Study on the Effect of Anti-galloping Device with Cubic Stiffness and Particle Damping

The galloping of iced conductors is a common disaster of overhead conductors. Most traditional anti-galloping devices lack a nonlinear dynamic vibration absorber; hence, they have a limited anti-galloping effect. In this study, we designed, optimized, and tested an electric power fitting and an anti-galloping device with cubic stiffness and particle damping for ultra-high-voltage (UHV) transmission lines. We built a nonlinear dynamic model of the coupling galloping system comprising split conductors and anti-galloping devices. The harmonic balance method obtained the steady-state analytical solutions and corresponding averaged equations. The test spans in the laboratory were used to design the test program and equipment. We verified the accuracy of the nonlinear dynamic model and the harvesting effect of the anti-galloping device with cubic stiffness and particle damping. The theoretical and experimental results were highly consistent, and in the range of (0, 2] Hz, the anti-galloping device reduced the galloping amplitude. Therefore, cubic stiffness and particle damping can effectively improve the anti-galloping ability of UHV transmission lines, prolonging their service period.


Introduction
Overhead transmission lines operate all year round under various weather conditions, which may cause wind-induced vibrations, such as in buildings [1], bridges [2,3], and other long, large flexible structures.Recent abnormal global climatic changes and the expansion of the operational scale of power grids have contributed to natural disasters that threaten the safe and stable operation of power grids by galloping overhead transmission lines with eccentric ice.These disasters typically occur in winter and spring when the frequency ranges from 0.1 to 5 Hz [4][5][6][7].Common measures against galloping are using additional fittings, such as rotary clamp spacers, damping detuning pendulums, and rotary clamp spacer double-pendulums [8][9][10][11][12].Thus, an anti-galloping design based on theory is crucial, even for anti-wind-induced vibration works for test spans.Existing design concepts of anti-galloping fittings are mostly derived from Den Hartog and Nigol mechanisms.Based on this linear theory, the design principle and anti-galloping fittings have been effective in single galloping frequencies and withstood adverse weather.The anti-galloping effect has been fully verified in practical engineering.Meanwhile, practical engineering experiences have shown that a reasonable theoretical analysis, structural design, and spatial distribution of these fittings are vital for improving transmission line reliability and prolonging the service life of the conductors and towers.
However, the galloping of actual transmission lines has become more complicated because of the influence of special structural parameters, sporadic icing conditions, microtopography and micrometeorological factors, and, most importantly, the rapid expansion of power grids.A foremost distinctive feature is that the dominant frequency point of a galloping line in a single span varies, and common fittings cannot effectively prevent galloping at every point of the frequency range.Because of the purpose of adapting to these dynamic properties and meeting the requirements of the UHV line [12], nonlinear energy sink (NES) theory and the theory of particle damping are used for designing antigalloping fittings of UHV transmission lines.
The NES [13][14][15][16][17][18][19][20][21][22][23][24] is a new type of absorber technology whose development is based on recent linear dynamic vibration absorbers, and considerable research has been conducted on its engineering application.Theoretically, the absorbing frequency range can be broadened, and the damping effect improved by changing the stiffness or damping of a traditional linear vibration absorption system (as a nonlinear spring is introduced in a conventional damping spring system).After optimization analysis and design, the nonlinear dynamic vibration absorber can harvest the vibration energy of the main structure in a wide frequency range.The absorber has excellent properties, such as low mass, high adaptability, and improved reliability.For instance, Vakakis [13][14][15] proposed NES theory, theoretically applied the technique to a linear oscillator system, and proved that an NES could irreversibly capture the energy within the coupling system.Oueini [16,17] and Zhao [18] achieved the effect of energy transfer by introducing cubic nonlinear stiffness and nonlinear damping.Guo [19] applied an NES device containing cubic stiffness to suppress the limit cycle oscillation (LCO) of the coupling system comprising the two-degree-of-freedom (DOF) airfoil and NES and described the influence of nonlinear stiffness and other coefficients on supercritical and subcritical Hopf bifurcation behavior.Bichiou [20] demonstrated that, as a nonlinear aeroelastic system, the LCO of an airfoil could be limited to an extent by an NES.Yang [21] designed an NES damping device for spacecraft and used the finite element method to study the effect of the NES key parameters on the system's target energy transfer property.Wang [22] designed a track-type NES to protect building models.Likewise, considering the nonlinear density properties of a magnetic field, Kremer [23] designed an absorber to effectively capture energy in the coupling system and avoid damaging the main structure.However, the NES theory is seldom applied in electric power transmission.
Therefore, we conducted theoretical and applied research to combine NES with particle damping technology [25,26] and developed a nonlinear dynamic vibration absorber with cubic stiffness and particle damping.The absorber was an anti-galloping device used to achieve anti-galloping in transmission lines.First, we established a 2-DOF coupling model comprising multi-bundle conductors and NES under a low-frequency forced vibration along the vertical direction.The NES was provided with cubic stiffness and linear damper.Second, we designed the anti-galloping device test according to NES theory.Finally, we combined experimental and theoretical analyses to validate the device's performance in the laboratory.This study provides a theoretical basis for applying NES theory and particle damping technology in the anti-wind-induced vibration of UHV transmission lines.

Theoretical Model
Fig. 1 illustrates the oscillator system with geometrically nonlinear characteristics.According to the geometrical and constitutive relations between the displacement of the oscillator and the deformation of the elastic rope at both ends, the following factors are necessary to obtain the cubic stiffness properties: • In the static equilibrium state, the free elastic rope on both sides must stay on a horizontal line, and the oscillator must be at the midpoint of the line determined by articulated joints on both sides; • The elastic rope must bear only axial expansion and lack mass or bending rigidity properties.Under these two conditions, when the oscillator system moves near the original point, the relation between the external force F and static displacement v can be approximately expressed as where k a is the cubic nonlinear stiffness coefficient of the anti-galloping device, which is constant and determined by the original rope length and axial stiffness.In Fig. 1, m a and c a are the mass of the oscillator and linear damping coefficients of the antigalloping device, respectively.The absence of linear stiffness implies the absence of a natural frequency for the oscillator in the ideal state, which indicates that the system's vibration can respond to excitation with any frequency.In the initial state, the oscillator placed at the center of the cross-section of the multi-bundled conductors is at the origin O; in the vibration state, we obtain the following relation: where u 1 (s,t) is the traverse displacement function; s is the location in line coordinate; t is time; v(t) and Δd(t) are the absolute and relative displacement functions of the oscillator, respectively.As a typical example of UHV lines, the installation of an eight-bundled conductor-damper system is illustrated in Figure 2. Here, l denotes the conductor length; l a and l f denote the corresponding line length coordinates where the spacer (and the anti-galloping device) and exciter are located, respectively.
Considering a curved beam model and inducing a nonlinear coupling effect between different internal strains [27][28][29], we derived mathematical equations to describe the 3-DOF movement of galloping iced conductors.We only focus on the vertical direction in the coupling system by using theoretical analyses and tests to demonstrate the absorbing effect of the anti-galloping device with a single oscillator.Therefore, we express the kinetic energy T l , potential energy V l , and dissipated energy W l of the conductor's 1-DOF (same as the normal degree of freedom here) as follows: ) We denote the differentiation concerning t and s by symbols "′" and "•," respectively.The initial conditions considered were ρ = 1.65 kg/m, c = 0.17 kg/s, E = 63 kN/mm 2 , A = 531.37mm 2 , and T 0 = 40 kN.
W f represents the external force doing work: ( ) where f and Ω 0 are the amplitude and frequency, respectively.Each energy expression of the anti-galloping device with cubic stiffness and particle damping is shown below: Substitute (3) to ( 5) in (6).By using the Galerkin discrete method, we establish the function that describes the former Nth-order model of the single-span conductor: , sin Let N = 1, and the anti-galloping device be placed at the midpoint of the span.We can obtain the ordinary differential equations for the nonlinear dynamics description of the coupling system motion.
( ) Seeing the common 1:1 internal resonance as a typical example, the first-order harmonic solution is obtained using the harmonic balance method.Thus, the solution and the main frequency are expressed as follows: where a i and φ i are the amplitudes and initial phase angles of the first-order harmonic solution, respectively; i = 1 (conductors), 2 (anti-galloping device).Equations ( 10) and ( 11) are substituted into Equation (8) to obtain an expression for the first-harmonic term.
a i , and φ i could be solved by Equation ( 12) to obtain the first-order harmonic solution of the system.As the above equations contain higher powers of unknown quantities, it isn't very easy to obtain a direct solution for a i and φ i .We programmed the corresponding process in symbolic computation on software to analyze and solve the equations when the relative parameters are constant.

Device prototype
Optimization measures are required to address the inefficiency of the traditional ungrounded NES structure for nonlinear dynamic control [30].To harvest and exploit more energy while increasing the damping structure as much as possible, we applied the particle damping technique to design the oscillator's structure in a "box + particle" form.This design increases friction to convert mechanical energy into heat energy.The test showed that when the total mass of the oscillator was 0.52 kg, the antigalloping device damping coefficient was 0.17 kg/s.The main structural design is shown in Fig. 3.
(a) Anti-galloping device prototype (b) Internal particles (steel balls, Φ 6 mm) Fig. 3. Anti-galloping device with cubic stiffness and particle damping The key components of the anti-galloping device are an elastic rope and oscillator-containing particles.We investigated whether a geometrically nonlinear characteristic and particle-damping structure could widen the frequency band to harvest and consume energy efficiently.The investigation was performed to prevent the UHV conductors and fittings from being damaged by drastic vibrations under eccentric ice and aerodynamic loadings.Under gravity, the initial axial tension of the elastic rope must be greater than 0; hence, the anti-galloping device we developed must demonstrate quasi-zero linear natural frequency.To counteract the influence of gravity on the oscillator, actual length, stiffness, and initial axial tension must be set to approximately 0.084 m, 212.76 N/m, and 0.01 N, respectively.

Laboratory experiments
Test data and quantitative analysis showed that the linear elastic restoring force was much lower than other forces; hence, we neglected it in the subsequent modeling and analysis.To verify the anti-galloping effect of the device, we performed a harmonic excitation test using eight-bundled LGJ-500/35 conductors.We compared the response amplitude of the conductors before and after the installation of the anti-galloping device.During the test, a frequency range of 0-2 Hz was considered for the exciter.We obtained a steady response amplitude at each frequency to determine the desirability of the anti-galloping effect of the proposed device.Fig. 4 illustrates the installation of the test span in the laboratory.The test steps are outlined as follows: 1) A group of eight-bundled LGJ-500/35 conductors was installed in the test system with a 140-m span; 2) The spacer was positioned, and an acceleration sensor was attached to it (see Fig. 4); 3) The three-phase motor was adjusted, and the force amplitude of the exciter was 80 N in the vertical DOF on the whole span before changing to 100 N. The exciting force frequency ranged from 0 to 2 Hz at 0.2 Hz steps.The amplitude of each frequency was maintained at 20 s to obtain stable data; the exciting frequency was quickly adjusted to the subsequent point; 4) The displacement waveform of the measured point was recorded on a computer; 5) The anti-galloping device with cubic stiffness and particle damping on the spacers was installed, and another acceleration sensor was attached to the oscillator.Step 3 was repeated at this point; 6) The data were then processed.Subsequently, the response amplitude data of the conductors before and after installing the anti-galloping device were compared to verify the harvesting and energy consumption-ability of the anti-galloping device in the multi-bundled conductors coupling system.
Approximate analytical solutions and amplitude-frequency response curves of the coupling system equations were obtained by Equation ( 12) after substituting the parameters of the multi-bundled conductors.

Discussion
By comparing the results presented in Fig. 5 and Fig. 6, we obtain the following conclusions: 1) The results of the mathematical model we established are consistent with the experimental results, demonstrating the validity of the model and indirectly validating the accuracy and credibility of the nonlinear equations that describe the motions of conductors based on the curved beam model; 2) If we consider cubic stiffness as the main stiffness, the anti-galloping device with particle damping has no obvious resonance peaks in the frequency range from 0 to 2 Hz, and each frequency effectively harvests and consumes energy within the coupling system; 3) The damping of the anti-galloping device cannot be ignored because of the introduction of the particle damping technology, which plays a significant role in reducing the steady-state response amplitude of the eight-bundled LGJ-500/35 conductors to protect the transmission line system; 4) At an appropriate installation position (the midpoint of an antinode), the anti-galloping device can significantly inhibit the vibration phenomenon of UHV multi-bundled conductors, regardless of the galloping or type of vibration.
The structure of the anti-galloping device has excellent properties, such as low additional mass, good economy, and absorption in wide-frequency broadband; hence, the device is suitable for transmission lines in regions with frequent high-level galloping and power lines that are threatened by other wind-induced vibrations.

Conclusions
We established a vertical mathematical model coupled with a conductor and an oscillator with cubic stiffness for applying the cubic stiffness and particle damping technologies to transmission engineering.The vibration properties of an anti-galloping device with cubic stiffness and particle damping were analyzed in detail by combining experimental and theoretical analyses.The results showed that in a wide frequency range, the design of the anti-galloping device could significantly inhibit stable-state amplitude, indicating that the axial dynamic tension of conductors can also be reduced.The advantages of the anti-galloping device designed in this study were verified in a test line experiment in the laboratory.
This study provides theoretical and engineering bases for applying NES theory and particle damping technology in the anti-wind-induced vibration of overhead transmission lines.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 2 .
Fig. 2. Distributions of eight-bundle conductors and anti-galloping devices in the test span (all devices are under the following test as in )

Fig. 4 .
Fig. 4. Installation of test span in the laboratory