The study of the self-damping properties of overhead transmission line conductors subjected to wind-induced oscillations.

Shindin "Finite Element Analysis of the Dynamic Behavior of Transmission Line Conductors Using MATLAB" Journal of Mechanics Engineering and Automation Published: February 25, 2014. Due to the spiral structure of the conductor strands, this gives rise to inter-strand contacts.

General Background

Furthermore, the analysis of the stick-slip behavior between the strands under dynamic conditions can be performed to generate the hysteresis phenomenon in order to evaluate the damping of the conductor. This can then be used to determine how much Coulomb friction energy is dissipated by the contact regions of the spiral threads, exhibiting relative motion in the dynamic behavior of the conductor.

Problem Description

The process of energy dissipation due to periodic bending of strands caused by dynamic loading. The finite element method can produce a more realistic representation of the physical model of the conductor.

Purpose of this Study

Consequently, to improve this developed linear model to some extent, a FEM model was developed in this study, which was of adequate representation of the conductor. The study presented in this thesis was used to develop a computational model of the conductor as a composite structure.

Research Hypothesis

Aims and Objectives

Scope and Assumptions

Although, the input power is distributed, an equivalent load point on the conductor is used to simulate the dynamic response of the conductor. This assumption was applied when modeling the conductor or its strands, using the Euler-Bernoulli beam theory.

Contribution of this Study to the Body of Knowledge

For the contact conditions, the core and helical strands interaction and interaction between helical strands of different layers are considered in the analysis, but those interactions between strands of the same layer are neglected. The successful geometric formulation paved the way for the realization of the model and it enabled the explanation of the mechanism responsible for damping and its numerical evaluation.

Organization of the Thesis

The FEM formulation procedures for the conductor follow the normal basic finite element type formulation and implementation of the solution of the resulting equations in a computer environment. Derivation of characteristic finite element equations for mass and stiffness matrices and load vectors are obtained using energy approaches.

Publications

Also, the program was designed to receive signals from accelerators located on the conveyor. The natural frequencies (resonances) from the first test serve as input frequencies in the second test to vibrate the conductor at a constant amplitude.

Introduction

These vortices induce an aerodynamic instability on the conductor, and the conductor then oscillates tangentially to the direction of the prevailing wind. For the aeolian vibration, the vibration amplitude is relatively small with a maximum amplitude of the order of one diameter of the conductor.

Figure 2.1: Overhead Transmission Line [22]

Review of Previous Analytical Models

The free and forced vibrations of the conductor system are analyzed to determine the various stresses placed on the structure of the conductor. The new approach was to develop an analytical formula that could be used to determine the power distribution within the conductor as a function of the conductor parameters.

The Power Line Bare Conductors

The model established the fact that bending stiffness of the conductor is not constant, it varies along the conductor due to curvature. Stranded conductors allow more bending motion before breaking, compared to solid conductor of the same diameter.

Power Line Conductors Geometric Analysis

The Conductor Geometry
Lay lengths and Lay angles
Geometric Description of a Conductor Cross-Section

A typical cross-section of the conductor consists of strands arranged as shown in Figure (2.8). In the top corner of figure (2.8), define the coordinate system used as a reference for the geometric description of the conductor.

Parameters Associated with the Conductor Geometry

The Analysis of Conductor Inter-Strand Contacts
Number of Contacting Points

The second form of contact within the conductor is due to the close arrangement of the conductor. In modeling, this form of contacts is the combination of radial and lateral contacts.

Figure 2.9: Inter-strand contacts in helical stands [12]

Contact Mechanics

Point Contact Mechanics
Line Contact Mechanics

The analysis of two cylinders crossing each other can be used to analyze the contact that occurs between two strands due to the angle of lie between them. As can be seen from Figure (2.9), point contact voltages can only apply to multilayer conductors.

Pressure between Strands of Different Layers

It should be noted that stress formulations along the contact areas between the strands (except those between the core and the first layer) appear at a point contact, but in most analytical models an equivalent line contact was used [19, 58]. ]. This was used to estimate the bending stiffness of the conductor, as only stresses from equivalent line contacts were used due to the cumbersome use of point contact stress analysis.

Evaluation of Inter-strand Contact Force

In accordance with the above, considering the outer layer forces of the single layer conductor (i1 or N. The solution to the above differential equation consists of the homogeneous and the particular solutions given as. 2.43).

Figure 2.13: The normal and frictional forces in a single-layered conductor In line with the above, considering the outer layer forces of the single layer conductor ( i  1 or N

Characterization of the Conductor Cross-Sectional Parameters

Description of the Conductor Cross-Section
Tensile Analysis of the Conductor

Tensile Analysis of Conductor for only Strands Axial Displacement
Tensile Analysis of Conductor for Strands Axial Extension and Rotation

Flexural Analysis of the Stranded Conductor

Pure Bending Action for Individual Strand
Conductor Bending Stresses and Moments
Conductor Flexural Rigidity

Analysis of Conductor for Combined Effect of Axial and Bending Loads

This stress is a function of the distance of the strands from the neutral axis of the conductor. The bending moment acting on a conductor as a function of bending can be obtained as:.

Figure (2.16) shows the strands along the helical path, subjected to loads, and this causes the conductor individual helical strands to experience a tensile force

Fluid-Solid Interaction

Vibration of a Cylinder in a Fluid
Vortex Induced Vibration

As the wind flows past the cylinder, two force components are generated by vortices: the vertical lift force and the horizontal drag force. The vertical vibration occurs due to the vertical force component induced by the airflow as it flows along the cylinder and is responsible for the small amplitude transverse vibration, i.e.

Conductor Wind-Induced Vibration

The Aeolian Vibration
Conductor Excitation
Resonance and Lock-in Effect

This happens due to the phenomenon of resonance, when the natural frequency of the cylinder coincides with the Strouhal frequency [1, 2]. This phenomenon means that the frequency of the excitation force due to the wind load and any natural frequency of the conductor are approximately the same.

Figure 2.18: Vortex wake shedding from a conductor

Analytical Evaluation of Wind Loading

Energy Balance Principle

Figure (2.21) shows the graph of power versus relative amplitude (A/D) obtained from the research conducted by some researchers in a wind tunnel as documented in [1, 2]. Each of these terms in equation (2.90) is a function of the frequency and amplitude of the guide oscillations.

Figure 2.21: The Graph to determine empirically the input power on a conductor [1, 2]

Conductor Modelling

These model shapes are used to obtain the conductor natural frequencies and mode shapes. An equivalent linear damping model is usually included mathematically in the formulation of the damping equation for a conductor.

Conductor Static Profile

This can be used to determine the cost saved by power suppliers in terms of cost contributed by the conductor to build the power line. The clearance which is the difference between the conductor length and the span length can be evaluated by.

Figure 3.1: The conductor static profile

Analytical Modelling Approach

The Global Approach
The Local Approach

The rigid beam or strained string model is commonly used for analytical modeling of the conductor. This approach gives a more accurate representation of the structure of the conductor and the results for the conductor.

Forms of Analytical Modelling

The position of the strand with x (t), its position relative to the lower strands with the assumption that the strand starts from rest at x (t) = 0 at t = 0. The value for the conductor damping was obtained as equal to the area of the loop.

The Analytical Continuum Structure Model

Linear Analytical Modelling of Conductor Vibration
Non-Linear Analytical Modelling of Conductor Vibration

The Semi-Continuous Model

The semi-continuous structure model involves the development of homogeneity parameters for each layer of the conductor using orthographic concepts. Using the 'semi-continuous' model, each layer of wires/strands is mathematically represented by an orthotropic circular cylinder using the 'average' mechanical properties.

The Composite Structure Model

Curved Beam Theory
Mechanics of Helical Curve Rod
Thin Rod Kinematic Analysis
The Conductor Composite Structure Model

Consider a spiral elastic strand as shown in Figure (3.3). The position of the helical strand in the composite conductor is considered in terms of the arc length. Solutions to these constitutive differential equations are expressed for the dynamic behavior of the helix.

Analytical Evaluation of Conductor Self-Damping as a Composite Structure

Conductor Damping

Because this parameter is central to this study, this chapter was specifically used to address concepts related to conductor self-turn-off. The chapter elaborates on the comprehensive concepts that describe damping as a function of its contact between strands, as well as the sticking and sliding phenomenon within the conductor as a composite structure.

Damping Models

Viscous damping
Proportional Damping
Material Damping

Viscoelastic damping
Hysteretic damping

Structural damping
Fluid damping

Using this damping model for a system, the damping is developed as a function of the mass and stiffness of the system. The viscoelastic form of damping is commonly modeled by the “Kelvin-Voigt” model for viscoelastic materials.

Mechanisms Responsible for Conductor Self-Damping

This type of damping thus takes place throughout the length of the conductor, and the distributed element damping model can be used for the evaluation. The third form of energy dissipation is the fluid dynamic (aerodynamic) damping, which results from the interactions between the conductor and the fluid in which it was immersed.

Analytical Determination of Conductor Self-Damping

This damping mechanism exhibits a hysteresis effects, the activities of friction between the individual strands serve as the memory of the system and the output is the displacement of the conductor strands as they undergo the bending deformation. In this second form of energy dissipation, part of the external energy is converted into molecular energy and this form of damping is known as material damping.

Table 4.1 [2] shows the comparison of some of the values of the exponents’ l, m and n obtained through experimental research of some authors [1, 2, 50]

Stick-Slip Model

If the imposed deformation on the string, which is a function of the force applied to the conductor structure, exceeds the breaking friction force acting at the contact area, the system is considered to be in a stick-to-slip transition. The motion was necessary for the calculation of the string displacement and the subsequent evolution of energy dissipation as the conductor goes from stick to slide and from slide back to stick.

Hysteresis Damping

Figure (4.4) shows the transition from stick to total slip as a function of the bending stiffness and the imposed curvature. The bending stiffness on the conductor at a particular point depends on the rate of application of the bending load.

Figure 4.5: The formation of hysteresis Loop under periodic loading The model constitutive relation is expressed as a linear first-order differential equation:

The hysteretic behaviour of Conductors

This assumption is sufficiently correct if the radius of curvature is much larger than the radius of the conductor. This non-linear transition is physically related to the development of the inter-strand relative displacement.

The Bending Moment-Curvature Relations

This is because the radius of curvature of each string as a beam depends on its position relative to the origin of the string axis relative to the radius of curvature. It is necessary to know the radius of curvature and that the axis of the conductor assumes a deformed state when stretched on the towers.

Figure 4.7: The graph of bending moment versus curvature [57]

The Formation of the Hysteresis Loop

To develop the hysteresis loop as a function of the bending moment, in [57] the relationship between the bending moment and the curvature was used to generate the hysteresis loop, illustrated by the diagram shown in Figure (4.9). In phase 3, the bending moment will have a slope as in phase 1, in order for there to be a reversal.

The Bouc-Wen Hysteresis Model

The application of the Bouc-Wen model to model the hysteresis behavior of the conductor in relation to the energy dissipation was done as a function of the bending moment and curvature, denoted by M and κ, respectively. The shape of the Bouc-Wen model was related to the moment-curvature relationship, and the model was considered as the parallel coupling of a linear and nonlinear element, as shown in Figure (4.12).

Introduction

The basic technique of the finite element method is the discretization of the structure into its substructure, i.e. in chapter 3, the developed analytical equations used to model the vibrations of the conductor were explained and the limitations of these models in terms of geometrical representation were discussed.

Conductor Finite Element Analysis

FEM for the conductor was used to determine parameters such as natural frequencies, mode shapes, and vibration amplitude. For an example of a dynamic conductor problem, an expression for the problem was developed as a function of space and time.

FEA Solution Concepts

The Conductor 3D Geometric Formulation

The Cylindrical Core Strand
The Cylindrical Helical Strands

The discretization of the geometric domain of the conductor for strands in different layers was determined by the distance from the neutral axis of the core strand. The position of the strands in a given layer of conductor will be governed by the rule of concentric alignment.

Finite Element Modelling

FEM Modelling Concept
The Curved Beams Model
The Beam Constitutive Equation
Curved Beam Strain–Displacement Relationships

This FEM composite structure concept for the conductor was achieved through the iso-parametric interpolation of the 2D curve beam for each individual strand along the defined geometric path. The y distance from the central axis of the cross section to the point (x, z) and is equal to the.

Figure 5.1: Forces and moments acting on the beam 5.5.4 Curved Beam Strain–Displacement Relationships

Finite Element Formulation

2, is the moment of inertia of the cross-section, A is the cross-sectional area. 5.18) When using the Euler-Bernoulli curved beam, shear deformation is neglected. The total energy and work for the equations of the curved beam element were therefore obtained by substituting the equations into equation (5.23).

Discretization using Shape and Trial Functions

The Matrix Formulation
Finite Element Analysis of Straight versus Curve Beam Element
Numerical Integration

For the curved beam shown in figure (5.3), there are three variables per node that are used to define the behavior of the element. In the iso-parametric formulation for the curved beam, the displacements were expressed in terms of natural coordinates.

Figure 5.3: The curved beam finite element

Formulation of Equation for the Un-Damped System

This was achieved by using a transformation matrix and then the Jacobin method to perform the numerical integration. This numerical integration for natural coordinates ranges from -1 to +1, for a function f(x) was obtained as a definite integral:.

FEM Modelling of Conductor Damping

Inter-strand Contact Patches
The Friction contact Model
The Lagrange multiplier

As shown in Figure (5.4), the contact point between the strands is comparable to two contacting inclined cylinders. This allows to distinguish between two friction states: the stick state, which is characterized by no relative displacement between the bodies, and the slip state, where the relative displacement was in the form of sliding.

Figure 5.6: The forces at the inter-strand contact 5.9.2 The Friction Contact Model

Finite Element Modelling of Inter-Strands Contact

Discretization of Inter-strand Contact Interface
Finite Element Implementation for Inter-strand Contact

The discretization of the contact inter-string patches was done by the normalized two-dimensional bilinear element type. As an example, depicted in figure (3.9), the elements are quadrilaterals, the shape of which is completely determined by the contact domain spots.

Figure 5.7: Bilinear two-dimension finite element

FEM modelling of Stick-Slip regime

To generate the hysteresis loop, the force model is implemented with the effect of the friction force fS, which determines the state of the strings. But when the speed of the string changes sign, the string will actually stop and start sticking to the surface with a static frictional force.

Figure 5.11: Contact points and its rheological representation [89]

The Formation of Conductor FEM Model

Dimensional Reduction from 3D solid element to 2D Line element
Iso-parametric Mapping
Geometric Mapping for the Conductor
The System Equation

The design of the composite structure of the conductor was achieved by isoparametric mapping of its substructure. The displacements and rotations at the nodes of the 2D beam element, as expressed in the global coordinate system for the conductor and the transformation matrix, are defined as:.

Figure 5.15: The global discretize Model of the Conductor

Numerical Computation for System Response

In solving the FEM equation for the system, the direct and iterative method was used and the explanation for the numerical scheme used is given in the next section. The derivations for the central difference method and the various responses are presented in Appendix C.

Experimentation
Experimental Investigation of Wind-induced Vibration
The Indoor Laboratory Testing Methods
Description of Test Set-up

Shaker and Shaker Conductor Connection
The Span End Conditions
Accelerometers and Force Transducers

Description of Experimental Methodology

The Sweep Test

Method for testing Conductor Self-damping

Experimental Design Philosophy for Generating Hysteresis Loop

Experimental Evaluation of Damping: Hysteresis Loop

Test Procedures and Data Acquisition using Labview
Data Acquisition and Display

Test Conductors

This figure represents a schematic representation of the actual experimental setup of the test span shown in Figure (6.1), the setup of the experimental setup at the VRTC. The area of the experimentally obtained hysteresis loop is then used to calculate the self-damping of the conductor.

General Remarks
The Conductors Natural Frequencies
The Analytical Results
FEM Model

The FEM Computer Implementation
FEM Dynamic Response
FEM Damping Results

Experimental results
Comparison of Self-damping Results and the Effects of Variable Tension
Conductors Self-Damping Characteristic

Retrospect on the Study Hypotheses
The Developed Algorithm to Evaluate Self-Damping

The formula for calculating RMSE is given as. 7.1) RMSE is concerned with deviations from the true value. The analysis was performed for the transverse vibration of the conductor as a simply supported beam.

Figure 7.1: Sweep test graph done Pelican conductor at 20% UTS

Conclusion

Recommendation

Physical Parameters for Rabbit Conductor

Physical Parameters for Pelican