling surface of bearing outer ring in horizontal rotor is

more precisely identified at horizontal plane which is perpendicular to

the defect. However, for vertical rotor bearings in both radial

directions defect is identified with no difference in accuracy.

5. After dynamics and diagnostic tests of the rotor with defect on

the inner surface of bearing ring under conditions of mismatch

between dynamic imbalance force and the phase angle of defect on

inner surface of bearing ring, it was found that the vertical rotors are

significantly more sensitive to the effects of imbalance, their defect

indications brightness, depends more on the magnitude of dynamic

imbalance force Fc. It was also found that the identification of defects

on inner surface of rolling bearing ring in horizontal rotors along the

direction of gravity force is uninformative. Defect indication does not

change even changing the dynamic imbalance force Fc.

6. During diagnostic investigation test of rolling bearings of

vertical rotor, which axis of rotation was tilted from the vertical the

following was found: when tilting of rotor system from the vertical is

up to 5˚ results measured in x and y directions are similar. The

difference is caused by the defect position in support (defect was

oriented in the y direction). At higher vertical rotor axis deviation

from the vertical (angle γ = 10˚ - 13,5˚), rotor dynamics makes more

influence to data measured in different radial directions. This

phenomenon is caused by variable dynamic stiffness of support during

test. It can be highlighted that under condition of larger (3,0 mm)

defect spall at surface of rolling bearing, outer ring angle β doesn’t

affect fault indication. It is caused by rolling elements which roll

48

through the defect and result increase in total dynamic force Fs.

Thus, defect indication decreases and also the parameter DVR

decreases.

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operating speeds of 1 500 r/min, 1 800 r/min, 3 000 r/min and 3

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50

with normal operating speeds of 1 500 r/min, 1 800 r/min, 3 000

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machine vibration by measurements on non-rotating parts –

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nominal speeds between 120 r/min and 15 000 r/min when

measured in situ“. 2009.

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vibration by measurements on non-rotating parts - Part 4: Gas

turbine sets with fluid-film bearings“. 2009.

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2009. PP 154 – 162.

LIST OF AUTHOUR’S SCIENTIFIC PUBLICATIONS

Articles in the Journals from the list of Institute of Scientific Information

(ISI)

1. Barzdaitis, Vytautas; Tadžijevas, Artūras; Mažeika, Pranas;

Grigonienė, Jurga; Modeling and diagnostics of vertical axis

rotary system powered by multi gear drive // Journal of

Vibroengineering. Kaunas: Vibrotechnika. ISSN 1392 – 8716.

2012, vol. 14, iss. 1, p. 171-178. [ISI Web of Science; INSPEC].

2. Artūras Tadžijevas, Vytautas Barzdaitis, Marius Vasylius, Pranas

Mažeika; The Comparison of Diagnostic Features between the

Vertical and Horizontal Axis Rotors // Journal of

Vibroengineering. Kaunas: Vibrotechnika. ISSN 1392 – 8716.

2013, vol. 16,[ISI Web of Science; INSPEC].

3. Artūras Tadžijevas, Vytautas Barzdaitis, Pranas Mažeika, Marius

Vasylius; Influence of Imbalance Force Angular Position to

Vertical and Horizontal Rotors Rolling Bearings Defects

Diagnostics // Journal of Vibroengineering. Kaunas:

Vibrotechnika. ISSN 1392 – 8716. 2014, vol. 16, iss. 3. [ISI Web

of Science; INSPEC].

52

4. Barzdaitis, Vytautas; Barzdaitis, Vytautas V.; Maskvytis,

Robertas; Tadžijevas, Artūras; Vasylius, Marius. "New deep

groove ball bearings high frequencies vibration testing" // ISSN

1392-1207. Mechanika. 2014 Volume 20(3), p. 287-293.

INFORMATION ABOUT AUTHOR

Artūras Tadžijevas was born in Klaipeda on November 12th,

1982. In 2001, he graduated from Klaipeda “Vetrunge” gymnasium.

During 2001-2005, he studied at Klaipeda University, Faculty of

Marine Engineering and in 2005 was awarded with a Bachelor’s

degree in mechanical engineering. During 2005-2007, he studied at

Klaipeda University, Faculty of Marine Engineering and in 2007 was

awarded with a Master’s qualification degree in mechanical

engineering. In 2010, started and in 2012 completed his doctoral

studies in a field of mechanical engineering at Kaunas University of

Technologies.

E-mail: tadzijevas@gmail.com

REZIUMĖ

Daktaro disertaciją sudaro įvadas, keturi skyriai, išvados ir cituotų

šaltinių sąrašas bei priedai. Bendra disertacijos apimtis142 puslapiai,

107 paveikslai ir 91 bibliografinė nuoroda.

Pirmajame daktaro disertacijos skyriuje analizuojama mokslinėje

periodikoje bei tarptautiniuose standartuose publikuotų rotorių

dinamikos ir jos sąryšis su rotorių elementų diagnostika. Skyriaus

pabaigoje pateikiamos skyriaus išvados.

Antrasis daktaro disertacijos skyrius dedikuotas analitinio

vertikalių rotorių bei baigtinių elementų modelio veikimui aprašyti.

Skyriuje taip pat pateikti skaičiavimų rezultatai. Skyriaus pabaigoje

pateikiamos skyriaus išvados.

Trečiajame skyriuje pateikiami trys nepriklausomi

eksperimentiniai tyrimai. Pirmasis tyrimas pateiktas pirmajame šio

skyriaus poskyryje ir yra skirtas ištirti ir palyginti vertikalių ir

horizontalių rotorių dinamikos įtaką šių rotorių riedėjimo guolių

diagnostikai. Skyriuje pateikiamas naujas, sukurtas statistinis

parametras, kurio dėka galime palyginti rotorių diagnostikos

ypatumus. Artajame šio skyriaus poskyryje pateiktas vertikalių ir

horizontalių rotorių eksperimentinis tyrimas su vidinio riedėjimo

guolio žiedo defektu, kai vidinio žiedo defekto ir dinaminės

disbalanso jėgos fazės kampas nesutampa. Trečiajame šio skyriaus

poskyryje pateiktas vertikalių rotorių pasvirusių nuo vertikalės,

dinamikos ir diagnostikos tyrimas. Poskyrio tikslas nusakyti, kaip

suminės dinaminės disbalanso jėgos bei ašinės jėgos dydis ir kampas

įtakoja vertikalių rotorių elementų gedimų diagnostiką. Skyriaus

pabaigoje pateikiamos išvados.

Ketvirtajame šio darbo skyriuje pateikiamas apibendrinto

vertikalių rotorių baigtinių elementų modelio verifikavimas. Skyriuje

aprašomas eksperimentinis tyrimas, kuris atliktas su skirtingą radialūjį

tarpą turinčiais slydimo guoliais, rezultatai lyginami su skaičiavimo

rezultatais. Skyriaus pabaigoje pateikiamos išvados.

Darbo pabaigoje pateikiamos bendrosios išvados, bibliografinių

nuorodų sąrašas, publikuotų, su šio darbo tematika susijusių, autoriaus

mokslinių darbų sąrašas, priedai.

INFORMACIJA APIE AUTORIŲ

Artūras Tadžijevas gimė 1982 metų lapkričio 12 dieną, Klaipėdoje.

2001 metais baigė „Vėtrungės“ gimnaziją Klaipėdoje. 2001 – 2005

metais studijavo Klaipėdos universitete, Jūrų technikos fakultete, įgijo

mechanikos inžinerijos bakalauro kvalifikacinį laipsnį. 2005 – 2007

metais studijavo Klaipėdos universitete, Jūrų technikos fakultete, įgijo

mechanikos inžinerijos magistro kvalifikacinį laipsnį. 2010 metais

pradėjo ir 2014 metais baigė Kauno technologijos universiteto

mechanikos inžinerijos mokslo krypties doktorantūros studijas.

El. paštas: tadzijevas@gmail.com

54

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