CC BY
46Electronic Journal «Technical Acoustics» http://webcenter.ru/~eeaa/ejta/
2004, 13
Sayah Houari1, Mostefa Rahli2, Ahmed Massoum3
lICEPS Laboratory (Power), Sidi Bel Abbes University, Algeria, e-mail: housayah@yahoo.fr 2Oran University (USTO), Algeria, e-mail: rahlim@yahoo.fr 3ICEPS Laboratory (Power), Sidi Bel Abbes University, Algeria, e-mail: ahmassoum@yahoo.fr
Aeolian vibrations of the overhead cables
Received 18.05.2004, published 01.10.2004
Wind induced vibrations (Aeolian vibrations) of overhead transmission lines and resulting conductors fatigue breakages constitute a problem for utility companies in maintaining uninterrupted supply of electrical energy. The aim is the study of Aeolian vibrations of overhead transmission lines conductors. The fractures caused by the wind vibrations often appear in front of the points of attachment, and with the junctions of the conductors. To this end one tried to give a mathematical model
of the forces acting on the cables, as well as a mathematical model of the cable.
This model permits to have curves of the amplitude according to the speed of the wind and the frequency. A simplified analysis of the partial differential equation describing these vibrations is presented using the concept of characteristic length as a single quantity describing the mechanical properties of a conductor.
INTRODUCTION
This work relates to the study of the dynamic behavior of the aerial lines subjected to the swirls detachment phenomenon. The action of the wind on a conductor involves various problems which are influenced by various factors, like the characteristics of the line, the geographical position, and the type of wind incident, for example:
- the phenomenon of the wind space-time turbulence,
- Aeolian vibrations, due to the alternative force which is generated on the conductor for
frequency range 5 - 100 Hz,
- vibrations related to the unstable form aerodynamics (galloping).
The problems related to the wind action on a conductor are dynamics; they are vibrations which can cause a rupture by the fatigue of the cable. To be able to estimate, with precision, the demand level and to possibly equip the conductor of appropriate equipment permits to reduce considerably the management costs of a high tension line. This can be done by avoiding the rupture of the conductors and the interruptions of the service which follow from there. Two approaches are mainly used to simulate the behavior of the cables subjected to the vibrations Aeolian:
- powers assessment, in the frequencies domain,
- finite elements, in the time domain.
This paper presents a model based on a modal approach concerning the schematization of the cable. This method permits to keep the advantages relating to an approach in the time domain and at the same time to reduce the computing times compared to finite elements model [1, 2].
1. THE SWIRLS DETACHMENT
Let’s consider a cylinder in a uniform viscous flow (fig. 1); the behavior of the air flowing around a smooth circular cylinder varies according to the value of Reynolds number:
Re = —, (1)
u
where Vis the fluid speed [m/s], D is the conductor diameter [m], u is the kinematic viscosity [m2/s].
Re<5 — undisturbed flow
40<Re<150 — laminar vortex street
150<Re<300 — transition to turbulence vortex
Fig. 1. Modes of fluid flows around a cylinder
Fig. 2. Swirls separation
For Re>40 this pair of vortex is unstable when it is exposed to small disturbances. The largest - vortex A - becomes enough strong to pull the vortex opposed across the wake fig. 2a, the vorticity from vortex A is in the clockwise direction, that of the vortex B is counterclockwise direction. The vorticity approach of opposite sign will cancel any supply of vorticity to vortex A, which comes from its boundary layer. It is at this time that vortex A falls apart. Following the separation of vortex A, a new vortex C will be formed next to the cylinder (fig. 2b).
2. MATHEMATIC MODEL OF FORCES
The object is to carry out an analysis of the phenomenon in the time domain, by carrying out an integration of the motion equations at every moment. It thus proves necessary to quantify the force transmitted to the structure by the fluid, at every moment of integration, according to the speed of the wind and other variables of the system [1, 3, 4]. This model must represent the resultant of the forces acting on the cables according to the flow as well as position and speed of each section of the cable: F = (v, X, X).
2.1. The equivalent oscillator
Let’s consider a rigid, free cylinder to oscillate in its plan vertical and subjected to a laminar flow. The equation to solve is
my(t) + ry (t) + ky(t) = F (t). (2)
That is to say a section of the conductor subjected to the swirls detachment; it can be represented by a rigid cylinder connected to the ground by a system of spring and damper which reproduces the structural characteristics (fig. 3). In the figure mcyl is the mass of the
cylinder of diameter D and length 1 meter, K st is the elastic structural constant, Rst is the damping structural constant, M aer is the aerodynamic mass which is connected to the ground by the nonlinear elements Kaer and Raer respectively elastic and of damping, and to the cylinder section through the nonlinear elements Kacc and Racc .
Fig. 3. Equivalent oscillator applied to a rigid cylinder
One can consider the component of the aerodynamic load acting on the cylinder according to direction X. This force will be expressed according to the forces of bearing pressure FL and trail FD due to the field fluido-dynamics according to the relation (fig. 4)
Fx = FL cos 9-FD sin 9.
(3)
If one expresses FL and FD according to the cylinder characteristics and the flow and under the assumption of small displacement, the force Fx is obtained as
Fx =PV2 LD —
2
D
H -— K D
sin (ft# + 9)- P V2 LD — CD
V
(4)
where V is the absolute velocity of the flow compared to the body; L is the length of the cylinder; H, K are the terms of quadratic dependence of the force compared to the amplitude of vibration; CD is the trail coefficient; —,X are the position and speed of the cylinder mass center.
It is thus noticed that the force is nonlinearly proportional to the amplitude of the cylinder variation through phase and quadratic components.
2.2. Choice of the oscillator parameters
The nonlinear part of the phenomenon is taken into account by springs and dampers which introduce forces that depend, in a cubic way, on displacements and relative speed. Equations of two systems motion taken into account are [5]:
mcilx - Rstx - KstX - K'acc (x-n)+ K'acc (x-^ - Kcc (X-'H) + Racc (x-11)3 = 0
for the cylinder, and
■ maern - Raer " + Koer^ - Kaer" + Ker ^ + Kcc (x - *1)- Kcc (x - *1)3 + Kacc (x - ^ ’
.X. - "fl P
acc '
- Kcc(x - n f = 0
for the aerodynamic cylinder.
(5)
(6)
The constant values are defined as follow:
Ka cc = AAK lacc—’ Ra cc = AAR
Kacc = AAK2acc D^l5 Kcc = AAR
Ke rr = AAK Ra rr = AAR!
Kerr = AAK 2aer^, Raer = AAR 2
1
oD
1
2acc o3 D3
1
1aer
oD 1
olD ’
where AA = (p/2)Vr2LD is the common term, p is the density of the fluid, Vr is the relative speed between the fluid and the cylinder length L and diameter D.
2.3. Application of the mechanical model to the cable
The model described reproduces the behavior of a rigid cylinder which has only one frequency and which vibrates at a single frequency. The extension to the real cable requires certain additional considerations. The continuous distribution of forces of fluid dynamic nature is replaced by a discrete and regularly spaced distribution of oscillators. The action of these oscillators on the cables is a distribution of forces which act at different nodes and which reproduce the same effects as the flow (fig. 5).
Fig. 5. Distribution of K oscillators along the cable, able to reproduce the effects of the forces
due to the detachment of swirls
The displacement components speed and acceleration will thus be relative to the section of the conductor corresponding to the node on which the oscillator acts. In the expressions (5) and (6) we can highlight the elastic terms [6, 7]:
’*aJXrnd = K 'jXrn,) +K acc(x-n)3
gkaer (n ) = kaer^ - karr^
and the viscous terms:
' " 3
gradX -*1) = Rac(X -*1) + Rac(X “*1)
graer (fli ) = Raer*1 i - Raer*1 3.
(7)
(8)
(9)
(10)
The values of the forces Fvorj and Foscj can be expressed according to these terms:
Fvo,l(xj , xj , n ,V) = gKacc (xj -% ) + gRac(x] -f\j ), (11)
Foscj (xj , xj , % , %,V) = g Kacc(xj - % ) + gRacc(xj - % ) - gKaeM) - gRaeMj ). (12)
3. MATHEMATIC MODEL OF THE CABLE
The mathematical model of the cable chosen for the description of the dynamic behavior of the cable is the reinforcing rod, because one takes into account the flexional and axial rigidities. The vibrations will be analyzed as small oscillations compared to the static equilibrium condition, that is to say the cable in horizontal position. One thus neglects the static deformations generated by the weight and the aerodynamic force of the trail acting on the same direction as the flow. Since the forces due to the detachment of the swirls acting in the vertical plane it is considered that the deformations of the cable take place in this plan.
3.1. Transversal vibrations in the reinforcing rod
The motion equations are found by considering the equilibrium equations dynamic for an infinitesimal element dx of the beam, in position x (Fig. 6).
u
Fig. 6. Deformation plan of a beam
The combination of the equilibrium equations in vertical direction and in the rotation gives the following relation [8]:
E d4u S d2u , . d2u ( )
- E,—r + S—2dx = pA—- (13)
j dx4 dx2 dt2
where E, is the flexional rigidity; S is the axial load; pA is the mass per unit of length.
The solution of the equation (11) is
u(x,t) = Y(x)G(t). (14)
The movement of the system is thus represented by the combination of a special term and a temporal term; the separation of the variables leads to the solution of two distinct equations (in t and x). The constant partners with the space variables x are calculated by imposing the conditions on the limits; in the case of the beam simply supported at the ends, the space terms become sinusoidal functions of argument growing linearly:
iK
(¥(x))o=o. = sin Lx, i = 1,2,3
iL
(15)
where L is the reinforcing rod length.
This expression thus represents the analyzed system modes, i.e. a deformation of the cable compatible with imposed constraints, each mode is associated to its own pulsation (fig. 7):
s + E,
22
i n
L
V_________IA
m
(16)
Fig. 7. Appearance percentage of the vibrations according to the wind speed: 1a - V = 1.45 m/s; 2a - V = 1.3 m/s; 1b - V = 2.14 m/s; 2b - V = 2.45 m/s
CONCLUSION
The problems, related to the cables fatigue, led to the realization of numerical models able to reproduce the behavior of the conductors subjected to the swirls detachment. A model based at the same time on a schematization of the cable of the modal type with integration in the time domain and on the modeling of the swirls detachment by a nonlinear system (equivalent oscillator) has been developed.
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