Transverse Vibration Amplitude Research Articles

Vibration characteristic of free-to-rotate elliptical cylinder at low Reynolds number

Investigating the vibration characteristics of elliptical cylinders is important to harvest useful energy or suppress harmful vibration, but there still exist insufficiently understood in freely rotating elliptical cylinders. Herein, this paper investigates the transverse vibration characteristics of a freely rotating elliptical cylinder at Re = 150. The incompressible Navier-Stokes equation and structural dynamics equation are solved separately, and the two-way fluid-structure interaction is performed by exchanging fluid forces and structural displacements. The elliptical cylinder, with varying axial ratios (AR = 1.0–2.5) and mass ratios (m* = 10 or 5), can vibrate transversely and rotate around its center freely. The established numerical model is verified by the published results, and the numerical values are in good agreement with the experimental results. The vibration response and flow field characteristics over a range of reduced velocities (2 ≤ Ur ≤ 14) are explored. The results demonstrate that both the axial ratio and mass ratio significantly affect the vibration characteristics. Increasing the axial ratio or decreasing the mass ratio lead to a wider synchronization region. Within the synchronization region, the elliptical cylinder undergoes significant and unstable rotation, and P + S mode can be observed. Outside the synchronization region, only a small amplitude rotation takes place near the initial position, the transverse vibration amplitude is consistent with that of a non-rotatable elliptical cylinder, and the vortex shedding mode demonstrates 2S. This work provides important guidance in investigating the vibration characteristics of the elliptical cylinder.

Wake-induced vibration and heat transfer characteristics of three tandem semi-circular cylinders

This paper studies the wake-induced vibration (WIV) and heat convection of the downstream tandem bluff bodies under the influence of an upstream bluff body at Re = 200 and Pr = 0.7. For the reduced fluidic velocity (Ur) in the range of 1–16 and different bluff body diameter ratios (d/D), three wake interference patterns, namely, the quasi-co-shedding (QCS) pattern, the co-shedding (CS) pattern, and the coupling between QCS and CS (QCS-CS) are formed due to the interference between the shear layer and vortex shedding process. The occurrence conditions of the three patterns are closely related to Ur and d/D. The vibration response of the midstream bluff body is roughly consistent with that of a heat-isolated semi-circular cylinder (HISC) if d/D is small. By increasing d/D, the vibration amplitude of the midstream bluff body first increases, then decreases with Ur. The vibration amplitude of the downstream bluff body starts to increase from small d/D and then can maintain a high value with the increase of Ur, even at large d/D. Compared with the HISC, the tandem configurations can help reduce the drag force. Because of the wake interference and the ‘blocking effect’ from the upstream bluff body, although the heat convection of the midstream and downstream bluff bodies can be effectively strengthened through WIV, the time-averaged Nusselt numbers (NuA) of the midstream and downstream bluff bodies are still lower than that of the HISC. Besides, heat transfer efficiency has been found to correlate with the transverse vibration amplitude of bluff bodies. Moreover, it is revealed that there is a trade-off between the heat transfer efficiency of the equipment and its service lifespan because increasing the vibration amplitude enhances convection but also increases the risk of potential damage to the structural integrity.

Torsional Optical Fiber Stress Analysis and Vortex-Induced Vibration Study of Three-Core Submarine Cable

Due to current scouring, submarine cables are prone to be exposed, suspended, and even vortex-induced vibration, which is not conducive to the safe operation of the power grid. In this contribution, the finite element simulation model of a 35 kV three-core optical fiber composite submarine cable with a suspended span length of 9.5 m is established. The natural frequency of the model is obtained through modal analysis. Then the vortex-induced vibration is simulated by the fluid–structure coupling method, and the stress distribution and change law of the torsional optical fiber is extracted. The results show that in the submarine cable, there appears to be a beating vibration and locking phenomena respectively, under two flow velocities. The transverse vibration amplitude of the latter increases significantly due to a resonance state. When the flow direction is perpendicular to the submarine cable, the stress distribution of the torsional optical fiber at the initial moment and at the 1/2 T moment of a vibration cycle approximately represents a mirror image relationship. In addition, the frequency of the stress change is the same as the frequency of the vortex-induced vibration, which can judge whether the vortex-induced vibration occurs. Moreover, the vortex-induced vibration range can be determined by the maximum stress change location.

Nonlinear free vibrations analysis of truncated conical shells made of bidirectional functionally graded materials

For the first time, present paper addresses the nonlinear vibrations analysis of a truncated conical shell made of bidirectional functionally graded (BDFG) materials whose mechanical properties varied continuously through the thickness and the length directions of the conical shell based on the power law distribution. Accordingly, the set of nonlinear equation of motions of the conical shell are derived based on first-order shear deformation theory (FSDT) and von Karman strain-displacement relations using Hamilton’s principle. System of partial differential equations of the BDFG conical shell are transformed into time-dependent ordinary differential equations using Galerkin approach. Afterward, the nonlinear equations are analytically solved by means of modified Poincaré–Lindstedt method, and the nonlinear frequency of the BDFG conical shell is obtained. For verification purpose, the present outcomes are compared with those available in previous researches. Eventually, the effect of longitudinal and transverse gradient indexes, vibration amplitude, and geometrical parameters on the nonlinear frequency of the BDFG conical shell is investigated. The present research may give benchmark solutions and some guidelines for designing BDFG truncated conical shells.

Simulation of Tthe Amplitude of Transverse Oscillations of the Rod System upon Impact of a Falling Load, Taking into account Deformation in the Contact Area

The oscillatory process of a rod system of arbitrary shape under shock interaction with a falling load is considered. The system may consist of a large number of rods connected to each other rigidly or pivotally, and striking is assumed to be one of the core elements, thus causing a complex oscillatory process. As an example, vibrations of a rigidly sealed statically indeterminate flat two-post frame experiencing a drop of a load of a given mass and pre-impact velocity are simulated. One of the vertical pillars of the frame has an initial curvature, the presence of which affects the maximum amplitude of transverse vibrations arising from impact. The impact of the load on the frame crossbar is modeled taking into account the deformation in the contact area, which is justified from the point of view of the accuracy of the calculations, because otherwise the magnitude of the impact force will be overestimated. When modeling the impact interaction of the load and the rod system under consideration, it is assumed that the falling load has the shape of a cylinder with a certain length of the generatrix. Linearization of the relationship between force and deformation of cylindrical surfaces is used. The proposed method of modeling the amplitude of transverse oscillations makes it possible to further study the characteristics of the oscillatory process depending on the mass of the falling load and its pre-impact velocity, as well as on the configuration of the rod system. The relevance of the work for calculations of structural elements of various purposes experiencing impact is emphasized, since the presented model can be used for engineering calculations of a wide class of core systems.

Одномерная математическая модель колебаний ствола с поперечным сечением произвольной формы

The problem of longitudinal and transverse vibrations of a barrel with arbitrary cross-sectional shapes is considered and solved in the framework of a one-dimensional model. The study shows that the amplitude of transverse vibrations in the vertical plane significantly exceeds that in the horizontal plane. This paper proposes to reduce the amplitude of vibrations by changing the shape of the barrel cross-section, namely by adding stiffeners. The numerical algorithm for solving the problem is developed on the basis of the integro-interpolation method. The verification of the numerical integration method is carried out, and the grid convergence is verified by means of the modeling of barrel vibrations for a 30 mm automatic cannon. The study of the impact of the barrel cross-section shape shows that the use of stiffeners can reduce the initial deflection and the amplitude of muzzle vibrations when firing in bursts. The obtained results demonstrate a narrow spread of projectile departure angles, and, consequently, the improved shooting accuracy of the automatic cannon.

Influence of Transverse Vibration Amplitude on the Loosening of Bolted Joint by Finite Element Simulation

A detailed finite element (FE) model of M20 bolted joint is established. Using the FE model, the loosening processes of bolted joint under different transverse vibration amplitude are simulated. The results show that there exists a critical transverse vibration amplitude. When the transverse vibration amplitude is smaller than the critical transverse vibration amplitude, the bolted joint will not loosen. When it is larger than the critical transverse vibration amplitude, the bolted joint will loosen. The larger the transverse vibration amplitude is, the faster the bolted joint loosens. Based on clamping force, nut rotation angle, bolt torsion deformation and slip of thread contact surface, the mechanism of the influence of transverse vibration amplitude on the loosening of the bolted joint is analyzed.

Modeling of the oscillation amplitude of a statically indeterminate flat frame subjected to the impact load with account taken of the contact deformation

Introduction. The article addresses transverse vibrations of a post in a rigidly fixed statically indeterminate flat frame subjected to the falling load, having pre-set mass and pre-impact velocity. The post of a vertical frame has initial curvature affecting the maximum amplitude of transverse vibrations triggered by the impact.
 Materials and methods. In the course of modeling, the frame beam impact takes account of deformations in the contact area to ensure the accuracy of calculations, because otherwise the impact force value will be overestimated. It is assumed that the falling load has the shape of a cylinder featuring a certain length, when the transverse load impact on a frame beam is simulated. The linearization of the relationship between the force and the deformation of cylindrical surfaces is used. The frame post, connected to the beam and the base, is a homogeneous rod rigidly fixed at both ends. Following the load impact on the frame beam, the top end of the post absorbs the initial bending moment and the shear force, whose values depend on the mass the and pre-impact velocity of the load, the beam attachment method, and the ratio between axial moments of inertia in beam and post sections in the plane of the frame. Using the method of initial parameters, the maximum deflection of a perfectly straight and curved post is calculated for different pre-impact velocity values of the load. The maximum deflection was modeled for curved and perfectly straight posts at different pre-impact velocities of a load having the same mass.
 Results. The dependence between the value of maximum deflection of a vertical frame post and the pre-impact velocity of the falling load was obtained.

Flow-induced vibration of a cantilevered cylinder in oscillatory flow at high KC

Flow-induced vibrations of a cantilevered circular cylinder are measured in sinusoidal, oscillatory, water flows with amplitude of reduced velocity in the range 1.9≤Ur≤4.4 and Keulegan–Carpenter number in the range 120≤KC≤900 respectively. Flow velocities are measured using laser Doppler anemometry, and forces and moments are measured using a 6-axis load cell; the two-degree-of-freedom (2-DOF) cylinder motions are determined from the measured moments. The dominant type of vibration occurring within the flow half-period is shown to depend mainly on Ur, with predominantly in-line vibration occurring for Ur⪅2.7, figure-8 vibration occurring for 2.7⪅Ur⪅4, and transverse vibration occurring for Ur⪆4. In-line vibration frequency, fx, is close to, or slightly higher than the cylinder’s natural frequency in still-water, while transverse vibration frequency, fy, is generally close to the vortex shedding frequency given by Strouhal number St=0.2. Some unsteadiness is seen in the transverse vibration frequency in that accelerating flow fy is consistently higher than decelerating flow fy for the same instantaneous reduced velocity ur. The most notable unsteady effect is seen in the in-line vibration amplitude, Ax, which is much higher during flow deceleration than during flow acceleration; maximum Ax occurs at decelerating ur≈2 for all three vibration types. Transverse vibration amplitude, Ay, increases with increasing ur and shows only slight asymmetry between accelerating and decelerating flow. Experiments with the cylinder placed within a large array of similar cylinders with a spacing between cylinders of six cylinder diameters, show that cylinder vibrations within the array are more variable than those of the isolated cylinder, but exhibit similar average vibration amplitudes and frequencies as the isolated cylinder. An empirical model for unsteady in-line vibration based on theoretical considerations and the experimental data is presented. Model-predicted and measured in-line vibration amplitudes through the flow half-period show good agreement for in-line, figure-8 and transverse vibrations.

Surface roughness effects on a tensioned riser vortex-induced vibration in the uniform current

Due to the long-term operation of marine riser, marine organisms will adsorb on its surface, thus affect its surface roughness. The surface roughness of the riser will affect the wake flow field of the riser, and then affect the dynamic characteristics of the vortex-induced vibration (VIV). The study of the influence of surface roughness on the characteristics of a rough riser VIV is helpful to improve our understanding of the mechanism of a rough riser VIV and ensure the safety of the riser during its service life. Based on the methods of computational fluid dynamics (CFD) and computational structure dynamics (CSD), this study presents the model considering tension variation with the structure vibration and the modified model of rough wall velocity gradient to construct the numerical calculation program of a rough riser VIV to study effects of various key parameters (surface roughness, inflow velocities) on the dynamic characteristics, vibration response, wake vortex shedding patterns and vibration trajectories of a rough riser. The results show that in the range of Reynolds number Re = 2.515 × 104∼1.258×105 and reduced velocity U* = 1.35–6.74, compared with a smooth riser, the streamwise vibration amplitude of a rough riser decreases, and with the increase of inflow velocity and surface roughness, the streamwise vibration amplitude of a rough riser decreases more and more obviously. In the range of reduced velocity U* = 2.70–5.39, the transverse vibration amplitude of a rough riser increases slightly. At a higher reduced velocity (U* = 6.74), the transverse vibration amplitude of the rough riser decreases. Larger surface roughness will enhance the multi-frequency and broadband vibration characteristics of the riser in the streamwise direction. With the increase of surface roughness, the multi-frequency vibration characteristics decrease gradually, but the broadband vibration characteristics increase gradually in the transverse direction. Surface roughness can suppress 2T and 2P wake vortex shedding pattern and enhance 2C wake vortex shedding pattern. This study will provide a profound understanding to guide the practical marine riser VIV research with the consideration of surface roughness.

Dynamic performance of engine timing transmission system with RU type synchronous belt

The dynamic characteristic of the timing system has a significant impact on the performance of the engine's NVH. With low contact stress and strong anti-interference ability, the arc tooth synchronous belt was adopted by engine timing transmission system. The multi-body dynamics model of the in-line four cylinder engine timing system was established. We explored the influence of crankshaft speed and initial tension on the dynamic performance of the engine timing system. The result shows when the crankshaft speed V= 2000 r/min and the initial tension of the synchronous belt F= 300 N, the load distribution of the RU type synchronous belt teeth root is more uniform. The stress is 0.2474 N/mm2 lower than the average value, and the transverse vibration amplitude decreases by 0.3067 mm. Under this circumstance, the fluctuation amplitude of camshaft angular velocity is 6.5312 rad/s lower than the maximum average amplitude, so the transmission error is reduced by 37.95 %. The fluctuation amplitude of interference value decreases by 0.3413 mm, and the fluctuation amplitude of interference velocity decreases by 263.8908 mm/s. The synchronous belt based engine timing transmission is more stable and the dynamic performance is better. It laid a solid foundation for the optimization of the tooth profile and load conditions of the engine timing system.

Comparative Analysis of Longitudinal and Transverse Vibration Characteristics of Ocean Mining Pipe

In this paper, the 5000 m mining pipe is taken as the research object, and the transverse and longitudinal vibration laws of the pipe under different working conditions are analyzed. Based on the finite element method (FEM), the pipe is discretized and calculated by the Wilson-θ Wilson - θ integral method; finally, the corresponding vibration laws of the mining pipe are obtained. The research shows that the mining pipe vibration responses are irregular motion, with the obvious oscillation phenomenon, and the overall vibration trend decreases first and then increases from the top to the bottom; the maximum vibration response occurs at the pipe top. Under the same working conditions, increasing the towing velocity will decrease the overall longitudinal vibration amplitude and increase the overall transverse vibration amplitude. While the ore bin weight will increase the longitudinal vibration amplitude and decrease the transverse vibration amplitude, increasing the mining pipe large diameter stepped section length and damping will decrease the longitudinal and transverse vibration simultaneously. When the towing velocity is between 0–2.8 m/s, the longitudinal vibration intensity is large, which is the main vibration mode. When the towing velocity is 2.8 m/s, the critical point is reached, and the longitudinal and transverse vibrations have the same intensity. When the towing velocity is greater than 2.8 m/s, the transverse vibration intensity is gradually greater than the longitudinal vibration intensity; at this time, the control of the transverse vibration should be appropriately increased.

NUMERICAL SIMULATION ON A DEEP-DRAFT SEMI-SUBMERSIBLE

A numerical study is conducted to determine the Vortex Induced Motion (VIM) effects on Deep-Draft Semi-Submersibles (DDSS). The VIM phenomena is a crucial problem that can cause severe impact on the fatigue life of mooring risers in DDSS. Therefore, a comprehensive numerical simulation is conducted using the Acusolve computational fluid dynamics (CFD) software. Five models of immersed columns with different aspect ratios (ie. 0.6, 0.8, 1.0, 1.2 and 1.4) are numerically investigated under two different incidence angles, which are 0° and 45°. The transverse and in-line vibration amplitude, amplitude of lift force coefficient and vortex shedding are analyzed. The numerical measurements are obtained to see the response of horizontal plane motions, which are transverse, in line and yaw motions. This study with detailed numerical results from parametric data will contribute future studies and the comparisons are made to demonstrate the capability of the present CFD approach.

Numerical investigation on two-degree-of-freedom vortex-induced vibration of a circular cylinder in power-law fluids

Abstract In this paper, the two-degree-of-freedom (2DOF) vortex-induced vibration (VIV) of a circular cylinder in shear-thinning and shear-thickening non-Newtonian power-law fluids are numerically investigated. A virtual spring-damper-mass system is employed to predict the VIV of the cylinder. A comprehensive analysis is conducted to study the influence of dimensionless parameters, including reduced velocity ( 3 ≤ U r ≤ 12 ) and power-law index ( 0.4 ≤ n ≤ 1.8 ) at a low Reynolds number ( R e = 150 ) and low mass ratio ( m r = 2 ) on the vortex shedding pattern, displacement frequencies, local apparent viscosity, friction coefficient, trajectories of the displacements, drag and lift coefficients, and the cylinder displacements. The results indicate that the apparent viscosity and the natural frequency of the system have a substantial effect on the characteristics of the VIV of the cylinder. By changing the values of the reduced velocity, various vortex shedding patterns such as 2S, 2P, S + P, and complex modes are observed for the non- Newtonian fluids, and several motion trajectories other than the typical figure “8″ are analyzed. By increasing the power-law index, the range of the lock-in region and the corresponding reduced velocity of the maximum transverse vibration amplitude change.

Hybrid model for the analysis of the modal properties of a ball screw vibration system

In accordance with the vibration characteristics of ball screw feed systems, a hybrid modeling method is proposed to study its dynamic behavior. Partially, the ball screw is modeled as a continuous body, and the remaining components are considered lumped masses, allowing for a realistic description of the dynamics of the feed system. The axial, torsional, transverse, and bending vibration models of a ball screw carriage system are established via the Rayleigh-Ritz series method based on the Timoshenko beam assumption. The established model that added the Timoshenko beam assumption obtains the coupling vibration displacement between the transverse and bending vibrations of the lead screw, which is close to real situations. Numerical simulations are conducted to investigate the changes of the natural frequency and modal shapes of ball screw systems with carriage positions. Results show that the carriage position has significant influence on the amplitude and direction of axial and transverse vibrations, substantial influence on the direction of the bending vibration, and minimal influence on the amplitude and direction of torsional vibration. These results indicate that the proposed hybrid model performs well to predict the vibration characteristics of the feed system. Moreover, the carriage position and carriage load also have a remarkable effect on the frequency response of the feed system. These results, along with the modeling approach, provide an important basis for the further study of in-machining monitoring and vibration controller design.

Numerical Study on Vibration Attenuation of Cylinder using Active Rotary Oscillating Controller

The present article is an attempt at utilizing a feedback control system based on cylinder rotary oscillations in order to attenuate the two-degree-of-freedom vibrations of an elastically-supported square-section cylinder in presence of flow. The control system benefits from the cylinder rotational oscillations about its axis that acts according to lift coefficient feedback signal of the cylinder. Based on the performed numerical simulations, it becomes clear that the active control system has successfully mitigated the two-degree-of-freedom vibrations of square cylinder both in the lock-in region and galloping zone. For a Reynolds number of Re = 90 located in the lock-in region, the active rotary oscillating (ARO) controller has achieved a 98% reduction in the cylinder transverse vibration amplitude, while the corresponding value for the in-line vibration is 88%. Moreover, for a Reynolds number of Re = 250 in the galloping zone, the ARO controller has successfully attenuated the cylinder transverse vibration amplitude by 72%, while the same value for the in-line vibration is 70%. One also observes that the ARO controller decreases the amplitude of lift and drag coefficients in the lock-in region by, respectively, 95% and 94%. In contrast, the corresponding percentages for the cylinder in the galloping zone are 24% and 39%, respectively.

Экспериментальное определение динамических нагрузок в приводной цепи при ее поперечных колебаниях как струны с закрепленными концами

Chain transmissions are widely used in drives of hoisting-and-transport and ag-ricultural machines, as well as in a number of machine tools and technological equipment in various branches of industry. However, a significant disadvantage of these transmissions is the high vibration activity under high dynamic loading. In this regard, the study of dynamic loads that occur during chain vibrations and affect the operability of the machine drive is one of the essential tasks of its dynamics. The article presents experimental studies performed in order to determine the maximum dynamic loads in a chain transmission of a machine drive with transverse vibrations of the chain. The most common case of transverse vibrations of a chain loop is studied as a string with fixed ends. This type of vibration is typical of chain transmissions with large masses of sprockets and parts attached to them. The article presents a description of an original test bench with an automated system for collecting and processing experimental data and a test procedure. The obtained experimental data are presented in the form of tables and graphical dependences of the maximum dynamic loads on the tension force of the chain average per cycle of vibrations and the amplitude of its transverse vibrations in the center of the span, related to its length. As a result of approximation of the experimental data array, an empirical formula is proposed that can be used in dynamic calculations of the drive. It is established that the maximum dynamic load and its amplitude increases with the increase of the ratio of the amplitude of transverse vibrations of the chain in the middle of its span to its length and the initial tension force of the chain. The ratio of the dynamic load amplitude to the tension force of the chain decreases with the increase of the latter.

Plate vibration suppression by optimizing friction threshold at its bolted supports

Hysteretic behavior due to some nonlinear sources is a common phenomenon in many mechanical systems. One of the sources of this behavior in such systems is dry friction in bolted or riveted joints. The dynamic response of a randomly excited thin rectangular plate with bolted supports at the boundaries is considered. Friction in bolted supports is modeled by Jenkins’ bilinear hysteresis element. Equivalent-Linearization technique, a Closed-Form technique and Moment-Closure technique are employed to obtain response statistics. Equivalent damping due to hysteretic supports is obtained versus instantaneous amplitude of the system and also the sliding threshold. Then optimum friction threshold for minimizing response amplitude is obtained versus other system parameters. Further, the Mont-Carlo simulation developed to verify the results. Results show that sliding at the bolted supports has significant effect on mean square value of transverse vibration amplitude and there is an optimum sliding threshold in which overall variance of system response is minimum.

Vortex-induced vibrations of an elliptic cylinder with both transverse and rotational degrees of freedom

Two-degree-of-freedom (2-DOF) vortex-induced vibration (VIV) of an elliptic cylinder is numerically investigated at a Reynolds number Re=150. The incompressible Navier–Stokes equations in arbitrary Lagrangian–Eulerian formulation are solved by a four-step fractional finite element method. The elliptic cylinder, with a variety of aspect ratios (AR=1.0–2.5), is free to vibrate in both transverse and azimuthal directions. At each AR, the computations are conducted within a wide range of reduced velocities (2≤Ur≤15). For comparison, the one-degree-of-freedom (1-DOF) responses of the cylinder vibrating only transversely are also provided. The results show that significant rotations occur under a condition of frequency synchronization similar to the lock-in condition of VIV in the transverse direction. Under the rotation effect, the transverse peak amplitude at AR=1.0 reaches a higher value about 20% larger than its counterpart of 1-DOF VIV, whereas it fells by more than half at AR=2.0 and 2.5, and the corresponding wake patterns present distinct characteristics. The typical phenomenon of phase switch can be captured successfully between the torque and rotation response, which occurs at the same reduced velocity to the phasing between the transverse force and motion. Our simulation results also elucidate that the free rotation may significantly affect the statistics of the drag, lift and moment coefficients. The most striking observation is that all of the force responses at AR=2.0 and 2.5 collapse onto a very low level compared to those of the transverse-only VIV. This is consistent with the dramatic drop in the transverse vibration amplitude at the same AR.

Numerical simulation of super upper branch of a cylindrical structure with a low mass ratio

SST (shear stress transport) k-ω and Newmark-β methods are used to comprehensively understand vortex-induced vibration (VIV) characteristics of a cylindrical structure with a mass ratio of 2.6 in a range of reduced velocity from 2.0 to 14.0. The details of drag and lift forces, cross-flow and streamwise displacements, vortex pattern, trajectory, and frequency of VIV are presented and compared systematically with the experimental work of Jauvtis and Williamson that first captured the super upper branch in VIV with the maximum value of 1.5 D (diameter). In this study, the numerical simulation results successfully captured the initial branch, the lower branch, and the super upper branch. Very few research studies have successfully simulated the super upper branch by numerical methods. The vibration amplitude corresponding to the super upper branch is stable and the maximum value of the super upper branch is 1.46 D, which is fairly consistent with the results of the Jauvtis and Williamson experiment. This research also successfully captured the law of trajectory under different reduced velocities. With the reduced velocity increasing, the trajectories switch from an irregular shape to a regular "Figure 8″ shape and then enter into an irregular movement, finally again into a regular movement of a Figure 8 shape or crescent. In the range of the super upper branch, the vibration trajectories gradually change from a Figure 8 shape to a crescent shape with the increase of the transverse vibration amplitude. This work has successfully captured the different vortex patterns corresponding to each branch under different reduced velocities, and found the transitional forms of 2S to 2T, 2T to 2P, and 2P to 2S, respectively.