Journal of Sensors

Volume 2015, Article ID 610416, 14 pages

http://dx.doi.org/10.1155/2015/610416

## A Comparative Study on Frequency Sensitivity of a Transmission Tower

^{1}Guangdong Power Grid Corporation Co. Ltd., Guangzhou 510080, China^{2}Key Laboratory of Roadway Bridge and Structural Engineering, Wuhan University of Technology, Wuhan 430070, China^{3}School of Transportation, Wuhan University of Technology, Wuhan 430070, China

Received 15 November 2014; Accepted 25 December 2014

Academic Editor: Fei Dai

Copyright © 2015 Peng-yun Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Sensitivity analysis can take structural parameters as variable and achieve the relationship only with one time analysis, which will dramatically reduce the analytical work especially for large scale structures. The comparative study on frequency sensitivity of a transmission tower is actively carried out in this study. The three-dimensional analytical model of a transmission tower is established by using the finite element (FE) method. The sensitivity coefficients to natural frequencies are deduced based on the equation of motion of the tower. In addition, the expression of the frequency sensitivity to Young’s modulus, density of material, the cross area of members, torsional stiffness, and bending moment inertia is proposed. A real transmission tower constructed in China is taken as an example to examine the feasibility and reliability of the proposed sensitivity computation approach. An intensive parametric study is conducted in detail in order to compare the sensitivity coefficients of different physical parameters. The work on an example structure indicated that the magnitudes of the sensitivity coefficients of Young’s modulus, the density, and the cross area are much larger than those of the torsional stiffness and the bending moment inertia.

#### 1. Introduction

To be a typical spatial steel structure, the transmission tower is widely used as electrical power infrastructures throughout the world. The transmission tower is a high-rise structure with small damping and is prone to strong dynamic excitations, such as earthquakes and wind loadings [1–3]. It is frequently reported across the world that the excessive vibration of a transmission tower under dynamic excitations may induce the structural damage and even failure [4–6]. Therefore, it is necessary to evaluate the structural performance of the transmission tower subjected to external dynamic excitations. Many theoretical and experimental investigations have been carried out during the past two decades for examining the performance of the transmission tower [7, 8]. With regard to the approaches and techniques used for performance evaluation, the transmission tower system is conventionally designed and constructed using appropriate design standards without considering the dynamic optimization effects. Therefore, the common approach does not provide deep insights into the structural transient behaviour under strong dynamic excitations, even though the consideration of dynamic effects may be important. Thus, the vibration-based structural health monitoring (SHM) approaches have been widely utilized in the performance evaluation of civil engineering structures across the world [9–11]. The SHM process needs to develop or improve a mathematical model of a physical system using measurement data to describe the input, output, and noise relationship [12, 13]. Various methods have been developed to improve the quality of the finite element model of a structure using measurement data [14]. Doebling et al. (1998) [15] gave a comprehensive review on SHM. With regard to the different SHM algorithm used, the effects of various physical parameters on the structural responses can be examined in detail to determine the crucial parameters for structural dynamic design and assessment.

For a transmission tower with determined parameters, it is troublesome to attain the relationship between dynamic characteristics and changed parameters by numerous recalculations which will be almost impossible for large scale structures, while sensitivity analysis as an alternative approach can take structural parameters as variable and achieve the relationship only with one time analysis, which will dramatically reduce the analytical work especially for large scale structures [16]. The sensitivity analysis is a reasonable and powerful tool for investigating the effects of physical parameters on static and dynamic responses. The sensitivity analysis concerns the relationship between parameters available to the structural responses under dynamic excitations. The dependence of response measures, such as displacement, velocity, acceleration, stress, strain, natural frequency, mode shape, and frequency response, is implicitly defined through the governing equations of structural mechanics [17]. Dynamic sensitivity analysis is used to compute the rate of dynamic property change with respect to changes of structural physical parameters.

Current studies on the dynamic assessment of the transmission tower focus on the response computation and mitigation. Up to now, the sensitivity analysis of the transmission tower for dynamic assessment has been investigated rarely. To this end, the comparative study on frequency sensitivity of a transmission tower is actively carried out in this study. The three-dimensional model of a transmission tower is established by using the FE method. The differential sensitivity analysis approach is presented based on the differential sensitivity coefficient, the absolute sensitivity coefficient, and the relative sensitivity coefficient, respectively. The sensitivity coefficients to natural frequencies are deduced based on the equation of motion of the tower. In addition, the expression of the frequency sensitivity to the Young’s modulus, density of material, the cross area of members, torsional stiffness, and bending moment inertia is proposed. A real transmission tower is taken as the example to investigate the effects of the structural parameters on the natural frequency through the detailed parametric study. An intensive parametric study is conducted to compare the sensitivity coefficients of different physical parameters. A real transmission tower constructed in China is taken as an example to examine the feasibility and reliability of the proposed sensitivity computation approach. The observations made indicate that the magnitudes of the sensitivity coefficients of the Young’s modulus, the density and the cross area are much larger than those of the torsional stiffness and the bending moment inertia. The Young’s modulus, the density and the cross area are important structural parameters affecting the dynamic properties of the transmission tower.

#### 2. Model of Transmission Tower

A transmission tower is a typical truss structure constructed by using steel members, which can be commonly modelled by using beam elements based on the finite element (FE) method [18, 19]. For an Euler-Bernoulli beam element, stiffness matrix of the th element of a transmission tower in the local coordinate system (LCS) is

The element stiffness matrix in the global coordinate system (GCS) can be expressed as the multiple of the element stiffness matrix in the LCS with the coordinate transformation matrix The mass matrix of the th element of a transmission tower in the LCS can be constructed based on lumped mass assumption [20]: where The element mass matrix of the th structural member in the GCS can be expressed as The global mass matrix and stiffness matrix of the transmission tower can be expressed, respectively, as follows: where is the number of elements of a transmission tower; is the freedom transform matrix from the LCS to the GCS.

#### 3. Frequency Sensitivity Analysis

##### 3.1. Differential Sensitivity Analysis

The performance assessment of complex structures to satisfy the dynamic responses restrictions is commonly hampered by the computational cost of dynamic analysis. There is a fundamental requirement for the information relating changes in the dynamic response quantities to changes in the structural parameters as well as a need for rapid reanalysis in structural performance evaluation. The sensitivity analysis of dynamic characteristics on structural parameters is very important and widely accepted in the structural design, optimization, and evaluation. Sensitivity coefficients are defined as the rate of change of a particular response quantity with respect to a change in a structural parameter . Sensitivity coefficients can be either evaluated as absolute, relative, or normalized relative values. Relative values are independent of the units of the parameter value while, in addition, normalized relative parameters are also normalized with respect to the response value.

A differential sensitivity coefficient is the slope of the response with respect to parameter , computed at a given state of the parameter. When this differential is computed for all selected responses with respect to the selected parameters, the element of the sensitivity matrix is obtained: In which is the number of the responses; is the number of the structural parameter. The absolute sensitivity coefficient is the (, )th element in the sensitivity matrix .

The absolute sensitivity coefficients are computed by using the units of the response and parameter value. If sensitivities for different types of parameters are to be compared, the relative sensitivities can be adopted: where is a diagonal, square matrix holding the parameter values. The relative sensitivity coefficient is given by Relative sensitivities can be normalized with respect to the response value where is a diagonal, square matrix holding the response values. The normalized relative sensitivity coefficient is given by

##### 3.2. Frequency Sensitivity

The eigenvalue equation of a MDOF transmission tower can be expressed as [21, 22] where , , and are the mass matrix, stiffness matrix and modal vector of the transmission tower respectively. The eigenvalue equation of the th mode vibration is The mass matrix, stiffness matrix, and modal vector are the function of physical parameters. Thus, the first derivative of 13 to the parametric of the th structural member for the th mode vibration results in in which and are the th circular frequency and modal vector of the system. Since the mass and stiffness matrices are a symmetric matrix, there exists To pre-multiply the vector to 14 will yield The relationship between the circular frequency and natural frequency of the th vibration mode can be written as The sensitivity of the frequency to the th structural parameter is Equation 18 provides a way to calculate the sensitivity of the th natural frequency to the change in the th structural parameter . Assuming that the transmission tower is linear and the change in natural frequency due to the change of structural parameter is small, the change in the th natural frequency , due to the variations of the structural parameter , can be expressed as

##### 3.3. Frequency Sensitivity of Structural Parameter

The sensitivity of the th natural frequency to the structural parameter can be rewritten by using the element stiffness matrix and mass matrix of the th element in the GCS: in which denotes the total number of the members of the transmission tower. The sensitivity of the th natural frequency to the Young’s modulus of the th structural element is given by where The sensitivity of the th natural frequency to the density of the th structural element is Substituting 3 into 23 yields where and are the cross area and the length of the th element, respectively. The sensitivity of the th natural frequency to the cross area of the th structural element is where The sensitivity of the th natural frequency to the moment of inertia , , and of the th structural element can be expressed as The sensitivity of the th natural frequency to the Poisson’s ratio of the th structural element is where

#### 4. Case Study

##### 4.1. Description of an Example Transmission Tower

To examine the frequency sensitivity based on the proposed approach, a real transmission tower constructed in China is taken as the example structure. Figure 1 shows the elevation of a large transmission tower used for electric power transfer. The transmission tower with a height of 84.5 m is located in the southern coastal area in China. The structural members used in the transmission tower are made of Q235 steel with a yielding stress of 235 MPa. Young’s modulus of the steel is 2.01 × 10^{11} N/m^{2} and the density is 7800 kg/m^{3}. The vertical major members, the skew members, the cross arms, and the platform of the tower are formed as a spatial truss tower as shown in Figures 2(a), 2(b), and 2(c), respectively. Six platforms are connected to the vertical major members to form the tower body and the skew members are incorporated to increase the vertical and lateral stiffness of the entire tower. Two cross arms are constructed on top of the tower body for the connection of the transmission lines. A three-dimensional FE model is constructed based on the FE method with the aids of the commercial package.