The document summarizes an investigation into the frequency mobility response of excitation techniques used to induce vibration on quad-bundled overhead transmission line conductors during indoor testing. Specifically, it examines whether using a rigid connection between the shaker and conductors interferes with the vibration response. Experimental frequency response functions were measured and compared to theoretical calculations. The results from different bundle configurations were also analyzed to assess fatigue and damping performance. The mathematical modeling of the bundle-excitation system is presented, including derivations of governing equations.

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77
Abstract:
In part A of this paper, the frequency mobility response
of the excitation technique used to induce vibration on
a quad-bundled conductor rigidly attached indoors was
investigated. Two reference inputs were used to link
the shaker to the quad bundle conductors via a square
rigid block made of welded square hollow steel. The
primary objective was to determine whether there
was interference due to the use of a rigid connection.
In the first instance, restraining the free motion of the
bundle conductors which would be disadvantageous
to the vibration response results. The interference
was evaluated in the form of mobility FRF responses
obtained through the direct measurements. These were
compared to the calculated theoretical results. Further,
the damping performance of various configurations of
quad bundle conductors was investigated by analysing
the vibration signatures of their FRF responses
when excited. The cross-spectrum analysis indicated
significant correlations between the experimental and
theoretical results for most of the bundle configurations
investigated, with no evidence of autocorrelation. Further
analysis of the data collected during the validation of
the excitation technique for the quad bundle conductors
led to a relevant assessment of the fatigue and damping
performances of different bundle span configurations.
This is presented in part B of this paper.
Aeolian ibrations of verhead
ransmission line undled onductors
uring ndoor esting, Part A:
Validation of xcitation echnique
Y. D. KUBELWA1
* A. G. SWANSON1
D. G. DORRELL2
1
Vibration Research and Testing Centre, School of Engineering,
University of KwaZulu-Natal, South Africa
2
School of Electrical and Information Engineering,
University of the Witwatersrand, South Africa
KEYWORDS
Aeolian vibrations, overhead lines, bundle conductors, frequency response function FRF, cross-spectrum analysis
* danielkubelwa2010@gmail.com
Nomenclature
and n
Angular vibration frequency and resonance angular vibration frequency at mode n (rad/sec)
Phase angle between two vectors :force and displacement (°)
and j
Damping coefficient ratio
El, k, kx
, and ky
Bending Stiffness or axial stiffness (Nm2
)
S and d Spacing of two sub-conductors in the bundle conductors (m)
P and Q Amplitude of the sinusoidal displacement (mm) and rotation (°), respectively
n and m Number of subconductors and spacer-dampers in the bundle conductors
mR
Mass of rigid block (kg)
kiC
Equivalent or result of left and right stiffness between two sub-conductors from either side of the
arm of the spacer-damper (Nm2
)
x,y, z, i, j, and k Reference coordinates
h Length of the rigid connection from one end to the centre of mass O (mm)
FRF Frequency response function (dB)
CLi Rigid clamp i
Ci Configuration of bundle conductors with i denoting the configuration code: in total ten
configurations were made
A and B Position of first spacer-damper and second spacer-damper from the rigid clamp (tension side),
respectively (m)

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78
is performed without any existing standard on the
indoor vibration of bundle conductors. Standards
on the condition and context of the set-up, testing,
measurements and analysis should be followed during
the indoor investigation. This validation may be
achieved by considering the physical impact of the
rigid connection dynamics of the bundle conductors
subjected to Aeolian vibrations. In signal processing, the
impact, known as interference in frequency response,
may be quantified from the experimental investigations.
Analytical methods based on experimental results are
used as direct measurements and calculations to resolve
the validation issue [4-13].
It is well known that indoor testing presents several
advantages over outdoor testing. It is less costly, time
saving, and more straightforward to move sensors
to different locations. If a structure is complex and
flexible, the input form (magnitude, phase, and position)
should be carefully chosen so that the obtained indoor
results effectively represent the actual results in such
way that there is insignificant discrepancy. As such,
the input energy has to be considered in the choice of
the excitation system. This depends on the position
of the shaker; for instance, the excitation of a flexible
structure at the position in which the damping is higher.
To this extent, the force and the velocity of the source
generates power instantaneously. However, the system,
b Position of the accelerometer on the rigid connection measured from its centre of mass (mm)
Vector product symbol
b,i
, b,j
, b,k
, Angular coordinates of displacement Yb
(°)
bi,E
, b,SD,i
and b,SD,j
Linear components of the vector receptance (m/N.sec)
xj
Displacement at position j of a sub-conductor with respect with x-axis (mm)
and Linear velocity (m/sec) and circular velocity (rad/sec)
ma
,mf
Mass arm and frame of spacer-damper (kg)
lk
Length of sub-span in a bundle conductor with k = n+1 (n denotes number of spacer-dampers) (m)
kiD
Equivalent stiffness in the rubber of the spacer-damper
Yb,i
, Yb,j
, Yb,k
Components of the vector displacement at any position in the bundle span in ( i, j, and k) (mm)
Mnn
Modal mass on a sub-conductor (kg)
Øbi
and ØDi
Respectively representing the rotation angle measured at any position i and position j of the
spacer-damper (°)
Vector force with the 3-D components according to the orthogonal axis i, j, and k (N)
1 Introduction
Mechanical oscillations of power line conductors
subjected to natural excitation, which leads to fatigue
failures, especially wind-induced motion, have been
studied for more than a century by engineers, companies,
and international institutions (e.g., CIGRE, EPRI and
IEEE). The main concerns of their investigations were
about the reliability and safety of the operating overhead
line, especially in terms of understanding the bending
stress mechanism at any attachment clamp where
damage to the conductors most often occurred [1-3]. To
investigate the behaviour of the conductors during wind-
induced Aeolian vibrations, outdoor and indoor tests
may be performed, and data of interest can be collected
and analysed accordingly. For the first excitation test
technique, the excitation is achieved naturally, and
on the actual structure (in situ). For the second test
technique, a shaker can be used. It is common to use an
electrodynamic shaker as a source of input force, where
the shaker is connected to a structure via an excitation
block. However, if the design is not suitable, this block
can modify the dynamics of the entire structure-shaker
system to a significant degree and affect the results.
The validation of the excitation technique becomes
important and indispensable for improving all aspects
of the data collection quality. The experimental work

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79
In such cases the cause-effect relationship or vice-versa
are determined in the frequency or time domain. It is
then convenient to evaluate the vibration performance of
a structure using the impedance response (as an effect),
or mobility response when evaluating the excitation
system (as a cause). The use of a mobility FRF response
to analyse or measure the angular vibration of a structure
is not new or recent [6-8, 13]. It has been used for many
years and often when appropriate sensors are lacking.
Angular vibrations are often generated by directing the
excitation moment [17-21].
In this work, the mobility FRF responses were
investigated at the excitation position in order to validate
the use of the indoor experimental test rig. The shaker
was placed at an identical position as prescribed by the
standards for single conductor indoor testing [22]. This
wascharacterisedbytheuseofthreedifferentcommercial
spacer-dampers attached as one and two on the span of
the model conductor. The bundle span was taken as a
black box: only the input and output were measured and
analysed at the excitation point, despite the control and
the observation of the experimental velocity (100 m/s)
being made at 20.125 m from the shaker location. In part
B of this paper, the fatigue and damping performances of
different bundle span configurations are presented.
2 Mathematical Modelling
of indoor bundle - excitation
system
2.1 General governing equation
A. Bundle conductors with one spacer
This study was carried out on a bundle conductor with
one spacer- and two spacer-dampers before being
extended into a general model with n and m numbers
of spacer-dampers and sub-conductors, starting from the
receptance approach as a governing equation in the form
of vectors. This describes the global receptance of the
system as being equal to the sum of all receptances in the
various sub-systems [23-26].
The significance of the receptance method is that, within
some limits, the full vector force is applied to the bundle
conductors (external and internal). The displacement
at any point resulting from those forces is taken as the
in response to the source, produces its own reciprocal
energy. Consequently, there is an interchange of energy
between source and system which, if positive, the source
is supplying power; if negative, the power has returned
to the source. The best way to understand this issue of
the position of the shaker is to call upon the mechanical
impedance notion of both shaker and system. The
driving-point was defined as the transfer-function
between the force and the velocity at the input point
(port), which was suggested as being lower in order to
generate an ideal motion source.
Over recent decades, modal analysis has been widely
used in structural dynamic investigation. This is instead
of using analytical models. This is due to a lack of
confidence in predicting the effective dynamic behaviour
of structures. Mechanical mobility is often used to predict
the behaviour of a structural assembly or its subsystem
from experimental or analytical measurements [14]. The
fundamental definition of mechanical mobility is either
an operational or a sinusoidal transfer function, in which
velocity is the numerator, and force the denominator. Its
inverse is termed the impedance. One of the advantages
presented by both mobility and impedance methods
is that the prediction of structural behaviour may be
conductedanalyticallyorexperimentallyonallindividual
subsystems before and after being connected to the entire
system. This way of analysing the dynamic response of
a structure is relevant. Correction or modification may
be made to save time and investment for such complex
structures as bridges, bundle conductors, and more.
Experimentally, excitation problems occur at the location
and in the model of the driving-input force, since the
form of force to be applied to a system requires careful
considerations of its physical parameters i.e., magnitude,
phase, and position.
Mechanical vibration problems are a complex physical
phenomena and often difficult to understand. They
require specialized tools and a skilled engineering
background for their investigation [15]. Often, the
purpose of vibration analysis is to: (i) find out their cause-
effect, (ii) evaluate the performance of the structure,
and (iii) monitor health. Therefore, it is necessary to
understand their mechanisms. Consequently, vibration
problems are often extended to a more comprehensive
form, such as an impedance or a mobility response [16].

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80
(3)
This implies, by the anti-commutativity of the cross
product, that:
(4)
The definition of the cross product also implies that the
product of the same vector axes gives zero vector:
(the zero vector) (5)
Applying the product of the Euclidean magnitudes of
two vectors, the force and receptance in (2) using the
vector properties given by expression (3), (4) and (5) can
be expressed as:
where Ybi
is the output displacement at any location of
the sub-span beside the position of the spacer-damper.
Applying the product of the Euclidean magnitudes of
two vectors – force and receptance at the spacer-damper
(between the excitation force and the arm of the spacer-
damper), (2) becomes:
function with a rotational frequency . To evaluate the
position of the excitation force in the bundle conductors,
(1)
where:
is the vector displacement
at 89 mm from the end-clamp;
is the vector force of excitation, and the
moment;
and
are, respectively, the receptance factors between the
excitation force and the vector displacement ;
is the dissipation force in the spacer-
damper;
denotes the vector product; and
is damping factor of spacer damper and the conductor
which depends on the mode of vibration and is discussed
further.
In (1), the first vector product term gives
Recalling the basics of vector algebra in which i, j, and k
are vector units and satisfy the following equalities in a
right-hand coordinate system:
Figure 1 Top view of quad bundled conductor bench with two spacer-dampers attached at positions A and B
Figure 2 Illustration of short-span bench of overhead line bundle conductors with two spacer-dampers
(2)
(6)
(7)

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81
the spatial and modal parameter of the sub-conductor in
the context represented by Figure 3. That is:
(13)
where n
is the modal damping factor , lk
denotes the
length of a subspan with the index k = n + 1; Mnn
is a
modal mass parameter; n
is the natural frequency, and
is a normal frequency. The expression in (13) does not
take into account the tension.
B. Bundle conductors with n sub-conductors and m
spacer-dampers
Importantly, the working range of the spacer-damper
is defined by the dependency of being in-phase or out-
of-phase with the excitation force on the coupled sub-
conductors. The governing equation or generalised
expression of n sub-conductors and m spacer-dampers
is given by:
(14)
where n is the number of sub-conductors, and m denotes
the number of spacer-damper positions, which is also the
index of the position parameter. The maximum existing
number of sub-conductors in the bundle is 12, and the
normal maximum span length of a conductor bundle is
500 to 600 m which carries on average 9 spacer-dampers.
2.2 Localised governing equation at the shaker
The shaker subjects the system to a vertical plane motion.
The centre of mass at the initial position is O1
while O2
, is
the position at time t, y is the displacement with respect
to the y-axis, and the rotational motion is described
The angle of rotation of the bundle conductors at the
chosen position can be determined using the system (6)
and (7) as the angle measured respectively at bending or
damper position:
- at any output position:
(8)
- and the spacer-damper position:
(9)
where s denotes the spacing between two consecutive
sub-conductors
From (8) and (9), the force generated by the arm of the
spacer-damper be expressed as:
(10)
and,
(11)
Alternatively, the force induced by the arm of the spacer-
damper may be written as in [27], and the dynamics of
the spacer-damper is displayed in Figure 3.
The harmonic damping force induced by the spacer-
damper at each arm is given by:
(12)
The governing equation of the indoor bench of quad
bundle conductors is given by a system of parametric
expressions: equation (6) is associated with the rotational
equation of the mode of vibrations in (8). The parameter
receptance at any point of a sub-span of the bundle
conductors in (6) has as alternative expression based on
Figure 3 Dynamic modelling of quad spacer-damper attached to bundle conductors

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82
The vertical and the rotational equation of motion with
respect to y-axis and the centre of mass can be expressed
as:
(15)
and
(16)
Since the vibration is in the range of small amplitude,
then:
sin (17)
around the centre of mass Oi
as the rigid connection is
moving along the y-axis. Since there is a rotation, the
moment of inertia is defined as , which depends on the
mass and the geometry of the rigid connection. Each
sub-conductor is represented by two damping elements
kx
and ky
, which are arranged in vertical and horizontal
directions, respectively. Both horizontal and vertical
consecutive spacings of a sub-conductor has an equal
distance (s). All details defined above are described in
Figures 4 and 5.
Figure 4 Mechanical modelling of the bundle conductor-shaker system
Figure 5 Illustration and mechanical modelling of bundle conductor-shaker system

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83
elements kx
and ky
representing each sub-conductor
may be taken as pure spring elements when the rigid
connection is located close to a clamp where the sub-
conductor is stiffer. This means that the viscous-
damping elements kx
and ky
become kx
= kx
* (1+i ) and
ky
= ky
* (1+i ) , with kx
* , ky
* and denoting the stiffness
components and damping factor, respectively. This
occurs, for instance when the rigid connection is moved
more towards the mid-span of the bundle conductors, the
damping factor tends towards its maximum.
3 FRF Mobility Response of
Bundle Conductors from the
Excitation Block
3.1 Analysis of the excitation block
The excitation of bundle conductors is unusual with
regards to the position of the shaker on an indoor test
bench. It is located close to the rigid clamp [28] and at
the middle of the span [29]. It is essential to analyse the
excitation system used for this investigation (Figures
6-A and 6-B). Two approaches can be used to assess the
response of the bundle conductors; owing to a range of
harmonic forces, either the impedance or the mobility
methods can be used. Since the force impartation
via a rigid excitation system is of importance, the
mobility approach is the method best suited compared
with impedance in terms of physical significance. The
main objective of this section is to assess whether the
vibration mode of the rigid connection interferes with
the results or is insignificant. If these interferences are
Considering the latter expression, re-arranging (15) and
(16) become, respectively:
(18)
and
(19)
Considering that y and are induced by sinusoidal force,
they can be given by:
(20)
and
(21)
Putting (20) and (21) into (18) and (19) gives:
(22)
and
(23)
To determine the frequency equation, the determinant of
the considered system (22) and (23) with coefficients P
and Q has to be taken as equal to zero. After arranging
different terms, the frequency becomes:
(24)
Eq.(24) is the general expression of the normal vibration
modes given by two real and two complex solutions for
Figure 6 Illustration of dynamic force/moment applied to the rig-quad bundled conductor system and excitation block moment of inertia calculus

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84
3.2 Excitation block reduced model
Experimentally, mobility measurements are analysed
and discussed on the single shaker via two attachment
connection points for various bundle configuration
scenarios given in Section 2.2. The force imparted
by the shaker is measured by two force transducers
LE1 (FA
) and LE2 (FB
) as a response (as a reaction force)
received from the total force of the bundled conductor.
The excitation system may be modelled as illustrated in
Figure 7 whereby two main parts are given: (i) the JIG
and (ii) the dynamic bar. The JIG consists of the base in
the squared plate where two parallel force-transducers
are mounted that connect the squared plate to the rigid
connection. However, a dynamic bar is a reduced
model of the rigid connection in which all the resultant
forces and moments acting on it; these are dynamically
represented. As such, those applied forces on the
excitation system are two forces given respectively,
by force at left side F(L)
, and at right side F(R)
. From
these forces, two associated moments were calculated
by a vector product between them and the distance d
(Figure 7). Arbitrarily, the positive sign was considered
for the left-side moment. The actual configuration of
the system excitation leads to so-called two-directional
mobility measurements deriving from external forces
and moments. The analysis of the rigid connection
corresponds with another type in previous research [4,
5, 8, 12]. This differs from this current work in that the
excitation force comes from the top. The force in both
cases was measured as the reaction force at two points
(force transducers) between the shaker table and the bar
of rigid connection.
For the two input reference forces A and B, the matrix
mobility response as well as the transfer matrix gives
different points, as described in Figure 6. Based on the
definition of the mobility response, and referring to the
illustration, the direct measurement scenario is given in
force and moment. Using the direct measurement, the
point mobility with two reference inputs (excitation)
and outputs (readings) is expressed as follows:
(27)
not insignificant, there is a possibility of comparing
them in the discussion section. Four different rigid
connections were developed in working towards the
ideal design. Various investigations were conducted
into the effect of accelerometer cross-sensitivity along
with an interaction between force and acceleration
sensors [30]. Those prototypes are all characterised by
the rectangular frame rigid connection with rectangular
hollows in the steel or aluminium bars. However,
the rigid frame connection with a rectangular hollow
section in steel was selected because it showed the most
encouraging results from all designed rigid connections.
Flexible connections are, however, not options. Previous
research [8, 12] concluded that they are a source of error
in the component data. Figures 6-B, 6-C and 6-D are
illustrations of the system excitation-quad-bundled
conductor characterised by the dynamics of forces and
moments. The forces and moments are respectively
given by even and odd numbers (0, 2, 4, 6 and 8) and
(1, 3, 5, 7 and 9) in Figure 6-C.
Table I Mobility and transfer matrices associated with the force dynamic of a
shaker input and the data output collected from the excitation block
Let the force and moment be analysed as illustrated in
Figures 6-C and 6-D by the expressions in (15) and (16).
Based on the data of the rigid connection structure, the
dynamic equations may be expressed as
(25)
and
(26)

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85
3.4 Sensitivity Analysis of Accuracy and Source of
Errors
A sensitivity analysis was performed on the expression
in (28) in order to evaluate the errors made by using the
mobility FRF response to measure the angular vibration
[31-34]. The excitation block is mainly characterised
by its mass, as well as its moment of inertia, which is
given by the product between its mass and the radius of
gyration. Sensitivity analysis is undertaken in order to
assess whether the mass of the rigid connection mR
or its
moment of inertia p interferes with the FRF response of
the bundle conductors. If this is the case, the sensitivity
indicates which, of the mass mR
or the moment of inertia
p, to optimise. The moment of inertia may be modified
by changing the geometry of the excitation block.
4. Experimental and Numerical
Method
4.1 Experimental description
The test bench consisted of four sub-conductors (ACSR
tern conductor) stranded and rigidly clamped. These
sub-conductors were numbered from 1 to 4. A load-cell
was connected to conductor 2, where the tension was
20 360 N. In Figure 4 (left) two force transducers, LE1
and LE2, were mounted between the shaker and the rigid
connection. These transducers had the same sensitivity
(load cell-excitation) when connected to the DAQ
system, thus allowing for the measurement of the induced
force, as well as the reactive force by the shaker to the
bundled conductors. It needed to be verified whether or
not the shaker was well balanced. Two accelerometers,
AE1 and AE2, were placed on the top-most part of
the rigid connections and connected the main DAQ.
Four accelerometers were placed between the shaker
and the spacer-damper position at the mid-span of the
bench; one accelerometer per sub-conductor, and at any
identified anti-node point for a frequency of interest.
The mobility and transfer matrices associated with the
force dynamic of the shaker input and data output are
given in Table I.
3.3Determinationoftheoreticalindirectmeasurement
The point mobility matrix of the excitation system
in this context, taking into account the mass mR
and
the global moment of inertia Ip
with respect to point
P (Figure 4), may be expressed as that developed in
references [4, 6, 8, 12]:
(28)
with the linear velocity expressed by
(29)
The angular acceleration is determined from
(30)
where
AL
and AR
are the accelerometers at the left and right
sides respectively and are measured on top of the bar at
the position of each force-transducer in (Figure 6) in m/
sec2
;
is the angular frequency given in rad/sec;
b is the distance between the orthogonal axis passing by
point p and an accelerometer position in mr
is the total
mass of the excitation block; and
Ip
is the moment of inertia of the excitation system.
Figure 7 Reduced model of the quad excitation system

10. ฀ ฀ ฀ ฀ ฀ ฀ ฀
86
spacer-damper attached at the mid-span. Figures 8-D and
8-E are the images of the excitation system. Figures 8-F
and 8-H display the instrumentation (force transducers
and accelerometers) located at the excitation.
Various configurations of quad-bundled conductors
were explored,: (i) without any spacer-damper attached
(Figure 9-A); (ii) with one spacer-damper at the mid-
span (Figure 9-B); (iii) with two spacer-dampers: one at
midspan, and the other at 56.5 m from the shaker (Figure
9-C);and(iv)twosimilarto(i)butthefirstspacer-damper
located at 30.25 m (Figure 9-D). In Figure 9-A, a is the
distance between the rigid clamp (tension side) and the
shaker connection is about 1 m; and b is the span length
(straight line) which is 84.5 m. Figure 9-B is described
by two equal sub-spans separated by one spacer-damper
at the mid-span (42.25 m): sub-span b and sub-span c.
Three sub-spans: b, c, and d, may be observed in Figures
9-C and 9-D. In the first configuration (Figure 9-C), b
This meant that the bundle conductor was vibrating first
at the frequency of interest. The anti-nodes of interest
were found using four accelerometers AFi,which were
moved and located at the antinodes (AFi: accelerometer
at the free span with i = 1, 2, 3, 4).
Other relevant test parameters were the sample rate NS
and sample size: these were set at 2600 samples per
second for 2600 samples. Virtual low-band filtering was
incorporated in the data-acquisition scheme to suppress
unnecessary noisy signals from the data collection. To
ensure consistent filtering, an automated condition was
set up in such way that each time the frequency fi
was
tuned, the cut-off frequency was automatically adjusted.
The data was captured when the average velocity of
0.1 m/s at a given frequency was reached at the control
points (free span). Figures 8-A and 8-B show the test
bench view taken from the shaker position and from the
loading arm position. Figures 8-C is an image of the
Figure 8 (A) Excitation of bundled conductors using a rigid connection attached at 1 m from the clamps: two excitation points
given by two force transducers; (B) and (C) are the force transducers and accelerometers at the excitation (rigid connection)

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87
4.2 Finite element analysis method inaccuracies and
uncertainties in evaluation of the excitation block
The FEA analysis was fixed in three different stages: the
pre-processing, the processing and the post-processing
stages. In the first stage, the model and the finite element
model that includes the meshing, material assignment
properties, and the boundary conditions, were developed
using the Ansys modeller. The boundary conditions
were defined for the elastic supports representing the
sub-conductors attached at the excitation block. The
elastic supports were characterized by the stiffness and
damping factors of each sub-conductors, and because
the shaker was placed at a short distance from the rigid
clamps, the damping factor was not taken into account.
Due to the complexity of its various components, it
is difficult to determine the moment of inertia of the
excitation block through calculus. A finite element
analysis approach was taken, using theANSYS package,
inordertoaccessthevariousmomentsofinertianecessary
in (28), as defined in Section 3.2. However, FEAanalyses
may result in some inaccuracies or uncertainties, e.g.
due to the choice of 3D design geometry (excluding
insignificant details of individual parts). Consequently,
the moment of inertia is characterised by its weight
and centre of gravity (radius). Errors in two elements
result in a significant error that may be corrected by
standard deviation between the measured (actual) mass
and the one obtained in ANSYS. The error indicator as
a percentage difference between the mass measured and
the mass in ANSYS was determined using
(31)
where, mM
is the mass measured
mA
is the mass determined in ANSYS
Equation (21) found that this is important and it is
addressed in the discussion section when the data results
from the simulations are compared with the direct
measurements.
is equal to 42.25 m, c is 24.25 m, and d is 28 m, while
in the second configuration (Figure 9-D) b is 31.125 m,
c is 25.375 m, and d is 28 m. The indoor span length
is about 85 m and it falls under the recommendation of
using only one spacer-damper to be attached at its mid-
span. Despite the recommendation, to give the minimum
length of the sub-span for this investigation then two
spacer-dampers were used as illustrated in Figures 9-C
and 9-D. The latter has as a practical objective, the
investigation of the participation factor of the spacer-
damper in the bundle conductors.
Three different types of quad spacer-dampers, SD1,
SD2, and SD3, were used for the experimental work.
The differences were given by the design aspect,i.e.
shape, weight, and damping mechanism. The damping
was the result of friction, either at the microscopic or
macroscopic level; this demonstrated the difference in
the selected spacer-dampers given by the rubber bushing
type. As such, the rubber bushing in SD1 and SD2 was
softer than the one used in SD3 (microscopic level),
whilst, even though the geometrical shape of the rubber
in SD1 and SD3 was the same (ellipsoidal), the friction
surface (macroscopic level) in the SD1 was larger than
in SD3. Specifications of different spacer-dampers used
in this investigation: SD1, SD2 and SD3 were discussed
in [35] and are given in Table II.
Table II Physical characteristics of spacer-dampers SD1, SD2, and SD3
Figure 9 Lateral view of various investigated indoor bundle span configurations: A) is a configuration without any spacer-
damper, B) has one spacer-damper attached at mid-span, C) has two spacer-dampers, respectively, attached at the mid-span
and 55.5m from the shaker, and D) is a configuration with two spacer-dampers, respectively, attached at 30.25 m and 55.5 m
from the shaker

12. ฀ ฀ ฀ ฀ ฀ ฀ ฀
88
Figure 10 Illustration of the Finite elements analysis (FEA) of the rigid connection at the frequency=60Hz corresponding to
the highest input-forces obtained (F1=136 N and F2=184 N)for the bundle with the configuration C2 (one SD2 attached at the
mid-span) i.e., A) Rigid connection Model, B) simple discretization Model, C) displacement with respect to x-axis (mm), D)
displacement with respect to y-axis (mm), E) rotation with respect to the z-axis (°), F) Tensile force (N), G) Shear Force V (N),
H) Bending Moment about V, I) longitudinal Stress at point 1(MPa), and displacement Magnitude (mm)

13. ฀ ฀ ฀ ฀ ฀ ฀ ฀
89
5.2 Assessment of the excitation block design via
direct FRF measurements
The point mobility of the excitation system with respect
to the centre P of the shaker table was evaluated for the
system alone, then compared with various configurations
of quad-bundle conductors (from C0 to C9). The
expression in (27) was used in this evaluation as a
numerical parameter of the excitation system. The total
mass, mR, and the moment of inertia, IP, are respectively
about 4.380 kg and 0.122 kg.m4. The moment of inertia
was determined both analytical and numerically using
the ANSYS package, which gave similar values. Table
III gives the physical parameters, i.e. moment of inertia,
and real mass (weighted). Figures 11-A, 11-B, 11-C and
11-D show different mobility matrix elements against
frequency obtained from various configurations tested.
These were compared with the results measured with the
rigid connection alone. Overall, the FRF responses of all
the configurations are closer to zero in magnitude for all
the measured frequency ranges than those with the rigid
connection alone, with some exceptions.
The difference in the results obtained on the rigid
connection alone is justified by the nature of the material
used for its design. However, the vibration of a steel
member depends on these factors:
(i) the stiffness (second moment of inertia): if it is stiffer
that implies that its resonance frequency is higher;
(ii) mass per length (kg.m-1
), if heavier = lower frequency;
(iii)length (m), if it is longer, lower in frequency;
(iv)the Young modulus (elastic strength, Pascal) is
stronger, i.e., higher frequency; and
(v) the method by which the beam is supported; a simple
support is fairly loose, and therefore lower in
frequency.
5 Results and Discussions
simulating rigid connection performance
The explicit dynamics finite element analysis using the
ANSYS and LISA packages for the excitation was used
to improve design performance in terms of preventing
permanent deformation or failure that may be caused
by an excessive excitation force if the excitation block
was not well designed. With two input forces at the rigid
connection, failures often begin at that weak point if not
well designed. The input forces at the rigid connection
were set for a short period of time at fixed frequency.
Figures 10-A and 10-B represent respectively the actual
and 1-D model with two input forces as well as the elastic
support at the sub-conductor location (represented here
by a node). The horizontal displacement towards the
x-axis is below 1 mm while the vertical displacement
about y-axis is equal to 8 mm which corresponds to
the frequency 60.5 Hz as shown in Figure 10-C. The
rotation with respect to z-axis of the rigid frame is given
in Figure 10. The focus of this analysis was based on the
design failure performance characterized by the fatigue
indicators: tensile force (strength), shear force, bending
moment and the longitudinal stress, respectively given
in Figures 10-F, 10-G, 10-H and 10-I. The fatigue limit,
endurance limit, and fatigue strength were used to
describe the amplitude (or range) of cyclic stress that
could be applied to the material without causing fatigue
failure. It remains too small compared to the result of the
limit of the steel used here = 340 MPa). The negative
tensile sign was an indication that the rigid block was
subject to compression.
Table III Moment of inertia, real mass and ANSYS mass of excitation block

14. ฀ ฀ ฀ ฀ ฀ ฀ ฀
90
mobility responses Yxx
and Yx
were significantly greater
than Y x
and Y . Figures 12-A, 12-B, 12-C and 12-D
represent the FRF mobility response of the configuration
C0 of the bundle conductors without any spacer-damper
attached. Each sub-conductor behaved as a simple
excited conductor whereas the control sensors were not
located at the mid-span but instead on the free span at
20.125 m from the shaker. Figures 13, 14, and 15 show
respectively the FRF mobility of configurations C1, C2
and C3, with one spacer-damper attached at their mid-
span. The parameters of the initial structure, considered
here as C0 (bundle without any spacer-damper attached),
were improved in overall weight, in sub-conductor
tension, and perhaps, the damping performance, for
one case (spacer-damper in phase with the shaker). The
experimental and theoretical FRF mobility responses of
configurations C1, C2 and C3 correlated better than the
responses from the configuration C0.
Figures 16, 17 and 18 give the FRF mobility responses
of configurations C4, C5, and C6 characterised by
the position of the first and second spacer-dampers,
respectively, at 41.5 m (mid-span) and 55.5 m from
the excitation position. The second spacer-damper
behaved in counter balance to the shaker. The overall
weight of the system C1, C2, and C3 was improved
with the weight of the spacer-damper according to each
configuration. The bench was characterised by 3 sub-
spans; the first sub-span between the shaker and the first
spacer-damper was much longer (42.5 m) than the two
others (32.5 m each). This method of placing spacer-
dampers on the bench generated different tensions in
The most influential among the listed factors is
the length of the structure on which the resonance
frequency depends. The expression below gives the
angular resonance frequency. From the direct FRF
mobility responses expressed in linear units, not much
evidence could be extracted besides the interference
noticed between the response of CR-C
(stand-alone
rigid connection) and all other configurations. These
interferences were observed at frequencies of 17.5 and 40
Hz in response to plots Yxx
, Y x
and Y respectively given
in Figures 11-A, 11-B, and 11-D, then only 40 Hz in Yx
(Figure 11-B). All response plots appeared significantly
lower compared to the FRF mobility responses in the
stand-alone testing of the rigid connection CR-C
. This was
understandable since the rigid connection was vibrating
even when unloaded.
of bundle conductors
The experimental and theoretical mobility FRF
response to the translational mobility Yxx
, semi-
translational-rotational mobility Yx
, semi-rotational-
translational mobility Yx
, and rotational mobility Y ,
were evaluated for different configurations of quad-
bundle conductors. The various mode shapes measured
on the rigid connection, were insignificant, and for
that reason, they were expressed as dB of the real part
response. In general, the normal vibration modes of the
bundle conductors were distinguished by the bending,
breathing, and torsional modes, which depend mainly
on the wind velocity. The range of Aeolian vibrations
was situated between 1 and 7 m/s. The two first FRF
Figure 11 Comparison of mechanical mobility between unconnected rigid connection and connection in various
configurations, as defined

15. ฀ ฀ ฀ ฀ ฀ ฀ ฀
91
Figure 12 Direct and analytical measurement of mobility characteristics of quad bundle conductors in configuration C0
Figure 13 Direct and analytical measurement of mobility characteristics of quad bundle conductors in configuration C1
Figure 14 Direct and analytical measurement of mobility characteristics of quad bundle conductors in configuration C2
Figure 15 Direct and analytical measurement of mobility characteristics of quad bundle conductors in configuration C3
Figure 16 Direct and analytical measurement of mobility characteristics of quad bundle conductors in configuration C4

16. ฀ ฀ ฀ ฀ ฀ ฀ ฀
92
Figure 17 Direct and analytical measurement of mobility characteristics of quad bundle conductors in configuration C5
Figure 18 Direct and analytical measurement of mobility characteristics of quad bundle conductors in configuration C6
Figure 19 Direct and analytical measurement of mobility characteristics of quad bundle conductors in configuration C7
Figure 20 Direct and analytical measurement of mobility characteristics of quad bundle conductors in configuration C8
Figure 21 Direct and analytical measurement of mobility characteristics of quad bundle conductors in configuration C9

17. ฀ ฀ ฀ ฀ ฀ ฀ ฀
93
mobility responses (dB) against frequency (Hz) for the
various configurations given in Figures 6-A and 9. For
configuration C7, the number of vibration modes in C0
was less or equal to other configurations investigated. In
practice, the vibration modes should be greater in a bundle
conductor with a spacer-damper than in one without
a spacer-damper. This indicated that possibly, some
vibration modes were missing or found at the frequencies
outside the experimental frequency range of 5 to 60 Hz.
5.5. Cross-spectral (Modal assurance criteria)
analysis and accuracy of the results
To evaluate the validity of the results obtained in the
previous section, the need for response similarity
or concordance analysis arises for the various curve
responses obtained from both the theoretical and the
experimental approaches [36]. This is a type of modal
assurance criterion, comparing the experimental and
the analytical or numerical data [37, 38]. The vibration
modes of the various approaches were compared in order
to evaluate how similar and consistent the theoretical and
experimental results were in this context, i.e., Yxx
, Y , Y ,
and Y
on the x-axis (frequency) and on the y-axis (magnitude),
giving perfect similarity when the frequency and
magnitude of the two curves corresponded. This was
givenasaratiobetween0and1toexpressthesignificance
of the similarities which were defined as non-significant
for a value below 0.5 and significant for values above
0.5. Tables V, VI, VII and VIII give a different spectrum
analysis of the various bundle configurations defined
those diverse sub-spans, and were certainly sources of
additional vibration modes. In general, the gap between
the experimental and theoretical FRF mobility responses
were insignificant and improved, compared with those
from the previous configurations.
Figures 19, 20, and 21 represent the configurations
C7, C8, and C9 respectively, the first and second were
located at 31.5 and 55.5 m from the shaker. The tensions
in different sub-spans were approximately the same;
there was not much difference in the length of their sub-
spans. Some particulars were observed in configuration
C7 (Figure 19), which gave only a few distinguished
mode shapes (2 modes). This could be explained as the
result of the weight of the spacer-damper SD1 compared
with two others, and its location close to the shaker. This
was the difference of the configuration C4. A significant
discrepancy could be observed in the rotational mobility
Y responses between the experiment and the theory of
C7 and C9, when this was compared with the previous
configuration.
5.4 Vibration modes of different FRF mobility
responses
By measuring the FRF mobility response from one bar
of the excitation block, three types of vibration were
identified: rigid, bending, and torsional modes. It was
clear that some missing modes could be included in the
collection of the data responses; or again, the missing
modes could be too insignificant to be detected by the
naked eye. Table IV gives numbers of various vibration
modes measured at the rigid connection as FRF
Table IV Number of different vibration modes of quad-bundle conductors subjected to Aeolian
vibrations; the value ( ), however, represents the theoretical value.
The shaker is placed at 1 m from the clamp side of the loading arm

18. ฀ ฀ ฀ ฀ ฀ ฀ ฀
94
taken into account. The sensitivity approach in (18) and
(19) was based on the overall errors of 3.74 % given in
Table II which indicate that the values of indirect results
have to be adjusted to the actual value of about 3 %.
6. Conclusion
This paper gives an experimental and numerical
analysis to validate the excitation technique used
for the indoor vibrations of bundle conductors. The
results show that there was a good agreement when
comparing the experimental and the theoretical results,
with the exception of the rotational mobility. The latter
indicated that the moment of inertia of the excitation
by the correlation between the direct and indirect
measurements. The analysis of the spectral plots showed
that there was no evidence of auto-correlation observed
in all the plots given. In general, there was significant
correlation between the experimental and the theoretical
FRF mobility responses, with the exception observed for
the correlation evaluation of Y of the configuration C0
which gave 0.33 as the ratio.
Uncertainty in the correlation assessments owing
to errors in designing of the block used in indirect
measurement (theoretical) can make a huge impact if not
Table V Cross-spectrum analysis between experimental and theoretical FRF mobility of various arrangements of bundle conductors
without spacer-damper C0, and with one spacer-damper placed at 41.25 (mid-span) of the shaker: C1 (SD1), C2 (SD2), and C3 (SD3)
Table VI Cross-spectrum analysis between experimental and theoretical FRF mobility of various arrangements of bundle conductors,
with two spacer-dampers placed, respectively, at 31.25, and 56.5 m from the shaker: C7 (SD1s), C8 (SD2s), and C9 (SD3s)

19. ฀ ฀ ฀ ฀ ฀ ฀ ฀
95
Table VII Cross-spectrum analysis between experimental and theoretical FRF mobility of various arrangements of bundle
conductors, with two spacer-dampers placed, respectively, at 31.25, and 56.5 m from the shaker: C7 (SD1s), C8 (SD2s), and
C9 (SD3s)
Table VIII. Experimental and theoretical cross-spectrum analysis of various arrangements of bundle conductors, with two spacer-
dampers placed, respectively, at 31.25 and 56.5 m from the shaker: C7 (SD1s), C8 (SD2s), and C9 (SD3s)

20. ฀ ฀ ฀ ฀ ฀ ฀ ฀
96
where {Q} is the column vector of response amplitudes;
{F} is a column vector of input force amplitudes;
r
is the rth
eigenvector;
{ r
} is the eigenvector of the rth
mode; and
mr
is the mass of the rth
mode.
It is particularly interesting to look at the response Qi
due to the single excitation Fj
:
(A.2)
and further, by dividing each side of (A.1) by the force Fj
:
(A.3)
If two responses are considered (X = Q1
, = Q2
),
their related two forces, i.e., Force (F) and Moment
(M) include the energy of the three types of motion
experienced: rotation, vibrations and the physical
movement from one point to another. This is obtained
by using a system called translation, and a bundle of
conductors experiencing Aeolian vibrations. Assuming
that the system is being considered continuous and the
ranges of finite number of modes need to be taken into
account at low frequencies, it can be written:
(A.4)
(A.5)
(A.6)
(A.7)
where Ar
= r1
2
mr
-1
; Br
= r2 r1
mr
-1
; Cr
= r2 r1
mr
-1
; and
Dr
= r2
2
mr
-1
Applying symmetry principle, we can say Br
=Cr
and
Dr
=Cr
2
Ar
It is necessary to study the geometry of the motion to
elaborate on the correction needed for the excitation
block, which has to be in the limit of mass and the
moment of inertia when interfering with the vibration
responses of the conductor bundle.
block must be improved, so that the results obtained
may be updated using the error difference between the
experimental and the theoretical measurements. The
number and the position of the spacer-damper in the
span were determined by the FRF mobility response,
and can be further explored in order to develop a
condition monitoring device. The interferences between
the motions of both the rigid connection and the bundle
conductors have a minor impact on the overall results
since they affect only a few frequencies. Further analysis
of several anti-resonance frequencies of the elaborated
FRF mobility responses could be validated the findings.
Such interferences could also be observed by comparing
direct and indirect FRF responses, as developed in [39].
The analysis in this work was conducted by considering
the system as a black box and evaluating only the
excitation force (input), the amplitude (output), and
the control point location, despite the theory on the
governing equation being developed in Section 2. For the
latter, a more explicit model could be determined. This
would take into account the current investigation and the
sophisticated model of the spacer-damper, conductor and
shaker, which would properly characterise the bundle
conductors.
7. Acknowledgments
This research was proposed by Professor Konstantin
O. Papailiou. The authors would like to acknowledge the
support the Electrical Power Research Institute (EPRI-
USA), The Eskom Power Plant Engineering Institute
(EPPEI) Specialisation Centre for HVDC and FACTS,
Pfisterer (South Africa), and African Cable (South
Africa).
8. Appendix
A. Determination of theoretical indirect measurement
(28) developed from the direct measurement (27)
The theoretical response of a n-degree of freedom
system subjected to a steady-state sinusoidal excitation
at an angular frequency can be represented using
generalized coordinates (q1
, q2
,…,qn
) as shown:
(A.1)

21. ฀ ฀ ฀ ฀ ฀ ฀ ฀
97
and
(A.13)
The denominator element of the matrix response can be
defined by
(A.14)
In order to extract the vertical for the reduced system
(Figure 7), let only the left force F(L) be active and the
right zero and similarly the right force F(R) was run to
obtain to response owing to the right force system
(A.15)
Now introduce the receptance matrix [Y] from the direct
measurement and replacing in (A.7) and (A.9) so that
(A.16)
where is the matrix transformation and
is the excitation block inertial parameter. The linear
and angular accelerations are respectively given by
, and , which are required to obtain
the mechanical mobility.
Based on Figure A-1 which illustrates the motion of the
system, let it be assumed that two motions are produced:
i) only rotation at centre of mass O1
from pure couple M,
and ii) both rotation and translation of O1.
Figure A-1 Analysis of couple and forces applied on the rigid connection
When the force generates the pure rotation on the angular
response, the couple M can be written as follows:
(A.8)
By bringing the length the expression of the angle intern
ofthedistancebetweenthecentreofmassandthefulcrum
of the excitation force, and also let it be underlined here
that the vibration remains in the small displacement
ranges. Then the couple M can be expressed as:
(A.9)
Analysing the force that produces the combined motion
in both translation and rotation, the displacements xa
and
xp
are
xa
= d + xp
(A.10)
and
xp
= xa
– d (A.11)
Thus, using the first of Newton’s laws and by exploring
the relationship of the torque, which is equal to the
couple, the reduced expressions are written as
(A.12)

22. ฀ ฀ ฀ ฀ ฀ ฀ ฀
98
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