Modelling damping in vibrating structures is a challenging task. Before analysing the conductor damping specifically, it will be imperative to have a general overview of the various damping models that can be used to represent the different damping mechanisms in different structures.
Most often, damping in a many mechanical systems can be modelled by one of the following damping models. This includes viscous damping, proportional damping, material damping, structural damping and fluid damping. All but some of these categories of damping are present in a conductor. An overview explanation of these damping models is imperative as a means of identifying which will be applicable in modelling the various mechanisms responsible for the conductor damping. The following subsections gives an overview description of these damping models:
4.2.1 Viscous damping
This damping model is the most commonly used and the simplest way to represent damping in systems. For a system descripted by figure (4.1), with the following variable; x the vector of generalized coordinates, K the stiffness matrix of the system, M its mass matrix and f (t) the forcing function vector.
Figure 4.1: Viscous damping model
The idea contained in the damping concept as applied to this system is to represent the damping capacity of the system by the so called equivalent viscous damping. The damping, which is a function of only acceleration, can be used to characterize the same amount of damping per cycle as compared to the real system. This linear viscous damping model can be introduced into the
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equation for the system by means of the damping matrix C, so that the damped equation of motion becomes:
x f Kx x C xM ....……….……. (4.1)
4.2.2 Proportional Damping
Mentioned already in chapter 3, is the proportional damping and this form of model used in modelling damping was formulated by Lord Rayleigh [69]. Using this damping model for a system, the damping is developed as a function of its systems’ mass and stiffness. Incorporating this damping model into equation (4.1), if proportional damping is used, the damping matrix C will be a linear combination of the mass and stiffness, defined in the form:
C = α M + β K ……….……. (4.2)
where and are the proportionality constants
Thus, the proportional damping is a special form of viscous damping.
4.2.3 Material Damping
The behaviour of systems on the basis of the so-called material damping or hysteric damping model is based on the formation of the stress-strain curve, as a tilted ellipse with average slope equal to Young's modulus. The material damping represents the energy dissipation that takes place within the micro-structure of the system. As explained latter, the energy dissipated over a cycle is given by the area covered by the stress-strain curve. This form of damping can be further classified into viscoelastic and hysteretic damping.
4.2.3.1 Viscoelastic damping
This damping is normally formulated by the relationship between stress and strain with respect to time. The viscoelastic form of damping is usually model by the “Kelvin-Voigt” model for viscoelastic materials. For this model, the damping capacity of the material is frequency dependent.
The linear differential equation for this form of damping is expressed as:
dt
E d
E
……….……. (4.3) 4.2.3.2 Hysteretic damping
The second form of material damping is the hysteretic damping. This damping model represents the energy dissipated in a structure over a cycle of deformation. The amount of energy dissipation is independent on the frequency, and also proportional to the square of the amplitude of vibration.
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In fact, all types of internal damping exhibiting hysteretic damping that produces a hysteresis loop, which will be used later in this chapter to evaluate energy dissipation within the conductor. The hysteretic damping as a function of the stress-strain relationship satisfies the equation:
dt
d E E
……….……. (4.4) 4.2.4 Structural damping
The structural damping is the energy dissipation mechanism which involves rubbing between components with friction acting at the areas of contacts in the mechanical system. The Coulomb friction model is the most commonly used model to represent this form of energy dissipation by rubbing or sliding i.e. Coulomb damping is shown in figure (4.2).
Figure 4.2: Coulomb friction model
This damping mechanism occurrence in the system are caused by sliding friction or dry friction and it is commonly modelled using the Coulomb damping model. The damping can be characterized by the relation:
x f
k sgn
……….……. (4.5)
Where f is the damping force, x is the relative displacement at the contact and 𝜇𝑘 is the friction parameter.
4.2.5 Fluid damping
The fluid damping is the last form of damping model. Fluid damping is produced when a body is immersed in a fluid and exhibits a relative motion with respect to the fluid flow. For a cylindrical system immersed in fluid, the drag force can be evaluated by:
x xd C
fd D sgn 2
1 2
……….……. (4.6) where 𝑥̇ is the relative velocity [m/s], CD the drag coefficient [L], ρ the fluid density [kg/m3] and d the cylinder diameter [m].
From these models explained above, in the case of a vibrating power line conductor three form of damping exist. The first is the material damping which occurs within the material of any strand.
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This corresponds to the energy dissipation that takes place in the micro-structure. The second is the structural damping which involves the rubbing friction between component strands of the conductor system, at contacts both at the clamp and along the entire conductor structure. Structural damping in conjunction with hysteretic forms of damping (material damping) occurs at the contact between two strands. The sliding at these contact points tends to exhibits the hysteretic phenomenon, because it is periodic. The form of damping mainly accounts for energy dissipation by the conductor. Finally, is the fluid damping which occurs due to the interactions between the vibrating conductor and the fluid in which it is immersed.