Content uploaded by Petro Pukach
Author content
All content in this area was uploaded by Petro Pukach on Jun 30, 2022
Content may be subject to copyright.
Applied mechanics
31
Copyright © 2022, Authors. This is an open access article under the Creative Commons CC BY license
ADVANCING
ASYMPTOTIC
APPROACHES TO
STUDYING THE
LONGITUDINAL
AND TORSIONAL
OSCILLATIONS OF
A MOVING BEAM
Andrii Slipchuk
Corresponding author
PhD, Associate Professor
Department of Robotics and
Integrated Mechanical Engineering Technologies*
E-mail: andrii.m.slipchuk@lpnu.ua
Petro Pukach
Doctor of Technical Sciences, Professor
Department of Computational
Mathematics and Programming*
Myroslava Vovk
PhD, Associate Professor
Department of Mathematics*
Olha Slyusarchuk
PhD, Associate Professor
Department of Mathematics*
*Lviv Polytechnic National University
S. Bandery str., 12, Lviv, Ukraine, 79013
This paper analyzes the influence of kinetic and
physical-mechanical parameters of systems on the
characteristics of dynamic processes in moving one-
dimensional nonlinear-elastic systems. Improved con-
venient calculation formulas have been derived that
describe the laws of changing the amplitude-frequen-
cy characteristics of systems for both a non-resonant
case and a resonant one. An important issue of study-
ing the influence of the speed of movement of elements
of mechanisms on the oscillations of one-dimensional
nonlinear-elastic systems has not been considered in
detail until now in the scientific literature. This issue
relates to the vibrations of shafts in gears, pipe strings
when drilling oil and gas wells, the oscillations of tur-
bine blades and rotating turbine discs, the longitudinal
vibrations of the beam as an element of structures. The
main reason for this in the analytical study of dyna mic
processes were the shortcomings of the mathematical
apparatus for solving the corresponding nonlinear dif-
ferential equations that describe the laws of motion of
those systems.
It was found that in the case of longitudinal oscil-
lations in the moving beam with an increase in the lon-
gitudinal speed of the medium to 10 m/s, the amplitude
of the oscillation also increases by 13.5 %. However,
when the longitudinal velocity of the beam is 5 m/s, the
amplitude will increase by only 3 %. It is established
that with the growth of the amplitude, the frequency of
longitudinal oscillations decreases sharply, and if the
system moves at a higher speed, for example, 20 m/s,
it reduces the frequency of oscillation by about 13 %.
The results reported here make it possible to assess
the effect of kinetic and physical-mechanical parame-
ters on the frequency and amplitude of oscillations. The
research that involved the asymptotic method makes
it possible to predict resonant phenomena and obtain
engineering solutions to improve the efficiency of tech-
nological equipment
Keywords: nonlinear oscillations, asymptotic me-
thod, elastic beam, longitudinal oscillations, torsional
oscillations
UDC 539
DOI: 10.15587/1729-4061.2022.257439
How to C ite: Slipchuk, A., Pu kach, P., Vovk, M., Slyu sarchuk, O. (2022). Advancing asymptotic approaches to study-
ing the longitudinal and torsional osc illations of a moving beam . Easter n-European Journal of Enterpr ise Technologies,
3 (7 (117)), 31–39. doi: https://d oi.org/10.15587/1729-4061.20 22.257439
Received date 20.04.2022
Accepted date 20.06.2022
Publi shed date 30.06.2022
1. Introduction
With the development of new technology, with increasing
speeds, that is, during the transition to high-speed engineer-
ing, the role of fluctuations in the elements of mechanisms has
become especially relevant. In particular, this concerns the
processes of oscillation of the drilling part of the rig (column)
when drilling oil and gas fields [1], fluctuations in the ring
combustion chamber in cars [2], vibrations in the elements
of power plants [3]. The stricter operational requirements for
the safe and efficient operation of modern machines lead to
the fact that increased attention in engineering calculations
is paid to solving problems associated with longitudinal and
torsional oscillations. Based on the theory of nonlinear oscil-
lations, important problems of machine building balancing
machines, torsional oscillations of shafts and gears, oscilla-
tions of turbine blades and rotating turbine disks, longitudi-
nal oscillations of the beam as an element of structures, etc.
were investigated. Important models relating to nonlinear
oscillations are considered in the problems of helioseismo-
logy [4]. Of practical importance are these studies in models
of nonlinear fluctuations of the railroad track [5]. Experimen-
tal research shows that even minor speeds of movement (in-
cluding angular ones) lead to changes in both quantitative
and qualitative characteristics of dynamic processes [6].
Such differences are considered between the nonlinear-elas-
tic one-dimensional system and their analogs, which are not
characterized by longitudinal (for longitudinal oscillations)
or rotational (for torsional oscillations) movements [7].
Only after numerous experiments related to the problems
of dynamic processes in mechanical systems, the difference
between the mechanics of nonlinear oscillations and the me-
Eastern-European Journal of Enterprise Technologies ISSN 1729-3774 3/7 ( 117 ) 2022
32
chanics of linear oscillations, which is fully preserved even
when considering weakly nonlinear oscillations, became
apparent [8]. Mathematical models of such oscillations are
described by differential equations, which differ from linear
equations with constant coefficients only by the presence of
sufficiently small terms [9]. Therefore, effective in the study
of models of such systems are asymptotic methods of non-
linear mechanics [10].
All real physical systems are nonlinear. The peculiarity
of nonlinear systems is the failure to fulfill the principle of
superposition in them. This means that individual harmonic
vibrations interact with each other. Significant difficulties
are that it is not always possible to use the Poincare method
to obtain results that would be suitable for studying move-
ment over a long enough period [11].
Therefore, deriving and using convenient calculation
formulas that describe the laws of changing amplitude-fre-
quency characteristics (AFCs) and take into consideration
kinematic and physical characteristics is a relevant task.
Such studies are a determining factor in investigating the
dynamics of moving environments. They are relevant both at
the design stage and in the operation of mechanisms.
2. Literature review and problem statement
Actual oscillatory systems are characterized by various
physical parameters, in particular, rigidity, mass, and charac-
teristics of damping. For such one-dimensional systems, dy-
namic processes are studied in detail (for practical purposes)
if the body material matches the linear or similar law of elas-
ticity [12]. If, in addition, such systems move at constant or
variable speeds along their geometric axis, then the study of
the corresponding oscillations, even for the case of linearly
elastic properties of the material, is associated with signifi-
cant mathematical difficulties.
There are several analytical methods for constructing
and studying solutions to nonlinear differential equations
describing the movements of mechanical systems [13]. Note
that they reject the assumption of the error in these solutions.
Methods that have already become classical are most
effectively applied to nonlinear systems with one degree of
freedom and are generalized to systems with a finite num-
ber of degrees of freedom [14]. However, with an increase
in the number of generalized coordinates of the system, the
possibility of obtaining analytical solutions is significantly
complicated. However, the use of computing equipment in
some cases makes it possible to overcome such difficulties.
Many approximate methods have been devised to calculate
the periodic movements of nonlinear systems. In particular,
quite often there is a method of harmonious balance [15]. How-
ever, the possibility of applying this method to stationary sys-
tems is determined by the proximity of the periodic movement
of the system to the harmonic one. This condition is usually
satisfied only when the linear parts of the system are low-fre-
quency filters, that is, they filter out high harmonics well.
Variational methods are also used. In particular, the Os-
trohrad-Hamilton principle is used to solve the equations
of longitudinal, torsional, and transverse vibrations of the
rod [16]. By establishing the equivalence of solving boundary
problems to solving problems about the extremum of the func-
tionality, this principle opens up the possibility for application
to vibration calculations of some special methods of varia-
tional calculus. These include, first of all, the so-called direct
methods of variational calculus, the use of which is effective
only in approximate calculations of their natural frequencies
and shapes of oscillations of rods of the variable cross-section
with an uneven distribution of rigidity and mass.
The most widespread in engineering computing practice
are the methods by Relay, Ritz, Galerkin [17]. The essence of
these methods is the use in the calculation of transverse oscil-
lations of the heterogeneous rod of the Ostrohrad-Hamilton
functionality. However, in this case, the influence of linear
velocity (with longitudinal fluctuations of the beam) and an-
gular (with torsional oscillations of shafts, pipe strings when
drilling oil and gas wells, etc.) was neglected.
These properties of nonlinear systems with concentrated
masses and distributed parameters greatly simplify their
research procedures. Underlying the study of such systems
is the principle of single-frequency oscillations in non linear
systems with many degrees of freedom and distributed
parameters [18], the asymptotic methods for constructing
solutions to some classes of differential equations with partial
derivatives. However, the presence of dissipative and other
nature of nonlinear forces, as well as external perturbing
forces in real-world systems, leads to the rapid disappearance
of higher harmonic oscillations. In addition, the dynamics of
movement with a frequency close to the frequency of external
perturbing force or basic harmonics are established.
Our review of literary sources [12–18] reveals that such an
important issue as the influence of the speed of movement of
elements of mechanisms on the oscillations of one-dimensional
nonlinear elastic systems was not considered in detail. The main
reason for this in the analytical study of dynamic processes was
the lack of a mathematical apparatus for solving appropriate
nonlinear differential equations that describe the laws of motion
of those systems. It is necessary to study both a non-resonant
case and a resonant one. This makes it possible to determine
the influence of perturbing force with a frequency close to the
frequency of oscillations of the system. In this regard, it becomes
necessary to conduct research and do something in this area.
3. The aim and objectives of the study
The purpose of our study is to determine the influence
of kinetic and physical-mechanical parameters of systems on
the characteristics of dynamic processes of moving one-di-
mensional nonlinear-elastic systems. This makes it possible
to predict resonant zones and establish the most effective
modes of operation of the equipment, to establish less strin-
gent requirements for the system and its elements.
To accomplish the aim, the following tasks have been set:
– to suggest a procedure for constructing mathematical
models that describe the dynamic processes of mechanical
systems characterized by longitudinal (for longitudinal oscil-
lations) movement;
– to offer a procedure for constructing mathematical
models that describe the dynamic processes of mechanical
systems characterized by rotational (for torsional oscilla-
tions) motion;
– to derive mathematical relations that determine the
laws of changing the amplitude, frequency (period) of oscilla-
tion, as functions from parameters that characterize the phy-
sical-mechanical and kinematic properties of the medium;
– to conduct numerical modeling of the influence of kine-
matic and physical-mechanical quantities on the nature of
changes in amplitude and frequency.
Applied mechanics
33
4. The study materials and methods
Using an example of longitudinal oscillations of the mov-
ing beam, the influence exerted on the dynamics of the os-
cillatory process by the physical-mechanical, kinematic, and
force factors is investigated. In particular, the speed of its lon-
gitudinal movement, nonlinearly elastic characteristics of the
beam material, external periodic perturbations are analyzed.
In the study of single- and multifrequency modes of lon-
gitudinal oscillations of the beam, practical tasks are often
encountered for the case of hinged ends [19]. In a linearly
elastic statement, the Fourier method can be applied to
them and reduce the initial problem to the study of ordinary
differential equations or a system of ordinary differential
equations. The current paper deals with more complex issues:
a) the beam moves along its undeformed axis at a con-
stant speed;
b) the material of the beam satisfies the elasticity close
to the linear law;
c) external periodic disturbances act on the beam.
The one-dimensional system (beam) is investigated,
which is described by the function of two variables u = u(x, t)
coordinates x and time t. Such oscillations can be described
by the differential equation [20]:
ρε
∂
∂−
∂
∂=
2
2
2
21
u
tEu
xX
,
(1)
where E is a module of elasticity of the first kind (Young
modulus); X1 is the first natural shape of oscillations (am-
plitude function, which is unchanged over time); ε is some
small positive parameter; ρ is the specific mass of the rod
material, kg/m3.
The oscillation equation (1) for individual practical cases
can be built by using kinematic hypotheses (flat cross-sec-
tions, straight normals, etc.). However, the disadvantage of
such methods is the inability to derive a solution for resonant
cases. These solutions can be obtained only taking into con-
sideration internal or external friction. Internal friction in the
case of linear nonstationary or stationary oscillations can be
taken into consideration using the Boca-Schlippe-Colar or
Kelvin-Foigt hypotheses. In addition, for a stationary case,
one can also use the method of complex elasticity mo dules (So-
rokin hypothesis). The Kelvin-Foigt hypothesis is employed
for stationary or nonstationary oscillations, even though
it is not experimentally confirmed for metallic materials.
Sorokin’s linear hysteresis hypothesis corresponds to experi-
mental results but is used only in the case of fluctuations that
have already been established. Its application on nonstatio-
nary oscillations is not mathematically correct because there
are both stable and unstable partial solutions to the equation.
When one discards unstable partial solutions based on phy-
sical reasons, the principle of superposition is violated.
The wave equation of the free longitudinal oscillations
of the beam, which has a constant cross-section, can be writ-
ten as [21]:
∂
∂−∂
∂=∂
∂
∂
∂
2
22
2
2
2
2
1u
xs
u
tfx u
x
u
x
εεη,,,, , (2)
where
sE=ρ
is the coefficient that determines the fre-
quency of the system, phase velocity (the rate of propagation
of longitudinal waves in the rod); ρ is the specific mass of the
rod material, kg/m3; E is the elasticity module; ε is the small
positive parameter; t – time; fx u
x
u
x
εη,,,,
∂
∂
∂
∂
2
2 is the func-
tion, infinitely differentiable in all its arguments, periodic
relative to η with a period of 2π;
ddttην=
()
is the positive
function. When there is no perturbation (ε = 0), we obtain
a purely harmonious oscillation uxtaXx t
,c
os
,
()
=
()
+
()
ωj
where ddt
jω=.
The shape of oscillations is determined by
the function X(x), then
du
xt dt aX
xt,s
in
.
()
=−
()
+
()
ωωj
In this case, a and j are some constants, the amplitude will
be constant da dt
=0
with a uniform phase angle j. Thus,
when fx u
x
u
x
εη,,,,
,
∂
∂
∂
∂
=
2
20 equation (2) can be solved by
standing waves (Fourier method) and running waves (d’Alem-
bert’s method).
To build an asymptotic solution to the perturbed system
in which single-frequency oscillations occur, it is necessary to
meet the following conditions:
1) in an undisturbed system, non-damping harmonic os-
cillations with a frequency ω1(τ) are possible, which depends
only on two free constants;
2) the only solution to the equation of an undisturbed
system is trivial;
3) there are no internal resonances in the undisturbed
system, that is, ω1(τ) ≠ ω k(τ) (k = 2, 3,...N);
4) initial conditions ensure the existence of a single-frequen-
cy oscillation mode, that is,
upXx
t
i
==
()
0,
∂∂ =
()
=
ut qX x
x
i
0,
where p and q are real numbers, Хі(x) is a fundamental
function of the undisturbed boundary problem, which is
described by a wave equation and linear homogeneous
boundary conditions (ε = 0). Under such assumptions, the
solution to the equation of the disturbed system is to be de-
rived in the form [22]:
uxtatXx
ua xu
ax
,cos
,,
,,
,, ,
()
=
() () ()
+
+
()
+
()
ψ
εηψε ηψ
1
2
2 (3)
where a(t) is the amplitude of single-frequency oscillations;
u1(a,η,ψ,x), u2(a,η,ψ,x) are the 2π-periodic functions with
variables ψ and η.
Only those cases for which the length of the longitudi-
nal waves of oscillations is large compared to the size of the
cross-sections of the beam are considered. In such cases, it is
possible to neglect the influence of transverse movements on
the nature of longitudinal movements.
5. Results of studying the longitudinal and torsional
oscillations of the moving beam
5. 1. Differential equation of the longitudinal oscilla-
tions of the moving beam
Among the different types of natural fluctuations arising
in the nonlinearly elastic rod, longitudinal fluctuations oc-
cupy a significant place. They are the easiest to investigate.
We assume that the cross-section of the beam is flat and
each point of such a cross-section executes only axial move-
ments (moving only along the axis). Longitudinal stretching
and compression, which occur with such fluctuations in
the beam, are accompanied by the occurrence of transverse
deformations. Fig. 1 shows a load-free prismatic beam with
a length of l, an infinitesimally small element of which is
equal to dx and placed at a distance of x from the left end.
Through u, the longitudinal movement of the cross-sectional
point with the coordinate x is indicated.
Eastern-European Journal of Enterprise Technologies ISSN 1729-3774 3/7 ( 117 ) 2022
34
S+dS
x
dx
dx
x
S
F
0
V
ρ
S
u
Fig. 1. Diagram of forces acting on a beam element
moving along its axis
The beam moves along its axis at a constant speed, so:
d
dt x
dx
dt tV
xt
=
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
, (4)
d
dt x
dx
dt
dx
dt xt t
V
x
V
xt
2
2
2
2
2
22
2
2
2
2
2
22
=∂
∂
+∂
∂∂+∂
∂=
=∂
∂
+∂
∂∂
++ ∂
∂
2
2
t
. (5)
When the beam fluctuates longitudinally, the sum of the
longitudinal forces acting on an infinitesimal element of the
beam (Fig. 1), in accordance with the d’Alembert’s principle,
takes the following form:
∂
∂−∂
∂+∂
∂∂+∂
∂
=
=∂
∂
∂
∂
∂
∂
S
xFu
tVu
xt Vu
x
fu u
t
u
x
u
ρ
ε
2
2
222
2
2
2
,,,xx 2
, (6)
where S is the equivalent to internal stresses arising in
cross-section with the coordinate x, which is directed along
the axis; ρ is the density of material; F is the cross-sectional
area of the beam; V is the speed of moving the beam along
its undeformed axis; εfu u
t
u
x
u
x
,,,
∂
∂
∂
∂
∂
∂
2
2 is the function that
takes into consideration the nonlinear elastic properties of
the beam material, as well as dissipative forces and resistance
forces. Below it is accepted that they are small compared to
nonlinearly elastic forces.
The internal force appearing in equation (6) is equal to
the product of the material density by the volume of the small
segment Fdx. Using Hooke’s law, the longitudinal force S can
be expressed through longitudinal stress and through axial
deformation in the form of:
Sg EF
xx
=
()
=+
()
σεδσεε
1,,
,
(7)
where E is the Young module; δ1 is a function that charac-
terizes the nonlinearity of the system. Substituting expres-
sions (4), (5) in equation (6), taking into consideration (7),
we obtain after transformations:
∂
∂−
∂
∂+
∂
∂∂+
∂
∂=
=∂
∂
∂
∂
∂
∂
2
2
2
2
2
2
2
2
2
2
2
2
u
t
u
xVu
xt Vu
x
fu u
t
u
x
u
x
α
ε,,,
, (8)
where
αρ
=E
.
Equation (8) describes the longitudinal
oscillations of the moving beam. It can be called a one-dimen-
sional wave to indicate that during longitudinal oscillations,
the contour of movements spreads in the axial direction, that
is, at the speed of sound propagation in the material.
It can be noted that equation (8) is similar to the equa-
tion of transverse oscillations of the string. The difference is
only in the physical content of some coefficients while the
principle remains the same. After all mathematical transfor-
mations, the solution to equation (8) is defined as follows:
– for a non-resonant case:
da
dt pFaxl
xx
l
=
()
∫∫
εωπ ψπ
ψψ
π
11
42
0
2
0
,, sinsin ,dd (9)
d
dt
k
l
V
pa
Faxl
xx
l
ψωπ
ω
εωπ ψπψ
π
=−
+
+
()
∫∫
22
2
0
2
0
11
4,, sincos dddψ
;
– for a resonant case
ων jω
j
π
ψθ π
j
π
−
()
∂
∂∂ −
∂
∂=×
×
()
∑∫∫
2
2
0
2
0
211
4
a
tatp
eFax k
l
is
s
l
,,,sin xxe x
is−j
ψψ
co
s,
dd (10)
ataa
t
V
lp
eFax
is
s
∂
∂∂ −
()
−
∂
∂+=×
×
()
∑∫
222
22
0
2
0
211
4
ψ
jων ω
π
π
ψθ
j
π
,,,
ll
is
k
lxe x
∫−
sincos
,
π
ψψ
jdd
where ddt
ψω=
and
ddttην=
()
are the positive frequen-
cies of natural and perturbing oscillations, respectively.
5. 2. Differential equation of the torsional oscillations
of the moving beam
Fig. 2 illustrates the torsional oscillations of a recti-
linear shaft, which rotates around its axis. Through θ, the
angle of torsion (around the axis of the shaft) of an arbitrary
cross-section is indicated.
2
p
Id t
∂δγ
γ∂
M
M+
val
M
2
Fig. 2. Diagram of forces acting on the element
of the shaft
With torsional oscillations, the equilibrium condition of
elastic and inertial moments acting on a small element of the
shaft, in accordance with the d’Alembert’s principle, is writ-
ten in the form of:
Applied mechanics
35
∂
∂
∂
∂
−∂
∂+∂
∂∂ +∂
∂
=
xGJ xJtt
xx
pp va
lv
al
θρθωθωθ
2
2
222
2
2
== ∂
∂
∂
∂
∂
∂
εθ
θθ θ
ftxx
,,,,
2
2 (11)
where
ωval
dx dt
=
is the angular speed of rotation of the
shaft around its axis; G is the elasticity module of the second
kind; Jp is the polar moment of inertia of the cross-section;
M is a moment that is equivalent to the internal forces act-
ing in the cross-section;
∂∂
22
θt is the angular acceleration.
According to the introduced designations, the moment of
inertia of mass is equal to ρξ
θ
Id
t
p
∂
∂
2
2. From the theory of sim-
ple torsion, we obtained the ratio:
MxGJ x
p
=
∂
∂
∂
∂
θ.
(12)
Substituting expression (12) in equation (11), after the
transformations, we obtain:
∂
∂+∂
∂+∂
∂∂+∂
∂=∂
∂
∂
∂
2
2
22
2
222
2
2
2
2
θλθωθωθεθ
θθ
tx xt xftx
valval ,,
,
λ
ρ
2=
G
. (13)
Equation (13) is a one-dimensional wave equation of
torsional oscillations of the rotating shaft. Again, there is
a certain similarity between equations (13) and (8).
Our equations coincide in form with similar equations
and formula for longitudinal oscillations of the prismatic
beam (8), if in the latter the values u, α, and Е are replaced
by θ, λ, and G, respectively. Therefore, all the results for the
problem of longitudinal oscillations of prismatic beams can
be extended to the problems of torsional oscillations of shafts
of the circular cross-section by simply replacing the desig-
nations. Therefore, the solution to equation (13) is defined
similar to (9), (10) as follows:
– for a non-resonant case:
d
dt pFaxl
xx
l
θεωπ ψπ
ψψ
π
=
()
∫∫
11
42
0
2
0
,, sinsin
,
dd (14)
d
dt
k
pa
Faxlx
val
l
ψωπ
θ
ω
ω
εωπ ψπ
π
=−
+
+
()
∫∫
22
2
0
2
0
2
11
4,, sincos
s;ψψ
ddx
– for a resonant case:
ων
θ
jθω
j
π
θψθπ
j
π
−
()
∂
∂∂ −
∂
∂=×
×
()
∑∫∫
2
2
1
0
2
0
211
4ttp
eFxk
is
s
l
,,,sin ll xe x
is−j
ψψ
co
s,
dd (15)
θψ
jων θθωπ
θπ
θψθ
j
∂
∂∂ −
()
−∂
∂+=×
×
()
∑
222
22
1
0
2
211
4ttp
eFx
val
is
s
,,,
ππ
j
π
ψψ
∫∫ −
0
l
is
k
lxe xsincos ,dd
where θ1 is the frequency of the perturbing force that acts
on the shaft.
Equations (14), (15) are similar in structure to equa-
tions (9), (10). The only difference is to change some coeffi-
cients but the very nature of the dynamic process is similar
to the longitudinal oscillations of the beam. This is due to
similar laws that describe the vibrations of the shaft or beam,
and the same nonlinear differential equations that characte-
rize this system.
5. 3. Laws of changing the amplitude and frequency
of oscillation as functions of parameters that characterize
the properties of the environment
With the help of the theory of nonlinear oscillations
and the asymptotic method used in the current work, it is
possible to establish the optimal characteristics of nonlinear-
ly-elastic systems. Employing mathematical models (9), (10),
and (14), (15), it is possible to expand the operating condi-
tions of machines. Our results make it possible to utilize the
equipment more efficiently in the event of fluctuations that
almost always occur during operation.
Increased attention in engineering calculations is paid to
solving problems associated with longitudinal and torsional
oscillations. This is due to an increase in size and an increase
in the speed of operation of modern machines. It is known
that such important problems were investigated on the basis
of the theory of nonlinear fluctuations. These include ba-
lancing machines, torsional vibrations of shafts and gears,
the vibrations of turbine blades and rotating turbine discs,
the longitudinal oscillations of the beam as an element of
structures, etc. As a rule, with the help of this theory, it is
possible to establish the optimal characteristics of nonlinear
elastic systems. Such characteristics expand the operating
conditions of machines, make it possible to use the equipment
more efficiently in the event of fluctuations that almost al-
ways occur during the operation of the equipment.
An example of using the above defining ratios for de-
termining the AFC of a dynamic process is considered. The
longitudinal vibrations of the movable prismatic beam on
which the harmonic force acts are investigated, provided that
the beam material matches the nonlinear technical law of
elasticity. The differential equation of motion of such a sys-
tem is written in the form of:
∂
∂−∂
∂=
=−∂
∂
−∂
∂∂
−∂
∂
∂
∂
+
2
2
22
2
2
2
223
2
u
t
u
x
u
xVu
xt
Vxbu
xH
α
εε
sin
n,
νt
where [b] = m2/s2 is the coefficient having the dimensio-
nality of the velocity square; the value of H is defined as
the maximum value of the perturbing force per unit of mass
of the beam.
If we consider that the boundary conditions for equa-
tions (6), (13) correspond to the hinged ends, then a single-
frequency oscillatory process under a mode close to the
frequency of external disturbances can be described as the
following dependence:
uxta
lxt
,sin co
s,
()
+
()
=
πνj
moreover, the parameters a and j for a resonant case are de-
termined by a system of differential equations (10).
The law of changing the frequency of longitudinal oscilla-
tions of the beam (in accordance with equation (9)) is found
from the following system:
Eastern-European Journal of Enterprise Technologies ISSN 1729-3774 3/7 ( 117 ) 2022
36
da
dt
=0, (16)
d
dt l
a
ml
Vjωε βπ
ω
π
ω
=− +
3
82
4
4
222
.
A resonant case similar to the previously obtained system
of equations (16) was obtained in a similar way:
da
dt
H
m
=− +
()
ε
πω νjco
s,
(17)
d
dt
a
ml
V
l
H
ma
jωνεβπ
ω
π
ωεπω νj=−−+
++
()
3
82
42
4
22
2si
n.
System (17) demonstrates that as the speed increases,
the frequency of longitudinal oscillations of the beam falls on
the parabola.
5. 4. Numerical modeling of the influence of kinemat-
ic and physical-mechanical quantities on the nature of
changes in amplitude and frequency
For the study, the following parameters are adopted: l = 2 m,
F = 0.12·0.085 m2, H = 500 Н, ρ = 7.900 kg/m3, E = 2.06·1011 N/m2,
a = 0.02 m, m = 80.54 kg/m, I0 = 6.1·10–6 m4, and the frequency
of the perturbing force
ωπ
ρ
1
8017==lE ra
ds
is the an-
gular frequency of the first mode of longitudinal vibrations
of the rod with fixed ends.
Fig. 3 shows how longitudinal velocity reduces the fre-
quency of longitudinal oscillations. This decrease follows
the parabolic law because in formula (16) there is a square of
magnitude V, and, therefore, the effect of speed will be signi-
ficant at high speeds (already at a speed of 20 m/s – the fre-
quency of oscillations decreases by 7 %). Similarly, the initial
amplitude affects such fluctuations. If the longitudinal velo-
city reaches 30 m/s, and the initial amplitude is 1 cm, then
the oscillation rate of the dynamic system will be 4.5 kHz.
This is 44 % less than with the natural oscillations of the
beam, which does not move along its axis.
Fig. 4 shows a 3D plot of the dependence of the oscil-
lation frequency on the length and initial amplitude at
a speed of 5 m/s.
Fig. 3. Dependence of the frequency of oscillation
of the system on speed and amplitude
Fig. 4. Dependence of the frequency of longitudinal
oscillations on the length of the beam and the initial
amplitude at a speed of 5 m/s
Fig. 4 shows how, as the amplitude increases, the fre-
quency of longitudinal oscillations decreases sharply. The
length of the beam does not affect the oscillations so much
if it is greater than 2 m; with a decrease in the length of the
beam, the frequency also drops sharply and there may even
be a breakdown of oscillations at a length of 0.5 m and an ini-
tial amplitude of 7 mm. If the system moves at a higher speed,
for example, V = 20 m/s (Fig. 5), then such a failure will oc-
cur even earlier. After all, speed also reduces the frequency
by about 13 %.
Fig. 5. Dependence of the frequency of longitudinal
oscillations on the length of the beam and the initial
amplitude at a speed of 20 m/s
The system of equations (16) demonstrates that the con-
stant speed of the environment affects only the frequency of
its transverse oscillations since the system is conservative.
As one can see from Fig. 6, the change in amplitude de-
pends on the longitudinal speed of the beam. Although its
impact at low speed is not very significant. When the lon-
gitudinal speed increases to 10 m/s, the amplitude increases
by 13.5 %. However, when the speed is equal to 5 m/s, the
amplitude value will increase by only 3 %, that is, there will
be no such tangible impact. Consequently, with a further
increase in the speed of the beam, the amplitude increa-
ses sharply.
Applied mechanics
37
Fig. 6. Transient processes that occur during longitudinal
oscillations of the beam for different speeds
As one can see from Fig. 7, the increase in longitudinal
velocity has almost no effect on changing the phase of the
beam oscillations. The nature of change in j remains the
same, the speed increase only slightly shifts the curve to the
right, that is, the value of the j changes later by 0.2 s.
Fig. 7. Dependence of the phase of oscillations
of the system for different speeds in a resonant case
(V = 10 m/s, V = 5 m/s, V = 0 m/s)
For such a system (17), one can build different graphic
dependences.
6. Discussion of the influence of kinematic and physical-
mechanical quantities on the amplitude-frequency
characteristics of oscillations
In contrast to [17] where models of oscillatory processes
are analyzed without taking into consideration the influence
of linear velocity (with longitudinal oscillations of the beam)
and angular velocity (with torque oscillations of shafts), our
mathematical dependences (16), (17) make it possible to
determine the change in frequency and amplitude depending
on the kinematic parameters. This is made possible using the
asymptotic method. Built on the basis of this method, com-
putational algorithms make it possible to more accurately
describe the dynamic process, in contrast to numerical ap-
proaches [14] where the derivation and analysis of solutions
is much more complicated. Asymptotic approaches make it
possible to establish an approximate solution with sufficient
accuracy for engineering calculations, to determine various
dynamic oscillation modes, in particular, resonant (17). This,
in turn, makes it possible to avoid resonantly dangerous
zones and set the optimal values of the parameters of the
moving element, as well as reduce strict requirements for the
system and its elements.
Our results illustrate the following properties of the con-
sidered oscillatory systems:
1. As follows from (16), the constant longitudinal velo-
city of the environment affects only the frequency of its
transverse oscillations, since the system is conservative.
2. With increasing speed, the frequency of longitudinal
oscillations of the beam falls according to the parabolic
law because in formula (16) there is a square of the velo-
city value. Therefore, the impact of speed will be significant
at high speeds (already at a speed of 20 m/s – the frequency
of oscillations decreases by 7 %). This result follows from
Fig. 3, which is a graphical representation of the solution
to the equation system (16).
3. For the resonant case described by system (17), the
change in amplitude depends on the longitudinal velocity.
With an increase in the longitudinal speed of the environ-
ment to 10 m/s, the amplitude also increases by 13.5 %. How-
ever, when the longitudinal velocity of the beam is 5 m/s, the
amplitude will increase by only 3 %, that is, there will be no
such tangible effect. Consequently, with a further increase
in the speed, the amplitude increases sharply. This is due to
the form of the approximate solution obtained, as well as
its graphic representation in Fig. 6.
4. The effect of longitudinal speed of movement on the
change in amplitude and frequency of longitudinal oscil-
lations of the beam is not so noticeable. However, in engi-
neering and design calculations, even for such fluctuations,
it is impossible to neglect such a kinematic quantity as the
longitudinal speed of movement. The result is explained by
the form of the right-hand part (16).
5. The approaches and results in our work can be exten-
ded for the case of mathematical models of torsional oscil-
lations represented by dependences (14), (15). Similarly, it
is easy to construct approximate solutions with the desired
accuracy and corresponding graphical dependences.
The proposed approach has limitations related to the
possibility of its use in the study of tasks with a sufficiently
low speed of movement. In addition, the problems under
consideration imply the presence of «small» nonlinear terms
in mathematical models (the right-hand part (6)). Subse-
quently, our results and proposed approaches can be used to
analyze fluctuations in nonlinearly dissipative systems.
7. Conclusions
1. With the help of the asymptotic method, functional
dependences have been derived that determine the influence
of physical and kinematic parameters on longitudinal oscilla-
tions for a beam that moves along its axis. Unlike the earlier
reported results in [17], the mathematical models discussed
in the current paper make it possible to take into consider-
ation the influence of these parameters on a change in the
amplitude and frequency of oscillation. The effectiveness
Eastern-European Journal of Enterprise Technologies ISSN 1729-3774 3/7 ( 117 ) 2022
38
of the suggested procedure is, in particular, in more precise
comparison with numerical methods [14] for predicting res-
onant modes of oscillatory process.
2. Mathematical models have been constructed that de-
scribe the dynamic processes of mechanical systems cha-
racterized by rotational motion oscillations for torsional
oscillations. The peculiarity of our result is the ability to
take into consideration the influence of angular velocity,
shear module, material density, and diameter on the ampli-
tude-frequency characteristics of torsional oscillations. This
makes it possible to more accurately establish the oscillation
amplitude for nonlinear-elastic moving systems for resonant
and non-resonant cases.
3. We have derived dependences, convenient for engi-
neering practice, which are more informative, compared to
those reported earlier [17]. Such ratios make it possible to
investigate the influence of the parameters of the moving
environment on the nature of changes in the frequency and
amplitude of oscillations and with the necessary accuracy to
predict the dynamic phenomena in them. With appropriate
use in engineering calculations of industrial equipment, our
dependences can become the basis for the synthesis and op-
timization of the parameters of the screw and other similar
structural elements.
4. Numerical simulation was carried out in the MAPLE
15 programming environment, as a result of which it was
found that at a speed of 20 m/s, the frequency of oscillations
decreases by 7 %. The initial amplitude similarly affects such
fluctuations. If the longitudinal velocity reaches 30 m/s, and
the initial amplitude is 1 cm, then the frequency of oscilla-
tions of the system is 4.5 kHz. This is 44 % less than with its
the natural oscillations of the beam, which does not move
along its axis. When the longitudinal speed increases to
10 m/s, the amplitude increases by 13.5 %. However, when
the speed is equal to 5 m/s, the amplitude value will increase
by only 3 %, that is, there will be no such tangible impact.
Consequently, with a further increase in the speed of the
beam, the amplitude increases sharply.
References
1. Andrukhiv, A., Sokil, B., Sokil, M. (2018). Resonant phenomena of elastic bodies that perform bending and torsion vibra-
tions. Ukrainian Journal of Mechanical Engineering and Materials Science, 4 (1), 65–73. doi: https://doi.org/10.23939/
ujmems2018.01.065
2. Humbert, S. C., Gensini, F., Andreini, A., Paschereit, C. O., Orchini, A. (2021). Nonlinear analysis of self-sustained oscillations in an
annular combustor model with electroacoustic feedback. Proceedings of the Combustion Institute, 38 (4), 6085–6093. doi: https://
doi.org/10.1016/j.proci.2020.06.154
3. Haris, A., Alevras, P., Mohammadpour, M., Theodossiades, S., O’ Mahony, M. (2020). Design and validation of a nonlinear vi-
bration absorber to attenuate torsional oscillations of propulsion systems. Nonlinear Dynamics, 100 (1), 33–49. doi: https://
doi.org/10.1007/s11071-020-05502-z
4. Pipin, V. V., Kosovichev, A. G. (2020). Torsional Oscillations in Dynamo Models with Fluctuations and Potential for Helioseismic
Predictions of the Solar Cycles. The Astrophysical Journal, 900 (1), 26. doi: https://doi.org/10.3847/1538-4357/aba4ad
5. Barbosa, J. M. de O., F r g u, A. B., van Dalen, K. N., Steenbergen, M. J. M. (2022). Modelling ballast via a non-linear lat-
tice to assess its compaction behaviour at railway transition zones. Journal of Sound and Vibration, 530, 116942. doi: https://
doi.org/10.1016/j.jsv.2022.116942
6. Hatami, M., Ganji, D. D., Sheikholeslami, M. (2017). Differential Transformation Method for Mechanical Engineering Problems.
Academic Press. Available at: https://www.sciencedirect.com/book/9780128051900/differential-transformation-method-for-
mechanical-engineering-problems
7. Lavrenyuk, S. P., Pukach, P. Y. (2007). Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect
to space variables. Ukrainian Mathematical Journal, 59 (11), 1708–1718. doi: https://doi.org/10.1007/s11253-008-0020-0
8. Chen, G., Deng, F., Yang, Y. (2021). Practical finite-time stability of switched nonlinear time-varying systems based on ini-
tial state-dependent dwell time methods. Nonlinear Analysis: Hybrid Systems, 41, 101031. doi: https://doi.org/10.1016/
j.nahs.2021.101031
9. Bayat, M., Pakar, I., Domairry, G. (2012). Recent developments of some asymptotic methods and their applications for nonlinear
vibration equations in engineering problems: A review. Latin American Journal of Solids and Structures, 9 (2), 1–93. doi: https://
doi.org/10.1590/s1679-78252012000200003
10. Sokil, B. I., Pukach, P. Y., Sokil, M. B., Vovk, M. I. (2020). Advanced asymptotic approaches and perturbation theory methods in
the study of the mathematical model of single-frequency oscillations of a nonlinear elastic body. Mathematical Modeling and Com-
puting, 7 (2), 269–277. doi: https://doi.org/10.23939/mmc2020.02.269
11. Andrukhiv, A., Sokil, B., Sokil, M. (2018). Asymptotic method in investigation of complex nonlinear oscillations of elastic bodies.
Ukrainian Journal of Mechanical Engineering and Materials Science, 4 (2), 58–67. doi: https://doi.org/10.23939/ujmems2018.02.058
12. Andrukhiv, A., Sokil, M., Fedushko, S., Syerov, Y., Kalambet, Y., Peracek, T. (2020). Methodology for Increasing the Efficiency of
Dynamic Process Calculations in Elastic Elements of Complex Engineering Constructions. Electronics, 10 (1), 40. doi: https://
doi.org/10.3390/electronics10010040
13. Le van, A. (2017). Nonlinear Theory of Elastic Plates. ISTE Press – Elsevier. Available at: https://www.sciencedirect.com/
book/9781785482274/nonlinear-theory-of-elastic-plates
Applied mechanics
39
14. Hashiguchi, K. (2020). Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity. Elsevier. doi: https://doi.org/10.1016/
c2018-0-05398-0
15. Jauregui, J. C. (2015). Parameter Identification and Monitoring of Mechanical Systems Under Nonlinear Vibration. Woodhead
Publishing. doi: https://doi.org/10.1016/c2013-0-16479-3
16. Kazhaev, V. V., Semerikova, N. P. (2019). Self-modulation of quasi-harmonic bending waves in rods. Bulletin of Science and Techni-
cal Development, 13–19. doi: https://doi.org/10.18411/vntr2019-142-2
17. Babenko, A. Ye., Boronko, O. O., Lavrenko, Ya. I., Trubachev, S. I. (2020). Kolyvannia nekonservatyvnykh mekhanichnykh
system. Kyiv: Nats. tekhn. un-t Ukrainy «KPI imeni Ihoria Sikorskoho», 153. Available at: https://ela.kpi.ua/handle/123456789/
38187?locale=uk
18. Sokil, B. I., Khytriak, O. I. (2009). Vibrations of drive systems flexible elements and methods of determining their optimal
nonlinear characteristics based on the laws of motion. Military Technical Collection, 2, 9–12. doi: https://doi.org/10.33577/
2312-4458.2.2009.9-12
19. Chen, L.-Q. (2005). Analysis and Control of Transverse Vibrations of Axially Moving Strings. Applied Mechanics Reviews,
58 (2), 91–116. doi: https://doi.org/10.1115/1.1849169
20. Marynowski, K., Kapitaniak, T. (2014). Dynamics of axially moving continua. International Journal of Mechanical Sciences,
81, 26–41. doi: https://doi.org/10.1016/j.ijmecsci.2014.01.017
21. Nazarov, V. E., Kiyashko, S. B. (2020). Stationary and Self-Similar Waves in a Rod with Bimodular Nonlinearity, Dissipation, and
Dispersion. Technical Physics, 65 (1), 7–13. doi: https://doi.org/10.1134/s106378422001020x
22. Pukach, P., Beregova, H., Slipchuk, A., Pukach, Y., Hlynskyi, Y. (2020). Asymptotic Approaches to Study the Mathematical Models
of Nonlinear Oscillations of Movable 1D Bodies. 2020 IEEE 15th International Conference on Computer Sciences and Information
Technologies (CSIT). doi: https://doi.org/10.1109/csit49958.2020.9321908
