Galloping Mode Research Articles

Mass ratio effects on flow-induced vibrations of an equilateral triangular prism

In this paper, we numerically studied the effects of mass ratio (m∗) on three identified responses of a transversely oscillating equilateral triangular prism in Chen et al. (J. Fluids Struct., vol. 97, 2020, 103099). Three angles of attack, i.e. α=10°, 40° and 60°, were selected, with α=0° being the configuration that one vertex faces upstream. Our results indicated that m∗ exerts significant influences on both responses and wakes. At α=10°, a higher m∗ leads to a significant shrink of the large-amplitude vibration region while the largest amplitude occurs at a higher reduced velocity. At α=40°, increasing m∗ leads to the appearance of three distinct responses successively. At α=60°, soft galloping dominates at m∗=2–10, while hard galloping where an initial disturbance is required occurs at m∗=20. For the wake, only the 2S mode is observed at α=10° owing to the small amplitude, while depending on the response, P+S, 2S, competition, and mS+nS (miS+niS) modes can dominate at α=40°. At α=60°, 2S and mS+nS (miS+niS) modes occur. Furthermore, dynamic mode decomposition is applied to extract the vortex structures of different frequencies. Results indicate that von Karman and galloping modes coexist in the region where the galloping phenomenon appears. With increasing amplitude, the scale of the galloping mode increases, while the von Karman mode is almost unaffected. Flow physics for identified responses disclose that both vortex-induced vibration and galloping are closely related to the vortex formation process. The sustenance of the large amplitude in galloping is associated with the continuous shear layer reattachments since multiple vortices are shed from the prism.

Strong flame acceleration and detonation limit of hydrogen-oxygen mixture at cryogenic temperature

A series of experiments were carried out in a closed tube at cryogenic temperature (77 K) for hydrogen-oxygen mixtures. Flame propagation speed and overpressure were measured by optical fibers and pressure sensors, respectively. The first and second shock waves were captured in the cryogenic experiments, although the shock waves always precede the flames in all cases indicating the absence of stable detonation. However, strong flame acceleration was observed for all situations, which is consistent with the prediction by expansion ratio and Zeldovich number. Besides, the tube diameter and length are also critical for flame acceleration to supersonic. All the flames in this work accelerate drastically reaching the C-J deflagration state. But at 0.4 atm, only fast flame is formed, while at higher initial pressures, the flame further accelerates to a galloping mode manifesting a near-limit detonation, which could be indicated by the stability parameter χ.

Mode-based energy transfer analysis of flow-induced vibration of two rigidly coupled tandem cylinders

Two rigidly coupled cylinders are commonly available in engineering practices, which are always subjected to flow-induced vibration (FIV) with large oscillating amplitude due to the complex gap flow. However, their FIV features and the underlying mechanism have not been clearly understanded. In this study, the FIV of two rigidly coupled square cylinders in a tandem arrangement was numerically investigated for Reynolds numbers 100 and 200 and gap L/D = 2.0 and 6.0 in a two-dimensional framework. The dynamic response and flow structures are first studied. Mode-based energy transfer analysis was developed based on proper orthogonal decomposition. The energy transfer between the cylinders and the coherence modes was then analyzed to uncover the underlying mechanism of the FIV. The results reveal that the soft lock-in phenomenon was observed for all cases. When the gap L/D = 2.0, the dynamic response is always smaller than a stationary cylinder and all the wake structures show “2S” modes, which results from the stabilized effects of the upstream cylinder. At L/D = 6.0, the oscillating amplitude is much higher than that of a single square cylinder, due to the vigorous interaction between gap flow and the cylinders. As Reynolds number increased from 100 to 200, both vortex-induced vibration (VIV) and galloping co-exist, while the VIV response was reduced. Concerning the energy transfer, for L/D = 2.0, the first mode pair induced by Karman vortex shedding dominated the wake flow. The primary mode governed by UC stabilized the vibration, while the second mode tended to excite the FIV. For L/D = 6.0 in the galloping branch, the first two modes belonged to the vortex-induced modes, and the third mode represented the galloping mode. The galloping mode tended to excite the flow-induced vibration, while it was contrary for the vortex-induced modes.

Research on Monitoring the Transmission Line Tension and Galloping Based on FBG Fitting Sensor

A method of monitoring the transmission line galloping based on fiber Bragg grating (FBG) fitting sensor is proposed and verified by video monitoring method in the overhead transmission line experiment. The proposed method can be used to measure the line tension, the galloping mode and amplitude in horizontal and vertical directions in real-time, simultaneously. The functions of this method were confirmed by the 92 hours transmission line galloping results and the influence of the wind speed and direction to the galloping measurement were also analyzed. Compared with the video monitoring method, it has the advantage of being able to measure the tension and galloping mode of line, higher resolution of galloping amplitude monitoring and real-time monitoring regardless of weather condition. This method can be widely used in transmission line monitoring and has positive significance to promote smart grid.

Fluid–structure interactions of a circular cylinder in a stratified fluid

Abstract

Flow-induced vibrations of an equilateral triangular prism at various angles of attack

Flow-induced vibrations (FIV) of an equilateral triangular prism are numerically studied using the immersed boundary method in a parametric space of α=0°–60° and U∗=U∕fnD=1–20, with α=0°(60°) representing the configuration of one vertex facing upstream (downstream). The Reynolds number based on the effective edge length perpendicular to the incoming flow is Re = 200 and the mass ratio is m∗=4m∕3ρD2=2.0. In above, U is the incoming flow velocity, fn is the natural frequency of the prism, D is the edge length of the prism, m is the prism mass and ρ is the fluid density. Three different vibration regimes, i.e., VIV at α=0°–25°, combined VIV and galloping at α=30°–40°, and galloping at α=45°–60°, are confirmed. In the VIV regime, the vibration amplitude is relatively small and the vibration frequency linearly increases with U∗, signifying the non-locked response. In the combined VIV and galloping regime, the amplitude is determined by the competition of the VIV and galloping modes, with large amplitude appearing when the galloping mode dominates. In the galloping regime, the vibration amplitude monotonically increases with U∗, with a lower-than-unity vibration frequency. Moreover, seven wake modes are observed in the examined α and U∗ ranges: 2S mode, 2S(P) mode, P+S mode, P+S/2S mode, the competition mode (CM), and the galloping modes (miS+niS and mS+nS). In CM, the VIV and galloping modes compete each other. In the galloping modes, the vortices are arranged in a ‘V’ shape. The effects of the streamwise vibration on the transverse responses is significant only in the combined VIV and galloping regime.

Prediction method for galloping features of transmission lines based on FEM and machine learning

Parameter study on galloping of iced transmission lines with different structural, ice and wind parameters is carried out by means of the finite element method (FEM), and a dataset is then created based on the numerical results. The random forests algorithm is used to set up a classification model to predict the galloping mode and vertical vibration mode. Using the span length, initial conductor tension, ice thickness, initial wind attack angle, wind velocity and the galloping mode as well as the vertical vibration mode obtained by the classification model as the input variables, a back-propagation (BP) neural network model is created to predict the galloping features including the frequencies, vibration amplitudes and the maximum conductor tension. The dataset created based on the parameter study is divided into two sub-datasets for training and testing of the two machine learning models respectively. The effects of the input variables on the output variables are quantified with a feature importance analysis for the classification model and a global sensitivity analysis for the feature prediction model respectively. The two models are finally integrated into one to predict galloping features efficiently and quickly. Making use of the proposed method, an early warning system for galloping of transmission lines may be able to be set up in the future.

Near-limit detonations of methane–oxygen mixtures in long narrow tubes

The near-limit gaseous detonation behavior of three different methane–oxygen mixtures was investigated. Experiments were performed in transparent tubes of three different inner diameters (d). Due to the relatively large tube length l ( $$\frac{l}{d} > 2500$$ except $$\frac{l}{d} > 1000$$ for the largest diameter), the tube was arranged in a spiral configuration for the convenience of testing. Photodiodes were spaced at uniform intervals along the tube to provide a high-resolution velocity measurement, such that up to eight cycles of the galloping mode were registered. From the velocity histogram and the probability distribution function, a bimodal behavior was observed in all galloping regimes for different unstable mixtures, with dominant modes near 70% of the Chapman–Jouguet detonation velocity ( $$0.7D_{\mathrm{CJ}}$$ ) and $$D_{\mathrm{CJ}}$$ . With decreasing pressure, the lower-velocity mode became more prevalent until the failure of detonation. The range of initial pressures, within which galloping detonations were observed, decreased rapidly with increasing tube diameter and with increasing mixture stability. These results suggest that both the instability and the boundary effect influence the existence of galloping detonations. The normalized wavelength of the galloping cycle ( $$\frac{L}{d}$$ ) was in the range of 200–450 for the three different mixture compositions and exhibited a general trend that the wavelength increased with decreasing initial pressure.

Nonlinear aeroelastic behavior of a base-isolated beam under steady wind flow

A homogeneous continuous visco-elastic shear-beam, describing the dynamics of base-isolated tall buildings exposed to a uniformly distributed steady wind flow, is studied. The shear beam is constrained at the bottom end by a nonlinear visco-elastic device and free at the top end. The aeroelastic effects, responsible for self-excitation, are evaluated via the quasi-static theory. The occurrence of Hopf bifurcation is detected. Critical and post-critical behavior is analyzed by applying a perturbation scheme. The critical wind velocity and the associated complex galloping mode are found. The steady value of the oscillation amplitude on the stable limit-cycle and its stability are evaluated as function of the mean wind velocity. The mechanical performances of the structure are investigated in according to the effectiveness of the visco-elastic isolation system. A passive controller is proposed to increase the galloping wind velocity and to reduce the amplitude of the limit-cycle.

Mode competition in galloping of a square cylinder at low Reynolds number

Galloping is a type of fluid-elastic instability phenomenon characterized by large-amplitude low-frequency oscillations of the structure. The aim of the present study is to reveal the underlying mechanisms of galloping of a square cylinder at low Reynolds numbers ($Re$) via linear stability analysis (LSA) and direct numerical simulations. The LSA model is constructed by coupling a reduced-order fluid model with the structure motion equation. The relevant unstable modes are first yielded by LSA, and then the development and evolution of these modes are investigated using direct numerical simulations. It is found that, for certain combinations of$Re$and mass ratio ($m^{\ast }$), the structure mode (SM) becomes unstable beyond a critical reduced velocity$U_{c}^{\ast }$due to the fluid–structure coupling effect. The galloping oscillation frequency matches exactly the eigenfrequency of the SM, suggesting that the instability of the SM is the primary cause of galloping phenomenon. Nevertheless, the$U_{c}^{\ast }$predicted by LSA is significantly lower than the galloping onset$U_{g}^{\ast }$obtained from numerical simulations. Further analysis indicates that the discrepancy is caused by the nonlinear competition between the leading fluid mode (FM) and the SM. In the pre-galloping region$U_{c}^{\ast }<U^{\ast }<U_{g}^{\ast }$, the FM quickly reaches the nonlinear saturation state and then inhibits the development of the SM, thus postponing the occurrence of galloping. When$U^{\ast }>U_{g}^{\ast }$, mode competition is weakened because of the large difference in mode frequencies, and thereby no mode lock-in can happen. Consequently, galloping occurs, with the responses determined by the joint action of SM and FM. The unstable SM leads to the low-frequency large-amplitude vibration of the cylinder, while the unstable FM results in the high-frequency vortex shedding in the wake. The dynamic mode decomposition (DMD) technique is successfully applied to extract the coherent flow structures corresponding to SM and FM, which we refer to as the galloping mode and the von Kármán mode, respectively. In addition, we show that, due to the mode competition mechanism, the galloping-type oscillation completely disappears below a critical mass ratio. From these results, we conclude that transverse galloping of a square cylinder at low$Re$is essentially a kind of single-degree-of-freedom (SDOF) flutter, superimposed by a forced vibration induced by the natural vortex shedding. Mode competition between SM and FM in the nonlinear stage can put off the onset of galloping, and can completely suppress the galloping phenomenon at relatively low$Re$and low$m^{\ast }$conditions.

A Continuum Approach to the Nonlinear In-Plane Galloping of Shallow Flexible Cables

The aeroelastic stability of horizontal, suspended, shallow, iced cables is studied via a continuum model. Both external and internal damping, consistent with the Rayleigh model, are taken into account. The quasi-static theory of the aerodynamic forces is applied. An in-plane nonlinear model of galloping is formulated, displaying the importance of internal damping, both on the critical velocity and on the limit-cycle amplitude. A perturbation procedure is developed for nonlinear analysis in nonresonant conditions (monomodal galloping). The modification of the galloping mode due to quadratic nonlinearities is studied, and its real or complex character is discussed.

Galloping behavior analysis of transmission line with thin ice accretions

A dynamic model of a galloping transmission line able to describe for the coupling of its longitudinal, in-plane, out-of-plane and torsional vibrations is established. It also considers the effects of geometrical nonlinearity and aerodynamic nonlinearity. By the static configuration, the reduced model is obtained. Then, the equations of motion are obtained through the Galerkin method. It contains two in-plane, two out-of-plane and two torsional components. By numerical calculation, the maximum amplitudes at wind speeds are drawn and the galloping behavior of transmission line with thin ice accretions is analyzed. The obtained results show that the second galloping mode is more triggered. The double-mode galloping occurs in all motions, in which the maximum amplitude is bigger than in single-mode galloping. And the double-mode galloping presents the track of inclined ‘8’ in longitudinal direction.

Mitigation of conductor line galloping by a direct cable-connection to non-conductive composite power pylons

Steel lattice towers with suspended insulator strings are typically used to carry high-voltage overhead transmission lines. The installation of non-conductive power pylons made of glass fibre reinforced plastics enables a direct cable-pylon connection, as the composite structure acts as an unibody insulator. At the same time, wind-induced vibrations, such as the severe cable vibration phenomenon galloping, will consequently be directly transferred to the slender composite mast structure, potentially leading to extensive damage. The aim of the study is therefore to investigate the galloping behaviour of iced conductor lines with regard to different cable support conditions. Furthermore, additional damping in the composite power pylon structure is assumed to mitigate conductor line galloping and therefore reduce the risk of phase flash-overs between adjacent conductor lines. A numerical galloping simulation is carried out in order to evaluate the effect of a rigid cable-pylon connection with enhanced damping properties on the cable vibration amplitudes. A pylon-cable system, consisting of 3×300 m spans, is investigated. It was found that the support conditions of the conductor lines have a significant influence on the galloping mode, the vibration amplitudes and the orientation of the characteristic galloping ellipse. The addition of damping to the pylon decreases the vibration amplitudes slightly and leads to a re-orientation of the galloping ellipse.

Experimental investigation of near-limit gaseous detonations in small diameter spiral tubing

The near-limit propagation of gaseous detonations in seven explosive mixtures with different reaction sensitivities is investigated. Experiments were performed in transparent tubing of four different inner diameters with relatively long tubing length (l/d > 2500 except l/d > 1000 for the largest diameter) arranged in a spiral configuration. Up to 83 fiber optics spaced at regular intervals along the tube were used to provide high resolution velocity measurement. Up to 8 cycles of the galloping mode were recorded, and the spiral boundary did not influence the persistence of galloping detonations. Results confirm that for mixtures with increasing argon dilution, making the detonation more stable with regular cellular pattern, the occurrence of galloping detonation diminishes. For stable mixtures with sufficiently large amount of argon dilution (e.g., stoichiometric C2H2/O2 with 70%Ar), the galloping mode was not observed in all tested tubing. For unstable mixtures, smaller diameters were necessary to achieve the galloping mode. The range of initial pressures, within which galloping detonations were observed decreases rapidly with increasing tubing diameter. These results suggest that both the instability and the boundary effect are essential for galloping detonations. From the velocity histogram and the probability distribution function, a bimodal behavior was also observed in all galloping regimes of different unstable mixtures, with dominant modes near half of the Chapman-Jouguet detonation velocity (DCJ) and DCJ. With decreasing pressure, the lower velocity mode became more prevalent until no more galloping detonation occurred. The normalized wavelength of the galloping cycle (L/d) ranges from 250 to 450 within experimental variation. Nevertheless, few results show a clear minor trend that the wavelength increases with decreasing initial pressure. By looking at the velocity amplitude in the galloping cycle, the lower value as well as the average is relatively constant, while the upper peak has larger fluctuations.

The three-dimensional structure of a detonation wave propagating in a round tube with orifice plates

Experiments were performed to study detonation propagation in a round tube equipped with repeating orifice plates, equally spaced at two tube diameters. Tests were performed with stoichiometric hydrogen–oxygen over a range of initial pressures up to 60 kPa absolute. The self-luminous high-speed video was used to visualize the detonation for two sizes of orifice plate, i.e., 50 and 75% cross-sectional area blockage. The high-speed videos were used to obtain the average combustion wave velocity, from which the detonation propagation regime was determined, and the propagation mechanism for each orifice plate geometry was identified. For the 50% blockage ratio orifice plate tests, the detonation limit was measured to be 6 kPa, and detonation propagation was characterized by cycles of detonation failure and re-initiation after each orifice plate, resulting in a CJ detonation velocity deficit of the order of 20%. The high-speed video, captured at an oblique camera angle, showed that the detonation wave forms at hot spot(s) on the tube wall just downstream from the orifice plate, following shock reflection. For the 75% blockage orifice plates, the detonation propagation limit was measured to be 20 kPa. For initial pressures between 20 and 40 kPa, a galloping detonation mode was observed, which was not observed in a previous study performed in the same tube and orifice plates but with one-tube-diameter orifice plate spacing. The galloping mode consists of repeating cycles of fast-flame propagation between successive obstacles, followed by detonation wave propagation between the following pair of orifice plates. The experimental results were used to explain why the well-established requirement that the orifice accommodates one cell for detonation propagation (i.e., $$d/\lambda >1$$ ) is sufficient for low blockage ratio orifice plates; however, for high blockage ratio plates it is necessary but not sufficient.

Electromechanical decoupled model for cantilever-beam piezoelectric energy harvesters with inductive-resistive circuits and its application in galloping mode

The electromechanical decoupled distributed parameter model for cantilever-beam piezoelectric energy harvesters with inductive-resistive circuits is proposed. The modified natural frequency and electrical damping for series and parallel inductive-resistive cases are derived, which are functions of the first natural frequency of the cantilever beam, capacitance of the piezoelectric layers, load resistance, inductance and electromechanical coupling term. As a demonstration, we apply the decoupled model to cantilever-beam piezoelectric energy harvesters operated in the galloping mode. The average harvested power is derived as an algebraic expression of the electrical damping in addition to the wind speed, aerodynamic parameters of the bluff body and mechanical properties of the cantilever beam. Besides these impacts, the amplitude of the tip displacement also depends on its modified natural frequency. The electromechanical decoupled model and its analytical solutions are confirmed by the numerical solutions of the coupled model for the galloping mode. The theoretical expressions for the maximal harvested power and corresponding tip displacement at the optimal electrical damping are then developed. The cantilever-beam piezoelectric energy harvester with the inductive-resistive circuit has multiple solution sets of the modified natural frequency and electrical damping. Any optimal electrical damping can be realized by series or parallel inductive-resistive circuit, which cannot be accomplished by pure resistive circuit. By introducing the inductance to the circuit of the galloping piezoelectric energy harvester, the performance of such a system is improved with larger maximal harvested power at high wind speed and smaller amplitude of the tip displacement. This study provides a theoretical approach to capture the intrinsic effects of the inductance in addition to the load resistance on the performance of cantilever-beam piezoelectric energy harvesters.

Investigation into galloping characteristics of iced quad bundle conductors

A nonlinear finite element method for galloping of iced bundle conductor transmission lines is presented. Wind tunnel tests for galloping of quad bundle conductor segments with crescent-shaped and sector-shaped ice models were carried out, and validity of the finite element method is demonstrated by the consistency of the galloping responses determined by the numerical method and those measured by the wind tunnel tests. Numerical models of four sections of iced transmission lines with different Irvine parameter λ2, which determines if internal resonance exists, are set up. Galloping of the iced transmission lines in steady wind flow and stochastic wind field is numerically simulated by the presented finite element method. Based on the numerical results, galloping characteristics of the iced transmission lines with and without internal resonance in steady and stochastic wind fields are discussed. It is found that the galloping mode and frequency of an iced quad bundle conductor transmission line in a stochastic wind field are similar to those in a steady wind flow, but the galloping amplitude may be different. Moreover, the Irvine parameter λ2 may be used to estimate some galloping characteristics, such as galloping mode and coupling movement induced by internal resonance.

Propagation of near-limit gaseous detonations in small diameter tubes

In this study, detonation limits in very small diameter tubes are investigated to further the understanding of the near-limit detonation phenomenon. Three small diameter circular tubes of 1.8, 6.3, and 9.5 mm inner diameters, of 3 m length, were used to permit the near-limit detonations to be observed over long distances of 300 to 1500 tube diameters. Mixtures with high argon dilution (stable) and without dilution (unstable) are used for the experiments. For stable mixtures highly diluted with argon for which instabilities are not important and where failure is due to losses only, the limit obtained experimentally appears well to be in good agreement in comparison to that computed by the quasi-steady ZND theory with flow divergence or curvature term modeling the boundary layer effects. For unstable detonations it is suggested that suppression of the instabilities of the cellular detonation due to boundary conditions is responsible for the failure of the detonation wave. Different near-limit propagation regimes are also observed, including the spinning and galloping mode. Based on the present experimental results, an attempt is made to study an operational criterion for the propagation limits of stable and unstable detonations.

Der Zusammenhang zwischen Lokomotionsweise, Begattungsstellung und Penislänge bei Säugetieren1

Penis length, copulation and locomotion: Their relationship to each other in Mammals A relationship between the mode of locomotion, copulatory position and length of the male copulative organ has been found in all groups of Mammalia. In this paper particular emphasis is given to the orders of the Testiconda, while the Testiphaena receive only brief account (for explanation of terms see Frey 1991 a). Results from my previous antomical study are utilized that are essential for assessing the respective modes of locomotion (Frey 1991b). Information on penis length and copulatory position are taken from the literature and evaluated. Small and middle-sized Testiphaena, which represent the majority of mammals, are capable of a dynamic sagittal bending in the trunk, which is evidenced both in the galloping mode of locomotion and in the usual mammalian copulatory position (mounting). A sagittal bending of the trunk enables the male to bring his genital region into close proximity of the female's genital opening. In this case, a short penis is sufficient to ensure sperm transfer. The construction of the trunk in the Testiconda (excepting the Hyracoidea) entirely or nearly entirely prevents the sagittal bending. This is largely due to rigidity of the lumbar region or insufficient (dynamic) muscular control. The immobilization and/or the functional weakness of the lumbar region are derived from different anatomical conditions. 1. Shortening of the lumbar region, e.g., Tachyglossidae, Elephantidae, Sirenia; 2. Tightening of the lumbar region by tendons and muscles, e. g., Macroscelididae, Cetacea; 3. Xenarthry of the lumbar vertebrae and a double connection of the pelvis on the vertebral column, e. e., Bradypodidae, Myrmeco-phagidae; 4. The lumbar region may be flexible but the hypaxial muscles too short and weak, e. g., Tenrecinae, Soricidae, Erinaceidae. Lumbar rigidity or insufficient muscular control in the lumbar region, resulting from any of these conditions, makes copulation difficult by restricting close approximation of the male and female genitals. Most Testiconda compensate for this by having a long penis. The same also applies for large-sized Testiphaena. Although these animals have retained the ability to gallop, the flexibility of their trunk is restricted and mostly localized in a single joint due to the great body mass and the constructive constraints necessary for increased stability. A long penis is not the only way to compensate for the absence of sagittal flexion. A phylogenetic change in the copulatory posture also solves the problem as seen in the Myr-mecophagidae and Bradypodidae (both Testiconda) which are equipped with a secondarily short-enedpenis. The permanently aquatic testicondid mammals (Sirenia and Cetacea) cannot copulate in the usual mammalian mount position. This is largely due to the phylogenetic reduction and reconstruction of the extremities as well as the secondary evolution of a powerful tail. Sagittal movements of the heavily musculated tail act upon the more or less rigid trunk to provide for a “rear drive” locomotion. Both a change in the copulatory position and a long penis were necessary for these aquatic mammals. The only Testiconda in which a marked dynamic sagittal flexibility of the trunk is developed -the Hyracoidea - are characterized by a short penis. On the whole, the relationship between the mode of locomotion, copulatory position and penis length within the mammalia is confirmed. A rigid lumbar region as opposed to a sagittally flexible, presumably represents the primitive condition among mammals. All Testiconda have

Nonlinear numerical simulation method for galloping of iced conductor

Based on the principle of virtual work, an updated Lagrangian finite element formulation for the geometrical large deformation analysis of galloping of the iced conductor in an overhead transmission line is developed. In numerical simulation, a threenode isoparametric cable element with three translational and one torsional degrees-offreedom at each node is used to discretize the transmission line. The nonlinear dynamic system equation is solved with the Newmark time integration method and the Newton-Raphson nonlinear iteration. Numerical examples demonstrate the efficiency of the presented method and the developed finite element program. A new possible galloping mode, which may reflect the saturation phenomenon of a nonlinear dynamic system, is discovered under the condition that the lowest order of vertical natural frequency of the transmission line is approximately two times of the horizontal one.