Modeling and modal analysis of the structure of long-span transmission tower

Abstract

The structure of the long-span transmission tower is a typical nonlinear structure with the characteristics of great height, large line span, heavy overall weight and flexible tower body. The current design code only analyzes the traditional tower types, but the analysis of the truss structure of transmission tower is limited. Aiming at improving the design defects of the structure of long-span transmission towers, this paper uses the finite element software APDL to build the three-dimensional finite element model of a long-span transmission tower, to carry out the modal finite element analysis as well as to extract the specific parameters of each modal finite element mode: Modality, Natural frequency of vibration, Periodicity. The results show that the natural vibration period of the main machinery of this type of steel transmission tower is about 0.37–1.37 s; The structure of the long-span transmission tower has certain displacements in six degrees of freedom, in which the value of the X-dimensional displacement is the largest. There are some large displacements and local torsion in the high-order mode, combined with the results of modal analysis, so it is suggested to consider the structural improvement or external reinforcement of the weak parts of the long-span transmission tower.

Keywords

Long-span transmission tower modeling modal analysis APDL

1. Introduction

The truss structure of a long-span transmission tower plays a pillar role in the national grid transmission link, because of its towering and flexible structure, it is extremely sensitive to the influence of wind loads, and a series of static and dynamic characteristic response s will occur, Especially, the severely special case will be the destruction of the truss structure. In the face of this problem, requiring the analysis of the static and dynamic characteristics of the transmission tower beam structure. For the analysis of the static dynamic characteristics of transmission tower structures, scholars at home and abroad have conducted a series of in-depth studies: Bhowmik et al. investigated the dynamic characteristics and behaviors of the tower structure by analyzing and testing modal analysis with Mild-Carbon Steel Strip welded together to create a prototype model of a scaled-down (1:15) transmission tower structure for modal testing [1]; Doohyun et al. studied the force performance of connecting foundation of transmission tower structure based on field test, focusing on the influence of load direction, and confirmed that the connecting foundation of transmission tower structure is equally effective under different load directions through the prototype proportional model load test [2]; Zou et al. studied the effect of earthquake on the tower acceleration, which concluded that the vibration of the tower top incoming direction is influenced by turbulent wind, and the vibration energy caused by earthquake excitation can be dissipated by aerodynamic damping, while the lateral vibration energy of the tower top can only be dissipated by structural damping, and the lateral tower top vibration frequency is mainly the first-order inherent frequency of the wind turbine structure [3]; Li et al. used a multi-mass point model to study the seismic response of the long-span transmission tower-wire system, simplifying the transmission line as multiple concentrated masses connected by rigid rods, simplifying the transmission tower as a multi-degree-of-freedom system with multiple concentrated masses, and simplifying the wire as a vertical chain system for out-of-plane transverse vibration, and simplifying the wire as a fixed suspension cable at both ends for in-plane longitudinal vibration. [4]; Zhao et al. The frequency and vibration mode of the single tower and the tower line system were calculated by using ANSYS finite element software, and the influence of the lead wire on the mode of the transmission tower line system was studied. [5]. However, there is a lack of research on such thin-walled truss structures connected by bolted flanges for long-span transmission towers. Therefore, it is important to analyze the dynamic characteristics of the beam frame structures of long-span transmission towers by modal analysis. In this paper, the finite element software APDL is used to establish the three-dimensional finite element model of the structure of the long-span transmission tower through command flows, to extract the first 10 orders of modalities of the finite element model, and to analyze their vibration patterns and frequencies to provide reasonable suggestions for the structural design of the long-span transmission tower and the reinforcement of the weak links.

2. Modal analysis fundamentals

Modal analysis is the study of the dynamic properties of linear structures through structural tests or simulations which is based on finite element analysis and is generally used in the field of engineering vibration [6]. Among them, the modal results refer to the inherent vibration characteristics of the structure of the object itself, and the modal analysis results of each section of the model include specific parameters such as inherent frequency, damping ratio and modal vibration pattern [7]. The modal analysis and its extracted modal parameters can predict the dynamic vibration pattern and the inherent frequency of the structure before the installation of construction, which saves a lot of manpower and material resources, and can help to find out the “disaster” caused by the resonance of the transmission tower structure during its lifetime in time and can also provide some scientific basis for strengthening the weak position of a transmission tower and improving its design.

At the same time, modal analysis can also be defined as transforming the physical coordinates in a linear time-invariant system of vibration differential equations into modal coordinates, so that the set of equations are decoupled and become a set of independent equations described by modal coordinates and related modal parameters, and the transformation matrix of coordinate transformation is the vibration matrix, each column of which is the vibration of each order [8]. The finite element method of modal analysis uses mathematical analysis to construct a modal model, and the essence of modal analysis is to study the modal vector of the structural finite degree of freedom problem, i.e. the equation of motion without load and damping [9]. The equation of vibration is

$\displaystyle\left\{{P(t)}\right\}=[M]\left\{{\ddot{u}\left(t\right)}\right\}+% [C]\left\{{\dot{u}\left(t\right)}\right\}+[K]\left\{{u\left(t\right)}\right\}$ (1)

In the formula, $\{P(t)\}$ – External load vectors; $[M]$ – Mass Matrix; $[C]$ – Damping Matrix; $[K]$ is the Stiffness Matrix (nth order Square Matrix); $\left\{{\ddot{u}(t)}\right\}$ , $\left\{{\dot{u}(t)}\right\}$ , $\left\{{u(t)}\right\}$ is respectively the acceleration vector, velocity vector, and displacement vector of the linear structural system [10]. When $\{P(t)\}=$ 0, that is, the equation of modal analysis is obtained, and the modal analysis of the structure is realized by using ANSYS finite element software, meanwhile, when the effect of structural damping on the vibration pattern and the inherent frequency of the structure is neglected, Eq. (2) can be obtained:

$\displaystyle\left[M\right]\left\{{\dot{u}\left(t\right)}\right\}+\left[K% \right]\left\{{u\left(t\right)}\right\}=0$ (2)

For all nodes within the structure the frequency is the same when doing simple harmonic motion, i.e.:

$\displaystyle\left\{{u\left(t\right)}\right\}=\left\{u\right\}\sin\left({% \omega t+\theta}\right)$ (3)

In the formula: $\{U\}$ is the Displacement Amplitude.

According to the above equation, the chi-square equation for the displacement amplitude is

$\displaystyle\left({\left[K\right]-\omega^{2}\left[M\right]}\right)\left\{u% \right\}=0$ (4)

When the coefficient determinant is zero, a non-zero solution for the displacement amplitude $|U|$ can be obtained

$\displaystyle\left({\left[K\right]-\omega^{2}\left[M\right]}\right)=0$ (5)

Equation (5) is called the frequency equation of the system, solved to obtain the system of n natural frequency $\omega_{i}\left({i=1,2,3\ldots n}\right)$ , Substituting into Eq. (4), n principal mode vectors can be obtained $u(1),u(2),\ldots u(n)$ .

3. Transmission finite element model

3.1 Engineering background

This paper is a study of the 500 KV long-span transmission towers constructed by the State Grid in Chongming Island, where transmission line construction requires multiple crossings of power lines, highways and across the Yangtze River. The background data of the long-span transmission tower project are shown in Table 1. The towers are built in accordance with “GB 50135-2019 Structural Design Code for Towering Structures”, with Q420 as the main steel material and Q345 as the auxiliary steel material, and 6.8 as the bolt grade.

Table 1
Background data of long-span transmission tower project

Crossing gear distance/m	Tower heights/m	Roots open/m	Base pile height/m
1609	204	42	50

3.2 Establishment of three-dimensional finite element model for long-span transmission towers

The CAD two-dimensional wireframe diagram of the long-span transmission tower is used to get the coordinates of each key node and the size of each rod. In the APDL finite element analysis software, the command flows are used to build a three-dimensional finite element model of the large-span transmission tower truss structure, as shown in Fig. 1.

Transmission tower poles are modeled with beam units and based on the main condition of not changing the structural dynamic characteristics of transmission towers, the transmission towers are simplified as follows [11].

The main and auxiliary parts of the transmission tower are simplified to beam unit steel frame structure, which can accurately and comprehensively reflect the structural dynamic characteristics of the transmission tower.

Set the transmission tower model beam unit link node as the ideal articulation point.

According to the transmission tower field installation mode, full displacement constraints are set for the tower foot nodes.

After meeting the above conditions, the relevant parameters of the long-span transmission tower are shown in Table 2.

Table 2
Correlation coefficient of transmission tower

Roots open/m	Elastic modulus/Pa	Density/(kg $\cdot$ m ${}^{-3}$ )	Poisson’s ratio
42	2.06 $\times$ 10 ${}^{11}$	7850	0.3

Figure 1.

Finite element model of long-span transmission tower.

According to the actual tower model, the beam unit, Beam188, is used to simulate the rod of the transmission tower. The Beam188 unit used in the modeling has the following characteristics: In the process of tower design, the rod can be considered as a two-force rod that is not subject to shear and bending moments but only to axial forces, and the end nodes of the rod unit only have degrees of freedom in the X, Y, Z displacement directions [12]. Beam188 beam unit contains 6 degrees of freedom at each end node X, Y, Z, RX, RY, RZ, etc. [10]. The geometry of Beam188 unit, nodes of unit (I, J, K), pressure directions ([baseline=(char.base)] [shape=circle,draw,inner sep=0.2pt] (char) 1;–[baseline=(char.base)] [shape=circle,draw,inner sep=0.2pt] (char) 5;), as shown in Fig. 2.

Figure 2.

Beam 188 unit.

3.3 Optimization of finite element model for the structure of the long-span transmission tower

The main parameters obtained for the modal analysis of the long-span transmission tower structure are the self-oscillation period and the inherent frequency [13]. The size of the self-oscillation period depends on the mass and stiffness of the transmission tower, but the modeling process is ideal modeling, without considering the flange and other details, the mass of the transmission tower should be multiplied by the mass increase factor, and the formula for calculating the mass increase factor is shown in the following formula:

$\displaystyle\textit{MIC}=\frac{M_{1}}{M_{2}}$ (6)

$M_{1}$ : Calculate the Mass of construction drawings, Weight of transmission towers in actual construction condition, $M_{2}$ : Calculated Mass, Weight calculated by APDL simulation.

4. Analysis of dynamic characteristics of long-span transmission towers

The modelling of the long-span transmission tower is based on the APDL software platform, and the tower is modeled by using command flows. In the modal analysis of the long-span transmission tower, the Block Lanczos method is chosen as the modal extraction method for the transmission tower model with solid units and more degrees of freedom, and the extracted modal order is 10. The extraction conditions are: (1) the transmission tower model cannot be loaded; (2) the material density and related parameter data need to be input. The parameters of the transmission tower material are shown in Table 3.

Table 3
Pole’s material parameters of the long-span transmission tower

Material	Tensile strength/MPa	Modulus of elasticity/(N*m ${}^{2}$ )	Density/(kg*m ${}^{-3}$ )	Poisson’s ratio
Q420	500–680	2.06*10 ${}^{11}$	7850	0.3
Q325	450–600	2.06*10 ${}^{11}$	7850	0.3

After post-processing by APDL software, 10 vibration patterns and inherent frequencies of the long-span transmission towers are obtained, and the vibration patterns are shown in Fig. 3.

Figure 3.

Modal shape diagram of large span transmission tower.

Figure 3.

continued.

Also, modal analysis is obtained about the transmission tower X, Y, Z, RX, RY and RZ direction of each order modal participation factors and the specific parameters are shown in Tables 4–9.

Table 4

Modal participation factors of X-direction

Modal order	Participation factors	Participation Ratios	Effective mass
1	$-$ 13.165	0.008564	173.328
2	1537.4	1	2.36E+06
3	7.4371	0.004838	55.3109
4	4.5464	0.002957	20.6696
5	$-$ 1511.3	0.983056	2.28E+06
6	0.21883	0.000142	4.79E-02
7	$-$ 2.5222	0.001641	6.36163
8	$-$ 0.1528	0.000099	2.33E-02
9	2.8804	0.001874	8.29652
10	193.48	0.12585	37434

Table 5

Modal participation factors of Y-direction

Modal order	Participation factors	Participation ratios	Effective mass
1	1509	1	2.28E+06
2	13.146	0.008711	172.811
3	0.5724	0.000379	0.327645
4	$-$ 1312	0.869447	1.72E+06
5	$-$ 5.1641	0.003422	26.6678
6	$-$ 5.268	0.003491	27.7523
7	989.19	0.655513	978504
8	$-$ 8.2064	0.005438	67.3447
9	$-$ 42.943	0.028457	1844.13
10	$-$ 31.769	0.021053	1009.27

Table 6

Modal participation factors of Z-direction

Modal order	Participation factors	Participation ratios	Effective mass
1	$-$ 0.13216	0.003263	1.75E-02
2	$-$ 0.25205	0.006224	6.35E-02
3	$-$ 0.32901	0.008124	0.108246
4	1.7156	0.042362	2.94328
5	$-$ 0.93418	0.023067	0.872698
6	$-$ 1.0059	0.024838	1.01183
7	12.207	0.301423	149.016
8	1.0368	0.0256	1.0749
9	29.11	0.71878	847.365
10	40.499	1	1640.13

Table 7

Modal participation factors of RX-direction

Modal order	Participation factors	Participation ratios	Effective mass
1	$-$ 2.32E+05	1	5.40E+10
2	$-$ 2013.6	0.008665	4.05E+06
3	$-$ 74.423	0.00032	5538.72
4	90187	0.388118	8.13E+09
5	315.22	0.001357	99364.2
6	234.12	0.001008	54810.9
7	$-$ 38808	0.167008	1.51E+09
8	299.55	0.001289	89728.9
9	1234.7	0.005313	1.52E+06
10	710.19	0.003056	504364

Table 8

Modal participation factors of RY-direction

Modal order	Participation factors	Participation ratios	Effective mass
1	$-$ 2005.6	0.008578	4.02E+06
2	2.34E+05	1	5.47E+10
3	1093.8	0.004678	1.20E+06
4	351.35	0.001503	123446
5	$-$ 92955	0.397554	8.64E+09
6	3.8136	0.000016	14.5432
7	9.416	0.00004	88.6614
8	5.4006	0.000023	29.1669
9	265.61	0.001136	70547.3
10	6983.7	0.029868	4.88E+07

Table 9

Modal participation factors of RZ-direction

Modal order	Participation factors	Participation ratios	Effective mass
1	$-$ 1.973	0.000077	3.89282
2	$-$ 98.259	0.003839	9654.8
3	18004	0.703447	3.24E+08
4	16.022	0.000626	256.698
5	2.9843	0.000117	8.90602
6	$-$ 246.07	0.009614	60549
7	$-$ 53.956	0.002108	2911.24
8	$-$ 25594	1	6.55E+08
9	4234.8	0.165458	1.79E+07
10	$-$ 197.52	0.007717	39013.8

Table 10

Frequency and amplitude of the first ten modes

Modal order	Self-oscillation frequency/Hz	Periodicity/s	Amplitude/mm
1	0.727212	1.3751	1.28
2	0.731113	1.3678	1.22
3	0.815197	1.2267	2.4
4	1.63608	0.61122	1.7
5	1.79621	0.55673	1.2
6	2.17074	0.46067	6.72
7	2.29183	0.43633	2
8	2.3554	0.42456	1.25
9	2.53147	0.39503	4.63
10	2.66751	0.37488	2.87

From the above vibration diagram and the modal participation factors in each direction, we can obtain: the first order vibration pattern is the overall round-trip flat motion of the transmission tower in the Y direction; the second order vibration pattern is the overall round-trip flat motion of the transmission tower in the X direction; the third order vibration pattern is the twisting of the cross-arms in the RZ direction; the fourth order vibration pattern is the round-trip flat motion of the cross-arms in the Y direction; the fifth order vibration pattern is the round-trip flat motion in the X direction and the twisting of the cross-arms in the RY direction; the sixth order vibration pattern is the twisting in the middle of the tower; the seventh order vibration type for the tower as a whole in the Y direction of the round-trip flat and cross-arms in the RY direction of the torsion; the eighth order vibration type for the tower as a whole in the RZ direction of the torsion; the ninth order vibration type for the cross-arms in the RZ direction of the local torsion; the tenth order vibration type for the cross-arms in the RZ direction of the local serious torsion. At the same time, through the analysis of the first ten orders of the transmission tower, the following conclusions can be obtained: the fixed frequency of the first, second and third order vibration patterns are less than 3 Hz, the vibration pattern of the translation and torsion in the X, Y, RX, RY direction, proving that the transmission tower structure stiffness, in line with the design standards; other vibration patterns are shown in the middle of the tower, the cross-arms appear local torsional vibration patterns.

Figure 4.

Displacement components in X, Y and Z directions.

The inherent frequency, period and vibration amplitude of the first ten modes of the long-span transmission tower are shown in Table 10, from the above data we can know: the main vibration period of the long-span transmission tower structure is 0.37–1.37 s. In order to prevent the resonance of the long-span transmission tower operation, the transmission tower should not be built on the geology whose main vibration period is between 0.37 s and 1.37 s.

Meanwhile, the modal vibration pattern of the long-span transmission tower is mainly divided into Z-direction vibration and XY-plane flat motion. By post-processing the vibration amplitude, the displacement components of the transmission tower as a whole in X, Y and Z directions are obtained, and the displacement component curve is drawn with the modal order as the horizontal coordinate and the displacement component as the vertical coordinate, as shown in Fig. 4.

From the figure, it can be obtained that the long-span transmission tower structure will produce displacement in three dimensions with the action of load excitation, in which the displacement in X dimension is larger compared with the other two dimensions, and the XY plane advection is much larger than the Z direction vibration.

5. Conclusion

From the above 3D modeling, model optimization and modal analysis of the structure of the long-span transmission towers, the following conclusions can be drawn.

The 3D modeling of the transmission tower: After the improvement of the structure and quality of the long-span transmission tower, the 3D model of the transmission tower basically conforms to the actual design standards.

The first three orders of vibration of the long-span transmission tower structure are dominated by the translational motion in the XY plane, and the displacement in the Z-dimensional direction is negligible, which proves that the design of the long-span transmission tower structure is very reasonable, and at the same time the structure is designed in such a way that it can perfectly resist the impact of external loads.

The local torsion and overall large deformation in the high-order vibration pattern of the structure of the long-span transmission tower proves that the weak part of the tower is the local torsion area of the structure of the long-span transmission tower, and structural improvement or external reinforcement is needed for the weak part.

The above modeling optimization and modal analysis of the long-span transmission tower, the structural design defects of transmission towers can be discovered in advance, which can provide data support for the subsequent study of the capacity of load bearing and dynamic characteristics of long-span towers, and then provide data support for the structural optimization of long-span transmission line towers.

Footnotes

Acknowledgments

This work was financially supported by the Science and Technology Project of State Grid Zhejiang Electric Power Co., LTD. (Project No. 2021ZK32).

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Modeling and modal analysis of the structure of long-span transmission tower

Abstract

Keywords

1. Introduction

2. Modal analysis fundamentals

3.1 Engineering background

Table 1 Background data of long-span transmission tower project

Table 2 Correlation coefficient of transmission tower

Table 3 Pole’s material parameters of the long-span transmission tower

Footnotes

Acknowledgments

References

Table 1
Background data of long-span transmission tower project

Table 2
Correlation coefficient of transmission tower

Table 3
Pole’s material parameters of the long-span transmission tower