*Corr. Author’s Address: Faculty of Industrial Engineering Novo mesto, Šegova ulica 112, 8000 Novo mesto, Slovenia, anatolij.nikonov@fini-unm.si 421
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 421-432 Received for review: 2021-03-21
© 2021 Journal of Mechanical Engineering. All rights reserved.  Received revised form: 2021-06-24
DOI:10.5545/sv-jme.2021.7178 Original Scientific Paper Accepted for publication: 2021-09-17
Multi-parametric Dynamic Analysis of a Rolling Bearings System
Kydyrbekuly, A. – Ibrayev, G.-G.A. – Ospan, T. – Nikonov, A.
Almatbek Kydyrbekuly
1
 – Gulama-Garip Alisher Ibrayev
 1
 – Tangat Ospan
1
 – Anatolij Nikonov
2,*
1 
Al-Farabi Kazakh National University, Kazakhstan 
2 
Faculty of Industrial Engineering Novo mesto, Slovenia
A method for calculating amplitudes and constructing frequency characteristics of forced and self-excited vibrations of a rotor-fluid-foundation 
system on rolling bearings with a non-linear characteristic based on the method of complex amplitudes and harmonic balance has been 
developed. Non-linear equations of motion of the rotor-fluid-foundation system are derived, and analytical methods of their solution are 
presented. Frequencies of fundamental and ultra-harmonic resonances are determined. The intervals between self-oscillation frequencies are 
estimated. The dependence of amplitudes on the amount of fluid in the rotor cavity, the mass of the foundation, linear imbalance, the value of 
the stiffness coefficient, and the damping coefficient is shown.
Keywords: forced vibrations, resonance, fluid, rolling bearing, self-oscillations, ultra-harmonic
Highlights
• 	 A dynamic model of the rotor-fluid-foundation system on rolling bearings with a non-linear characteristic has been constructed.
• 	 The specific features of the system with many degrees of freedom are defined.
• 	 The prerequisites under which the fluid stabilizes the behaviour of a non-linear system are determined.
• 	 The optimal parameters of the system, providing its stable operating mode, are determined. 
0  INTRODUCTION
One of the main tasks of dynamics of rotors and design 
of rotary machines is the choice of parameters that 
provide their stable operation. The damping of forced 
and purely non-linear self-excited rotor vibrations by 
selecting parameters of the system, taking into account 
vibrations of the foundation, elasticity of supports, 
imbalance and other parameters, is of great interest 
from both technical and economic points of view.
Currently, many works on the study of non-
linear oscillations of rotor systems exist [1]. However, 
their dynamics is insufficiently studied because of 
difficulties arising in taking into account the mutual 
influence and combined actions of factors such as 
the presence of fluid in the rotor cavity, external non-
conservative forces [2], different types of linear and 
angular imbalances and nonlinearities [3] and [4].
In the design of rotary machines, one of the most 
important mechanical components described by non-
linear models is elastic support, which ensures the 
operability and reliability of the system [5] to [7]. 
In this paper, rolling bearings act as elastic supports 
[8]. The neglect of non-linear properties of bearings 
prevents correct qualitative and quantitative results 
for rotor systems from being obtained [9] to [11]. This 
phenomenon can be explained by the fact that when 
analysing linear rotor systems with rolling bearings, 
an approximate estimate of stiffness and damping 
properties of bearings is most often used, whereas in 
reality the bearing stiffness heavily depends on the 
load, i.e., on the operating mode of the rotor system, 
on the geometry and size of the bearing clearances, on 
the size of the fit of the inner and outer rings in the 
bearing, and other factors [12] and [13].  
To obtain a detailed description of the process, 
it is necessary to consider the influence of such 
factors as imbalance, asymmetry of the rotor on the 
shaft, external friction, changes in inertial parameters 
and positional forces of various kinds [14] to [16]. 
Such complications of the model in the analysis of 
dynamics make it possible to study the influence of 
the gap size, rotation frequency on frequency spectra, 
and amplitude-frequency characteristics for any rotor 
system on rolling bearings.
Mathematical models of bearings that take 
into account nonlinearity factors are distinguished 
by complexity and primarily by the loads that they 
consider. In our case, to describe the bearing model, 
the Hertz contact theory is used, which connects the 
radial loads acting on the bearing and the deformation 
at the contact points between the rolling element 
and the bearing rings [17]. In the bearing model, it is 
assumed that there are no types of sliding of bodies 
and rolling surfaces. Damping is considered in terms 
of equivalent viscous and linear friction.
1  STATEMENT OF PROBLEM AND EQUATIONS OF MOTION
A symmetrical vertical rotor of mass m with a 
cylindrical cavity of radius R, partially filled with 
an ideal fluid, is rotating on rolling bearings with a 
constant angular velocity Ω
0
. The angular velocity is 
considered sufficiently large so that the fluid in the 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 421-432
422 Kydyrbekuly, A. – Ibrayev, G.-G.A. – Ospan, T. – Nikonov, A.
rotor cavity takes the form of a cylindrical ring. Due 
to the different positions of the centre of mass and 
the geometric centre, the rotor has a static imbalance 
(or linear eccentricity) e. The foundation of mass M 
rigidly connected to the outer ring is mounted on 
elastic supports with an equivalent linear stiffness 
coefficient c
2
 (see Fig. 1); below, all the dimensional 
parameters shown in this figure are appropriately 
dimentionalized, see Section 3. The radial compliance 
of bearings occurs due to deformation of the rolling 
elements and raceways at the contact points. In this 
case, the non-linear restoring force in bearings can be 
generally described using the Hertz contact formula 
[14] and [18]:
 FC
Cb r
 
3
2
. (1)
However, for the sake of simplicity, in what 
follows, the restoring force in rolling bearings is 
approximated by a power series according [16]:
 Fc c
Cr r
 
01
3
 , (2)
where F
C
 is a component of the restoring force in the 
radial direction, δ
r
 is the deformation in the radial 
direction, C
b
 is the stiffness coefficient, c
0 
and c
1
 are 
stiffness coefficients for the linear and cubic terms, 
respectively. This expansion at δ
r
 < 1000 µm agrees 
with the experimental results in [18].
Fig. 1.  Scheme of a rotor on rolling bearings, with a cavity partially 
filled with an ideal fluid and mounted on a foundation
The motion of the system is considered 
concerning the fixed coordinate system Oxyz. The 
coordinates of the centre of mass of the rotor are 
denoted as A( х, у), and the coordinates of the centre 
of mass of the foundation are denoted as A
2
( х
2
, у
2
), χ 
and χ
0
 are the external friction coefficients, О ξη the 
coordinate system associated with the rotor, η is its 
polar axis, and axis ξ is arbitrarily drawn through the 
eccentricity vector of the rotor mass.
Let us introduce complex variables in the form:
 xi yzxi yz   ,
22 2
, (3)
for analysing plane-parallel motion of the rotor and 
foundation. Then, assuming that damping force acting 
on rotor depends on velocities of the rotor itself, and 
similarly, damping force acting on foundation depends 
on the velocities of the foundation itself, we present 
the equations of motion of the system in question 
in the form, here and below see [8] and references 
therein,
 
 

znzz nz zk z
ei t
F
m
zn z
r
 


0
2
21 2
3
0
2
0
22
2
2
2 () ()
exp( ), 
    nzzn zz kz
01
2
21 02
3
02
20 () () ,  (4)
where
n
c
m
n
c
m
k
m
n
c
M
n
c
M
nn
c
M
n
0
2 0
1
1
2
2 2
01
2 0
0
2
10
1
2 2
2
2
2 2

 
,, ,,
,


1 10
0
2 ,, , k
M
m
M



and F
r
 is a complex expression for the reaction force 
of a fluid, which is defined as:
 FR hP ed
rr R
it




|,
()
0
2
0




 (5)
where h is the rotor cavity height, P|
r = R
 is the fluid 
pressure on the rotor wall.
The equations of motion for fluid are written as:
 


 






 
u
t
P
r
xt
yt
t
u
2
1
2
1
00
0
0










cos( )
sin( ),
 


r
P
xt
yt






sin( )
sin( ),


0
0
 (6)
where u and υ are the radial and tangential components 
of the velocity of the fluid particle, P and ρ are 
pressure and density of the fluid.
The continuity equation at ρ = const is given by:
 






()
.
ur
r


0 (7)

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 421-432
423 Multi-parametric Dynamic Analysis of a Rolling Bearings System
The boundary conditions for the Eqs. (6) and (7) 
can be written as:
 u
rR
|,
=
= 0 (8)
and
 





P
t
ru
rr
rr
0
0
0
2
0
 |, (9)
where r
0 
is the radius of the free fluid surface. These 
relations express the impenetrability condition at the 
rotor wall and the conditions on the fluid pressure 
along the free surface, respectively.
Eqs. (3) to (7) with boundary conditions Eqs. (8) 
and (9) in combination with the continuity equation 
form a closed system of equations.
2  NON-LINEAR FORCED VIBRATION AND SELF-
OSCILLATIONS OF THE ROTOR-FLUID-FOUNDATION SYSTEM
Assuming that the rotor and the foundation perform 
harmonic oscillations, we seek the solution of the 
system of equations Eq. (4) in the form [20] and [21],
 zZ it Ai t
rr
  
  expe xp , 
0
 (10)
 zZ it Bi t
ff 20
  
  expe xp ,   (11)
where ω is natural frequency, Z
r
 and Z
f
 are forced 
vibration amplitudes of rotor and foundation, while 
A
r
 and B
f
 are natural vibration amplitudes of rotor and 
foundation, respectively.
The presence of an angle ϕ i.e., a phase lag, is 
caused by the presence of resistance forces in the 
system.
As an expression describing the action of the 
hydrodynamic force on the system, which arises due 
to the presence of fluid in the cavity, we start from 
formulae in [8] and [19], taking into account Eqs. (10) 
and (11):
 
FFiF Zm it
Am
rxyr L
rL
   








0
2
0
2
2
00
2
2
0
2
2
exp 


  




0
2
exp( ), it (12)
where
 q
R
r
 
0
0
,,  (13)
m
L
 =  π ρ R
2
h is the mass of fluid required to 
completely fill the rotor cavity, σ vibration frequency 
of the free surface of the fluid.
Substituting Eqs. (10), (11) and (12) in Eq. (4), 
we obtain a system of algebraic equations for the 
unknowns A
r
, B
f
, Z
r 
and Z
f
. Its solution is given by:
 Z
f
f
Z
f
f
rf
= =
0
11
,, (14)
 Bp ip A
fr
 () ,
12
 (15)
 ZZ
ff
f
f
rf



0
1
2
, (16)
 A
p
pp
ip
pp
B
rf










1
2
12
2
2
2
12
2
, (17)
where
 
fr pp rf pq qp fr qq r
qp er
L
0000 01 00
0
2
0
2
1
  
 
,, ,
() ,c os ,     
 
n
qk pe rk
2
2
0
2
00 00
2
00 0
22

 
,
,s in ,,   
 
p
n
D
D
kk
m
p
D
D
L
L
1
2
2
22 3
4
0
2
2
3
4
14
21
1



















  

,
  
2
02
22
1
kn k
m










, 
 
mn kimn k
mn kiD
02
2
0
2
00 12
22
2
0
22
22
22
03
24
2
 



 ,,
,

 
2 2
00
2
4
2
00
2
2
2

 



 
,
. D 
µ
L
 = m
L
/m is the ratio of the mass of fluid required 
to completely fill the rotor cavity to the mass of the 
rotor.
Thus, to determine the unknown amplitudes, 
we may derive two non-linear equations from the 
relations above:
 
mZ ZZ
nn ZZ AB
fr f
rf rf
0
01 10
22
3
4
3
2
0

 













()
() (), (18)
 
mB AB
nn ZZ AB
rf
rf rf
2
01 10
22
3
2
3
4
0

  













()
() (). (19)
After substituting Eqs. (14) to (16) into Eq. (18), 
a quadratic equation for A
r
 can be obtained. Using the 
values of the amplitudes Z
r
, Z
f
, A
r
 and B
f
 calculated 
from Eq. (19), we can determine the phase angle ϕ.
The linearized form of Eqs. (18) and (19) 
becomes:

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 421-432
424 Kydyrbekuly, A. – Ibrayev, G.-G.A. – Ospan, T. – Nikonov, A.
 
 
 
znzz kz
zn zn zz kz
 
 
0
2
2
22
2
20 1
2
20 2
20
20
() ,
() . (20)
Since the resistance forces do not strongly affect 
the value of the natural frequency [22], in this case, it 
is sufficient to find the frequency equation for k = k
0
 = 
0, i.e., for the equation in the form:
 
10
01
2
0
2
0
2
0
2
2
2
0
2
2




























z
z
nn
nn n
z
z 
 







0
0
. (21)
To obtain a frequency equation, we seek the 
solution in the form:
 zz it zz it
ss
  
  
exp, exp.  
22
 (22)
Substituting Eq. (22) into Eq. (21), we obtain:
nn
nn
nn n
nn n
0
22
0
2
0
22
0
2
0
2
2
2
0
22
0
2
2
2
0
2
00
00
00
00


 
 


 
  
2
0
0
0
0








































A
B
C
D
s
s
s
s
, (23)
where
 zA Bz CD
ss ss ss
 
22
2
22
,. (23)
Then, to find the values of the natural frequencies 
ω, it is sufficient to find the determinant of system Eq. 
(23), i.e.:
det
nn
nn
nn n
nn n
0
22
0
2
0
22
0
2
0
2
2
2
0
22
0
2
2
2
00
00
00
00


 



 

0 0
22
0



. (24)
Then, the frequency equation is written as:
    
8
1
6
2
4
3
2
4
0   , (25)
where
 


11 42 1
2
23 14 4
2
31 41 44 1
2
24
2
    
   
   
  
,,
,   
 
4
2
2
2
3
2
10
2
20
2
30
2
42
2
0
2

   
,
,, ,. nn nn n 
Thus, four natural frequencies are obtained, where 
the lowest frequency is taken as the first frequency. 
The accuracy of the approximate formulae for natural 
frequencies are discussed in [19], [21], and [22].
It should be noted that the search for a solution 
in the form of Eqs. (10) and (11) makes it possible 
to study both forced and self-excited vibrations in 
general; however, at the same time, this approach 
excludes the possibility of studying steady-state 
vibrations (which is typical for the case of an empty 
or a completely filled rotor), since the partial filling 
of the cavity with fluid causes the appearance of self-
oscillations.
Thus, by slowly changing the angular velocity 
of the rotor, it is possible to construct the amplitude-
frequency characteristics of the system while varying 
the parameters of the rotor, foundation, and fluid.
3  RESULTS AND DISCUSSION
To analyse the system and determine the optimal 
operating mode, the amplitude-frequency 
characteristics of the rotor and foundation were 
constructed. They were obtained from Eqs. (18) 
and (19) and numerical solution of the differential 
equations of motion of the system for different 
degrees of filling of the rotor cavity with fluid, 
different values of the damping coefficient ( χ), non-
linear stiffness (c
1
), imbalance (e) and different values 
of the foundation mass (M). For qualitative analysis 
of the system, the following dimensionless parameters 
were introduced:
 sE
e
R
K
k
C
ne
 

 
0
0
1
2
0
2

,, ,, (26)
where s is a dimensionless frequency, E is a 
dimensionless parameter of linear imbalance, Z
r
 
is a dimensionless amplitude of the rotor, Z
f
 is 
a dimensionless amplitude of the foundation, K 
is a dimensionless damping coefficient, C is a 
dimensionless coefficient of stiffness at the cubic 
term, R is a cavity radius (see Figs. 2 to 11).
When an insignificant amount of fluid is present 
in the cavity, e.g. r
0
 = 0.93R, (see Figs. 2 and 3), 
three self-oscillation zones and three resonance 
amplitudes are observed for both the rotor and the 
foundation. All figures show examples when the 
system parameters take the values: χ = 4200 kg/s, 
χ
0
 = 6.59 kg/s, c
0
 = 1.1∙10
7
 kg/s
2
, c
1
 = 0.87∙10
7
 kg/m
2
s
2
, 
c
2
 = 3.26∙10
5
 kg/s
2
, M = 25 kg, m = 2.4 kg, e = 0.001 m. 
The amplitude-frequency characteristic, in this case, 
demonstrates a monotonic increase over the interval 
0 < s < 0.065 up to the first resonance peak. The first 
resonance occurs at s = 0.065, where the amplitudes 
are Z
r
 = 0.083 and Z
f
 = 0.0091. Then, over the interval 
0.065 < s < 0.65, a self-oscillating mode occurs (see 
Figs. 2 and 3) with maximum amplitudes of 0.028 for 
the rotor and 0.001 for the foundation, respectively. 
At s = 0.65, the second resonance occurs with the 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 421-432
425 Multi-parametric Dynamic Analysis of a Rolling Bearings System
amplitudes 1.72 and 0.79 for the rotor foundation, 
respectively. Further, over the interval 0.65 < s < 1, 
the vibration amplitudes decrease monotonically to 
the values 0.07 and 0.013. Next, over the interval 
1 < s < 1.35, the amplitude of vibrations grows rather 
intensively up to the third resonance, where s = 1.35, 
and the amplitudes are Z
r
 = 2.23 and Z
f
 = 1.38. 
Further, in the interval 1.35 < s < 3.41, the amplitudes 
monotonically decrease to the third self-oscillation 
zone, which occurs at s = 3.41, where the amplitudes 
take constant values Z
r
 = 0.057, Z
f
 = 0.0003, which 
is caused by the effect of self-centring and non-
conservative circulation-type forces arising due to the 
presence of fluid in the rotor cavity. Here, the self-
oscillation zones are presented as a curve showing 
the oscillatory process. In all other cases, the curves 
corresponding to self-oscillations are smoothed.
With the increase of the fluid volume in the cavity 
at r
0
 = 0.8R, it can be seen that the first and second 
resonances, as well as the first self-oscillation zone, 
occur at the same frequencies as at r
0
 = 0.93R, but with 
larger amplitudes. For example, at the first resonance, 
i.e., at s = 0.065, the amplitudes are Z
r
 = 0.15 and 
Z
f
  = 0.015, that is, a 4.74-times increase of the fluid 
volume leads to 1.82- and 1.65-times increase in the 
amplitudes of the rotor and foundation, respectively. 
The maximum amplitudes of self-oscillations, in this 
case, also have higher values. For example, they reach 
             
Fig. 2.  Amplitude-frequency characteristics of the rotor at different degrees of the rotor cavity filling, r
0
             
Fig. 3.  Amplitude-frequency characteristics of the foundation at different degrees of the rotor cavity filling, r
0

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 421-432
426 Kydyrbekuly, A. – Ibrayev, G.-G.A. – Ospan, T. – Nikonov, A.
0.088 for the rotor, and 0.0097 for the foundation, 
which is 3.14- and 9.7-times higher than in the 
previous case, respectively. At the second resonance 
(s = 0.65), the amplitudes are Z
r
 = 1.5 and Z
f
  = 0.59, 
which practically does not differ from the previous 
case. The third resonance peak in this case shifts to 
higher frequency range and occurs at s = 1.75, where 
Z
r
 = 2.36, Z
f
  = 2.83. Then, for s > 3.41, a self-oscillation 
zone is formed, where the amplitudes take on constant 
values Z
r
 = 0.139 and Z
f
  = 0.0006.
When the fluid volume in the rotor cavity is one 
third of the total volume, i.e., r
0
 = 0.667R (see, e.g., 
Figs. 2 and 3), the maximum resonance amplitudes 
are observed. The first and second resonances, as 
well as the first self-oscillation zone, occur at the 
same frequencies as in the two previous cases, the 
amplitudes are, respectively, Z
r
 = 0.198, Z
f
 = 0.033 
and Z
r
 = 1.9, Z
f
  = 0.38; the maximum amplitudes of 
the first self-oscillation zone are Z
r
 = 1.9, Z
f
  = 0.38. 
In this case, the third resonance peak also appears 
at a greater angular velocity occurring at s = 2.12, 
with amplitudes Z
r
 = 2.93, Z
f
  = 2.99. Then, as before, 
at s > 3.41, a self-oscillation zone is formed, where 
the amplitudes take on constant values Z
r
 = 0.15 and 
Z
f
  = 0.0008. The maximum amplitudes when one third 
of the cavity is fluid-filled and are associated with a 
specific behaviour of a “coupled” solid-fluid system 
[8], [19], and [23].
Furthermore, for cases r
0
 = 0.5R, r
0
 = 0.333R 
and r
0
 = 0.125R, frequencies of the first and second 
resonances, i.e., s = 0.065 and s = 0.65, as well as the 
frequencies of formation of self-oscillation zones, i.e., 
0.065 < s < 0.65, 0.65 < s < 3.41 and s > 3.41, do not 
change. The values of the first and second resonance 
amplitudes are damped. The third resonant amplitude 
is also damped, while the related resonance frequency 
is shifted to the right (see, e.g., Figs. 2 and 3), due to 
the increase of the fluid volume. At the same time, 
the fluid volume does have a significant effect on the 
amplitudes over the first, second and third zones of 
self-oscillations. 
For an empty rotor, a breakdown of the 
amplitude-frequency characteristics of both the rotor 
and foundation, corresponding to the fundamental 
resonance, is typical for the studied cubic nonlinearity, 
as for a Duffing oscillator. This phenomenon, which 
arises as a result of a monotonic increase in the 
amplitudes and their sharp decrease after a certain 
frequency, has been studied in detail by many authors, 
e.g. [24] to [26]. In this case, the breakdown of the 
amplitude of the empty rotor with a gradual increase 
in frequency occurs at Z
r
 = 1.05 and s = 1.33, then the 
amplitudes of the rotor decrease to 0.15. A similar 
behaviour of the amplitude at s = 1.33 is observed for 
the foundation; in this case, the maximum amplitude is 
Z
f
  = 0.115, which is almost an order of magnitude less 
than the rotor amplitudes. The foundation amplitudes 
after the breakdown decrease to 0.01. This difference 
between the amplitudes of the forced vibrations of 
the rotor and the foundation is primarily due to the 
ratio of the masses of the rotor and the foundation, 
as well as the value of linear eccentricity. As might 
be expected, an additional (first) linear resonance 
appears at s = 0.065, according to the general theory 
in the book [27]. It should be noted that the third 
resonance is a specific feature of a non-linear system; 
this feature is addressed in greater detail below. For a 
fully fluid-filled rotor, the first and third resonances 
are suppressed, while the second resonance looks 
similar in a sense to the linearized case.
Thus, maximum amplitudes of forced oscillations 
of the rotor and the foundation are observed when one 
third of the rotor cavity is filled with fluid. In addition, 
a decrease in the fluid volume leads to a shift of the 
third resonance towards lower angular frequencies.
Decrease of the damping coefficient (K = 0.01) 
causes a shift of the third resonance to higher 
angular velocities (see Figs. 4 and 5). The first and 
second resonances are observed at virtually the 
same frequencies as for r
0
 = 0.667R and K = 1, i.e., 
at s = 0.065 and s = 0.65 with amplitudes Z
r
 = 0.62, 
Z
f
  = 0.33 and Z
r
 = 0.52, Z
f
  = 0.07. The first resonance 
at K = 0.01 coincides in magnitude with the resonance 
at K = 1. The second resonance of the rotor and 
foundation for K = 0.01 is 6.35 and 54.28 times lower 
than the corresponding resonances for K = 1. The 
number of self-oscillating zones does not change, i.e., 
three self-oscillation zones are observed, the first in 
the interval 0.065 < s < 0.65, the second in the interval 
0.65 < s < 3.41, and the third at s > 3.41 with maximum 
amplitudes Z
r
 = 0.43, Z
f
  = 0.043, Z
r
 = 0.9, Z
f
  = 0.19 and 
Z
r
 = 0.104, Z
f
  = 0.017, respectively. The shifted third 
resonance appears at s = 3.4 with amplitudes Z
r
 = 8 and 
Z
f
  = 13.3, which is 2.76 times and 1.34 times less than 
the amplitudes of the third resonance of the rotor and 
foundation, respectively, at K = 1.
With a significant increase in the damping 
coefficient, i.e., for K = 0.1, the amplitudes of self-
oscillations are damped and coincide with those at 
K = 1. The resonance amplitudes of the rotor and the 
foundation increase in magnitude. The first, second 
and third resonances appear at the same frequencies 
as for K = 1, that is, at s = 0.065, s = 0.65 and s = 2.12, 
with amplitudes Z
r
 = 0.9, Z
f
 = 0.3, Z = 2.9, Z
f
  = 4.1 and 
Z
r
 = 25.5, Z
f
  = 9.1, which coincides in magnitude with 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 421-432
427 Multi-parametric Dynamic Analysis of a Rolling Bearings System
the first, second, and third amplitudes of both the rotor 
and the foundation for K = 1.
Further increase in the damping coefficient (cases 
K = 5 and K = 10) leads to significant damping of self-
oscillations and resonance amplitudes, for example, 
for K = 5, the amplitudes of the first, second, and third 
resonances are equal to Z
r
 = 0.02, Z
f
  = 0.003, Z
r
 = 1.15, 
Z
f
  = 0.4 and Z
r
 = 1.7, Z
f
  = 3.6, respectively. For 
K = 10, the amplitudes of the first, second, and third 
resonances are Z
r
 = 0.02, Z
f
  = 0.003, Z
r
 = 0.66, Z
f
  = 0.1 
and Z
r
 = 1.1, Z
f
  = 2.5, respectively.
In addition, with a significant decrease of the 
damping coefficient, a shift of the third resonance to 
higher angular velocities occurs. In particular, for a 
third fluid filled rotor, the aforementioned resonance 
Fig. 4.  Amplitude-frequency characteristics of the rotor Z
r
 for 
various damping coefficients K
Fig. 5.  Amplitude-frequency characteristics of the foundation Z
f
 
for various damping coefficients K
Fig. 6.  Amplitude-frequency characteristics of the rotor Z
r
 for 
different values of the foundation mass, µ
Fig. 7.  Amplitude-frequency characteristics of the foundation Z
f
 for 
different values of the foundation mass, µ

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 421-432
428 Kydyrbekuly, A. – Ibrayev, G.-G.A. – Ospan, T. – Nikonov, A.
resonance for s = 0.065 are equal to Z
r
 = 0.03 and 
Z
f
  = 0.01, the amplitudes of the second resonance for 
s = 0.65 are equal to Z
r
 = 0.07 and Z
f
  = 0.37, and the 
amplitudes of the third resonance for s = 2.125 are 
Z
r
 = 0.31 and Z
f
  = 0.83, i.e., with an increase in the 
stiffness coefficient by two orders of magnitude, the 
resonance amplitudes of the rotor and foundation 
decrease in 54.78 and 15 times for s = 0.065, in 266.87 
and 10.37 times for s = 0.65, and in 76.21 and 16.97 
times for s = 2.125. Any significant shift of resonant 
frequencies and self-oscillation zones is observed 
when the stiffness coefficient changes. In this case, 
as in linear cases, an increase in stiffness leads to 
damping of all vibrations. Analysis of greater values 
is observed at the same frequency as that of an empty 
rotor (s = 3.4), which is a distinctive feature of non-
linear vibrations.
To assess the effect of the foundation mass 
on self-oscillations and resonance amplitudes at 
r
0 
 = 0.667R, various ratios of the rotor and foundation 
masses µ = 0.96, µ = 0.192, µ = 0.096, µ = 0.048 and 
µ = 0.0096 were considered and for each case the 
amplitude-frequency characteristics were constructed 
(see Figs. 6 and 7).
In all of the above cases, the first, second, and 
third resonances were observed at s = 0.065, s = 0.65, 
and s = 2.12, respectively. With an increase in the mass 
of the foundation (cases µ = 0.096, µ = 0.048), there is 
an increase in the amplitudes of the rotor at the third 
resonance (at s = 2.12, Z
r
 = 5.6, Z
f
 = 7.53) and self-
excited vibrations in general, whereas the resonance 
amplitudes and amplitudes of self-oscillations of the 
foundation decrease in magnitude. In the case in which 
the mass of the foundation is slightly greater than the 
mass of the rotor (cases µ = 0.192 and µ = 0.96), the 
minimum amplitudes of the third resonances of both 
the rotor and the foundation are observed (Z
r
 = 1.94 
and Z
f
  = 3.39, Z
r
 = 0.34 and Z
f
  = 1.01). It should be 
noted that at µ = 0.96 in the interval 0.65 < s < 2.125, 
a zone of intense self-oscillations of the foundation 
with a maximum amplitude of 0.28 appears (see Fig. 
7). Damping the rotor amplitudes with a decrease in 
the mass of the foundation is a specific feature of the 
non-linear system. In the linear case, a decrease in the 
rotor amplitudes is observed with an increase in the 
mass of the foundation, as in this case the foundation 
itself acts as an anti-load.
To study the influence of nonlinear effects 
arising from nonlinearity of the restoring force, 
different values of the stiffness coefficient at the cubic 
term were considered, i.e., С = 0.01, С = 0.1, С = 1, 
С = 10 and С = 100, and for each case the amplitude-
frequency characteristics with a rotor fluid filled by 
one third, i.e., for r
0
 = 0.667R, are obtained (see Figs. 
8 and 9).
An increase in the stiffness coefficient at the 
cubic term leads to a decrease in both the resonance 
amplitudes and the amplitudes of self-oscillations 
of the rotor and the foundation. The maximum 
amplitudes are observed in the case of С = 0.01, 
for which the amplitudes of the first resonance for 
s = 0.065 are Z
r
 = 1.86 and Z
f
  = 0.15, the amplitudes of 
the second resonance for s = 0.65 are Z
r
  = 19.215 and 
Z
f
 = 3.84, and the amplitudes of the third resonance for 
s = 2.125 are Z
r
 = 23.47 and Z
f
  = 14.09. The minimum 
resonance amplitudes are, respectively, observed in 
the case of С = 100, where the amplitudes of the first 
Fig. 8.  Amplitude-frequency characteristics of the rotor Z
r
 for 
different values of the stiffness coefficient of the cubic term, С
Fig. 9.  Amplitude-frequency characteristics of the foundation Z
f
  
for different values of the stiffness coefficient of the cubic term, С

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 421-432
429 Multi-parametric Dynamic Analysis of a Rolling Bearings System
of the stiffness coefficient is not of practical interest 
for the original engineering system.
Due to the imperfection of equipment and other 
factors at high operating speeds, it is important to 
determine the range of critical values of inbalance 
e, and to identify its impact on the behaviour of the 
system. In this work, the cases E = 0.5, E = 1, E = 5 and 
E = 10 were considered; for each case the amplitude-
frequency characteristics were constructed (see Figs. 
10 and 11).
Fig. 10.  Amplitude-frequency characteristics of the rotor Z
r
 for 
different values of imbalance, E
Fig. 11.  Amplitude-frequency characteristics of the foundation Z
f
  
for different values of imbalance, E
For Е = 0.5, Е = 1 and Е = 5, as before, three 
resonance amplitudes are observed for s = 0.065, 
s = 0.65, and s = 2.125. In this case, the change in the 
imbalance value practically does not affect the values 
of the resonance amplitudes and amplitudes of self-
oscillations.
As the value of imbalance increases by an order 
of magnitude, i.e., at E = 10, the amplitude-frequency 
characteristic of the rotor and the foundation is given 
by the nonlinearity of a hardening (stiff) type. Thus, 
with an increase in linear imbalance, self-oscillations 
arising from the action of the fluid are suppressed 
by forced vibrations. In this case, the characteristic 
breakdown of the amplitudes is shifted towards 
higher frequencies of the system. Further, after the 
breakdown, due to the self-centring effect, operation 
of the system is stabilized, and vibrations occur 
with an amplitude of about Z
r
 = 1.26, which slightly 
exceeds the amplitude of self-oscillations. During 
rotor vibrations, one self-oscillation zone is observed 
in the interval 0 < s < 0.346 with maximum amplitudes  
Z
r
 = 0.08. During vibrations of the foundation, a 
sharp variation of the amplitude up to the value 
Z
f
  = 0.09 is observed over the interval 0 < s < 0.112. 
Further, with an increase in frequency, a gradual 
increase in foundation vibrations is observed until 
its amplitude breaks down, and, then, as in the case 
of the rotor, vibrations stabilize with an amplitude of 
about Z
f 
 = 0.03. In both cases, for the rotor and for the 
foundation, breakdown is observed at s = 2.817. 
To understand the nature of the first and third 
resonances, we construct the amplitude-frequency 
characteristics of the linear and non-linear “rotor-
foundation” system without fluid (see Figs. 12 and 
13), which can be obtained from the equations of 
motion, Eq. (13) for F
r
 = 0 as
 
 

znzz kz ei t
zn zn zz k
 

0
2
20
2
0
22
2
20 1
2
20
2
2
() exp( ),
()

  z
2
0  , (27)
and
 

znzz nz zk ze it
zn zn
 

0
2
21 2
3
0
2
0
22
2
20 1
2
2 () () exp( ), 
( () () . zz nzzk z   
21 02
3
02
20  (28)
The linear system of equations, Eq. (27) can be 
solved with sufficient degree of accuracy by numerical 
methods (see Figs. 12 and 13, black (solid) curve).
A cubic nonlinearity supports ultra-harmonic 
(Ω
0 
 = 3 ω, or s = 3) -harmonic resonances (Ω
0 
= ω/2 
and Ω
0
 = ω/3, or s = 0.5 and s = 0.333). The frequencies 
of the latter are multiples of the fundamental 
one (Ω
0
 = ω, or s = 1). The solution of non-linear 
equations in Eq. (28) is found analytically, in order 
to incorporate explicitly the contribution of higher 
harmonics corresponding to sub- and ultra-harmonic 
oscillations. 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 421-432
430 Kydyrbekuly, A. – Ibrayev, G.-G.A. – Ospan, T. – Nikonov, A.
The restoring force F
c
 approximated by Eq. 
(2) does not support sub-harmonic resonance in 
the system. Then, to find third-order sub-harmonic 
resonances, we introduce a notation Ω
0
 = 3 ω
f
, ( ω
f
 is 
sub-harmonic frequency) and search for a solution in 
the form [28]:
 zB it Bi t
ff

 

 

13 1
3
/
expe xp ,   (29)
and
 zD it Di t
ff

 

 

13 1
3
/
expe xp .   (30)
Substituting Eqs. (29) and (30) into Eq. (28) 
and using the harmonic balance method, we obtain a 
system of algebraic equations:
 
nB nD nB D
nB D
f 0
22
13 0
2
13 11 31 3
3
11 31
9
3
4
3
4


 




 

// //
/ / /
//
,
3
2
11
11 31 311
2
2
3
2
9











BD
nB DB De
f
 (31)
nB nD nB D
nB D
f 0
22
10
2
11 13 13
11 31 3
81
1
4
3
2


 





 

//
//
2 2
11 11 1
3
3
4
0
 
BD nB D 



 , (32)
 
nn Dn Bn BD
n
f 2
2
01
22
13 01
2
13 10 13 13
3
10
9
3
4
3
4


 




 

// //
B BD BD
nB DB D
13 13
2
11
10 13 13 11
2
3
2
0
//
//
,










 

 (33)
nn Dn Bn BD
nB
f 2
2
01
22
10 1
2
11 01 31 3
10 13
81
1
4
3
2


 




 

//
/
 






 
DB Dn BD
13
2
11 10 11
3
3
4
0
/
. (34)
Solving Eqs. (31) to (34), we obtain the 
amplitudes of the steady-state forced vibrations  
 
BB D
13 11 3 //
,, and 

D
1
, which can be used to construct 
the amplitude-frequency characteristics of the rotor 
and foundation in the case of sub-harmonic vibrations.
To determine ultra-harmonic resonances, the 
solution is sought in the form:
 zZ it Bi t
r
  
  
expe xp , 
00
3  (35)
and
 zZ it Di t
f 20 0
3   
  
expe xp .   (36)
In this case, the force of reaction of the fluid on 
the rotor wall has the form:
 
Fm Ai t
mB it
rL
L
  
 


0
2
0
0
2
0
93
exp( )
exp( ).

 (37)
Substituting Eqs. (35) and (36) into Eq. (4) and 
using the harmonic balance method, we obtain a 
system of non-linear algebraic equations resulting in
Fig. 12.  Amplitude-frequency characteristics of the rotor Z
r
 in the 
linear and non-linear cases
Fig. 13.  Amplitude-frequency characteristics of the foundation Z
f
   
in the linear and non-linear cases

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 421-432
431 Multi-parametric Dynamic Analysis of a Rolling Bearings System
 Bd D =
0
, (38)
and
 Dd B = , (39)
where
 
daia dd d
ai a
aa
a
bc bc
b
a
bc
0010
01
0
2
1
2
0
00 11
3
1
0
1  






,, ,
()
,
(
1 11 0
3
30
2
1
2
00
2
0
10 0
91
6
1
43

  


bc
b
bbbb D
bk D
L
)
,
,( ),
,,





c cn
ck bk
02
2
0
2
10 01 0
9
66




,
,. 
To determine the unknowns, we also have a 
system of two equations, which takes the form:
eC nZ Zn ZZ ZZ
BD BD
rf rf rf 00 11 0
2
2
3
4
3
2
3
4
   




() () ()
() () (Z ZZ
eD nBDn BD
ZZ BD
rf
rf





  



 
),
() ()
() ()
0
3
4
3
2
10 11 0
3
2
 




1
4
0
3
() , ZZ
rf
 (40)
where
 en ik en ik
02
2
0
2
00 12
2
0
2
00
29 6      ,.
From the system in Eq. (40), taking into account 
Eqs. (38) and (39), we have:
 Bg Bl
2
0   , (41)
where
 g
f
d
l
ef ff nn f
ff nd



 

2
01 12 01 10 2
3
2101
2 21
3
4
3
2
1
()
,
()
()
.
Solving Eq. (41), we obtain the values of the 
amplitudes B
1,2
,
 
and using Eqs. (38) to (40), we can 
find all the amplitudes of ultra-harmonic oscillations 
and the phase angle.
Thus, by comparing the amplitude-frequency 
characteristics obtained from the system in Eq. (27) 
and the amplitude-frequency characteristics obtained 
from systems in Eqs. (4) and (28), it is possible to 
determine whether sub-harmonic and ultra-harmonic 
resonances are present.
As sub-harmonic and ultra-harmonic resonances 
are observed only in non-linear systems, and (as in 
the linear case) there is no third resonance, which 
is observed at s = 3.24 (see Figs. 12 and 13, black 
(solid) curve), we can conclude that this resonance is 
ultra-harmonic, which is also confirmed by the fact 
that Ω
0
 ≈ 3 ω; in doing so, a shift in frequency is due 
foundation vibrations, whereas the presence of the 
first resonance is typical for a multi-degree of freedom 
system. 
The numerical solution of the non-linear system 
in Eq. (28) does not indicate the presence of sub-
harmonic resonances (see Figs. 12 and 13, purple 
curve).
To design rotor installation with these parameters, 
the optimal value of the foundation mass has to be up 
to 1.042 of the rotor mass, with linear eccentricity not 
exceeding E = 10. It is recommended to increase the 
stiffness coefficient at the cubic term by five orders of 
magnitude relative to the linear term and to decrease 
the damping coefficient by two orders, which will 
result in smaller amplitudes and will increase ultra-
harmonic resonances.
Thus, for stable operation of the system at any 
filling of the cavity, it is necessary that the operating 
speed is in the range 0.065 < s < 0.65 or s > 3.41.
4  CONCLUSIONS
A method for calculating amplitudes and constructing 
frequency characteristics of forced and self-excited 
vibrations of the “rotor-fluid-foundation” system on 
rolling bearings with a non-linear characteristic based 
on the method of complex amplitudes and harmonic 
balance has been developed. The specific features of 
the non-linear dynamic behaviour characterised by 
ultra-harmonic vibrations are studied. The optimal 
parameters of linear imbalance, the mass of the 
foundation, the fluid volume in the rotor cavity, the 
stiffness and damping coefficients, for which the 
amplitudes take optimal values, have been determined.
5  ACKNOWLEDGEMENTS
This work has been supported financially by the 
research project AP08856167 of the Ministry 
of Education and Science of the Republic of 
Kazakhstan and was performed at Research Institute 
of Mathematics and Mechanics in Al-Farabi Kazakh 
National University, which is gratefully acknowledged 
by the authors. The corresponding author gratefully 
acknowledges financial support from the Slovenian 
Research Agency (ARRS), Grant Ref. J2-9224.

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)9, 421-432
432 Kydyrbekuly, A. – Ibrayev, G.-G.A. – Ospan, T. – Nikonov, A.
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