# Design and analysis of the characteristics of a new vibration reduction system for an in-service long-span transmission tower

### Nonlinear finite element model

The 500 kV nonlinear finite element model ISLSTT with the new vibration reduction system for dynamic simulation is established in the finite element software ANSYS, as shown in Fig. 6. According to the geological and meteorological data of Jiamusi area of Heilongjiang province in China, the design reference wind speed is 27m/s and the ground roughness index is 0.12. The values of the structural parameters are as follows: height of 500 kV ISLSTT *H*_{1} is 122.0 m, height of the transom *H*_{2} is 97.0 m, height of vibration reduction system *H*_{3} is 35.0 m, width 500 kV ISLSTT *B*_{1} is 51.0 m, main chord width *B*_{2} is 23.0 m, width of vibration reduction system *B*_{3} is 45.0 m.

The stress condition is that he imposed full 6 degrees of freedom stresses at the four fulcrums at the bottom of the 500 kV ISLSTT. Since the main chord, diagonal web and cross web of the long span steel pipe tower are steel pipe structures, the beam188 element is selected for the finite element model of the steel pipe. steel, link10 is selected for the steel cable, shell181 is selected for the rectangular plate, and Q345 steel is used for the material of all steel structures.

### Random wind load simulation

In order to simulate the wind load on the 500 kV ISLSTT, it is assumed that the wind speed (v(z,;t)) at any height is a time variable, which is composed of the average wind speed and the fluctuating wind speed, i.e.:

$$v(z,;t) = overline{v}(z) + v_{d} (z,;t)$$

(1)

where (overline{v}(z)) is the average wind speed at height Z can be expressed by Davenport’s formula as follows: (overline{v}(z) = overline{v}_{10} (z/10)^{alpha })in which (overline{v}_{10}) is the average wind speed at a height of 10 m, *α* is the soil roughness index; (v_{d} (z,;t)) is the fluctuating wind speed at height Z.

Taking the Davenport spectral function as the power spectral density function of the fluctuating wind speed at any point in space, there is:

$$S(f) = 4Koverline{v}_{10}^{2} frac{{x^{2} }}{{f(1 + x^{2} )^{4/3} }}$$

(2)

In which, (x = 1200;f/overline{v}_{10})where *F* is the wind speed frequency, (Hz); *K* is the ground roughness coefficient.

Given the spatial correlation of the fluctuating wind speed, and according to the theory of random processes, the power spectral cross-density function of the fluctuating wind speed at any point *I* and *I* in space is:

$$S_{i,j} (f) = sqrt {S_{i,i} (f) cdot S_{j,j} (f)} coh(f)$$

(3)

where the exponential form given by Davenport is:

$$coh(f) = exp left[ { – frac{{2fsqrt {c_{y}^{2} (y_{1} – y_{2} )^{2} + c_{z}^{2} (z_{1} – z_{2} )^{2} } }}{{overline{v}(z_{1} ) + overline{v}(z_{2} )}}} right]$$

where (S_{n,n} (f)) is the power spectral density function of the fluctuating wind speed at *not* points in space, *not*= *I*, *I* ; (coh(f)) is the square root of the coherence function; *vs*_{there} and *vs*_{z} are attenuation coefficients of orthogonal direction in any two-point plane; *there*_{I} and *z*_{I} are the coordinates of the point *I*by plane, *I*= 1, 2.

According to Bernoulli’s theorem, the wind load at any point in space can be expressed by the wind pressure*P* ( *you* ), as shown in formula (4):

$$P

(4)

where *γ*is the apparent density of air; *g*is the gravitational acceleration; (mu_{s} (z)) is the shape factor at the height Z; (A(z)) is the effective area upwind at height Z.

The Spectral Density Matrix*S* ( *ω* ) of ISLSTT 500 kV structure under random wind load can be obtained by combining formula (2) and formula (3).

Assuming there is *not*the load points on the structure, then the wind speed at any point in space can be obtained by the method of harmonic superposition of the multidimensional stationary random process:

$$v_{d,j} (z,t) = 2sqrt {Delta omega } sumlimits_{m = 1}^{j} {sumlimits_{l = 1}^{N} {left| {H_{j,m} (omega_{m,l} )} right|} } cos (omega_{m,l} t + phi_{m,l} )$$

(5)

where,

$$Delta omega = (omega_{u} – omega_{d} )/N,quad omega_{m,l} = omega_{d} + (l – 1)Delta omega + frac{m}{n}Delta omega ,quad (l = 1,;2, ldots ,;N),$$

(H_{j,m} (omega_{m,l} )) is the spectral density matrix, then the Cholesky decomposition method is used, i.e. ({mathbf{S}}(omega ) = {mathbf{H}}(omega ) cdot {mathbf{H}}^{*} (omega )). Where *NOT*is the large enough positive integer; (phi_{m,l}) is the uniformly distributed random phase angle in[02π);[02π);[02π) ;[02π);(omega_{u}) and(omega_{d}) are the upper and lower limits of the frequency band.

### Wind-induced vibration response simulation analysis

The wind pressure on the tower is calculated according to Eq. (4), the 500 kV ISLSTT is discussed in sections of this article, and wind pressure is assumed to act on the discrete nodes shown in Fig. 7a. The magnitude and direction of the wind speed are*v*( *z* , *you* ) and *α* as shown in Fig. 7b, in which the *there* direction is the connection direction of conductor and ground wire. Considering the natural mode of the steel pipe tower, when simulating the wind pressure history, the time step is 0.5s and the total duration is 6000s. When the basic wind speed is 27 m/s, the first tower section, the third tower section and the wind speed time fluctuating curves of the seventh tower section and the eleventh tower section are illustrated in Fig. 8. The central heights of each section are 8 m, 38 m, 74.1 m and 106.5 m, respectively. The accuracy of each simulated wind speed fluctuating section in this article is verified by comparing the simulated wind speed power spectrum with the Davenport power spectrum. The results of the comparison are shown in Fig. 9.

It can be seen from Figure 8 that the wind speed at different heights is not only different in magnitude, but also in phase, and the correlation of different wind speeds decreases with increasing distance. It can also be observed from Fig. 9 that the simulated power spectrum in each section of the 500 kV ISLSTT is very consistent with the target power spectrum, demonstrating the accuracy of the fluctuating wind speed established in the paper. Applying the simulated fluctuating wind speed to the nonlinear finite element model of the 500 kV ISLSTT with vibration reduction system shown in Fig. 7, and taking into account the geometric nonlinearity of the structure, the four fluctuating wind speed directions *α* equal to 0, 45°, 60° and 90° are resolved by transient dynamic analysis. Therefore, the time evolution curve of the root point stress *A* and the curve of time evolution of the acceleration of the highest point *B* 500 kV ISLSTT are obtained in Figs. 10 and 11, respectively.

It can be seen in Figs. 10 and 11 that compared with the 500 kV ISLSTT without any vibration reduction system, the new vibration reduction system based on the damping structure of wire rope established in this article can greatly reduce the root point stress *A* under the action of a basic wind speed of 27 m/s and the direction of the wind speed *α*is 0, 45°, 60° and 90°, and the average vibration reduction efficiency is 66.67%, 65.32%, 61.27% and 68.23%, respectively. High point acceleration*B*is also greatly reduced, and the average damping efficiency is 77.17%, 75.42%, 73.83% and 71.65%, respectively.

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