Beam-like damage detection methodology using wavelet damage ratio and additional roving mass

A BSTRACT . Early damage detection plays an essential role in the safe and satisfactory maintenance of structures. This work investigates techniques that use only damaged structure responses. A Timoshenko beam was modeled in finite element method, and an additional mass was applied along their length. Thus, a frequency-shift curve is observed, and different damage identification techniques were used, such as the discrete wavelet transform and the derivatives of the frequency-shift curve. A new index called wavelet damage ratio (WDR) was defined as a metric to measure the damage levels. Damages were simulated like a mass discontinuity (mass damage). It was evaluated for different damage levels and positions. Numerical results showed that all proposed techniques are efficient for damage identification in Timoshenko beams with low computational cost and practical application. using discontinuity with of deep for open and experimental associating frequency-shift technique to roving as a allow the monitoring of bridges in a more agile way. And a more detailed analysis of the influence of noise is suggested for future works.


INTRODUCTION
arly damage detection plays a vital role in the safe and satisfactory maintenance [1]. Structural damage is any punctual or generalized change in the structure material or geometric properties that may affect the durability. In practice, it is difficult to recognize most damages by using visual inspection techniques, for instance, in cases that E damage is in a position that is difficult to access. Besides, it can require long analysis time and high cost. In this way, they may be detected by acoustic emission, ultrasonic guided-waves, eddy current detection, liquid penetrant test, magnetic particle inspection, radiographic testing and vibration-based non-destructive techniques [2][3][4][5][6][7][8][9]. System identification is an essential tool for such a purpose. Even if some non-destructive methods require finding other techniques with greater efficiency and lower cost. Generally, techniques compare the intact and damaged response of the structures, which need the identification of both beforehand. Most of the time, the intact response is very difficult to be obtained. Hence, researchers have been testing some techniques using only the damaged response, as [10][11][12][13][14][15][16]. Structural health monitoring (SHM) consists of periodic evaluations, on a full scale, the dynamic behavior of structures using sensors installed in the structural system to detect environmental actions and determine proactive maintenance actions. Non-destructive methods and monitoring techniques have received particular attention among which the vibration analysis for damage detection has been applied for its simplicity of implementation and for presenting parameters sensitive to damage [17][18][19]. Reviews [17,18] present damage identification techniques based on (a) natural frequency, (b) modal shape, (c) curvature in the modal form, (d) measure of dynamic flexibility, (e) updating, (f) heuristic methods by specialized networks, among others. There are several methodologies in the literature for solving damage detection problems using energy-based techniques [20][21][22][23][24][25][26][27][28][29], some based on genetic algorithms [30][31][32][33][34][35] and elastic wave propagation at medium and high-frequency bands, such [36][37][38][39]. Structural damages may have a severe influence on the dynamic characteristics. It produces a local change in stiffness, changing dynamic characteristics such as mass distribution and damping properties. Therefore, reduction in stiffness is associated with decreases in the natural frequencies and modification of the mode shape of the structure [40]. The ease of identification of natural frequencies has motivated the dynamic analysis of cracked structures. Some researchers are focused on the computation of natural frequencies of the cracked structure. Salawu [41] has presented a review of damage detection methods using natural frequencies potentially useful for routine integrity assessment of structures. Frequency values obtained from periodic vibration tests can monitor structural behavior and assess the structural condition. An advantage of the approach is the global nature of the identified frequencies, thus allowing the selected 36 measurement points. Fan and Qiau [19] presented an updated version of the review later. Zhong et al [12,42,43] focused on investigating natural frequencies of cracked beams subjected to a roving mass that is stationary at each location considered. The roving of the mass enhances the crack's effects on the beam dynamics and facilitates the identification and location of damage in the beam.
Researchers have focused on the study of the vibration analysis with auxiliary masses. It consists of crossing the additional mass along the structure to magnify the effect of discontinuities on the dynamic response and, hence, to facilitate the identification and location of damage in structures. Palechor et al. [44] applied this technique using impact force excitation in supported beams and identified frequency perturbation by wavelet transform. Palechor et al. [45] developed a new spectral-element with additional mass and compared the spectral method, and Galerkin assumed-mode with experimental results of an I-shape simply supported beam, presented good experimental agreement with low computational cost. Mahmoud and Abou Zaid [46] have investigated mode shapes of structures, supported and cantilever beams, subjected to a moving mass of a fixed or different velocity. Eun et al. [47] presented the damage detection method using the variation of Frequency Response Function (FRF) measured by moving an additional mass in the structure. The results showed that the FRF curvature method could be used under external noise through a numerical experiment. Solís et al. [48] presented the damage detection methodology based on the wavelet analysis based on the variation in the mode shapes derived from the damage. An additional moving mass was used to emphasize the effect of the damage and reduce experimental noise. Wang et al. [49] presented the frequency-shift to detect local stiffness reduction. The authors claim this technique can be easily adapted to a given problem since the index sensitivity can be adjusted by changing the additional mass or excitation power. Besides, an algorithm was proposed to adjust the frequency and amplitude contribution in the method automatically. Therefore, a procedure based on the Discrete Fourier Transform was explored to extract precise frequency and amplitude. Lee and Eun [50] performed a numerical and experimental analysis in a damaged beam subject to a moving mass and presented that the strain data, the acceleration, the mass magnitude, and the velocity can affect the damage detection's viability. Damage detection of structures is an exciting field of research, and the use of additional mass is promising. This paper presents a study on an additional mass spatial probing identification technique as a preliminary step to apply Zhong et al. [12] damage identification technique. First of all, a numerical model on the Timoshenko finite element 2-nodes beam by FEM was implemented. Discrete wavelet transform [51] and derivatives of the frequency-shift curve [4] were used to locate the damage. Damages were simulated like a mass discontinuity [52][53][54]. Both models were correlated to ease future experimental tests since it is more practical to simulate damage as mass added which does not require the destruction of the sample under study. Also, it was evaluated for different levels and positions of damages, and the results were presented as maps.

TIMOSHENKO BEAM FORMULATION WITH MASS DISCONTINUITY
n the case of free vibration, the motion equation for uniform beams with a roving mass a m positioned at the damage position d L is described by [56][57][58]: where u and  are the beam's transverse displacement and rotation, respectivetly, E is the Young's modulus, I the second moment of area,  the density of material, A the cross-sectional area, k a shear correction factor (  5 / 6 k for rectangular sections), and G the shear modulus, L the length of the beam and  is de Dirac delta function. According to Lee and Park [58], the dynamic equilibrium equation based on the principle of virtual work can be written as which L the beam length and  denotes the virtual terms. The equation of motion in the frequency domain in matrix form is given by where Φ is a set of displacement-type amplitude at the control points otherwise known as the model vector,    / 2 n n f the natural frequency   Hz , K and M are global stifness and mass matrices that can be explicitly written as: where ab K and ab M are stiffness and mass matrices, respectively, to control points a and b ; and M a unitary acceleration is applied in the direction j . Thus, the inertia force that results in i is calculated to keep all the accelerations in other zero directions. Therefore, the consistent mass matrix is given by where e L is the length of the element.

Dimensionless equation
The Timoshenko beam design is performed to determine dimensionless variables and the parameters that govern the problem. Considering the equilibrium equations of bending and shear moments (1) of a Timoshenko beam with constant section, the equation of motion is given by: The second equation when taking derivatives with respect to space becomes where  is the material density, A the cross-section area, u the transversal displacement,  the rotation, k the shear correction factor, G the shear modulus, f the external force, I the second moment of area and E the Young's modulus. Then, Eqn. (8) can rewritten as, in which EI and  A are constant. Thus, Recall that, In this way, the equation of motion can be expressed as follows Considering this is a problem of free vibration with mass addition as discontinuity d m at d L , we have Dividing everything by   2 x n A f , Eqn. (16) is described as Furthermore, according to Blevins [59]: Thus, the equation of motion can be given by It is known that   / E Gk and  2 / I A R (radius of rotation of the cross section), we have the dimensionless motion equation of the Timoshenko beam: where Neglecting the terms underlined in Eqn. (20), the dimensionless beam motion equation is obtained on the Euler-Bernoulli theory that considers a kinematic hypothesis formulated without considering the shear and the rotational beam effects. In Eqn. (21), the motion of the Euler-Bernoulli beam is described.
Therefore, it is noted that the behavior of the beam is a function of its boundary condition  , the intensity of the mass discontinuity d m and its position d L .

Wavelet transforms
he one-dimensional wavelet transform projects a signal into two-dimensional space. Considering a signal   f x , the wavelet transform is defined as (22) where    * . indicates the conjugate complex of    . . The translation parameter  indicates the location of the rove wavelet window in the wavelet transform and the scale a indicates the width of the wavelet window. It is a mathematical operation that expands or compresses the signal.
To detect damages, large scales are used to ensure that the signals are dilated, to make easier the identification of discontinuities. Disregarding the average value of the function   The corresponding function for wavelet transform is given by where   , a is the generating (basis) functions in the spatial domain x from which the wavelet coefficients by translation and scale are generated. This function is also known as "mother wavelet".

Discrete wavelet transform
In the Discrete Wavelet Transform (DWT), the mother wavelet function can be generated by scale a and translation  in powers of two. In this context, it reduces the computational cost in calculating the respective coefficients. The scale parameter is defined as 2 a and the translation 2 a . This way, the wavelet functions are given by where a and  refers to the scale and translation parameters. Palechor [15] presents four steps to damage detection from the DWT: 1. obtain a signal associated with the complete structure's response or a specific area of the structure; 2. compute the wavelet coefficients, performing the signal's DWT at different levels or different scales; 3. plot the graph of the wavelet coefficients for each level of decomposition; 4. analyze the distribution of the wavelet coefficients for each level. A severe change (peak) in the distribution of the wavelet coefficients means a local disturbance. If the disturbance detected is not caused by a known source, such as geometric or material discontinuity, then this means that there is damage near to the location of the disturbance.

Wavelet damage ratio index
Four basis wavelet functions were tested in this work: db5, coif3, and sym6 and bior6.8. To define the most useful function to facilitate damage detection, it is proposed the wavelet damage ratio (WDR). This index is used to evaluate the damage level by the relation between the damage signal d S and the base signal b S : Sd WDR dB log Sb (26) in which d S corresponds to the damage position that is the maximum absolute value (peak) of the wavelet coefficients, desconsidering the high values in the ends caused by the geometric descontinuities of the beam and/or the boundary conditions. Thus, b S consists of the maximum local value of the region without singularities, as shown in Fig. 1 that presents the plots using the four functions for each DWT calculated.

NUMERICAL EXAMPLE
umerical analyses of an aluminum beam were carried out using the parameters estimated experimentally in the Systems and Vibrations laboratory of the University of Brasília, UnB. Tab. 1 presents the geometric and material properties of the beam used as a reference for the present study. Two different boundary conditions were applied: free-free (F-F) and supported-supported (S-S). Firstly, the damage was simulated as an additional mass positioned in different locations of the beam: L/2 and L/4. Fig. 2  Poisson's ratio 0.33 Table 1: Geometric and material properties of the aluminum beam.

Frequency-Shift
The frequency-shift technique was used for the first three vibration modes of the cases. An additional roving mass a m equal to 2% of the total beam mass was applied. The damage was modeled as an additional mass d m varied from 1% to 5% of the beam total mass t m . The beam was discretized into 100 elements for this analysis. The spatial evolution of the first three frequencies for the four cases is shown in Figs. 3-4. In all cases, the sensitivity of natural frequencies is noted when an additional mass was applied along the beam length. Besides, the influence of the structure's mode shape was observed since the natural frequency curve undergoes minor variations when the damage is located at a nodal point, as shown in Fig. 3 (b1 and a2), and Fig. 4 (b1). Next, the influence of modes is presented in more detail.

DWT
A convergence analysis was performed for the example of free-free beam (C1) and simply supported beam (C3) case with mass discontinuity (Fig. 5). Several simulations were performed for 10, 50, 100 and 500 finite elements to evaluate DWT of frequency shift curve, using bior6.8 mother wavelet. For better visualization, only the DWT result was plotted for higher ) and light mass additional ( ). However similar results were obtained for all set of damage levels and mass additional. The wavelet coefficients reach fully convergence for 500 finite elements. It is observed a tendency of improvement with damage severity. Therefore, for the present analysis, it is sufficient to use 100 finite elements to detect reasonably the damage position in the present cases with a considerable reduction of computational effort. In experimental point of view, Palechor et al. [15] carried out the need to apply interpolation methods to increase the amount of data. Experimental data is limited to the number of points that can be measured with available instrumentation. To apply the Wavelet Transform, it became necessary to increase the discrete data to obtain good results in damage localization. The interpolation method used was the Cubic Spline, which, presented the best results in the identification of damage in metal beams under static tests according to Palechor et al. [60].

Roving mass (ma) and damage mass (md) relation
Given a damage located in the center of free-free (C1) and simply supported (C3) beams, using the bior6.8 mother wavelet, the influence of the additional mass on the frequency shift curve is studied. Figs. 6 and 7 present the frequency shift curve for 1 and 10% damage relative to roving masses of 1, 2 and 5% additional mass. In the case of a free-free beam (C1), Fig. 6 presents a frequency shift curve for additional mass of 5% with a slight tendency to saturation. Fig. 7, for the simply supported case (C3), shows a clearer trend in this regard. In these cases, increasing the additional mass does not improve the perception of damage, that is, an increase in WDR.

RESPONSE MAP AND DISCUSSION RESULTS
fter Section Numerical Results, the presence of some dimensionless numbers is highlighted in Timoshenko beam dimensionless equation: additional moving mass ratio a m , damage mass ratio d m and the damage position / x L . The relationship between these variables and WDR was investigated. Figs. 8 and 9 present WDR for first and second natural frequency of free-free and simply supported beams, respectively, as a function of the ratio of damage mass and total mass / d T m m positioned at 1/ 8L , 1/ 4L , 3/ 8L and 1/ 2L . The symmetry of beam was considered. The WDR response map is represented for half beam. For both figures, the roving mass is 1% of total mass.  Fig. 9 shows better results of WDR for damage mass located at middle span for the simply supported beam. The index WDR associated with roving mass technique can be used as a metric to determine and to localize damage in beam-like structures. Associate to machine learning techniques, the interpretation of wavelet transform of frequency shift function can be automatized to be used as a damage index. However, more studies are necessary to be done.

CONCLUSIONS
his research paper presented a numerical study of additional mass spatial probing identification technique using discrete wavelet transforms of frequency shift curves of damaged Timoshenko FE 2-nodes beam. It was used a mass discontinuity to simulation damage, known as damage mass. Using a roving mass positioned along beam span, a modal frequency was monitored to determine the frequency-shift curve. And the discrete wavelet transform of the frequency-shift curve are used to locate the damage. Simply supported and free-free beams was analyzed for two damage mass, ¼ and ½ of beam span. WDR index is proposed to analyze the magnitude of the damage. Also, it was evaluated for different levels and positions of damages, and it was presented as maps. The frequency-shift curve makes it possible to detect and locate a damage using only damaged response, eliminating comparison to an intact response. The WDR index proposed a metric to reduce frequency-shift curve as a value. With WDR, a response map synthetizing the level of damage was produced. The node of modal shape has an influence in damage T detection for a determined modal frequency. This point-out to necessity to analyses more than one frequency for a good damage detection procedure using frequency-shift curve. Other essays are necessary to correlate mass discontinuity with damage and make a relationship of damage deep for open cracks. Numerical and experimental studies are necessary associating the frequency-shift technique to mobile roving mass, as a vehicle, to allow the monitoring of bridges in a more agile way. And a more detailed analysis of the influence of noise is suggested for future works.