Abstract
As to the transmission line accreted with ice,it is very likely to arise a kind of vibration with large amplitude and low frequency,which is the so-called galloping.The amplitude ranges typically from 0.1 to 1.0 times the sag of the line[1],and might result in break of conductors,damage of connectors or even collapse of the tower.Therefore,galloping of iced-conductors has been attached extensive attention in recent years.
The first galloping model was proposed in 1932 by Den Hartog[2],which revealed the classical vertical galloping mechanism.According to field measurement,however,the twist of the transmission line may also contribute to the initiation of galloping.The importance of twist was later emphasized by Nigol and Buchan,and the torsional galloping mechanism was then proposed[3].Besides,the horizontal oscillation along wind direction was proved to be essential to initiate the vertical galloping when it is coupled to the plunge[4].In this paper,a simplified three-degree-of-freedom model(3Do F model)for describing the galloping behavior involving plunge,twist and swing is developed,as shown in Figure 1.
Figure 1 Three-degree-of-freedom model representing iced conductors
In this model,the cross-section eccentricity caused by the coated ice is considered,and the interactions between the plunge,twist and swing of the conductor along wind direction are included as well.Based on Lagrange's equation,the equation of dynamic motion of the 3Do F model is derived as follows.
where m0 denotes the mass per unit length of iced conductor;Sy0 denotes the first mass moment of area with respect to y0 axis;Sz0 denotes the first mass moment of area with respect to z0 axis;I0 denotes the mass moment of inertia per unit length;ξy,ξz andξm denote the damping ratio along y,z and φ direction,respectively;ωy,ωz and ωm denote the natural frequency along y,z and φ direction,respectively;ky,kz and km denote the stiffness along y,z and φ direction,respectively;α0 denotes the static torsional angle;φ denotes the dynamic torsional angle;V is the wind velocity;ρ0 denotes the density of air;D denotes the diameter of conductor;CL,CD and CM denote the aerodynamic coef ficient of lift,drag and torsional moments,respectively.
The galloping behavior of a D-shaped iced conductor is simulated using the proposed 3Do F model,which was experimentally investigated in the previous study[5],as shown in Figure 2.
Figure 2 D-shaped iced conductor
Using the Runge-Kutta algorithm,the equation of motion of the model can be solved ef ficiently.The galloping displacements of the D-shaped iced conductor along directions of plunge,twist and swing are shown in Figure 3,respectively.It is seen that the amplitude of galloping is increasing with time going,and it eventually achieves at a steady-state value.
Figure 3 Galloping displacement along
Table 1 presents the numerical result and its error in comparison with the data from wind tunnel test.It is seen that the vertical galloping amplitude by the proposed 3DoF model shows a consistence with that from the experimental test.One might recognize that the proposed model remains the aerodynamic nonlinearity although it ignores the geometrical and material nonlinearities.Therefore,the proposed 3DoF model provides a reliable mean to understand the galloping behaviors of iced conductors as well as to underlie the refinement control of galloping motion.
Table 1 Comparison between numerical result and wind tu nnel test
