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Galloping of an iced transmission line subjected to a moderating airflow has been analysed in this study, and a new form of galloping is discovered both theoretically and experimentally. The partial differential equations of the iced transmission line are established based on the Hamilton theory. The Galerkin method is then applied on the continuous model, and a discrete model is derived along with its first two in-plane and torsional modes. A trapezoidal wind field model is built through the superposition of simple harmonic waves. The vibrational amplitude is generally observed to be more violent when the wind velocity decreases, except in the 2^{nd} in-plane mode. Furthermore, the influence of the declining wind velocity rates on galloping is analysed using different postdecline wind velocities and the duration of the decline in wind velocities. Subsequently, an experiment has been carried out on a continuous model of an iced conductor in the wind tunnel dedicated for galloping. The first two in-plane modal profiles are observed, along with their response to the moderating airflow. Different declining rates of the wind velocity are also verified in the wind tunnel, which show good agreement with the results simulated by the mathematical model. The sudden increase in the galloping amplitude poses a significant threat to the transmission system, which also improves the damage mechanism associated with the galloping of a slender, a long structure with a noncircular cross-section.

Long, slender structures are widely employed in engineering applications wherein the external environment plays a major role, including high-voltage transmission lines, suspended cables, inclined cables, and deep-sea mooring. Under adverse weather conditions such as rainfall or snowfall, the cross-section of these structures becomes noncircular, which causes instable aerodynamic forces in the airflow and leads to galloping. Since these structures exhibit a light mass, a small damping ratio, and nonlinear factors with regard to their geometry and aerodynamic forces, the galloping trend is observed to be diverse. The galloping of iced transmission lines is the main research topic in this paper.

Since wind flow is the main factor that results in galloping, the mean wind velocity has been extensively analysed to acquire aerodynamic coefficients [

Wind tunnel test is a reliable method to analyse the impacts of the wind loads acting on the conductor. On one hand, these tests mainly focused on the aerodynamic coefficients with different cross-sectional shapes [

Therefore, galloping of the iced transmission line subjected to a moderating wind field is analysed, and a wind tunnel test is performed on a continuous conductor model in this study. Theoretical analyses have been elucidated in Section

The transmission line is modelled as a flexible cable with length _{0} represents the initial configuration as per the following expression:

Theoretical model of the iced transmission line. (a) Configuration. (b) Cross-section.

Γ denotes the configuration when galloping occurs. _{r} represents the wind velocity relative to the conductor. The relationship between _{r} can be deduced as follows:

_{L} and _{D} are the aerodynamic lift and drag forces acting on the conductor under the relative wind, respectively; the sum of their projection on the _{y}, which is represented along with the torsional aerodynamic force _{y} and _{M} are the aerodynamic coefficients acquired from the wind tunnel test [_{0} is the initial angle of the ice accretion. The partial differential equations of the iced transmission line, which have been established based on the Hamilton theory, are of the following form [

Parameters of iced transmission lines.

Notation | Parameter | Value |
---|---|---|

Cross-sectional area | 423.24 mm^{2} | |

Diameter of a bare cable | 0.0286 m | |

Elasticity modulus | 4.78 × 10^{10} N·m^{−2} | |

Torsional stiffness | 101 Nm | |

Rotational inertia | 0.333 × 10^{−3}kg·m^{2} | |

Cable length | 125.88 m | |

Cable density | 2.379 kg·m^{−1} | |

_{0} | Initial tension | 30 kN |

_{0} | Initial angle of ice accretion | 0.698 rad |

Air density | 1.29 kg·m^{−3} | |

In-plane damping ratio | 0.01 | |

_{θ} | Torsional damping ratio | 0.04 |

The Galerkin method is employed to transform the partial differential equations into ordinary differential equations. Consequently, the following assumptions are made:_{i}(_{k}(_{θ1,} and _{θ2} are the natural frequencies of the first two in-plane and torsional modes, respectively. _{1}–_{4}, _{1}–_{4}, and _{1} and _{2} are the integral coefficients, as seen in Appendix.

Trapezoidal wind field is employed in this study to analyse the galloping feature, especially when the wind velocity declines. Consequently, the wind field used is expressed as follows:_{0} is the mean wind velocity shown in Figure _{1}–_{2}, and _{1} represents the previous mean wind velocity. The postdecline wind velocity is denoted by _{2}; _{d} is the elapsed time of the wind decline.

Model of trapezoidal wind field.

The Runge–Kutta method has been employed to simulate equations (^{st} in-plane mode takes about 1500 s to achieve its maximum values. This denotes that galloping requires energy accumulation along with a constant mean wind velocity.

Time histories (0–3000 s). (a) 1^{st} in-plane mode. (b) 2^{nd} in-plane mode. (c) 1^{st} torsional mode. (d) 2^{nd} torsional mode.

Nevertheless, all the modal amplitudes (except that of the 2^{nd} in-plane) sharply increase within 2 s when the wind velocity declines. A sudden increase in the amplitude poses a greater threat to the transmission system; however, no obvious impacts are observed in the 2^{nd} in-plane mode. As shown in Figure ^{st} in-plane mode comprises one crest, while the 2^{nd} in-plane mode has two crests and one node, which are also observed in the subsequent experiments. The blue lines in Figure ^{st} in-plane vibration, while smaller than those in the 2^{nd} in-plane vibration.

Time histories (1500–1650 s). (a) 1^{st} in-plane mode. (b) 2^{nd} in-plane mode. (c) 1^{st} torsional mode. (d) 2^{nd} torsional mode.

Galloping profiles. (a) 1^{st} in-plane mode. (b) 2^{nd} in-plane mode.

Since galloping exhibits a synchronous behaviour for each mode (except the 2^{nd} in-plane mode) during the decline in wind velocity, the 1^{st} in-plane mode is used in the subsequent simulation due to its significant threat to the transmission system. The influence of the declining wind velocity rate on the galloping behaviour is discussed in this section. Herein, the previous wind velocity _{1} and the elapsed time _{d} are kept constant, and the postdecline wind velocity _{2} is selected as 0 m/s, 0.5 m/s, and 1 m/s (Figure _{2} value leads to larger wind velocity differences, thereby eliciting larger amplitude increase. Subsequently, _{d} is selected as 0.5 s, 2 s, and 5 s for simulating the decrease in wind velocity from 8 m/s to 0 m/s. Figure

Time history of the first in-plane modal galloping with wind speed reduction at _{1} = 8 m/s and _{d} = 2 s. (a) _{2} = 1 m/s. (b) _{2} = 0.5 m/s. (c) _{2} = 0 m/s.

Time history of the first in-plane modal galloping with wind speed reduction at _{1} = 8 m/s and _{2} = 0 m/s. (a) _{d} = 0.5 s. (b) _{d} = 2 s. (c) _{d} = 5 s.

An iced conductor experiment is performed on the galloping platform (Figure

Galloping platform.

A steel wire rope is used to simulate the continuous transmission line with the length of 4.8 m. As shown in Figure

Cross-section of the model.

The wind tunnel is streamlined, as shown in Figure

Wind tunnel.

Contact sensors have significant impacts on the measurement due to the light mass of the conductor model. Therefore, two noncontact laser sensors (HL-G103-S-J) are used under the cable model. To verify the theoretical results mentioned above, the first two modal profiles should be measured. Based on the existing theory, one sensor should be placed at a point that represents 1/2 of the cable length, where the largest amplitude of the 1^{st} mode occurs. The other sensor is installed at a point that represents 1/4 of the cable length, where the maximum displacement of the 2^{nd} mode appears. Due to the limitations associated with the experimental conditions, the measurement of torsional vibration is yet to be attained.

When the mean wind velocity is 0.85 m/s and 1.207 m/s, the iced conductor model is observed to undergo the 1^{st} and 2^{nd} in-plane modal galloping, respectively, whose profiles are shown in Figure ^{st} in-plane modal galloping exhibits a crest and valley at a cable length of 1/2, which becomes a node in the 2^{nd} in-plane modal galloping. Further, the crest and valley of the 2^{nd} in-plane mode show their appearances at cable lengths of l/4 and 3/4. The profiles are consistence with those depicted by the theoretical equation in Figure

Galloping profile captured in the wind tunnel test. (a) 1^{st} in-plane mode (^{nd} in-plane mode (

The galloping data of the first two in-plane modes are then collected by the sensor at cable lengths of 1/2 and 1/4, which are shown in Figures ^{nd} in-plane mode is twice that of the 1^{st} in-plane mode (Figures ^{st} in-plane amplitude is observed when the wind velocity decreases to 0 m/s, which is 1.49 times the previous value (Figure ^{nd} in-plane mode decays with the decline in wind velocity (Figure

Galloping data obtained from the laser sensor at 1/2 cable length at _{1} = 0.85 m/s and _{2} = 0 m/s. (a) Amplitude-frequency curves. (b) Time history.

Galloping data obtained from the laser sensor at 1/4 cable length at _{1} = 1.207 m/s and _{2} = 0 m/s. (a) Amplitude-frequency curves. (b) Time history.

Since the control gear of the motor fans can only adjust the wind velocity, the durations for which the wind velocity changes are yet to be controlled and measured. Therefore, different values of the postdecline wind velocity _{2} are used to verify the theoretical results. During the test, the wind velocity is allowed to rapidly decline in order to avoid the influence of the elapsed time _{d}. As shown in Figure _{2} value causes a high declining rate of the wind velocity, which leads to a more violent galloping process. This characteristic of galloping is consistent with that obtained via numerical simulation.

Time history of the first in-plane modal galloping with wind speed reduction at _{1} = 0.85 m/s. (a) _{2} = 0.664 m/s. (b) _{2} = 0.533 m/s. (c) _{2} = 0 m/s.

In this study, a mathematical galloping model of an iced transmission line has been established under a moderating airflow, and its first two in-plane and torsional modes are elucidated. Numerical simulations are used to determine the response of galloping to the declining wind velocity. A wind tunnel test, which has been designed for a continuous model of the iced conductor, is conducted to validate the simulated results mentioned above. The theoretical analysis and experimental data are in accordance with each other qualitatively. The main results of this study are summarized as follows:

Galloping can only occur when energy accumulation takes place under a constant mean wind velocity. Meanwhile, a sharp decline in the wind velocity causes violent galloping, especially for the torsional modes; however, this decline in the wind velocity rarely influences the 2^{nd} in-plane mode.

Different postdecline wind velocities and the elapsed times corresponding to the decline in wind velocities are used to interpret that a higher declining rate of the wind velocity leads to a more violent galloping.

The simulated results qualitatively agree with the experimental observations for the first two in-plane modes, with regard to the galloping profile, vibrational frequency relationship, galloping feature of each mode under a moderating airflow, and the influence of the declining rate of the wind velocity on the galloping. Therefore, the mathematical model is verified to be feasible and can be used for simulating and predicting the galloping of long, slender structures with noncircular cross-sections.

The mechanism governing the increase in amplitude during the decline in wind velocity is still unclear. However, energy release has been concluded as the reason on account of the galloping trend responding to the wind velocity difference and elapsed time; this aspect will be further researched in subsequent studies.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (no. 51808389) and the Natural Science Foundation of Tianjin City (nos. 18JCQNJC08000 and 18JCQNJC75300).