he railway track is usually modeled as a continuous beam
on elastic support. Train circulation is a random dynamic
phenomenon and, depending on the different frequencies of
the loads it imposes, there is a corresponding response of the
track superstructure. At the instant when an axle passes from
the location of a sleeper, a random dynamic load is applied on
the sleeper. The theoretical approach for the estimation of the
dynamic loading of a sleeper requires the analysis of the total
load acting on the sleeper to individual component loads-
actions, which, in general, can be divided into: (a) the static
component of the load‚ and the relevant reaction/action per
support point of the rail (sleeper) and (b) the dynamic
component of the load, and the relevant reaction/action per
support point of the rail (sleeper). The static component of the
load on a sleeper, in the classical sense, refers to the load
undertaken by the sleeper when a vehicle axle at standstill is
situated exactly on top of the sleeper. For dynamic loads with
low frequencies the load is essentially static. The static load is
further analyzed into individual component loads: the static
reaction/action on a sleeper due to wheel load and the semi-
static reaction/action due to cant deficiency ([1]). The dynamic
component of the load of the track depends on the mechanical
properties (stiffness, damping) of the system “vehicle-track”
(Fig. 1), and on the excitation caused by the vehicle’s motion
on the track.
Fig. 1: (upper) A Railway vehicle running on a Track and the
static deflection of the Track; (lower) the complete System
“Vehicle-Track” in a semi-cross-section as an ensemble of
springs and dashpots.
T
Influence of Track Defects on the Track-Loads and Confidence
Interval of a Track Recording Car
KONSTANTINOS S. GIANNAKOS
Civil Engineer PhD, Fellow/Life-Member ASCE
108 Neoreion str., Piraeus 18534 GREECE
1. Introduction – the Track-vehicle System
Abstract: — The Railway track, is simulated as a beam on elastic foundation with damping. The motion of a
vehicle is simulated by the 2nd order differential equation of motion; the input to the system “vehicle-rail” is
the form of the Track Geometry which “acts” as a “signal”. The defects with short and long wavelength
influence the value of the dynamic component of the acting loads on the railway track and they are recorded in
the frame of the Quality Control of the Railway Track, which presents a determined Confidence Interval
depending on its “measurement-base”.
Keywords: Track Defects, Long and Short Wavelength, Fourier Transforms, Track Recording Car, Confidence
Interval.
Received: May 11, 2022. Revised: January 12, 2023. Accepted: February 7, 2023. Published: March 20, 2023.
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
16
Volume 18, 2023
The response of the track to the aforementioned excitation
results in the increase of the static loads on the superstructure.
The dynamic load is primarily caused by the motion of the
vehicle’s Non-Suspended (Unsprung) Masses, which are
excited by track geometry defects, and, to a smaller degree, by
the effect of the Suspended (sprung) Masses. In order to
formulate the theoretical equations for the calculation of the
dynamic component of the load, the statistical probability of
exceeding the calculated load -in real conditions- should be
considered, so that the corresponding equations would refer to
the standard deviation (variance) of the load ([1]]; [2]). The
track defects are classified in short wavelenth defects and long
wavelength defects. An article for “Modeling the Influence of
the Short Wavelength Defects in a Railway Track on the
Dynamic Behavior of the Non-Suspended Masses” appeared
([3]); it refers to the consecutive defects, not isolated. In the
present paper the dynamic component of the acting loads, for
the long wavelength defects, is – mainly – investigated
through the second order differential equation of motion of the
Non Suspended Masses of the Vehicle and specifically the
transient response of the reaction/ action on each support point
(sleeper) of the rail.
The railway vehicles consist of (a) the car-body, (b) the
primary and the secondary suspension with the bogie between
them and (c) the axles with the wheels (Fig. 1-upper). In
general the mass of the (c) case under the primary suspension
is the Non Suspended Mass of the vehicle. The heaviest
vehicles are the locomotives which are “motive units” and
have electric motors on the axles and/or the frame of the
bogie. In Fig. 2-left a locomotive with three-axle bogie is
depicted while the three axle bogie with the springs of the
primary and secondary suspensions is depicted in Fig. 2-
lower.
Fig. 2: (upper) Electropoutere type diesel locomotive of the
Greek railways, with three-axle bogies of 20,8 t/axle and NSM
5,08 t/axle, on track; the bogie is in the white ellipse; (lower) a
bogie in detail: the primary and the secondary suspensions.
Electric motors are either suspended totally from the frame
of the bogie or they are suspended on the frame of the bogie at
one end and supported on the axle at the other end. In the
second case the electric motor is semi-suspended (Fig. 3) and
a part of it is considered also as Non Suspended Mass, as it
will be clarified below.
If we try to approach mathematically the motion of a
vehicle on a railway track, we will end up with the model
shown in Fig. 1-lower, where both the vehicle and the railway
track are composed of an ensemble of masses, springs and
dashpots.
Fig. 3: Schematic depiction of an Electric Motor “semi-
suspended” from the bogie’s frame at one end (“Nose”
suspension) and supported on the vehicle’s axle at the other
end. A percentage/portion of its weight belongs to the SM and
the rest of it to the NSM.
As we can observe, the car body is supported by the
secondary suspension that includes two sets of “springs-
dashpots”, seated on the frame of the bogie (Fig. 1-left and
Fig. 3-right). The loads are transferred to the truss and the side
frames of the bogie. Underneath the bogie there is the primary
suspension, through which the bogie is seated onto the
carrying axles and the wheels. Below the contact surface,
between the wheel and the rail, the railway track also consists
of a combination of masses-springs-dampers that simulates the
rail, the sleepers, the elastic pad, the rail fastenings, the ballast
and the ground.
The masses of the railway vehicle located under the primary
suspension (axles, wheels and a percentage of the electric
motor weight in the case of locomotives) are the Non
Suspended Masses (N.S.M.) of the Vehicle, that act directly
on the railway track without any damping at all. Furthermore a
section of the track mass (mTRACK) also participates in the
motion of the vehicle’s Non Suspended Masses, which also
highly aggravates the stressing on the railway track (and on
the vehicle too) ([1]; [2]].
The defects of the rail running table, a wave in space of
random nature, impose a forced oscillation on the Non
Suspended Masses of the vehicle; their form constitutes a
forcing excitation. From the form of the rail running table the
forcing period or frequency can be calculated.
The remaining vehicle masses are called Suspended Masses
2. Suspended (S0) and Non-suspended
Masses (N60)
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
17
Volume 18, 2023
(S.M.) or Sprung Masses: the car-body, the secondary
suspension, the frame of the bogie, a part of the electric
motor’s weight and the primary suspension.
The theoretical analysis of a railway track is based mainly in
Winkler's theory ([4]), which models it as an infinite beam on
elastic foundation. In European literature it is also referred to
as Zimmermann's theory ([5]). The elastic foundation of the
railway track can be simulated by a large number of closely
spaced translational springs and the following equation is
valid ([3]; [6]):
(1)
in the absence of external force, or:
(2)
with the presence of external force.
In these equations y is the deflection of the beam, ρ1 is the
mass of the track participating in the motion, k1 the viscous
damping of the track, J is the moment of inertia of the rail, E is
the modulus of elasticity of the rail, Q the force/ load from the
wheel (when the force is present) and δ(x) the deflection of the
rail at the contact point between wheel and rail.
The solution of equations (1) and (2) becomes challenging
if we want to take into account all the parameters according to
professor J. Alias ([7]). However, if we make some
simplifying hypotheses we will be able to approximate the
influence of certain parameters provided that we will verify
the theoretical results with experimental measurements.
Apparently having in mind the tests on track under operation,
performed by the European Railways and the International
Union of Railways (U.I.C.), professor J. Eisenmann states
already since 1988 that, the -based on Zimmermann's theory-
methods give results correspondent at the average of the
measured on track values, for track's loading and stressing, as
well as track's deflection ([8]). Consequently, the level of
maximum values is dependent on the possibility of occurrence
–mainly– of the dynamic component of the acting load.
If the acting load is determined, then the Action-Reaction
on each support point (sleeper) of the rail will be determined
too. The system “railway vehicle-railway track” operates
based on the classical principles of physics: equivalence of
Action-Reaction between the vehicle and the track. It is a
dynamic stressing of random, vertical form.
The loading of the railway track from a moving vehicle
consists of:
(a) the static component of the load (static load of the
vehicle’s axle), as given by the rolling stock’s producer.
(b) the semi-static component of the load (due to cant or
superelevation deficiency at curves, which results in non-
compensated lateral acceleration).
(c) the component of the load from the Non-Suspended
Masses of the vehicle (the masses that are not damped by any
suspension, because they are under the primary suspension of
the vehicle), which is a dynamic load by its nature and
(d) the component of the load from the Suspended Masses
of the vehicle, that is a damped force component of the total
action on the railway track and it is also a dynamic load.
On each support point of the rail (sleeper) a reaction/action
is applied due to the distribution of the acting load to the
adjacent sleepers (support points of the rail) because of the
total elasticity/total static stiffness coefficient of the track. For
the static and the semi-static components of the load these
reactions/actions are given from the Eqns (2a) and (2b), as
derived from the solution of the differential equation of
motion. But for the dynamic component of the load a
modelling of the motion of the Non Suspended Masses should
be performed.
Finally the differential equation is transformed to the
following equation connecting the deflection of the continuous
beam and the bending moment ([3]; [6]):
(3)
where y is the deflection of the rail, M is the bending moment,
J is the moment of inertia of the rail, and E is the modulus of
elasticity of the rail. From the differential equation (3), it is
derived that the reaction of a sleeper Rstatic is (since the load is
distributed along the track over many sleepers):
(4a)
where Qwheel the static wheel load, ℓ the distance among the
sleepers, E and J the modulus of elasticity and the moment of
inertia of the rail, Rstat the static reaction/action on the sleeper,
and ρ reaction coefficient of the sleeper which is defined as:
ρ=R/y, and is a quasi-coefficient of the track elasticity
(stiffness) or a spring constant of the track. A
=A
stat equals to
Rstat/Qwheel, which is the percentage of the acting (static) load
of the wheel that the sleeper undertakes as (static) reaction. In
reality, the track consists of a sequence of materials –in the
vertical axis– (substructure, ballast, sleeper, elastic pad/
fastening, rail), that are characterized by their individual
coefficients of elasticity (static stiffness coefficients) ρi (Fig.
4).
(5)
where ν is the number of various layers of materials that exist
under the rail -including rail– elastic pad, sleeper, ballast etc.
Five main parameters of the layers are used: ρrail, ρpad, ρsleeper,
ρballast and ρsubgrade, that is finally the number of springs ν=5.
The semi-static Load is produced by the centrifugal
acceleration exerted on the wheels of a vehicle that is running
in a curve with cant deficiency, given by the following
equation ([1]; [7]; [9]):
(5-1)
42
11
42 0
+ + =
yy
EJ k z
xx
( )
42
11
42
+ + = −
yy
EJ k y Q x
xx
42
42
1d y d M
EJ
dx dx
= −
33
44
1
2 2 2 2
wheel stat
stat stat
wheel
QR
R A A
E J Q E J
= = = =
11
11
1 1 1
i i total i total
ii
i i i
total ii
i total i
R R R
y y y y
y
yR
==
==
= = = =
= =
2
2CG wheel
h
QQ
e
=
3. Application of the Second Order
Differential Equation of Motion in
a Railway Track
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
18
Volume 18, 2023
Fig. 4 Cross-section of Track; Characteristic Parameters.
where α is the cant deficiency, hCG the height of the center of
gravity of the vehicle from the rail head and e the track gauge.
The semi-static Action/Reaction is derived by the
multiplication of Qα by the A
stat. So equation (2a) is
transformed to:
(2b)
The Suspended (sprung) Masses of the vehicle –masses
situated above the primary suspension– create forces with very
small influence on the wheel’s trajectory and on the system’s
excitation. This enables the simulation of the track as an
elastic media with damping, as shown in Fig. 5 (see relevantly
[3]; [10]), and also the modelling of the motion of the Non
Suspended Masses of a railway vehicle on it. Forced
oscillation is caused by the irregularities of the rail running
table (like an input random signal) –which are represented by
n–, in a gravitational field with acceleration g, whilst the total
deflection of the rail’s running table, due to the wheel’s
passage, is y. As already described, there are two suspensions
on the vehicle for passenger comfort purposes: primary and
secondary suspension. Moreover, a section of the mass of the
railway track participates in the motion of the Non-Suspended
(Unsprung) Masses of the vehicle. These Masses are situated
under the primary suspension of the vehicle.
If the random excitation (track irregularities) is given, it is
difficult to derive the response, unless the system is linear and
invariable. In this case the input signal can be defined by its
spectral density and from this we can calculate the spectral
density of the response. The theoretical results confirm and
explain the experimental verifications performed in the former
British railway network ([11]; relevant results in [7, p.39, 71]
and also in [3]; [6]).
Fig. 5 (upper) A Single-Degree of Freedom (SDOF) Model of
Vehicle on a Rail running Table and the acting Forces.
(Lower) Model of a car-body on a bogie with one rolling
wheel, on the rail running table: n is the ordinate of the defect
of the rail running table, y is the deflection of the rail; the
Suspended and the Non-Suspended Masses are depicted.
If the random excitation (track irregularities) is given, it is
difficult to derive the response, unless the system is linear and
invariable. In this case the input signal can be defined by its
spectral density and from this we can calculate the spectral
density of the response. The theoretical results confirm and
explain the experimental verifications performed in the former
British railway network ([11]; relevant results in [7], p.39, 71
and also in [3]; [6]).
The equation for the interaction between the vehicle’s axle
and the track-panel becomes ([3]; [6]; [12]):
( )
( )
3
4
3
4
1
22
1
22
stat stat stat wheel
wheel
stat wheel stat
RA A R Q Q
Q Q E J
R Q Q A
EJ
= = = = +
+
= +
4. The Motion of the Non Suspended
Massesin the System “railway
Vehicle-railway Track”
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
19
Volume 18, 2023
(6)
where: mNSM the Non-Suspended (Unsprung) Masses of the
vehicle, mTRACK the mass of the track that participates in the
motion, mSM the Suspended (Sprung) Masses of the vehicle
that are cited above the primary suspension of the vehicle, Γ
damping constant of the track (for its calculation see [13];
[14]), hTRACK=ρdynamic the total dynamic stiffness coefficient of
the track, n the fault ordinate of the rail running table and y the
total deflection of the track.
The phenomena of the wheel-rail contact and of the wheel
hunting, particularly the equivalent conicity of the wheel and
the forces of pseudo-glide, are non-linear. In any case the use
of the linear system’s approach is valid for speeds lower than
the Vcritical≈500 km/h. The integration for the non-linear
model (wheel-rail contact, wheel-hunting and pseudoglide
forces) is performed through the Runge Kutta method ([7],
p.94-95, 80; [15], p.98; [16], p.171, 351). Consequently for all
operational speeds in High Speed Railway Lines up to now the
linear model, which we use, is quite reliable.
The defects of the rail running table are categorized in short
wavelength defects and long wavelength defects. The analysis
and investigation of the Eqn (6), for consecutive short
wavelength defects (e.g. the rail surfaces corrugation, of
wavelength of some centimeters), was presented in [3]. The
long wavelength defects are difficult to be measured, since
sometimes, their wavelength overpass the measurements’
base, which is determined by the distance between the
measuring vehicle’s axles. In this article the long wavelength
defects are modelled, analyzed and investigated.
The dynamic component of the acting load consists of the
action due to the Sprung or Suspended Masses (SM) and the
action due to the Unsprung or Non Suspended Masses (NSM)
of the vehicle. To the latter a section of the track mass is
added, that participates in its motion ([13]; [14]). The
Suspended (Sprung) Masses of the vehicle –masses situated
above the primary suspension (Fig. 1)– apply forces with very
small influence on the trajectory of the wheel and on the
excitation of the system. This enables the simulation of the
track as an elastic media with damping which takes into
account the rolling wheel on the rail running table ([3]; [17];
[18]). Forced oscillation is caused by the irregularities of the
rail running table (simulated by an input random signal) –
which are represented by n–, in a gravitational field with
acceleration g. There are two suspensions on the vehicle for
passenger comfort purposes: primary and secondary
suspension. Moreover, a section of the mass of the railway
track participates in the motion of the Non-Suspended
(Unsprung) Masses of the vehicle. These Masses are situated
under the primary suspension of the vehicle.
We approach the matter considering that the rail running
table contains a longitudinal fault/ defect of the rail surface. In
the above equation, the oscillation of the axle is damped after
its passage over the defect. Viscous damping, due to the
ballast, enters the above equation under the condition that it is
proportional to the variation of the deflection dy/dt. To
simplify the investigation, if the track mass (for its calculation
see ([13]; [14]) is ignored -in relation to the much larger
Vehicle’s Non Suspended Mass- and bearing in mind that y+n
is the total subsidence of the wheel during its motion (since
the y and n are added algebraically), we can approach the
problem of the random excitation, based on a cosine defect
(V< Vcritical=500 km/h):
(7)
The second order differential equation of motion is:
(8)
The complete solution of which using polar coordinates is
([6], p.199 and ch.3):
(9)
where, the first term is the transient part and the second part is
the steady state part.
The modelling described in paragraph 4, give equations to
calculate the actions on track depending on the parametrical
analysis of the conditions on the railway track. In order to
approach the long wavelength defects, we begin by trying to
relate the depth (sagittal) of an isolated defect to the dynamic
component of the load. We neglect the steady state part of (9):
(10)
We focus herein on the transient part of the load, that is the
term:
(11)
We investigate this term for ζ=0. The theoretical analysis
for the additional –to the static and semi-static component–
dynamic component of the load due to the Non Suspended
Masses and the Suspended Masses of the vehicle, leads to the
examination of the influence of the Non Suspended Masses
only, since the frequency of oscillation of the Suspended
Masses is much smaller than the frequency of the Non
Suspended Masses. If mNSM represents the Non Suspended
Mass, mSM the Suspended Mass and mTRACK the Track Mass
participating in the motion of the Non Suspended Masses of
the vehicle, the differential equation is (with no damping ζ=0):
(12)
( )
( )
2
2
2
2
NSM TRACK TRACK
NSM NSM SM
d y dy
m m h y
dt
dt
dn
m m m g
dt
+ + + =
= − + +
cos cos 2 Vt
aa
= =
( )
2
2
2cos
NSM TRACK NSM
d z dz
m h z m a t
dt
dt
+ + = −
()
( )
2
sin 1 cos
ntn
steady state part
transient part
z A e t a B t
−
−−
−
= − − + −
( )
cos −a B t
()
2
sin 1
ntn
A e t
−
− −
( )
2
2
2
2
NSM TRACK NSM
NSM TRACK TRACK NSM
dz
m h z m g
dt
dz
m m h z m g
dt
+ =
+ + =
5. The Specific Case of an Isolated
Track-defect
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
20
Volume 18, 2023
where the track mass mTRACK that participates in the motion
of the Non Suspended (Unsprung) Masses of the Vehicles,
ρtotal the total static stiffness coefficient of the track, ℓ the
distance among the sleepers, E, J the modulus of elasticity and
the moment of inertia of the rail, m0 the unitary mass of track
(per unit of length of the track).
For a comparison of the theoretical track mass to
measurement results refer to [13]; [14]. The particular solution
of the differential Eqn (12) corresponds to the static action of
the weight of the wheel:
We assume that the rolling wheel runs over an isolated
sinusoidal defect of length λ of the form:
where n is the ordinate of the defect. Consequently, the
ordinate of the center of inertia of the wheel is n+z. Defining
τ1 as the time needed for the wheel to pass over the defect at a
speed V:
, then:
Since:
Where:
and ω1 the cyclic frequency of the external force and ωn the
natural frequency. The additional dynamic component of the
load due to the motion of the wheel is:
(14)
To solve Eqn (12) we divide by (mNSM+mTRACK):
(15a)
The differential equation of motion, for an undamped forced
harmonic motion is ([19]; [20]):
(15b)
where:
Eqn (15a) is quite the same as Eqn (15b). The complete
solution is (see analysis in the Annex, at the end of the article):
(16)
when: k=hTRACK, m=mNSM+mTRACK, and:
The general solution of Eqn (15) is:
and
(17)
where, Tn=2π/ωn the period of the free oscillation of the wheel
circulating on the rail and T1=2π/ω1 the necessary time for the
wheel to run over a defect of wavelength λ: T1=λ/V.
Consequently, Tn/T1=ω1/ωn.
From Eqn (17):
(18)
We can investigate equation (18) after a sensitivity analysis
by variating parameters: for given values of Tn/T1=ω1/ωn and
for given value of V (for example equal to 1) the time period
T1 is proportional to μ=0.1, 0.2, … 1.0 of defect λ (where λ is
the defect’s wavelength). Equation (18) is transformed:
TRACK
TRACK
mg
zh
=
22
1 cos 1 cos
22
a x a Vt
n
= − = −
1V
=
( )
22
22
0 + + + =
NSM TRACK TRACK
d d z
m z n m h z
dt dt
( )
22
22
2
2
1
1
22
cos
NSM TRACK TRACK NSM
NSM
d z d n
m m h z m
dt dt
at
m
+ + = − =
= −
1
1
2 2 2 2
sin sin
22
22
sin
2
dn a V Vt a Vt
dt
dn a Vt
dt
= =
=
2
1
2
, 2 , TRACK
n
NSM
h
V
x V t Tm
= = = =
( )
NSM TRACK TRACK
m z n h z m z
− + = +
( )
( )
2
2
2
2
1
1
22
cos
TRACK
NSM TRACK
NSM
NSM TRACK
h
dz z
mm
dt
mat
mm
+ =
+
= −
+
( )
( ) ( )
01
2
00
11
cos
cos cos
n
m z kz p t
pp
k
z z t t
m m k
+ =
+ = =
2
2
= =
n
n
kk
m
m
( ) ( ) ( )
0
1
2
1
1cos cos
1
n
steady state transient part
n
p
z t t t
k
−−
= −
−
2
2
02
1
2
,
TRACK NSM
n
NSM TRACK
hm
p
mm
= = −
+
( ) ( ) ( )
( ) ( ) ( )
0
2
1
22
11
1
2
1
211
cos cos
1
1cos cos
21
NSM n
TRACK steady state transient part
pn
NSM n
NSM TRACK nsteady state transient part
m
z t t t
h
mtt
mm
−−
−−
= − − =
−
= −
+
−
( ) ( ) ( ) ( )
( ) ( ) ( )
2
1
2
1
2 2 2
11
1
2
1
1 4 1 cos cos
21
1cos cos
21
TRACK
NSM n
n NSM TRACK steady state transient part
hn
NSM n
NSM TRACK nsteady state transie
m
z t t t
mm
mtt
mm
−−
−
= − − =
+
−
= −
+
−
nt part−
( ) ( )
( ) ( )
1
2
1
11 cos cos
21
NSM TRACK
NSM
n
nsteady state transient part
mm zt
m
tt
−−
+=
= −
−
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
21
Volume 18, 2023
(19)
where n=ωn/ω1, ω1=λ/V and we examine values of μ·λ=0,
0.1λ, 0.2λ,...., 0.8λ, 0.9λ, λ, for discrete values of n=ωn/ω1
Fig. 6a Measurement of the ordinate d of a longitudinal
vertical defect at the position x on a measurement-base (chord)
of 2a length: 2a is much smaller than the defect’s wavelength
2a1, consequently the measured value d is smaller than the real
defect’s depth (sagitta) d1.
It should be clarified that different wavelengths address
different vehicles’ responses depending on the cord (different
from 10 m). This is of decisive importance for the
wavelengths of 30 – 33 m, which are characteristic for very
High Speed Lines ([7], p.30, [21], [15], p.342). In the real
tracks, the defects’ forms are random with wavelengths from
few centimetres to 100 m. Thus we should pass from the
space-time domain to the frequencies’ domain through the
Fourier transform, in order to use the power spectral density of
the defects (about spectral density of defects in a railway track
see [3]). The total procedure for this is out of the scope of the
present paper since it belongs to the domain of the
measurements of track defects via track recording vehicles.
Some interesting articles on the subject include but are not
limited to [22]; [23]; [24]; [25].
The defects of short wavelength exert important dynamic
increments of the load on the track. If the speed V increases,
the T1/Tn decreases and the supplementary subsidence, owed
to the dynamic increase of the load, decreases too. We observe
that when the speed overpasses the critical speed, for the
defect that we investigate, when the rigidity of the track
(stiffness) increases, Tn decreases and T1/Tn increases and we
approach the maximum-maximorum. For a defect of short
wavelength the high rigidity (stiffness) is disadvantageous.
In the real tracks, the defects are random with wavelengths
from few centimetres to 100 m. The defects constitute the
“Input” in the system “Vehicle-Track” since the deflection y
and the Action/Reaction R of each support point of the
rail/sleeper are the “Output” or “Response” of the system. The
accuracy of the measurements of the defects is of utmost
importance for the calculation of the deflection y and the
Reaction R; this accuracy, due to the bases of the measuring
devices/vehicles, is fluctuating. Thus we should pass from the
space-time domain to the frequencies’ domain through the
Fourier transform, in order to use the power spectral density of
the defects (for the power spectral density see [3]; [6]),
especially for defects, with (long) wavelength, larger than the
measuring base of the vehicle.
We remind that for a function f(x) or y(x) [let’s say the
vertical defects’ function, the input], its Fourier transform
F(ω) is given by:
(20)
and:
(21)
where the graphic representation of the absolute value is cal-
led the Fourier spectrum of the f(t) and is called the
energy spectrum with φ(ω) the phase angle. F(ω) or F(ν) is
called the Fourier transform of the function f(t) or y(t).
In the case of random defects then we do not use the
function f(x) but its Fourier transform:
(22)
In practice we don’t know the function of real defects y(x)
but the measured values f(x) (see Eqns 23 below), from the
recording vehicle, and we imply that:
(24)
where SZ(Ω) is the spectral density of the Fourier transform of
the real defects (input in the track recording vehicle), SF(Ω) is
the spectral density of the Fourier transform of the measured
values (output) and K(Ω) is a complex transfer function, called
frequency response function, transforming the measured
values of defects to the real values. For very High Speed Lines
we should analyze the system “railway track – railway
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
11
2
2
1 1 1 cos cos
21
11 cos 2 cos 2
21
NSM TRACK
NSM steady state transient part
steady state transient part
mm z t t n t
mn
n
n
−−
−−
+
= − =
−
= −
−
( ) ( )
( ) ( )
, 1,
1
2
it
it
F f t e dt with i and
f t F e dt
+
−
−
+
−
= = −
=
( ) ( )
( )
[]i
F A e dt
=
( )
2
F
( ) ( )
ix
F f x e dx
+
−
−
=
( ) ( ) ( )
( )
( )
( )
22
F OUTPUT
Z INPUT
SS
SS KK
= = =
(=Τ1/Τn) and μ a percentage of the wavelength λ. In Fig. 6 the
equation (19) is depicted.
6. Confidence Interval of Track
Recordingvehicles/Cars and Control
of the Geometry of the Track
The measurement-base (chord) of 10 m length, along a
railway line, is very important because it includes the
wavelengths of the vehicles’ hunting, since normally the
measurements’ base in railway measuring vehicles is
approximately 10 m; consequently for larger wavelengths the
measured defects’ values f(x)=d [from the measuring vehicles
or track recording cars] are smaller than the real ordinates
z(x)=d1 of the defects (Fig. 6a).
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
22
Volume 18, 2023
vehicle”. The calculation of the spectrum of track defects is
described in [3] (paragr. 6) and [6] (p.155-158).
Every point on each rail (at the rail running table) of a
track can be defined by its three coordinates: at the position x
and the functions y(x), in the horizontal alignment, and z(x), in
the vertical alignment. The measurements of the defects are
performed with track geometry cars, the contact cars, which
are made by actual contact with the rails, movable feeler
points (transducers) that touch the rails to measure the
parameters, e.g. the profile. The cars use the position of the
car-body, its yaw and roll and, consequently their axles as the
reference base for a relative measurement ([22, p.678-679]).
We examine the vertical defect, the dip between two bumps,
of an oscillograph recording ([22, p. 684]), as in Fig. 1
(upper left), with a reference chord of length 2a (normally 10
m) and the reliability of the measurements. The value of d at
the position x is:
(23a)
This transformation cannot be easily reversed: If we know
z(x) in every x easily f(x) can be derived, but if we know f(x)
it is not easy to calculate z(x). Thus we try to approach the
matter for the case of a sinusoidal defect:
, we can derive that:
(23b)
with Ω=2π/ℓ. J(Ω) is a real function and it is the transfer
function, permitting to pass from z(x) to f(x). If the chord 2a is
the chord used as base from the track recording vehicle then
J(Ω) is the transfer function of the vehicle. The track recording
vehicles measure the defects, the track displacement, “under
load”, that is under their axle load, which is usually smaller
than the maximum axle load of the Railway-Line but enough
for the measurement of the gaps under the seating surface of
the sleepers, if any. Normally the chord used as reference base
both for the vertical and horizontal defects is the 10 m length
and the axles of the vehicle are used for that. Regarding the
reliability of the measurements, three cases are distinguished:
(a) The vehicle’s reference base 2a (chord of 10 m) is
smaller than the defect’s wave-length ℓ=2a1 (Fig. 6a). In that
case the measured defect’s ordinate f(x)=d is much smaller
than the real defect’s ordinate z(x)=d1.
(b) The chord 2a is equal to the defect’s wavelength
ℓ=2a1, and f(x)=d=z(x)=d1 (Fig. 6d ).
(c) The chord 2a is larger than the defect’s wavelength
ℓ=2a1, with the reliability fluctuating. The most characteristic
case happens when the defect’s wavelength (2a1) equal to ½ of
the chord’s length (2a) and the measured ordinate f(x)=d=0
instead of the real defect’s ordinate z(x)=d1 (Fig. 6c).
Fig. 6b Measurement of a longitudinal vertical defect at the
position x on a chord of 2a length: the reliability of track
recording vehicle’s measurements is depicted through its
Transfer Function J(Ω).
Fig. 6c Measurement of a longitudinal vertical defect at the
position x on a chord of 2a length: 2a is double of the defect’s
wavelength 2a1.
In order to approach the matter of the reliability of the
measured values, by the track recording vehicle, we examine
its transfer function J(Ω), presenting minimums, zero, for:
(25)
with k integer and maximums equal to 2 for:
(26)
( ) ( ) ( ) ( )
1
2
d f x z x a z x a z x= = − + + −
( )
2
sinz x b x
=
( ) ( )
( ) ( )
1 2 2 2
( ) sin sin sin
2
1 2 1 2 2
sin sin sin
22
1
2
2 2 2 2 2 2 2 2
sin cos cos sin sin cos cos sin
2
sin
f x b x a b x a b x
b x a b x a b x
b
x a x a x a x a
bx
= − + + − =
= − + + − =
=
− + + −
−
( )
( )
( )
22
( ) sin 1 cos
2
1 cos ( )
J
f x b x a
z x a z x J
= − − =
= − − = −
22
aa
kk
= =
( )
22
12 12
aa
kk
= + = +
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
23
Volume 18, 2023
Fig. 6d Measurement of a longitudinal vertical defect at the
position x on a chord of 2a length: 2a is equal to the defect’s
wavelength 2a1.
(27)
(28)
as depicted in Fig. 6b. The 10 m chord is very important
because it includes the wavelengths of the vehicles’ hunting,
but for larger wavelengths the measured values f(x)=d are
smaller than the real ordinates z(x)=d1 (Fig. 6a). It should be
clarified that different wavelengths address different vehicles’
responses depending on the cord (different from 10 m). This is
of decisive importance for the wavelengths of 30 – 33 m, that
are characteristic for very High Speed Lines ([7, p.30], [21],
[15, p.342]): for 30 m, J(Ω)=0.5 and for 50 m J(Ω)=0.2 (as we
saw in the previous paragraph. In any case, for each vehicle its
reference chord should be used instead of the 10 m chord.
In the real tracks, the defects are random with wavelengths
from few centimeters to 100 m. Thus we should pass from the
space-time domain to the frequencies’ domain through the
Fourier transform, in order to use the power spectral density of
the defects. We remind that for a function f(x), its Fourier
transform F(ω) is given by:
(29)
(30)
where the graphic representation of the absolute value
is called the Fourier spectrum of the
f(t) and is called the energy spectrum with
φ(ω) the phase angle.
In the case of random defects then we do not use the
functions f(x) and z(x) but their Fourier transforms:
(31)
Applying the Fourier transforms in the above equations (1)
and (2) we find:
(32)
similar to equation (23b), but for the Fourier transforms of f(x)
and z(x). In practice we don’t know the real defect’s function
z(x) but the measured values f(x), from the recording vehicle,
and from the equation (32) we imply that:
(33)
where SZ(Ω) is the spectral density of the Fourier transform of
the real defects (input in the track recording vehicle), SF(Ω) is
the spectral density of the Fourier transform of the measured
values (output) and K(Ω) is a complex transfer function, called
frequency response function, transforming the measured
values of defects to the real values. As it was mentioned in the
previous paragraph, for the very High Speed Lines we should
analyze the system “railway track – railway vehicle”.
Before we examine any signal, waveform, random
excitation, we should first bear in mind that there are two
possible mathematical representations of this signal:
(a) a representation of the form y = f(t) for which the
independent variable is the time t which elapses, and
(b) a frequency representation of the form Y = F(ν) for
which the independent variable is frequency ν, whose unit is
sec-1, that is inverse of time.
These two mathematical representations are related through
the Fourier transform and its inverse transform as given by the
Eqn (20) and (21).
Fig. 7 below gives an idea of the physical realization of
the Fourier transform.
As analyzed in [6] (chapter III, paragraph 9), we can move
( )
( )
1
0,
5 , 2.5 , 1.67 , 1,25 , 1,00
fx dfor
z x d
m m m m m and
==
=
( )
( )
1
2,
10 , 3.33 , 2 , 1, 42857 , 1,1111
fx dfor
z x d
m m m m m
==
=
( ) ( )
( ) ( )
, 1,
1,
2
it
it
F f t e dt with i and
f t F e dt and
+
−
−
+
−
= = −
=
( ) ( )
( )
[]i
F A e dt
=
( ) ( )
FA
=
( )
2
F
( ) ( )
,
ix
Z z x e dx and
+
−
−
=
( ) ( )
( )
( ) ( ) ( )
1
12
1 cos ( )
i a i a
F Z e e
Z a Z K
−
= − − + =
= − − = −
( ) ( )
ix
F f x e dx
+
−
−
=
( ) ( ) ( )
cos 2R F y t t dt real part
+
−
= −
7. The Fourier Transform and the
Spectrum of Amplitudes and Phases:
General Application in Solving the 2nd
Order Differential Equation
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
24
Volume 18, 2023
from periodical to non-periodical excitation functions by
«extension» of the period of the periodic (or harmonic)
function to infinity: T → ∞. Hence, the interval of frequencies
ν=1/T tends to zero and the spectral frequency function can be
a continuous function.
F(ω) or F(Ω) or Y(ν) is a function of frequencies ν, which is
a complex function in its general form, and contains a real and
an imaginary part.
Fig. 7. Physical realisation of the Fourier transform ([6]).
The spectrum of the amplitudes is:
The spectrum of the phases are:
From the above, we may conclude that in the case of non-
periodic functions, the Fourier transform is a continuous
frequency function. In Εqns (20) and (21), functions F(ν) or
F(ω) and f(t) or y(t) represent the same physical phenomenon,
but through different presentation. If we consider the function
f(t), the point of the system that is being examined moves
Where V the speed of the vehicle, T=2π/ω→ ωt=2π/(Τt)=
2πVt/λ where λ the length of the defect, run by the wheel in:
( ) ( ) ( )
( )
( )
( )
22
F OUTPUT
Z INPUT
SS
SS KK
= = =
( ) ( ) ( )
22
F R F I F
=+
( ) ( )
( )
arctan IF
RF
=−
( ) ( ) ( )
sin 2I F y t t dt imaginary part
+
−
= −
( ) ( ) ( )
cos 2R F y t t dt real part
+
−
= −
cos cos 2 Vt
a t a
= =
within the domain «amplitude-time». If we consider the
function F(ω), the point of the system that is being examined
moves within the domain «amplitude-frequency». The first
function gives the image of the object under investigation (e.g.
structure) in dimensions of space and time and the other one in
dimensions of frequencies and amplitudes.
When we investigate F(ν) for a specific value νi of ν, this
means that we examine its whole history, up to the present, but
also to the future of f(t), to find whatever corresponds to
frequency νi. This corresponds to an infinitesimally selective
filtering. This filtering cannot take place under a physical
process. Hence we cannot determine F(ν) with perfect
accuracy for a specific νi, in the frequency axis.
In the same way, if we go on to f(t), starting from F(ν), we
have to know the spectrum for all the frequencies up to
infinity and the formula shows that the same operation of
infinitesimally selective filtering intervenes, since the
variables of time and frequency are interchangeable. This
means that in order to determine the exact value of f(t) at an
instance t, we must have and examine a frequency zone that
tends to infinity. This is another form of the uncertainty
principle that expresses «the weakness of man as an observer
to understand reality without decomposing it or rendering it, in
any way, “blurred” (see [23, p.23]). We remind the uncertainty
principle of Heisenberg, according to which, neither the
momentum nor the position of a particle inside the nucleus of
an atom can be determined at the same time, with any great
accuracy. On the contrary, the uncertainties for the two
quantities have complementary roles. If position coordinate x
of the particle has uncertainty Δx, as defined in this way, and
if the corresponding linear momentum coordinate px has
uncertainty Δpx, then the two uncertainties relate to each other
through the inequality Δx ⋅ Δpx ≥ R/2π ([24, p.1149]). The
more precisely we determine the position, the more uncertain
is the determination of the linear momentum, and vice versa.
8. Relation of the Track Measured
Profile Defects To the Dynamic
Component of Loads
In Fig. 5-Lower the rail running table depicts a
longitudinal fault/defect of the rail surface, in the profile as
recorded by the Track Recording Car. In the above Eqn (6),
the oscillation of the axle is damped after its passage over the
defect. As it is mentioned above, viscous damping, due to the
ballast, enters the above equation under the condition that it is
proportional to the variation of the deflection dy/dt. To
simplify the investigation, if we ignore the track mass (for its
calculation Giannakos, 2010a [2]) in relation to the much
larger Vehicle’s Non Suspended (Unsprung) Mass and bearing
in mind that y+n is the total subsidence of the wheel during its
motion (since the y and n are added algebraically), we can
approach the problem of the random excitation, from cosine
de-fect (V<< Vcritical=500 km/h):
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
25
Volume 18, 2023
, if we set:
where the quantity:
represents the subsidence due to the static loads only, and z
absolutely random ([25, p.6]) due to the dynamic loads.
Equation (6) becomes:
(6a)
(6b)
Since, in this case, we are examining the dynamic loads
only, in order to approach their effect, we could narrow the
study of Eqn (6b), by changing the variable:
Eqn (6b) becomes:
(6c)
(6d)
Where, u is the trajectory of the wheel over the vertical
fault in the longitudinal profile of the rail.
If we apply the Fourier transform to equation (6a) (see
relevantly [26] or solving second order differential equations
with the Fourier transform):
H(ω) is a complex transfer function, called frequency response
function [26], that makes it possible to pass from the fault n to
the subsidence Z. If we apply the Fourier transform to Eqn
(6c):
(6c-1)
G(ω) is a complex transfer function, the frequency response
function, that makes it possible to pass from Z to Z+n. If we
name U the Fourier transform of u, N the Fourier transform of
n, p=2πiν=iω the variable of frequency and ΔQ the Fourier
transform of ΔQ [he dynamic component of the Load due to
the Non Suspended (Unsprung) Masses] and apply the Fourier
transform at equation (6d):
B(ω) is a complex transfer function, the frequency response
function, that makes it possible to pass from the fault n to the
u=n+z. Practically it is verified also by the equation:
passing from n to Z through H(ω) and afterwards from Z to
n+Z through G(ω). This is a formula that characterizes the
transfer function between the wheel trajectory and the fault in
the longitudinal level and enables, thereafter, the calculation
of the transfer function between the Dynamic Component of
the Load/Action and the track defect (fault). The transfer
function of the second derivative of (Z+n) in relation to time:
that is the acceleration γ, will be calculated below (and is
equal to (iω)2∙Β(ω)). The increase of the vertical load on the
track due to the Non Suspended (Unsprung) Masses,
according to the principle force = mass x acceleration, is
given by:
If we apply the Fourier transform to this equation:
The transfer function B(ω) allows us to calculate the effect of
a spectrum of sinusoidal faults, like the corrugations
(undulatory wear) as described by professor Hay ([22, p.521-
523]). If we replace ω/ωn=ρ, where ωn= the circular
eigenfrequency (or natural cyclic frequency) of the oscillation,
and:
where ζ is the damping coefficient. Thus the above formula of
calculation of B(ω) is transformed:
T T V
V
= =
22
22
SM NSM
TRACK
mm dy dz d y d z
y z g and
h dt dt dt dt
+
= + = =
SM NSM
TRACK
mmg
h
+
22
22
22
22 0
NSM TRACK NSM
NSM TRACK
d z dz d n
m h z m
dt
dt dt
d z d n dz
m h z
dt
dt dt
+ + = −
+ + + =
2 2 2
2 2 2
d u d n d z
u n z dt dt dt
= + = +
( ) ( )
2
2
2
2
0
0
NSM TRACK
NSM TRACK
d u dz
m h z
dt
dt
d u n
du
m h u n
dt
dt
+ + =
−
+ + − =
( ) ( ) ( ) ( ) ( ) ( )
22
() TRACK
NSM NSM
ih
i Z Z Z i N
mm
+ + = −
( ) ( )
( ) ( )
( )
24
2
2
2 2 2
,NSM
NSM TRACK
Zm
HH
Nmh
= = − +
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( )
20
2 2 2
2
24
0
,
TRACK
TRACK
NSM
i U i Z h i Z
Uh
GG
Zm
+ + =
+
= =
( )
( )
2
2
2
.(6 ) NSM TRACK TRACK
NSM TRACK TRACK
d u du dn
Eq d m h u h n
dt dt
dt
m p p h U p h N
+ + = +
+ + = +
( )
( )
( )
( )
( )
2
2 2 2
2
2
2 2 2
,
TRACK
NSM TRACK
B
TRACK
NSM
ph
U N and
m p p h
h
Bmh
+
=
+ +
+
= − +
( ) ( ) ( )
( )
2 2 2
2 2 2
2
2 2 2
TRACK
NSM TRACK
h
B H G mh
+
= = − +
( )
2
2
d Z n
dt
+
( )
2
2
22
NSM NSM
d n Z
du
Q m m
dt dt
+
= =
( ) ( )
22
ˆ
ˆNSM NSM Z n
Q m p U m p f
+
= =
( ) ( ) ( )
222
ˆNSM NSM n
Q m p B m B N
= =
22
, , 2 ,
TRACK
nn
NSM NSM n
hV
mm
= = = =
( ) ( )
( )
22
22
2
2 2 2
14
14
n
BB
+
==
− +
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
26
Volume 18, 2023
that is the complex transfer function, the frequency response
function, that makes it possible to pass from the fault n
(measured) to the u=n+z.
The first term in the bracket of Eqn (19) is depicted on the
horizontal axis while on the vertical axis the percentages of
the wavelength μ·λ are shown. We observe that z(x) has its
maximum value for T1/Tn=0,666667=2/3, equal to 1,465:
(34)
for x=0,91λ. The relation T1/Tn represents the cases for short
and long wavelength of the defects. For T1/Tn=2-2,5 the
wavelength is long and for T1/Tn << the wavelength is short
([7, p.49]). The second derivative of z(x) from Eqn (17), that
is the vertical acceleration that gives the dynamic overloading
due to the defect (i.e. the dynamic component of the total Load
of the wheel), is calculated:
(35a)
(35b)
Fig. 8. Mapping of Eqn (19). On the Vertical Axis the
percentage of the wavelength λ of the defect is depicted. On
the Horizontal Axis the first term of Eqn (19), inside the
brackets, is depicted.
for discrete values of n=ωn/ω1 (=Τ1/Τn) and μ a percentage of
the wavelength λ, and Tn=0,0307 sec as calculated above. The
additional subsidence of the deflection z at the beginning of
the defect is negative in the first part of the defect. Following
the wheel’s motion, z turns to positive sign and reaches its
maximum and possibly afterwards z becomes again negative.
After the passage of the wheel over the defect, one oscillation
occurs which approaches to the natural cyclic frequency ωn
(this oscillation is damped due to non-existence of a new
defect since we considered one isolated defect) in reality, even
if in the present analysis the damping was omitted for
simplicity. The maximum value of ζ is given in Table 1 below,
as it is –graphically– measured in Fig. 8.
Table 1: Maximum Values of ζ
It is observed that the maximum value:
is shifted towards the end of the defect as the ratio T1/Tn
decreases, that is when the defect’s wavelength becomes short.
The maximum is obtained for T1/Tn = 0,666667 = 2/3. For
each combination of “vehicle + track section”, the critical
value of the speed V, for which the 2/3 are achieved is a
function of the wavelength λ. Since:
we can calculate the critical speed Vcritical for any combination
of track layers and their corresponding stiffness. As a case
study we use the ballasted track depicted in Fig. 4, for high
speed, equipped with rail UIC60 (ρrail = 75.000 kN/mm),
monoblock sleepers of prestressed concrete B70 type (ρsleeper =
13.500 kN/mm), W14 fastenings combined with pad Zw700
Saargummi (ρpad fluctuating from 50,72 to 48,52 kN/mm),
ballast fouled after 2 years in circulation (ρballast = 380 kN/mm)
and excellent subgrade/substructure for high speed lines:
ρsubgrade = 114 kN/mm [e.g. the New Infrastructure (NBS) of
the German Railways]. These parameters’ data are the ones
measured on track by the German State Railways (DB), as
cited in [7] and [18]. The calculation of the static stiffness
coefficient of the subgrade ρpad for a high speed line of this
type, as it is derived from the load-deflection curves of the
pads provided by the producer, by the trial-and-error method
described in [6, ch. 9]; see also [27] and [28]. For this cross
section of ballasted track, hTRACK is equal to 85,396 kN/mm =
8539,6 t/m and mTRACK is equal to 0,426 t (for the calculations
see [14]; [13]). If we consider an average mNSM=1,0 t, then:
T1/Tn
2,5
2
1,5
1
0,8
0,6667
0,6
0,5
ζ
0,18
0,335
0,65
1,205
1,415
1,47
1,43
1,34
where: ζ=[(mNSM+mTRACK)/mNSM]·[zmax/α]
( ) ( )
1,465
NSM
NSM TRACK
m
zt mm
=
+
( ) ( ) ( ) ( )
11
2
1
1sin sin
21
−−
= − +
+
−
NSM nn
NSM TRACK nsteady state transient part
m
z t t t
mm
( ) ( ) ( ) ( )
22
11
2
1
1cos cos
21
−−
= − −
+
−
NSM nn
NSM TRACK nsteady state transient part
m
z t t t
mm
( )
+
NSM TRACK max
NSM
mm z
mα
1
1
3
2n
TV
V T T
= = =
1,0 0,426 0,145
9,81
NSM TRACK
m m tons mass
+
+ = = −
9. Sensitivity Analysis for Track-defects
in a High Speed Line
9.1 General
9.2 The Case of Short-wavelength Defects
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
27
Volume 18, 2023
Where g = 9,81 m/sec2, the acceleration of gravity. The
period Tn is given by:
where an average hTRACK=8539,6 t/m is used and Vcritical is
given in [m/sec], λ in [m]. For a wavelength of 1,0 m, Vcritical =
57,69 m/sec = 207,7 km/h.
If, now, we consider a long wavelength defect with a
wavelength that produces a forced oscillation with:
we calculate (in Fig. 8 is 0,19, for x=0,41·λ):
with the values calculated above: Tn = 0,026 sec, T1 = 0,065
sec, the wavelength λ equals:
This value represents a defect of adequately long
wavelength. The static deflection due to a wheel load of 11,25
t or 112,5 kN is equal to:
Consequently, for α=1 mm, that is for every mm of vertical
defect, the dynamic increment of the static deflection is equal
to (0,133/0,606)=21,9% of the static deflection (for every mm
of the depth of the defect).
If we examine the second derivative (vertical acceleration)
as a percentage of g, the acceleration of gravity, then [from
Eqns (35)]:
(36)
and the result is a percentage (%) of g.
Equation (36) is plotted in Fig. 9.
The first term in the bracket of Eqn (36) is depicted on the
horizontal axis while on the vertical axis the percentages of
the wavelength μ·λ are shown. For the case calculated above
in Fig. 9, at the point x=0,41·λ the term in bracket has a value
of -0,332:
Eqn (12) (its second part corresponds to the static action of
the wheel load) has as particular solution:
Fig. 9. Mapping of the Eqn (36), for the vertical
acceleration due to a defect of long wavelength. In the Vertical
Axis the percentage of the wavelength λ of the defect is
depicted. In the Horizontal Axis the first term of Eqn (36), in
the brackets, is depicted.
, and abandoning the second part leads to the
classic solution where z is the supplementary subsidence owed
to the dynamic increase of the Load. The dynamic increase of
the Load is equal to:
1,0 0,426
2 0,026 sec
9,81 8539,6
33 57,69
2 2 0,026
n
critical
n
T
VT
+
= =
= = =
1
1
2,5
n
n
T
T
==
( )
max
0,19 0,133
NSM
NSM TRACK
m
zmm
= =
+
12,5 0,065 57,69 3,75
n
V T V T m
= = = =
3
43
112.500
2 2 2 2
wheel
static
total
QN
zEJ
= =
5
112.500 1,524228617 10 0,606
22
−
= =
N mm mm
N
( ) ( )
2
1
2
1
1
21
NSM TRACK
NSM n
m m z t
m
+
= −
−
( ) ( ) ( ) ( )
2
2
2
2 1 1 cos 2 cos 2
21
nsteady state transient part
nn
n T g n
−−
= − −
−
NSM
TRACK
mg
zh
=
33
3
47 4 3
23
600
210.000 3,06 10 85.396
mm
NN
mm
mm mm
=
( ) ( ) ( )
0.33204
11
NSM TRACK
NSM
m m z t zt
mg
+
= =
−
1,0 0,233
1
0.332041 0,0 , 426 gg
=− =
+
2
2
1
22
cos cos
n
steady state transient part
VV
n
VV
−−
− =
9.3 The Case of Long Wavelength Defects
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
28
Volume 18, 2023
(37)
where, from the analysis above: hTRACK = 8539,6 t/m = 85.396
N/mm, mTRACK = 0,426 t = 426 kg. Consequently, for arc
height (i.e. sagitta) α=1 mm of a defect of wavelength λ, that
is for every mm of vertical defect, the dynamic increase of the
load is equal to (1,04/ 11,25)=9,24% of the static load of the
wheel (for every mm of the depth of the defect). Apparently
the increase of the static stiffness coefficient and of the
inferred dynamic stiffness coefficient of track leads to lower
values of Qdynamic since the hTRACK is in the denominator in the
equation for calculation of z, consequently the first term of the
Eqn (35b) for the dynamic overloading due to a defect (i.e.,
the dynamic component of the total Load of the wheel) will be
reduced. The same happens for the track mass participating in
the motion of the Non Suspended Masses of the wheel. Thus
finally the Qdynamic will be reduced when the ρtotal and the
hTRACK are increased.
In the case of a defect of long wavelength, when the speed
V increases, then T1 decreases and the supplementary
subsidence, owed to the dynamic increase of the load,
increases; consequently the dynamic component of the load
due to the Non Suspended Masses increase more rapidly since
it depends on (ω1)2, that is on the square of the speed V. When
the dynamic rigidity hTRACK=ρdynamic increases, then the
eigenperiod Tn decreases and T1/Tn increases and the
supplementary subsidence, owed to the dynamic increase of
the load, decreases for the same speed V; one higher rigidity
(stiffness coefficient) is still advantageous. For defects of
longer wavelength, the oscillations of the Suspended Masses
become predominant since the oscillations of the Non
Suspended Masses decrease.
The results presented so far and the arithmetic comparisons
give an idea and enlighten the influence of some kinds of track
defects as well as of several parameters, but the calculations
do not take into account the amortization of the oscillations
due to the damping of the track and mainly of the ballast,
consequently the derived arithmetic values are larger than the
real values. For example in the case of the theoretical
calculation of the track mass which participates in the motion
of the Non Suspended Masses of the railway vehicles without
damping give results 33% larger than the real ones since if we
take into account the damping coefficient of the track the
variation between the results of the theoretical calculations
and the real values measured on track fluctuates between 0,5
and 4% ([14]; compared to [13]), fact depicting the accuracy
of the theoretical calculations, if the totality of the parameters
is taken into account.
In order to approach the matter of the reliability of the
measured values of the track-defects by the track recording
vehicles/cars and their Confidence Interval, we should
examine the transfer function of the recording vehicle which
presents minimums and zero-points. In the real conditions, the
defects are random with wavelengths from few centimeters to
100 m. Since the length of the vehicle’s measuring base is
much shorter than 100 m, we should pass from the space-time
domain to the frequencies’ domain through the Fourier
transform, in order to use the power spectral density of the
defects. Furthermore, in the case of random defects, we cannot
and do not use the functions f(x) and z(x) but we can use their
Fourier transforms.
For a defect of long wavelength λ and sagitta of 1 mm
(depth of the defect), the dynamic increase of the acting load –
compared to the static wheel load– is equal to 9,24%.
Furthermore from Fig. 8 and Fig. 9, it is verified that when the
speed increases, the period T1 decreases and the
supplementary sagitta (depth of the defect) increases.
Supplementary (sagitta), since it is added to the static
deflection and it is owed to the dynamic component of the
load. The increase of the dynamic component of the load
increases faster since it is dependent on the square of the speed
(ω1)2. When the dynamic stiffness coefficient hTRACK
increases, Tn decreases, T1/Tn increases, the supplementary
sagitta decreases (for the same V), and the dynamic
component of the action decreases also. Furthermore, in the
case of longer wavelengths the oscillations of the Suspended
Masses become predominant since the oscillations of the Non
Suspended Masses decrease. Consequently, the softer the pad
and/or the subgrade (subgrade and prepared subgrade) then the
higher percentage of the load is transmitted through the
sleeper to the substructure of the railway track under the
running load/axle. Finally in total, the reaction per support
point of the rail/sleeper, in the case of softer pads and more
resilient fastenings, is smaller due to a distribution of the load
along the track in more support points of the rail/sleepers, as it
can be derived from literature ([1]; [6]; [2]). In the case of the
short wavelength defects this is more clearly verified.
It should be clarified that different wavelengths address
different vehicles’ responses depending on the measurement-
base/cord (different from 10m). This is of decisive importance
for the wavelengths of 30 – 33 m, which are characteristic and
of very high-impact in the case of very High Speeds.
For defects of very long wavelength, the oscillations of the
Suspended Masses (strongly influencing the feeling of
“passenger-comfort”) become predominant, since the
oscillations of the Non Suspended Masses decrease (strongly
influencing the Loading of the Track): in the case of long
wavelength defects, when the speed V increases, then T1
decreases and the supplementary subsidence, owed to the
dynamic increase of the load, increases; consequently the
dynamic component of the load due to the Non Suspended
Masses increase more rapidly since it depends on (ω1)2, that is
on the square of the speed V.
A next step of the analysis is to investigate the influence of
the defect’s dip [in mm] on the magnitude of the acting Loads
on the Railway Track and – due to the principle “Action =
Reaction” – on the Vehicle’s wheels and bogies; but this is
beyond the frame of the present article.
2
0,133
426 0, 233 9,81 1,04
se
85.396
c
dynamic TRACK TRACK
Q h z m z
m
kg t
= + = −
− =
10. Conclusions – Perspectives of Research
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
29
Volume 18, 2023
[1]. K. Giannakos, A. Loizos: Evaluation of actions on
concrete sleepers as design loads – Influence of
fastenings, International Journal of Pavement
Engineering (IJPE), Vol. 11, Issue 3, June, 197 – 213,
(2010).
[2]. K. Giannakos: Loads on track, Ballast Fouling and
Life-cycle under Dynamic Loading in Railways,
International Journal of Transportation Engineering –
ASCE, Vol. 136, Issue 12, 1075-1084, (2010a).
[3]. K. Giannakos: Modeling the Influence of Short
Wavelength Defects in a Railway Track on the
Dynamic Behavior of the Non-Suspended Masses,
Journal Mechanical Systems and Signal Processes
(jmssp), Elsevier, 68-69, (2016), 68-83,
http://dx.doi.org/10.1016/j.ymssp.2015.07.020, (2015).
[4]. E. Winkler: Die Lehre von der Elastizität und Festigkeit
(The Theory of Elasticity and Stiffness), H. Dominicus,
Prague, (1867).
[5]. H. Zimmermann: Die Berechnung des
Eisenbahnoberbaues, Verlag von Wilhelm Ernst &
Sohn, Berlin, (1941).
[6]. K. Giannakos: Actions on the Railway Track,
Papazissis publications, Athens, Greece, http://www.
papazisi.gr, (2004).
[7]. J. Alias: La Voie Ferree, IIeme edition, Eyrolles, Paris,
(1984).
[8]. J. Eisenmann: Schotteroberbau – Moglichkeiten und
Perspektiven fur die Moderne Bahn, Die Naturstein-
Industrie, Heft 3, (Isernhagen, Germany), 6-11 (1988).
[9]. K. Giannakos: Ties’ Design in High -Speed and Heavy
Haul Railroads: Ultimum Strength vs Actions on
Track, presented in the workshop for sleepers and
fastenings of TRB (2013a), 92nd Annual Meeting,
proceedings.
[10]. K. Giannakos: Selected Topics in Railway
Engineering, University of Thessaly, Dpt. Civil
Engineering, Greece, www.uth.gr, (2013b).
[11]. H. Jenkins, J. Stephenson, G. Clayton, G. Morland, D.
Lyon: Incidences des parametres carac-teristigues de la
voie et des vehicules sur les efforts dynamigues
verticaux gui se developpent entre rail et roue, Rail
International 10/(1974), 682-702.
[12]. SNCF: Mecanique de la voie, (1981).
[13]. K. Giannakos: Theoretical calculation of the track-
mass in the motion of unsprung masses in relation to
track dynamic stiffness and damping, International
Journal of Pavement Engineering (IJPE) - Special Rail
Issue: “High-Speed Railway Infrastructure: Recent
Developments and Performance”, Vol. 11, number 4,
p. 319-330, August, (2010b).
[14]. K. Giannakos: Influence of the Track's Damping on
the Track Mass Participating in the Motion of the Non
Suspended Masses of Railway Vehicles- Theoretical
Calculation and Comparison to Measurements,
volume published in honor of professor G.
Giannopoulos, Aristotle University of Thessaloniki,
(2012).
[15]. J. Fortin: La Deformee Dynamique de la Voie Ferree,
RGCF, 02/1982.
[16]. D. Thompson: Railway Noise and Vibration, Elsevier,
2009.
[17]. SNCF/Direction de l’ Equipement: Mecanique de la
Voie, Paris, (1981).
[18]. A. Prud’homme: Sollicitations Statiques et
Dynamiques de la Voie, SNCF/ Direction des
Installations Fixes, Paris, (1969/ 1966).
[19]. A. Chopra: Dynamics of Structures – Theory and
Applications to Earthquake Engineering, Prentice
Hall, second edition, USA, (2001).
[20]. J. Argyris, H.-P. Mlejnek: Texts on Computational
Mechanics, Volume V Dynamics of Structures, North-
Holland, Elsevier, New York, USA, (1991).
[21]. OSE/SNCF, Cooperation: Comportment des traverses
en relation avec les charges dynamiques, June, (1988).
[22]. W. Hay: Railroad Engineering, John Wiley & Sons,
USA (1982).
[23]. J. Max: Methodes et Techniques de Traitement du
Signal et Applications aux Measures Physiques,
Masson, Paris, (1985).
[24]. Hugh J. Young: Physics, Greek edition, vols I, II,
Papazisis publications, Athens, 1994; Addison-
Wesley, 1992.
[25]. L. Gent and G. Janin, La Qualite de la Voie Ferree –
Comment la definir Quantita-tivement?, RGCF
12/1969, SNCF’s reprint.
[26]. C.R. Wylie and L.C. Barrett, Advanced Engineering
Mathematics, sixth edition, McGraw-Hill, Inc., USA,
(1995).
[27]. K. Giannakos, S. Tsoukantas, A. Sakareli, C. Zois:
Schwankung des Steifigkeit zwischen Fester Fahrbahn
und Schottergleis - Beschrieben wird die
parametrische Analyse der Schwankung des
Steifigkeitsfaktors im Übergangsbereich zwischen
Fester Fahrbahn und Schottergleis, EI -
Eisenbahningenieur, Germany, February 2011, 12-19,
(2011).
[28]. K. Giannakos, S. Tsoukantas: Requirements for
Stiffness Variation between Slab and Ballasted
Railway Track and Resulting Design Loads,
ICASTOR (Indian Centre For Advanced Scientific
and Technological Research) Journal for Engineering,
January 2010, Vol.3, No 1 (2010), p. 21 - 35, (2010).
References
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
30
Volume 18, 2023
For the free oscillation (without external force) the equation is:
(XI-1)
The general solution is:
(XI-2)
Where:
(XI-3)
If we pass to the undamped harmonic oscillation of the form:
(XI-4)
where:
(XI-5)
The particular solution of the linear second order differential
equation (1.4) is of the form:
(XI-6)
Substituting equation (XI-6) to equation (XI-4) we derive:
(XI-7)
The general solution for the equation (XI-4) is the addition of
the solution (XI-2) and of the solution of the equation (XI-6)
combined with equation (XI-7):
(XI-8)
(XI-9)
Calculating the values of equation (1.8) and (1.9) at t=0:
(XI-10)
(XI-11)
(XI-12)
and for initial conditions z(0)=(0)=0:
(XI-13).
2
0 0 0
n
k
m z k z z z z z
m
+ = + = + =
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
cos sin
0
0 cos sin
nn
nn
n
z t A t B t
z
z t z t t
= +
= +
( ) ( )
0
0,
n
z
A z B
==
( ) ( )
( )
20
0
22
0
cos cos
cos
n
nn
p
m z k z p t z z t
m
p
z z t
k
+ = + =
+ =
2
2
n
n
kk
m
m
= =
( ) ( ) ( ) ( )
( ) ( )
2
cos sin
cos
pp
p
z t C t z t C t
z t C t
= = −
= −
( ) ( ) ( )
22 0
2 2 2 0
cos cos cos
n
nn
p
C t C t t
m
p
CC k
− + =
− + =
( ) ( )
2
2 2 2 00
22
0
2
1
1
n
nn
n
n
pp
CC
kk
p
Ck
− = =
−
=
−
( ) ( ) ( ) ( )
0
2
1
cos sin cos
1
nn
n
p
z t A t B t t
k
= + +
−
( ) ( ) ( )
sin cos
n n n n
z t A t B t
= − + −
( )
0
2sin
1
n
pt
k
−
−
( ) ( ) ( ) ( )
( )
0
2
0
2
1
0 cos 0 sin 0 cos 0
1
1
0
1
n
n
p
z A B k
p
Az k
= + +
−
= −
−
( ) ( ) ( ) ( ) ( )
( )
0
2
0
2
0
1
0 cos sin
1
1cos
1
nn
n
n
transient part
n
z
p
z t z t t
k
pt
k
−
= − + +
−
+
−
( ) ( ) ( ) ( )
( )
0
2
0 sin 0 cos 0 sin 0
1
0
nn
n
n
p
z A B k
z
B
= − + −
−
=
( ) ( ) ( )
0
2
1cos cos
1
n
steady state transient part
n
p
z t t t
k
−−
= −
−
1$11(;
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.3
Konstantinos S. Giannakos
E-ISSN: 2224-3429
31
Volume 18, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
