The interactions of contact forces between internal structures play an important role in many mechanical systems such as conductors. In most models used to analyse transmission line conductor’s oscillation treats, the conductor as a continuous homogenous system [5-7]. These models completely neglect internal phenomenon such as slippage at contacts point. The modelling of contact problems of helical strand of the conductors’ structures have been considered by some authors in terms of using the conductor discrete geometry [9-13]. However, these analyses are restricted to modelling the strands as rods in contact at discrete points as shown in figure (4.3).
The rod theory was based on the concept of modelling the helical strand as a long, slender object, which are then regarded as substructures of the conductor. This method has been used by many researchers to investigate many phenomena that are linked with the internal structure.
As explained in earlier section, one of the damping mechanisms is characterized by energy dissipation due to frictional effect brought about by the relative motion between strands at the points of contact. This type of frictional sliding at contacts, the vibration exhibits a stick-slip regime during the harmonic oscillation and the contact-friction model can be used to model and simulate this stick-slip vibration. Stick-slip vibration is characterized by a displacement-time history. The history of the vibration, function as a memory stored in the conductor and this can be characterized by four phases of the vibration history. These are clearly defined as; the stick, the transition from stick to slip, the slip phases and finally the transition from slip to stick. The vibration is governed by a static friction force in the stick phase and a velocity dependent kinetic friction force in the slip phase.
The wind input energy, due to lock-in effect produces a harmonic induced vibration that has a nearly sinusoidal displacement-time pattern. As the motion is initiated the conductor operates from stick to the slip phase in a cyclical manner. As the conductor undergoing the alternating bending between the stick and slip regime, it thus exhibits flexural hysteresis phenomenon. The term hysteresis implies that the relation between the curvature that causes the bending and the bending moment is not unique and the bending moments do not depend on the absolute value of curvature but only on its sign. Therefore, for any given motion the bending moment depends not only on the actual value of curvature but also on sign of the curvature; more generally, the moment depends on the history of the deformation due to the imposed curvature.
During the bending, the conductor experiences damping due to internal friction and this form of damping exhibits significant flexural hysteresis, resulting from inter-strand friction slip. The
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flexural hysteresis is due to continual changes between the maximum and minimum bending stiffness. The evaluation of this type of hysteresis damping of mechanical vibrations such as the conductor is very difficult. The damping mechanism in the conductor due to hysteresis resulting from this regime is mostly modelled, using the coulomb dry friction model. This model was applied between the individual strands accounting for inter-strand friction during bending deformation.
The adequate model this stick-slip phenomenon in conductors was achieved by evaluating all possible phases of the hysteresis history and incorporated them into one conditional equation. This follows that during the stick-slip motion two different mechanisms take place, the first is the static friction mechanism and the second is the kinetic friction mechanism. The major problem of the formation of stick-slip for the hysteresis process is the determination of the transition between these two states.
For this sinusoidal motion, the equation for this motion requires a periodic solution with the combination of conditions to describe the periodic stick-slip regime solutions. This phenomenon is an example of nonlinear dynamic system that can be represented by the equation of the contact friction model:
0
1 sgn
0 sgn
, min
) (
rel rel
rel slip
rel
rel stick
s stick
rel slip condition withv
v v v F
F
v with condition stick
F F
x F x
F x
v F
….……. (4.9)
This equation can describe the motion of the hysteresis phenomenon of the conductor system as a four different sets of ordinary differential equations: the first is for the stick, the second is for the transition from stick to slip, the third for the slip phase and the fourth back from slip to stick phase.
Using quasi-static analysis, this process starts from an initial state which after a certain time step, then due to slippage the conductor goes into slip mode and thus there will be transitions from stick to slip with intermediate process. This process is similar to switching operation. Having then evaluated the first switching point, a new process is then initiated with a modification to the set of differential equations. In this state, the conductor initial condition is identical to the state at the switching point or transition point. At end of slippage there is the second switching as the transition from slip to stick. This process continues as a ‘to and fro’ motion for the periodic motion.
The periodic solution of the hysteresis time-dependent with contact friction model was then formed. The conditions for changing from the stick mode to the slip mode and vise-vasa are operated as switches between the systems equation as described by sets of condition equations as applied to equation (4.9) are given by the following condition equations.
If Fstck Fthen
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x The equation of solid beam is used stick f
x
………….……. (4.10a)Else if Fs F
x The equation of composite beam is used stick to slip
f
x
…. (4.10b)Else Fs F
slipmove to
free completely are
strands the
which
in used is beam composite of
equation x The
f
x
...……. (4.10c) Else if Fs F
x The equation of composite beam is used slip to stick
f
x
.……. (4.10d)Firstly, the stick states where the system was considered to be at equilibrium. In imposing bending, during the sticking, the friction force rises until a maximum value is reached and this is the static friction force also known as breakaway friction force. The time-dependent static friction model, considers the static friction force to be dependent on the stick time. The static and kinetic friction forces can be evaluated by equations (2.46) and (2.47) and now expressed as:
stick stick T F
……….……. (4.11a)
slip
slip T
F
……….……. (4.11b) The externally applied force on the conductor system, is then transferred to the strands causing its displacement. If the conductor state lies within the stick regime the conductor assumes a solid mass. If the imposed deformation on the strand which is a function of the force applied forces on the conductor structure exceeds the breakaway friction force acting at the contacts area, the system is considered to be experiencing a transition from stick to slip. The system is considered to be in the slip mode if there is relative velocity as a result of inter-strand motion. When in the slip mode, the slippage is defined by a sliding velocity which is defined by the wave speed across the conductor.
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Figure 4.4: The graph of bending stiffness and curvature [57]
The motion was necessary for the computation of the strand displacement and the subsequent evolution of energy dissipation as the conductor goes from stick to slip and from slip back to stick.
Figure (4.4) shows the transition from stick to total slip as a function of the bending stiffness and the imposed curvature. The switch mode alternating between the stick and slip regime gives rise to the hysteresis phenomenon as explained in the next section.