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Study on Wind-induced Vibration and Fatigue Life of Cable-stayed Flexible Antenna

The cable-stayed flexible antenna is a large-span space structure composed of flexible multibody, with low frequency of vibration, vortex-induced resonance can occur under the action of Stochastic wind, and a larger amplitude is generated when resonance occurs. To solve this problem, based on the theory of vortex-induced vibration, this paper analyzes the vortex-induced vibration of a cable-stayed flexible antenna under the action of Wind. Based on the sinusoidal force model and Autoregressive Model (AR) method, the vortex-induced force is simulated, then the fatigue analysis of the structure is based on the linear fatigue cumulative damage principle and the rain-flow method. The minimum fatigue life of the structure is calculated to verify the vibration fatigue performance of the structure.


Introduction
The cable-stayed flexible antenna is a kind of signal transmitting antenna. when it under the action of wind load, the cable-stayed flexible antenna can produce a vortex-induced resonance with a transverse direction, which produces large amplitude when resonance occurs, and the structure is easily destroyed.
The structure of cable-stayed flexible antenna is similar to the cable-stayed bridge and the spatial latticed structure, so the analysis method of these structures is used for reference. Qingkuan Liu introduced wind load calculating methods, types of wind induced vibration and mechanisms of cables and summarized the research achievements of vibration control methods designed from aerodynamic perspective [1]. Tingting Liu studied on the wind-induced vibration response and wind-induced fatigue damage of two typical flexible structures of super high-rise buildings and Long-span bridges under vortex-induced vibration and buffeting-induced vibrations [2]. Based on the results of the above references, the vortex-induced vibration response and fatigue life of the cable-stayed flexible antenna are studied on the basis of the theory of vortex-induced vibration, the sinusoidal force model, the fatigue cumulative damage theory and the rain-flow method.

Vortex-induced Vibration Principle
When the wind passes through the structural surface at a certain speed, the vortices are periodically alternately formed on both sides of the back surface and shedding from the structural surface at a definite frequency, producing a pair of opposing vortices in the left and right sides of the tail flow that are paired, alternately arranged, and opposite to the rotation direction, that is, Karman vortex. When the vortices alternately appear on both sides of the structure, a across-wind load with the same frequency as the vortex is formed, resulting in the lateral vibration of the structure perpendicular to the wind direction, namely vortex-induced vibration.

Requirements for Vortex-induced Vibration Generation
Vortex-induced resonance occurs when the vortex shedding frequency and the natural frequencies of the structure occur simultaneously. When resonance occurs, the vortex shedding frequency is controlled by the vibration frequency of the structure, even if the wind speed is increased, the vortex shedding frequency remains unchanged, this particular phenomenon is called "lock" [2]. In this locked area, the vortex shedding frequency is a constant rather than a linear function. From the above phenomenon, it is known that resonance occurs in a certain range of wind speed when the induced resonance begins, and the initial wind speed of the resonance is called the critical wind speed. Its equation is: In the formula: i cr ,  is the critical wind speed corresponding to the ith order vibration mode of the structure, St is the Strouhal number, for the round section to take 0.2, the rest of the section takes 0.15, i T is the oscillation period of the ith order vibration mode of the structure, L is the diameter of structure section. The condition for the structure to produce the Carmen Vortex is that the value of the resonant critical wind speed is within the range of the possible wind speed of the structure, that is

Dynamic Equation of Vortex-induced Vibration of Flexible Antennas
The direction of the wind is relative to the horizontal plane, but the angle is usually less than 5 °, so the horizontal wind is usually considered. The simplified model diagram and the main coordinate system of the cable-stayed flexible antenna are shown inFiguer2.1and Figuer2.2. The midpoint of the bottom of the main pipe is the coordinate origin, the axis of the base is the x axis, the y-axis is perpendicular to the head axis of the base, and the positive direction of the wind speed is the z-axis positive direction.  The vortex-induced vibration of flexible structures is more complex, vortex shedding is not only affected by the structure shape, but also considering the coupling of wind and structure, the simplified empirical mathematical model is often used in engineering, the parameters are determined by experiments, the vortex-induced force is obtained, and the vortex-induced force is added to the motion equation to solve the structural vibration [3].
The basic form of the sinusoidal force model is: In the formula: is the mean wind speed, B is the structure reference width,  is the damping ratio, n  is the structure vibration angular frequency, s  is the vortex shedding angular frequency,  is the initial phase angle, L  is the lift coefficient of across-wind .
The vortex-induced force formula at height z is:

Dimensionless Parameters Related to Vortex-induced Vibrations
The Navier-stokes equation of one-dimensional flow of air is: According to the literature [2], the dimensionless relation of force F is: In the formula, x4,x5 ,x6 is the undetermined coefficient.

Reynolds Number
In the experiment, the karman-dunn of smooth cylindrical body in viscous fluid is studied, and the effect of Reynolds number on the flow around the smooth cylinder is pointed out. , which is the supercritical range, the vortex is irregular motion; When the Reynolds number is within the range of 6 10 3 Re   , which is the critical range, the vortex gradually becomes a regular movement, with periodic deterministic vibrations.
The expression of Reynolds number in the air is： In the formula: is the wind speed used for calculation, and its value is critical wind speed cr  .

Calculation of Natural Frequency and Reynolds Number of Cable -stayed Flexible Antenna
The free vibration equation of a cable-stayed flexible antenna is： In the formula:   M ,   K are the mass matrix and stiffness matrix of flexible antennas.
Therefore, the natural frequency values of the cable-stayed flexible antennas are shown in Table 2.3.

Wind-induced Structural Vibration Frequency
The wind-induced vibration frequency is the structure vibration frequency of the structure under the wind action, and the vibration of the structure is caused by the vortex shedding. Therefore, by the definition of the Strouhal Number, the expression of wind-induced vibration frequency s n is: In the formula: z  is the average wind speed of the structure at a certain time (generally taking 10min) in height z, L is the diameter of the flexible antenna transverse rod, 1 z is the standard height (10m), 1  is the average wind speed under the standard height,  is the index related to the roughness of the ground, its values are 0.12, 0.16, 0.22, 0.3.
The cable diameter of cable-stayed flexible antenna is 8mm, The shortest length of cable is 10.59m, according to the fundamental frequency formula of cable [4] F 1 is the tensile force of the cable, the fundamental frequency of the cable is 0.047Hz, According to the formula (2-1), the critical wind speed of the vortex-induced resonance of the cable is 0.002m/s, and the Vortex force produced by the low wind speed will be difficult to provide the energy to excite the large amplitude vibration of the cables, so the vortex-induced vibration of the cable is not considered. In summary, the wind-induced structural vibration frequencies of transverse rod at wind levels of 2 to 9 are shown in Table 2 , it's in the subcritical range and the natural frequencies of flexible antennas and wind-induced vibration frequencies are not equal, so the flexible antenna does not occur in subcritical range of the wind resonance.

Wind speed and vortex simulation based on auto regressive model
The autoregressive model of the column vectors of the space related time series of fluctuating wind speed can be expressed as: In the formula: Thus we can get M discrete pulsating wind speed process vectors with time and space correlation and time intervals of t  [5]. Davenport spectrum is simple and easy to use and does not change with height, therefore, this paper uses autoregressive model (AR) into wind speed simulation and confirms the rationality of wind speed simulation method.
According to China's norms on the definition of basic wind pressure, the length of simulation time is 600s, the wind power spectral density curve and the Davenport Power spectral density curve in the center of the transverse rod under the action of the 9-level wind and the time series of vortex-induced force are shown from

Fatigue life analysis of cable-stayed flexible antennas
The linear fatigue Cumulative damage theory is that under the action of the regular variable amplitude cyclic stress, the damage caused by each stress to the structure is independent and the damage linearly accumulates into total damage, and when the life damage rate of each stress is equal to 1, the structure breaks down. The expression of the total damage D is: Under the action of stochastic wind load, the stress response of cable-stayed flexible antenna belongs to stochastic stress time series, so the stress is calculated by rain-flow method. In this paper, the finite Element software (Workbench) is used to analyze the fatigue life of flexible antennas, the shortest fatigue life of structure is 2 years, the overall stress response and fatigue life isoline of flexible antenna is shown in Figure 3

4.Conclusion
Based on the theory of vortex-induced vibration, the analysis process of vortex-induced vibration of a cable-stayed flexible antenna under low speed wind is presented. The base frequency of the cable of the flexible antenna is small, so the critical wind speed of the resonance is small, the vortex force under the wind speed is not enough to provide the large vibration energy of the cable, so the vortexinduced vibration of the cable is not considered, However, based on fatigue cumulative damage principle and rain-flow method, the fatigue life of cable is short and fatigue damage is easy to be found.