Application of the Discrete Wavelet Transform to Damage Detection in a Guy Cable of Guyed Antenna Mast

1. Introduction

Early detection, localization and estimation of structural damage is nowadays one of the most important engineering problems that has focused much attention over the last years. When a defect occurs in the structural element, it can seriously disrupt its functioning. There are different non-destructive techniques which enable the identification of defective part of a structure, most of them base on the analysis of the structural response signals. Different approaches and many advanced methods have been developed, e.g. radiography [1], ultrasonic [2], acoustic emission [3], magnetic field [4], and eddy current [5] methods. Damage identification method which combines empirical mode decomposition (EMD) and Autoregressive Integrated Moving Average (ARIMA) models was presented in [6]. Thermal testing [7] or making use of RGB-D camera [8] also proved to be an effective method in structural health monitoring. A great potential is assigned to soft methods, mainly evolutionary algorithms [9] and artificial neural networks [10].

Different types of structural response, namely eigenfrequencies, displacements, angles of rotation, velocities or accelerations can be useful to assess the reliability of the structure. The method which enables to extract the desired detailed information from a numerous data representing the global response of a defective structure is called Wavelet Transform (WT). In the paper it is applied in its discrete form (DWT). An indicator of the damage presence and location in this type of analysis are strong disturbances of the transformed signal. Previous studies of the authors [11,12,13] proved that, in the class of considered problems, the most effective appeared to be the Daubechies wavelet [14]. Similar study has been carried out for damage detection in beams [15] and mining machines [16]. The comparison between the efficiency of Haar, Daubechies and Coiflet wavelet in damage detection was discussed in [17]. Wavelet functions proved to be highly useful in many applications as well, including their implementation to the theory of homogenization [18].

Steel masts are the lightweight structures, which behaviour has been discussed in many papers, e.g. the dynamic analysis was presented in [19] and analysis taking into account the geometric and physical non-linearity of the guys in [20]. In the present work, the authors took up the topic of analyzing a mast with guy cables, taking into account their weakening, the location of which is made possible by the use of the Discrete Wavelet Transform in the analysis of the structure’s response signal.

2. Theoretical foundations

2.1. Discrete Wavelet Transform (DWT)

At the beginning of theoretical considerations, let there be a given function $\psi \left(t\right)$, which is continuous and belong to the field of $L^2\left(\mathit{R}\right)$. This function may be called the wavelet function (mother function) and must satisfy the condition of admissibility [17]. The mother function may be real- or complex-valued. The real-valued wavelets will be applied in the considered cases. For signal decomposition the set of wavelets (wavelet family) is needed. This set of functions is obtained by translating and scaling the function ψ, what can be written by means of the relation:

$${\psi }_{a,b}=\frac{1}{\sqrt{\left|a\right|}}\cdot \psi \left(\frac{t-b}{a}\right)$$

(2.1)

where t is a time or space coordinate, a and b are the scale and translation parameters respectively. The parameters a and b take real values $\left(a, b\in \left(\mathit{R}\right)\right)$ and additionally $a\ne 0$. The element ${\left|a\right|}^{-1/2}$ expresses the scale factor which ensures the constant wavelet energy regardless of the scale, i.e. $‖{\psi }_{a,b}‖=‖\psi ‖=1$.

In the present analysis, Discrete Wavelet Transform (DWT) plays the leading role. The wavelet family can be obtained by substitution $a=1/2^j$ and $b=k/2^j$ in Equation (2.1). This yields the relation:

$${\psi }_{j,k}\left(t\right)=2^{\left(j/2\right)}\cdot \psi \left(2^j\cdot t-k\right)$$

(2.2)

in which k and j are scale and translation parameters, respectively.

The Discrete Wavelet Transformation (DWT) of the signal (e.g. the response function of the structure) $f\left(t\right)$ is expressed by the equation:

$$Wf\left(j,k\right)=2^{j/2}\cdot {\int }_{-\infty }^{\infty }f\left(t\right)\cdot \psi \left(2^j\cdot t-k\right)\cdot dt=⟨f\left(t\right),{\psi }_{j,k}⟩$$

(2.3)

The scalar product of the response function $f\left(t\right)$ and the wavelet function allows to find the set of wavelet coefficients $d_{j,k}=⟨f\left(t\right),{\psi }_{j,k}⟩$.

In identifying of singularities of the tested signals the vanishing moments of the wavelet function play a significant role. A wavelet has k vanishing moments if the subsequent relation is fulfilled:

$${\int }_{-\infty }^{\infty }{t}^k\psi \left(t\right)dt=0 k=0, 1, 2, \dots , \frac{n}{2}-1$$

(2.4)

The wavelet of order n has $\frac{n}{2}$ vanishing moments and is orthogonal to polynomials up to $\frac{n}{2}-1$.

In the case of above mentioned Daubechies wavelets which are of order from 2 to 20, always even numbers, a number of vanishing moments equal to half of the order, e.g. for Daubechies 4 there are 2 vanishing moments. Vanishing moments restrict the capability of wavelets to represent polynomial behavior or information in a signal, meaning that, e.g. Daubechies 6 encodes constant, linear and quadratic signal components.

For 1-D DWT analysis, decomposition of the discrete signal is carried out according to the Mallat pyramid algorithm [21] which can be generally presented in the simple form:

$$f_J=S_J+D_J+...+D_j+...+D_1$$

(2.5)

where each component in signal representation is coupled with a specific range of frequency and provides information at the scale level $\left(j=1, ..., J\right)$. The discrete parameter $J$ describes the level of a multi-resolution analysis (MRA), $S_J$ expresses the smooth signal representation, $D_j$ is the detail of the transformed signal at the scale level $j$ and $D_1$ corresponds to the most detailed representation of the transformed signal. The function $f_J$ must be approximated by $N=2^J$ discrete values to fulfill the dyadic requirements of DWT.

2.2. Finite Element Method

The Finite Element Method (FEM) has been applied to the numerical, geometrically non-linear analysis of the antenna guyed mast. The finite element describing the behaviour of the lattice part of the entire mast is shown in Figure 1. It is a typical two-node element with six degrees of freedom per node. The general description of the guy (cable) element is shown in Figure 2.

The displacement formulation of the FEM carries a typical matrix notation:

$$\mathbf{K}\bullet \mathbf{q}=\mathbf{P}$$

(2.6)

where $\mathbf{K}=\mathbf{K}\left(\mathbf{q}\right)$ is the global structural stiffness matrix, $\mathbf{q}$ is the global structural displacement vector and $\mathbf{P}$ is the external loading vector.

The global stiffness matrix $\mathbf{K}$ is composed of three matrices:

$$\mathbf{K}=\mathbf{K}\left(\mathbf{q}\right)={\mathbf{K}}_{\mathrm{L}}+{\mathbf{K}}_{\mathrm{N}\mathrm{L}}^{\left(1\right)}\left(\mathit{q}\right)+{\mathbf{K}}_{\mathrm{N}\mathrm{L}}^{\left(2\right)}\left(\mathit{q}\right)$$

(2.7)

where the first one ${\mathbf{K}}_{\mathrm{L}}$, is the classic linear stiffness matrix of the whole structure, the second one ${\mathbf{K}}_{\mathrm{N}\mathrm{L}}^{\left(1\right)}\left(\mathit{q}\right)$ is the first non-linear stiffness matrix, commonly called geometric matrix, whose terms depend on the value of the axial force and only on structural geometry (and at the level of a finite element on its geometry) and finally, the last one ${\mathbf{K}}_{\mathrm{N}\mathrm{L}}^{\left(2\right)}\left(\mathit{q}\right)$ is the strongly non-linear stiffness matrix that depends simultaneously on the geometry of the structure and displacements. These matrices can be derived in an uncomplicated way by writing the expression for the potential energy and postulating the equilibrium condition by its stationarity. An comprehensive description of non-linear stiffness matrices for a simple flat two-node element is given by Rakowski and Kacprzyk [22] and similar procedures have been introduced here.

Cable non-linearity affects the calculation model and analysis by the initial geometry of the cable and, as a consequence, its finite element mesh, the initial tension of the cable, and finally linear and nonlinear behaviors of the element [23].