Experimental Verification of Effect of Different Fluid Properties on the Vibration Response of a Cantilever Rotor
The current analysis is an effort to obtain the vibration characteristics of the cantilever rotor shaft with an extra mass added at the free end of the rotor shaft partially immersed in the viscous medium. This work is concentrated on the theoretical analysis of the natural frequency and amplitude o...
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irk-123456789-1882792023-02-21T01:26:54Z Experimental Verification of Effect of Different Fluid Properties on the Vibration Response of a Cantilever Rotor Yadao, A.R. The current analysis is an effort to obtain the vibration characteristics of the cantilever rotor shaft with an extra mass added at the free end of the rotor shaft partially immersed in the viscous medium. This work is concentrated on the theoretical analysis of the natural frequency and amplitude of the spinning cantilever rotor shaft with addition mass using the influence coefficient method. The influence of fluid forces is studied using the Navier-Stoke’s equation. The gap ratio (ratio of the fluid-filled container radius to the shaft radius) and viscosity of the fluid are taken as the main variable parameters. MATLAB programming is used to obtaining vibration behavior from the theoretical expressions. Аn obtained result from the numerical analysis is validated by the comparison of the results of experimental analysis. Запропонований аналіз має метою отримати характеристики коливань валу консольного ротора з додатковою масою на вільному кінці валу, частково зануреного у в'язке середовище. Увага зосереджена на теоретичному аналізі власних частот та амплітуд обертового консольного валу з додатковою масою з використанням методу коефіцієнта впливу. Вплив сил рідини вивчається за допомогою рівняння Нав'є-Стокса. Коефіцієнт зазору (відношення радіуса ємності, наповненої рідиною, до радіуса валу) та в'язкість рідини приймаються за основні змінні параметри. Для отримання вібраційної поведінки з теоретичних виразів використовується програмування MATLAB. Отриманий результат чисельного аналізу підтверджується порівнянням з результатами експериментального аналізу. 2020 Article Experimental Verification of Effect of Different Fluid Properties on the Vibration Response of a Cantilever Rotor / A.R. Yadao // Прикладная механика. — 2020. — Т. 56, № 5. — С. 136-144. — Бібліогр.: 17 назв. — англ. 0032-8243 http://dspace.nbuv.gov.ua/handle/123456789/188279 en Прикладная механика Інститут механіки ім. С.П. Тимошенка НАН України |
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The current analysis is an effort to obtain the vibration characteristics of the cantilever rotor shaft with an extra mass added at the free end of the rotor shaft partially immersed in the viscous medium. This work is concentrated on the theoretical analysis of the natural frequency and amplitude of the spinning cantilever rotor shaft with addition mass using the influence coefficient method. The influence of fluid forces is studied using the Navier-Stoke’s equation. The gap ratio (ratio of the fluid-filled container radius to the shaft radius) and viscosity of the fluid are taken as the main variable parameters. MATLAB programming is used to obtaining vibration behavior from the theoretical expressions. Аn obtained result from the numerical analysis is validated by the comparison of the results of experimental analysis. |
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Yadao, A.R. |
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Yadao, A.R. Experimental Verification of Effect of Different Fluid Properties on the Vibration Response of a Cantilever Rotor Прикладная механика |
| author_facet |
Yadao, A.R. |
| author_sort |
Yadao, A.R. |
| title |
Experimental Verification of Effect of Different Fluid Properties on the Vibration Response of a Cantilever Rotor |
| title_short |
Experimental Verification of Effect of Different Fluid Properties on the Vibration Response of a Cantilever Rotor |
| title_full |
Experimental Verification of Effect of Different Fluid Properties on the Vibration Response of a Cantilever Rotor |
| title_fullStr |
Experimental Verification of Effect of Different Fluid Properties on the Vibration Response of a Cantilever Rotor |
| title_full_unstemmed |
Experimental Verification of Effect of Different Fluid Properties on the Vibration Response of a Cantilever Rotor |
| title_sort |
experimental verification of effect of different fluid properties on the vibration response of a cantilever rotor |
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Інститут механіки ім. С.П. Тимошенка НАН України |
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2020 |
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http://dspace.nbuv.gov.ua/handle/123456789/188279 |
| citation_txt |
Experimental Verification of Effect of Different Fluid Properties on the Vibration Response of a Cantilever Rotor / A.R. Yadao // Прикладная механика. — 2020. — Т. 56, № 5. — С. 136-144. — Бібліогр.: 17 назв. — англ. |
| series |
Прикладная механика |
| work_keys_str_mv |
AT yadaoar experimentalverificationofeffectofdifferentfluidpropertiesonthevibrationresponseofacantileverrotor |
| first_indexed |
2025-07-16T10:16:31Z |
| last_indexed |
2025-07-16T10:16:31Z |
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1837798252761579520 |
| fulltext |
2020 П Р И К Л А Д Н А Я М Е Х А Н И К А Том 56, № 5
136 ISSN0032–8243. Прикл. механика, 2020, 56, № 5
A . R . Y a d a o
EXPERIMENTAL VERIFICATION OF EFFECT OF DIFFERENT FLUID
PROPERTIES ON THE VIBRATION RESPONSE OF A CANTILEVER ROTOR
Department of Mechanical Engineering G. H. Raisoni College
of Engineering and Management Pune Indi; email: adik.mech@gmail.com
Abstract. The current analysis is an effort to obtain the vibration characteristics of the
cantilever rotor shaft with an extra mass added at the free end of the rotor shaft partially
immersed in the viscous medium. This work is concentrated on the theoretical analysis of
the natural frequency and amplitude of the spinning cantilever rotor shaft with addition mass
using the influence coefficient method. The influence of fluid forces is studied using the
Navier-Stoke’s equation. The gap ratio (ratio of the fluid-filled container radius to the shaft
radius) and viscosity of the fluid are taken as the main variable parameters. MATLAB pro-
gramming is used to obtaining vibration behavior from the theoretical expressions. Аn ob-
tained result from the numerical analysis is validated by the comparison of the results of
experimental analysis.
Keywords: cantilever rotor shaft, dynamic response, Navier-Stoke’s equation, influence
coefficient method.
1. Introduction.
In the last few years, the researchers focused on the dynamic analysis of the critical
speed of the rotor, under several conditions, due to its significance in design. When a shaft
rotates in a viscous medium, the analysis of natural frequency and amplitudes becomes dif-
ficult. The effects of a crack in the shaft of rotating machinery associated with dynamic
analysis were determined by many investigators. The researchers are done a lot of work on
the vibration response of the shaft. But no particular analysis of the rotor shaft in a different
fluid medium is described. The current analysis is helpful for the vibration characteristics of
the rotating shaft in a viscous fluid such as high-speed turbine rotors, long spinning shaft
used for the extracting oil from the seabed by drilling, etc.
Wauer [1] has reported a review of the dynamics of a cracked rotor. Kedyrow et al. [2]
have studied the vibration analysis of the oscillation of cylinder immersed inside the viscous
fluid-filled tubular duct. They have used theoretical and mathematical significance for ob-
tained the frequency with consideration of different fluid parameter Walston et al. [3] have
investigated the vibration characteristics of a spinning shaft inside a viscous medium, but no
exact distinction was made among the damping effect and virtual mass effect on the spin-
ning shaft. Papadopoulos and Dimarogonas [4] have discussed the dynamic characteristics
of a spinning shaft with transverse crack, but they have not discussed the separately external
damping effect and virtual mass effect on the crack shaft. Ostachowicz and Krawczuk [5]
have studied the effect of transverse cracks on coupled torsional and bending vibrations of a
rotor. Gasch [6] has studied the dynamic response of a spinning shaft with a transverse
crack. They took the De-Laval rotor for the investigation and explored some prospects for
early crack identification. Fritz [7] has examined a spinning rod in a cylindrical vessel with
consideration of the effect of Taylor vortex and turbulence for a small rod cylinder gap.
Cho-Chung et al. [8] have investigated a simple technique to define the vibration mode
shapes and frequencies of cantilever plates which are submerged in the fluid medium based
on empirical added mass formulation. Sol [9] has analyzed the dynamic behavior of the
137
cracked shaft with critical speed. They have obtained analytical results compare with the
results of experimental analysis. Brenner [10] has investigated the theoretically the fluid
forces acting on spinning rod inside the circular cylinder for low and high Reynolds num-
bers. Walston et al. [11] have reported the vibration analysis of a rotating shaft submerged
inside the viscous fluid. The effect of the fluid forces on the dynamic response of the rotor
evaluate experimentally and by using the statistical regression analysis, obtained the equa-
tion of amplitude as a function of viscosity, velocity and mass. Achenbach et al. [12] have
presented the vibration analysis of an immersed beam of circular cross-section using math-
ematical methods. Han and Chu [13] have studied the dynamic instability of the breathing
cracked rotating shaft. Xiaohui et al. [14] based on the Rayleigh-Ritz formulation and green
function method, presented the dynamic analysis of the cracked rectangular plate with the
infinite water field. Awlad et al. [15] have investigated numerically the dynamic characteris-
tics of cantilever beam which is partially submerged inside the viscous fluid. Also they have
done the experimental verification by using polytech scanning vibrometer.
Hahgholi et al. [16] have studied the dynamic analysis of a nonlinear rotating simply
support shaft. In the modeling of the system rotary inertia and gyroscopic effect are consid-
ered. The equations of motion are derived with the help of the extended Hamilton principle.
Catherine et al. [17] have investigated the vibrations characteristics of a cantilever beam of
rectangular cross section immersed in a viscous fluid under harmonic base excitation. The
Interactions between fluid and structure are demonstrated by a complex hydrodynamic func-
tion which is defines additional mass and damping effect in dynamic response of large alter-
ation of amplitudes.
In this article, a systematic investigation for the vibration characteristics of a spinning
cantilever rotor shaft in viscous fluid medium is presented. The external fluid force is calcu-
lated with the help of Navier Stoke’s equation. Dynamic response (i.e. Natural frequency,
amplitude)of the rotor shaft is obtained using the influence coefficients method. Finally, the
numerical analysis results are compared with the developed experimental analysis results.
2. Theoretical analysis.
2.1 Equation of motion. Figure1 represents a fixed-fixed type rotating shaft in which
the X-Y coordinate system and the point ‘O’ indicate the center of spinning shaft. The point
‘ό’ rotating around the point ‘O’ with whirling speed ‘Ω’ and whirling radius ‘δ’.
Fig. 1. Whirling speed of the rotor shaft
U
ω
O
θ
Ωt
Ó
Y
R1
δ r
A
Χ
V
Ω
138
In polar coordinates of the Navier Stroke’s equation can be expressed as,
2 2
2 2 2 2 2 0;
1 1 2 1p u u v u u u
r r r r r r r t
(1a)
2 2
2 2 2 2 2 0.
1 1 2 1p v v u v v v
r r r r r r r t
(1b)
According to the non- stationary components of radial and tangential velocities of the
rotor shaft are given as follows:
2
1 1 1
1 1 ;
1 j t
nc
R R R
u j A B C I kr D K kr e
r r r r
(2a)
2
1 1 1
1 1 0 1 1 0 .
nc
j t
v
r
R R R
A B C I kr kR I kr D K kr kR K kr e
r r r
(2b)
2.2 Analysis of fluid forces. Substituting the velocities of external fluid flow into the
equation (1), the pressure (P) can be calculated as
.2 2 i(wt-q)
1
dp -A dy
P = dq = drw R + Br e
dq r dx
(3)
The direct stress (
rr
) and circumferential stress ( r ) can be expressed as,
2 i
rr
du
m p t
dr
and
1
.i i
rr
n dud
m r t
dr r r dq
(4)
External fluid force impact on the rotor shaft at the both directions (i.e. x and y) are de-
termined as
2
2
1 1 1
0
cos sin ;i t
x rr rF R d m A B CI DK e
(5a)
2
2
1 1 1
0
sin cos .i t
y rr rF R d im A B CI DK e
(5b)
2.3 Equation of rotor motion. In the current study, the cantilever rotor shaft with a disc
at the mid-span of the shaft which is immersed inside the fluid medium has been considered.
Firstly, here consider the equivalent lump mass of rotor shaft is determined. After that the
additional masses is added and calculate the overall lumped mass of the rotor system.
The equivalent lump, mass of the system can be expressed:
44
1 2
44 2
eq
s
K
M
and 55
2 2
55 2
.eq
s
K
a
M
The whole mass of the cantilever rotor shaft becomes
1 1 2 1 0s eq s sM M M M and 1 2 2 2 0s eq s sM M M M ;
139
2
44 1 2
cos
;x s
d X t
F K X M M
dt
(6a)
2
55 22
sin
.sy
d Y t
F K Y M M
dt
(6b)
The equation (6) can be rewritten as,
2
2
Im( ) Re( ) ;x
dx d x
F M H M H
dt dt
(7a)
2
2
Im( ) Re( ) .y
dy d y
F M H M H
dt dt
(7b)
Where 1 1 2 2Re( ) Re( ) Re( )eqM H M H M H
1 1 2 2Im( ) Re( ) Im( );eqM H M H M H
1 2 .eqM M M
Where M1= Fluid mass moved by the additional mass and M2= fluid mass moved by rotor
shaft.
2
2
2
( Re( ) ) Im( ) cos 0;s s s
d x dx
K x M H M M H M t
dt dt
(8a)
2
2
2
( Re( ) ) Im( ) sin 0.s s s
d x dy
K y M H M M H M t
dt dt
(8b)
Taking the eccentricity 1 and 2 . Primarily, 1 is considered for X- axis Dirn and Y-
axis Dirn is finding out in the subsequent method.
2
2* * * * * *2
1 1 1 1 1 1 1
1 1
1 Re( ) Im( ) cos( );
d d
M H M H
dt d
(9a)
2
2* * * * * *2
2 2 2 2 2 2 2
2 2
1 Re( ) Im( ) sin( ).
d d
M H M H
d d
(9b)
By solving equation (9), we can get a the non-dimensional form as
*
22 2 * 2
;
( ) ( )
A
C K
*
1
* 2
tan ;
( )
C
K
* *
*
Im
;
1 Re
m H
C
m H
* * 2
*
( )
.
Re 1
A
M H
Where δ* is the maximum non-dimensional amplitude
3. Numerical analysis and discussions.
Different parameters of the cantilever rotor shaft for numerical and experimental analy-
sis are given below in Table 1.
Table 1. Parameters related to vibration behavior of cantilever rotor shaft.
140
a b
a b
c d
e f
Fig.2. (a, c, e) Frequency (ω) Vs Non-Dimensional amplitude (δn*/ɛ*),
(b, d, f) Frequency (ω) Vs Phase angle (ϕ), Mild Steel Shaft, R1= 0,01m,
L= 0,8m, q=12, MD= 0,55kg, TD= 0,022m
S.N. Parameters Value (Unit)
01 Material of rotor shaft Mild steel
02 Density of shaft material ρ =7850 kg/m3
03 Radius of the rotor shaft R1 = 0,01m
04 Length of rotor shaft L = 0,8m
05 Gap ratio(R2-R1/R1) q = 6, 12, 16, 20
06 Mass of disc MDisc = 0,55kg
07 Radius of the disc RD = 0,035m
08 Thickness of disc TD = 0,022m
09 Three different type of viscous fluid
10 Fluid Name Kinematic viscosity v(stokes) Ratio of density (M*)
11 Water 0,0633 0,144
12 Palm oil 0,541 0,153
13 Lubricant oil 2T 2,4 0,158
540 560 580 600 620 640
0
50
100
150
200
250
300
Frequency ( RPM )
N
on
-D
im
en
si
on
al
A
m
pl
itu
de
v(stokes) M*
0.0633 0.144
540 560 580 600 620 640
0
50
100
150
200
Frequency ( RPM )
P
ha
se
A
ng
le
v(stokes) M*
0.144 0.0633
540 560 580 600 620 640
0
20
40
60
80
100
Frequency ( RPM )
N
on
-D
im
en
si
on
al
A
m
pl
itu
de
v(stokes) M*
0.541 0.153
540 560 580 600 620 640
0
50
100
150
200
Frequency ( RPM )
P
ha
se
A
ng
le
v(stokes) M*
0.541 0.153
540 560 580 600 620 640
0
10
20
30
40
50
Frequency ( RPM )
N
on
-D
im
en
si
on
al
A
m
pl
itu
de
v(stokes) M*
2.4 0.158
540 560 580 600 620 640
0
50
100
150
200
Frequency ( RPM )
P
ha
se
A
ng
le
v(stokes) M*
2.4 0.158
141
Fig. 3. Frequency (ω) Vs Non-Dimensional amplitude (δn*/ɛ*),
Mild Steel Shaft, R1= 0,01m, L= 0,8m, MD= 0,55kg, TD= 0,022m
3.1 Discussion. Fig. 2 shows the influence of different viscosity’s of viscous fluid on
the dynamic response of the cantilever rotor shaft. It is found that as the viscosity of fluid
medium increases the whirling speed of the rotor shaft and the amplitude of vibration fall
down due to increase in virtual mass. Fig. 3 shows the effect of different gap ratio on the
dynamic response of the rotor shaft. It is observed that the amplitude of vibration increases
due to increases the radius of the fluid filled container.
4. Experimental analysis.
To compare the obtained results of the theoretical analysis, experimental analyses have
been done in a spinning cantilever rotor shaft inside the various fluid medium. For this purpose
we have developed an experimental setup which comprises the rotor shaft with extra mass (i.e.
disc), variac, power motor, three ultrasonic sensors, microcontroller, bread board, USB con-
nection serial port and response display devices (i.e. Computer system) as shown in Fig. 4.
Fig. 4. Experimental Setup of cantilever rotor shaft system with viscous medium.
540 560 580 600 620 640
0
10
20
30
40
50
60
Frequency (RPM)
N
on
-D
im
en
si
on
al
A
m
pl
it
ud
e
q=6
q=12
q=16
q=20
v (stokes) M*
2.4 0.158
142
Experiments have been conducted on the rotating cantilever rotor shaft with attached
mass at the free end in the different viscous fluid medium. The inner radius and length of the
fluid-filled container are 260 mm and 400 mm respectively. The shaft of the power motor is
connected with the rotor shaft with the help of universal joint. Variac is used to control the
speed of the spinning rotor shaft. The rotating shaft is supported by ball bearing. In this
work the bearing effect is not taken into account in order to simplify for the rotor system.
The rotating speed of the rotor shaft is taken in the range of 300 to 700 rpm. The three ultra-
sonic sensors are arranged around the rotating cantilever rotor shaft in the viscous fluid is
connected to the Arduino microcontroller by using the bread board and jumper wire is used
to obtain the deflection of the rotor shaft from starting position. Each sensor has separate
deflection data of the rotor motion. For accuracy, reading has been taken for three times at
the same speed. Each set have 900 samples measured by sensors within 15 minutes (i.e.
sampling time 1sec/sample).
Table 2. Detail of parts of experimental test rig of cantilever rotor shaft system.
Table 3. Effect of the different viscosity of fluid medium on the amplitude of the rotor shaft.
S.N.
Numerical results Experimental results % Error
Three different viscosity (v) in Stokes
2,4 0,541 0,0633 2,4 0,541 0,0633 2,4 0,541 0,0633
1 4,704 4,445 4,328 4,887 4,582 4,522 3,9 3,1 4,5
2 7,383 6,808 6,546 7,597 7,114 6,801 2,9 4,5 3,9
3 14,896 13,124 12,239 15,673 13,465 12,899 5,2 2,6 5,4
4 40,244 68,160 57,932 41,491 71,022 56,309 3,1 4,2 – 2,8
5 38,359 91,848 88,496 40,123 95,614 92,478 4,6 4,1 4,5
6 33,782 96,964 230,615 34,998 94,443 224,157 3,6 – 2,6 – 2,8
7 30,062 74,084 207,944 31,475 77,492 217,509 3,1 4,6 4,6
8 26,673 55,157 100,519 27,927 57,143 104,439 4,7 3,6 3,9
9 11,112 13,425 14,687 11,489 13,035 14,305 3,4 – 2,9 – 2,6
10 7,122 7,931 8,333 7,421 7,724 8,566 4,2 – 2,6 2,8
S.N. Name of different parts of experimental test rig
1
Mild Steel cantilever rotor shaft
Shaft length (L) = 0,8; Radius of shaft (R1) = 0,01m
2 Radius of liquid filled vessel 13cm
3 Power supply, 50Hz , 230 to 240v AC
4 Taco meter: Telco Hand Taco meters, Type: H 10000 rpm
5 Driving unit ( Motor, Type AC/DC, 220 v , 125 watt ,6000 rpm )
6 Variac, Input: 230v, Output: 0 to 270 v, 50 to 60 Hz
7 Bread board
8 USB connection Serial Port
9
Microcontroller,
Arduino UNO A Tmega328,
Analog Input Pins 6,
Digital Input Pins 14,
Input voltage 6-20V,
Operating Voltage 5V.
10
3 Ultrasonic Sensors,Range of distance measure: 0.01m to 4.5 m
Operating Voltage: DC 5V, Operating Current: less than 15mA,Response Time: 100mS
143
Table 4. Effect of different gap ratio on the amplitude of rotor shaft.
S.N.
Numerical results Experimental results % Error
Three different gap ratio(ratio of vessel radius to shaft radius)
q1=12 q2=16 q3=20 q1=12 q2=16 q3=20 q1=12 q2=16 q3=20
1 4,704 7,165 7,117 4,887 7,458 7,302 3,9 4,1 2,6
2 7,383 14,269 14,130 7,597 14,768 14,723 2,9 3,5 4,2
3 14,896 51,735 56,122 15,673 53,907 57,974 5,2 4,2 3,3,
4 40,244 49,944 54,972 41,491 51,742 56,511 3,1 3,6 2,8
5 38,359 44,650 48,916 40,123 45,677 50,334 4,6 2,3 2,9
6 33,782 38,526 41,647 34,998 39,489 42,938 3,6 2,5 3,1
7 30,062 33,052 35,249 31,475 34,241 36,024 3,1 3,6 2,2
8 26,673 15,074 15,429 27,927 15,32 15,953 4,7 1,6 3,4
9 11,112 9,867 10,000 11,489 10,153 10,21 3,4 2,9 2,1
10 7,122 7,413 7,486 7,421 7,826 7,845 4,2 4,6 4,8
Table 3 shows the comparison of the obtained numerical and experimental result of am-
plitude of shaft due to influence of the different viscosity of the viscous fluid. Table 4 shows
the comparison of the obtained numerical and experimental results of amplitude of vibration
due to effect of different gap ratio (i.e. ratio of the radius of fluid vessel to the radius of rotor
shaft). The result of numerical analysis was compared with the obtained experimental re-
sults. It was found to be good agreement.
5. Conclusions.
In the current investigation, presented the dynamic analysis of cantilever rotor shaft
immersed in a fluid medium was firstly obtained by the numerical analysis which deter-
mined by the MATLAB programming. Then the result of the numerical analysis was com-
pared with the results of developed experimental analysis of test rig. It is found to be very
good agreement. The viscosity and density of the viscous fluid in which spinning cantilever
rotor shaft was immersed were observed to remarkable effect on its vibration characteristics.
The fundamental natural frequency of the rotor shaft and amplitude of vibration was ob-
served to be fall down with the increased fluid viscosity and density. The phase lag of the
rotor shaft is varied due to the existence of viscous fluid. An amplitude of vibration is in-
creased while increases the gap ratio of the cantilever rotor system.
Nomenclature
A = Cross-sectional area of rotor
D =Diameter of rotor shaft
E =Young’s modulus of elasticity
h= Height of rectangular strip
i = Variable
I = Moment of inertia
Ks= Stiffness of the shaft
L = Length of shaft
m = Fluid mass displaced by the
shaft per unit length
ms= Rotor shaft mass per unit length
M*= Dimensionless parameter (m/ms)
MD= Mass of disc
q = Gap ratio (R2-R1/R1)
RD = Radius of disc
R1 =Radius of the rotor shaft
R2 = Radius of the fluid vessel
TD = Thickness of disc
u = Radial flow velocity
v = Tangential flow velocity
ɛ*= ɛ1/R1
ɛ = Eccentricity of mass of disc
from its geometry center
Ω = Angular velocity of whirling
144
РЕЗЮМЕ. Запропонований аналіз має метою отримати характеристики коливань валу кон-
сольного ротора з додатковою масою на вільному кінці валу, частково зануреного у в'язке середови-
ще. Ця робота зосереджена на теоретичному аналізі власних частот та амплітуд обертового консоль-
ного валу з додатковою масою з використанням методу коефіцієнта впливу. Вплив сил рідини вивча-
ється за допомогою рівняння Нав'є-Стокса. Коефіцієнт зазору (відношення радіуса ємності, наповне-
ної рідиною, до радіуса валу) та в'язкість рідини приймаються за основні змінні параметри. Для
отримання вібраційної поведінки з теоретичних виразів використовується програмування MATLAB.
Отриманий результат чисельного аналізу підтверджується порівнянням з результатами експеримен-
тального аналізу.
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_____________________
*From the Editorial Board: The article corresponds completely to submitted manuscript.
Поступила 28.03.2019 Утверждена в печать 09.07.2020
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
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/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|