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Figure 2.20: The graph illustrating Lock-in Effects [70]
At this point the conductor is said to experience resonance or set into resonance. This produces a sensation that the natural frequency, w0 is being influenced by the external frequency w. It is assumed that there is a range of wind speeds over which the vortex shedding frequency and the conductor’s natural frequency will lock-in and this will result in large conductor vibration [70].
This occurrence indicates that the frequency of the exciting force due to the wind loading and any of the natural frequencies of the conductor are approximately equal. When this occurs, the frequency of vortex shedding envelops any of the natural frequencies of the conductor that coincide with, thus the amplitude of vibrations increases. This leads to a condition known as “Lock-in effect”.
The lock-in effect is a concept from nonlinear mechanics, it is also known as the synchronization effect. It occurs as a result of fluid-solid interaction. When the lock-in phenomenon occurs, in the case of power line conductor vibration, severe vibration can persist long enough to cause structural failure. The difficulty with lock-in phenomenon arises from finding the range of natural frequency of the structure, for each mode shape, over which lock-in can occur. To determine the frequency of wind that has the tendency of vortex shedding that can cause the vibration of conductor, with regards to lock-in effect can be computed as a range of frequencies around the shedding frequency as a comparison to modal vibration frequencies of the conductor.
The phenomenon of lock-in effect explains why during the conductor excitation by wind loading, the occurrence of lock-in means that changes in the wind speed at or near the resonant response frequency do not cause the vortex shedding frequency to change, but instead the response frequency will remain constant. After the initiation of resonance, the lock-in effect at resonance can stay for wind speed as large as 90 to 130% of the onset velocity [2]. Flow visualization as shown in figure (2.18) demonstrates the vortex-shedding as the formation of pressure fluctuation producing lifting force which equal to the Strouhal frequency, which can be sustained for a long period by the lock-in effect.
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The investigation of the motions of a dynamic conductor’s model is usually done in the proper scaled wind tunnel test. This wind tunnel tests is capable of duplicating reliably the motions of a convenient length of the conductors usually modelled as a cylinder. The wind forces and the rate at which they can build up energy of oscillation respond to the changing amplitude of the motion.
The rate of energy change can be measured as a function of its amplitude. Thus, the section-model test measures the one unknown factor, which can then be applied in the calculation of the variable amplitude of motion along the conductor to predict the full behaviour of the structure under the specific wind conditions of the test.
The maximum power brought into the system by aerodynamic forces as the stationary laminar air flow perpendicular to the conductor, can be determined using empirically derived equations. This empirical formula was formulated based on wind tunnel experiments.
The mechanical power transferred from the wind to a vibrating conductor may be expressed in the general form [1, 2]:
A D
fnc D Lf
P
3 4/
………..…….……. (2.88)Where fnc (A/D) is the power function (function of relative vibration amplitude A/D), L is the span length, D, is the conductor diameter and f is the vibration frequency.
Figure 2.21: The Graph to determine empirically the input power on a conductor [1, 2]
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The rate of power imparted on the conductor can be measured and plotted against amplitude.
Figure (2.21) shows the graph of the power that is plotted against the relative amplitude (A/D) as obtained based on the research conducted in a wind tunnel by some investigators as documented in [1, 2]. As it can be seen on the graph, the power functions vary significantly from researcher to researcher. This is because the different results obtained by the researchers are due to the different characteristics of wind tunnels and different test methods. The form measurements from these experiments are of small energy levels (fractions of a watt per meter of conductor) which are very sensitive to disturbances. Thus, the turbulence of the incoming flow is another parameter which will influence the value of the power imparted by wind to a vibrating conductor. The lower the turbulence, the higher the wind power that can be imparted. This factor in the form of the level of turbulence was investigated and documented in [31, 71]. This turbulence parameter, as related to a wind power input, is a function of reduced velocity, Vr, dimensionless amplitude [Ymax/D], and reduced decrement r. The reduced decrement can be used to convert input power into reduced wind power as given below.
2 max 2 4
3 4
D
Y D
f P
r
Wind
W.s3/m5
……….……. (2.89)Where mass density of the fluid medium; 1.2 kg/m3
rreduced decrement
The reduced wind power as a function of relative amplitude has been drawn for different level of turbulence. Increasing the turbulence level decreases the wind power input. An example of wind power curves for different turbulence levels is shown on figure (2.22). This graph illustrates how increase in turbulence decreases the energy imparted by wind on the conductor.
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Figure 2.22: The graph used to determine the reduced input power on a conductor [71]
2.12.1 Energy Balance Principle
The analysis of dynamic behaviour of conductor follows an energy flow pattern in terms of input power and the various energy dissipation mechanism i.e. self-damping and vibration absorbers.
But the analysis of the phenomenon of vortex shedding caused by wind loading and energy dissipation can be done by the simplified approach known as ‘Energy Balance Principle’ (EBP).
This method is based on some simplifications that give useful estimates of the maximum vibration levels [1, 2]. This means that the principle is used when the conductor undergoes maximum vibration amplitude due to Aeolian vibrations. The steady state amplitude of vibration for a single or bundle conductor due to Aeolian vibration is that for which the energy dissipated by the conductor and other devices used for its support and protection, equals the energy input from the wind. The vibration amplitude is determined by a power balance between what is provided by the wind and what is dissipated by the conductor self-damping and by any dampers. This can be mathematically formulated as [1, 2]
damp cond
wind P P
P
……….……. (2.90) This means, the power imparted by the wind equals the power dissipated by the conductor self- damping, and the power dissipated by the damper. The EBP as document in [72] has been used to develop an algorithm for analysing Aeolian vibration of a span of a single conductor with multiple dampers. Each of these terms in equation (2.90) is a function of the frequency and amplitude of the conductor oscillations. In chapter 4, the empirical formulae for evaluating power dissipated
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 1 2 3 4 5 6 7 8 9
Relative Amplitude[Ymax/D],pk-pk
Reduced Wind Power [Pw/f3 D4
1 % 5 % 10 % 15 % 20 %
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will be presented for the power line conductors. Most analysis are done by the use of the power law according to Noiseux’s exponents [50] to calculate the self-damping power for conductor.
The conductor vibration level (anti-node displacement) can be evaluated as function of frequency or the wind speed and this can be estimated by using this principle. This is based on presumed knowledge of energy:
(a) Imparted to the conductor by the wind
(b) Dissipated by the transmission line conductor (conductor self-damping) (c) Dissipated by the vibration absorbers (dampers)
The energy balance method has been used for studying the energy balance among the input power, the internal damping and the influence the section of dampers as a means for optimizing its placements on the span. The vibration level is determined by calculating the complex eigenvalues and eigenfunctions. These values are used to determine the amplitudes of vibrations at each resonance frequency. In case of laboratory experiments for fatigue failure, bending strains are estimated at the critical points, usually 89 mm from the suspension clamp.
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