The Assessment of the Condition of the Variable Stiffness Sections by
Determining the Frequency of Natural Vibrations
Zulfiya Telmonovna Fazilova and Aleksey Alekseevich Loktev
Russian University of Transport (MIIT), Moscow, Russia
Keywords: Variable stiffness, mechanical characteristics, the model of a flat element, dynamic loading, the model of the
"ground-plate" system, vibration frequencies.
Abstract: The presence of "barrier places" significantly limits the traffic capacity of railway lines. This situation can be
improved by increasing speeds and tonnage, but these measures require preliminary strengthening of the
existing track infrastructure. The increase in the efficiency of the infrastructure operation depends on the
improvement of infrastructure monitoring and diagnostics systems based on modern approaches, which allow
to predict the development of pre-failure conditions. This study is devoted to the analysis of the monitoring
peculiarities of variable stiffness sections, which are located in front of engineering structures, meanwhile the
formulation of the problem allows to take into account the possibility of defects at various stages of the life
cycle. The proposed method can be applied to the development of algorithms for monitoring and diagnostics
of transitional sections, which is relevant due to the difficulties of visual inspection of such structures and a
large range of dynamic loads and stiffness parameters in different seasons.
1 INTRODUCTION
Railway transportation safety is the main requirement
in the work of JSC "Russian Railways", which is
especially important when there is an increase in train
speeds, train weight, axle load (Hess, 2009).
Unfortunately, without any significant repairing
works, the current condition of the transportation
infrastructure does not allow to carry out the
measures which increase the traffic capacity of the
lines. The length of such sections is more than 10
thousand km (11% of the total length of the railway
network of the Russian Federation). The existing
regulatory and technical framework contemplates to
bring the track infrastructure into the condition of the
required level, using modern track structures, which
will significantly increase the operational life and
reduce the life cycle cost on the sections with high-
speed, heavy haul and especially heavy traffic due to
the overall repair of the first level tracks
(Organization of Railway Cooperation, 2016;
Technical standards and requirements for design and
construction, 2014; Ivanchenko, 2011).
The main reasons, causing the track deterioration,
are the increased dynamic impact on the railway
track. This problem is especially acute in the zone of
the so-called areas of variable stiffness, where track
deformations in the form of shaped depressions occur
not only due to over-compaction of the ballast, but of
the subgrade as well, and the depth of this
embankment layer can reach 2-3 m (Organization of
Railway Cooperation, 2016). These areas may also
include railway track sections on approaches to
engineering structures (bridge crossings, tunnels,
through bridges), boundary sections with various
track structures (ballast and ballastless) where there
are various vertical stiffness parameters of the rail
seat. The increase in train speeds and their weight
significantly affects the length of deformations along
such sections and the intensity of their manifestation.
Another negative factor, causing the increased track
deformability in the transition zone of bridge
crossings is the increased moistening of the
floodplain embankment soil, induced by the
hydrological regime of rivers, the increase in the
groundwater level, poor drainage on the bridge
structure (Figure 1) (Matsumoto, 2009; Fryba, 1996;
Poliakov, 2017).
Monitoring of the variable stiffness section
technical condition plays an important role in the
extinction of the entire engineering structure life
cycle. A distinctive feature of the current stage in the
development of transportation routes is that the
improvement of regulatory and legal documentation,
which is supposed to extend the life cycle of the entire
Fazilova, Z. and Loktev, A.
The Assessment of the Condition of the Variable Stiffness Sections by Determining the Frequency of Natural Vibrations.
DOI: 10.5220/0011576300003527
In Proceedings of the 1st International Scientific and Practical Conference on Transport: Logistics, Construction, Maintenance, Management (TLC2M 2022), pages 5-9
ISBN: 978-989-758-606-4
Copyright
c
 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)
5

structure and its individual elements, does not often
keep up with the change in speed modes and mass-
dimensional characteristics of vehicles (Barchenkov,
1976; Bolotin; Kogan, 1997).
2 METHODS
This study is aimed at the development of the
mathematical models of the variable stiffness sections
and their adaptation to diagnostic needs without
imposing restrictions on the facility operation.
The improvement of diagnostic algorithms will
minimize the possible rapid development of
significant defects on the approaches to the
engineering structure and on the structure itself
(Chen, 2000; Kawatani, 2001; Pan, 2002), which will
optimize repair and restoration processes. This
research work proposes to consider the behavior of a
variable stiffness section simulated by a plate element
and to consider it with the respect to its interaction
with other elements of the structure and the
environment, as the characteristic, the change of
which takes into account the change in the transition
section condition (Kou, 1997; Kurbatsky, 2018). It is
proposed to use the frequency of natural vibrations.
The determination of the structure vibration
frequency will allow to identify defects, damages and
deviations from the project, to assess the strength and
durability (Fryba, 1996; Kurihara, 1978; Loktev,
2012).
The used mathematical model of a flat element
allows to take into account the anisotropic properties
of the variable stiffness section and to present the
design scheme in the form of a plate (Loktev, 2011;
Glusberg, 2020; Vinogradov, 2018; Chistyy, 2018),
lying on the deformable roadbed, two edges of which
are pivotally supported, and the other two edges are
rigidly fixed.
3 RESULTS AND DISCUSSION
According to the proposed model, free vibrations of a
transversally isotropic plate of the constant thickness,
characterized by boundary parameters in a non-
deformable condition within limits
{}
hzhlylx ≤≤−≤≤≤≤ ;0;0
21
.
The approximate equation of transverse vibrations
of such a flat element in partial derivatives of the
fourth order has the form:
24 2
123
242
2
4
()0,
WW
AAAW
ttt
AWPW
∂∂∂
+−Δ
∂∂∂
+Δ + =
(1)
where
W
– the value of the transverse
displacement of points in the plate median plane;
Δ
– the Laplace operator.
The functional coefficients have the following
form
a) b)
Figure 1: Engineering structures: a) a bridge of the combined design scheme, b) an elevated structure for the urban rail
transport.
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CONSTRUCTION, MAINTENANCE, MANAGEMENT
6

()
()
{}
()
()
2
21 1
1 1 2 1 33 44
2
12 11
3 1 11 33 13 11 33 33 44
2
12
433113313
2
11
516114433
1
7 1 11 33
;3;
22 3 ;
2;
;3;
222
4;
2
h
AA AA
b
h
AAAAAAAA
b
h
AAAAA
b
SSh
AA AA
hh
S
AAA
h
ρρ
ρ
ρρρ
ρ
−−
−−−
−
−−
−
== +

=− − − −

=−
== +
=−
(2)
where
t
W
A
t
W
A
t
W
AWP
∂
∂
Δ+
∂
∂
+
∂
∂
=
7
3
3
65
)(
- the
support reaction from the subgrade soil;
ρ
- the density of the span material,
b – the velocity of the transverse (shear) wave,
nm
AAA === ...
1311
– anisotropy coefficients
(Loktev, 2012; Loktev, 2020; Lyudagovsky, 2018).
The boundary conditions for the problem in such
a formulation take the form:
2
1
2
2
,0;0
,0;0
ly
y
W
W
lx
x
W
W
==
∂
∂
=
==
∂
∂
=
(3)
The solution of the homogeneous equation is
presented in the following form:
()()






=
h
bt
yxWtyxW
ξ exp,,,
, (4)
here
−
ξ
the desired frequency of the plate
natural vibrations
With respect to (4) the equation (1) may be
presented in the following form
()
() 0,
21
2
=+Δ+Δ BByxW
, (5)
Here the following notations were adopted














+






+






+






=














+






−=
3
65
4
2
2
1
4
2
7
3
3
4
1
1
;
1
h
b
A
h
b
A
h
b
A
h
b
A
A
B
h
b
A
h
b
A
A
B
ξξξξ
ξξ
Considering dimensionless coordinates
() ()
4
12 1
4
;;, ,
ll l
xyWxyV α
β
α
β
ππ π
== =
,
;
2
1
l
l
=
η
From the equation (5) we obtain the following
functional relation
()
444
24
4224
2422
2
11
12
22 2 4
2
,0;V
ll
BB
ηη
ααββ
αβ
η
πα β π


∂∂∂
++


∂∂∂∂


=


∂∂

+++


∂∂


.(6)
The equation (6) can be represented as a set of
three auxiliary tasks, each of which has its own
relevant function
V (α, β),
()
βα
α
,
1
4
1
4
f
V
=
∂
∂
0
1
1
=
∂
∂
=
α
V
V
π
α
,0=
()
βα
β
η ,
2
4
2
4
4
f
V
=
∂
∂
0
2
2
=
∂
∂
=
β
V
V
π
β
,0=
(7)
24422
22
11
123
22 2 2 2 4
12
2
0;
ll
BBV
ff
ηη
αβ π α β π



∂∂∂
+++



∂∂ ∂ ∂



++ =
Due to the applied decomposition method, it can
be clearly seen that the following relations are valid
for the given points of a flat element
()
21321
2
1
; VVVVV +=≅
, (8)
()
()
()( )
() () () ()
()
()
()( )
() () () ()
1
,
1
4
,1
32
1234
2
,
2
44
,1
32
1234
, sin sin
;
62
, sin sin
;
62
nm
nm
nm
nm
a
Vnm
n
a
Vnm
m
αβ α β
αα
ψ
β
ψ
βα
ψ
β
ψ
β
αβ α β
η
ββ
φα φα βφα φ α
∞
=
∞
=
=+
+++
=+
+++


(9)
The Assessment of the Condition of the Variable Stiffness Sections by Determining the Frequency of Natural Vibrations
7

here
()
β
ψ
i
и
()
α
ϕ
i
arbitrary functions, and
() i
mn
a
,
- arbitrary constants, i = 1,2.
Substituting the possible solutions to the auxiliary
tasks into the expressions of boundary conditions (3),
we determine the functions
()
β
ψ
i
and
()
α
ϕ
i
. After
that, the determining relations for auxiliary tasks can
be reduced to the system of equations (Matsumoto,
2009;
Lyudagovsky, 2018)
() ( )
() ( )





=+
=+
0
0
2
1122
1
1121
2
1112
1
1111
acac
acac
(10)
in which the following notations are adopted
[]
1
2
1
2
2
4
4
1
2
2
2
2
1
111
++++−= η
π
η
π
l
B
l
Bc ,
2
1
12 1
22 2
4
1
2
44 2
21
11
24
112
111
24
l
cB
l
B
π
πη π η
π
πη η π


=−+−−+




 
−− −++
 
 
,
1
21
=c
,






−−=
4
1
1
4
22
π
η
c
.
The determining system of equations (10) has a
non-zero solution only when the main determinant is
equal to zero, revealing which we obtain a
characteristic equation in reference to the frequencies
of plate natural vibrations
0
54
2
3
3
2
4
1
=++++ ddddd ξξξξ
, (11)
here
4
4
21
1
44
4
3
4
6
1
2
44
4
3
2
2
2
1
3
22
2
4
1
1
22
7
2
1
4
22
4
1
4
1
4
11 11
1
242
1
4
113
1
22 2 4
Al
b
d
Ah
A
l
b
d
Ah
A
l
b
d
A
h
l
A
A
l
d
A
A
π
πη
π
πη
π
ηπ
πη
π
πη
π
π
πη

=− −



=− −






+−+−−









=





−






−+ − −




=
2
1
5
22
5
22
1
4
112
11 1 1
4
b
h
l
d
π
πη
π
ηηπ









−







=−+ +−+−






When solving the characteristic equation (11), it
is possible to obtain natural frequencies, with respect
to different values of the mechanical characteristics
of the variable stiffness section structures.
4 CONCLUSIONS
The proposed models of a plate element on the soil
subgrade can be used for the vibration diagnostics
systems of both railways and highways on approaches
to engineering structures. They will allow not only to
detect and identify defects and deviations from the
project, but also to predict the dynamics of changes in
the major characteristics, affecting traffic safety, as
well as considering the changes in the properties of
the subgrade soil due to moistening.
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The Assessment of the Condition of the Variable Stiffness Sections by Determining the Frequency of Natural Vibrations
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