Microsoft Word - numero_62_art_25_3670.docx J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25 349 Beam-like damage detection methodology using wavelet damage ratio and additional roving mass Juliana C. Santos, Marcus V. G. de Morais, Marcela R. Machado, Ramon Silva University of Brasilia, Brazil julianasaaantos@gmail.com, https://orcid.org/0000-0003-0459-8655, mvnorais@unb.br, https://orcid.org/0000-0003-4800-4513, marcelam@unb.br, https://orcid.org/0000-0002-7488-7201 ramon.silva@unb.br, https://orcid.org/0000-0003-2004-6236 Erwin U. L. Palechor Federal University of Cariri, Brazil erwinlopezpalechor@hotmail.com, https://orcid.org/0000-0003-1409-1882 Welington V. Silva University of Brasilia, Brazil welington.vital@gmail.com, https://orcid.org/0000-0001-5200-5995 ABSTRACT. 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. KEYWORDS. Structural Health Monitoring; Damage detection; Timoshenko beam; Frequency-shift; Wavelet; Additional Roving Mass. Citation: Santos, J. C., de Morais, M. V. G., Machado, M. R., Silva, R. Y. R. C., Palechor, E. U. L., Beam-like damage detection methodology using wavelet damage ratio and additional roving mass, Frattura ed Integrità Strutturale, 62 (2022) 349-363. Received: 12.07.2022 Accepted: 30.08.2022 Online first: 01.09.2022 Published: 01.10.2022 Copyright: © 2022 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 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 https://youtu.be/w0UlIG08et8 J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25 350 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 [2–9]. 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 [10–16]. 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 [17–19]. 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 [20–29], some based on genetic algorithms [30–35] and elastic wave propagation at medium and high-frequency bands, such [36–39]. 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 [52–54]. Both models were correlated to ease future J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25 351 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 am positioned at the damage position dL is described by [56–58]:                                               2 2 2 2             a d u u kGA A m x L x x t u EI kGA I x x x t (1) 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/ 6k 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                              0 0 L L u u EI dx kGA dx x x x x (2)            ¨ ¨ ¨ 0 0 0 , L L L d du A u dx I dx m u x L dx which L the beam length and  denotes the virtual terms. The equation of motion in the frequency domain in matrix form is given by  2 Φ 0nK M (3) where Φ is a set of displacement-type amplitude at the control points otherwise known as the model vector,   / 2n nf the natural frequency  Hz , K and M are global stifness and mass matrices that can be explicitly written as:                                 , , 0 , , , , 0 00 0 T a p b p L ab a p b p a p b p N N EIx xK K dx N kGA N N N x x (4)     , , , ,0 0 , L L ab a p a p a p a pM M N AN dx N IN dx (5) where abK and abM are stiffness and mass matrices, respectively, to control points a and b ; and ,a pN and ,b pN are the shape functions. The elementary mass matrix eM can be defined as consistent or concentrated. In the first case, it considers the acceleration effects in i and the inertia force in j . In other words, the mass matrix coefficients are not null when i j . Briefly, to I J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25 352 determine e ijM 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                 2 2 2 156 22 54 13 22 4 13 3 54 13 156 22420 13 3 22 4 , e e e e e ee e e e e e e e L L L L L LAL M L L L L L L (6) where eL 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:                     ¨ ¨ '' ' '' ' 0 A u kGA u f I EI kGA u (7) The second equation when taking derivatives with respect to space becomes                     ' 0 f Au I EI kGA u (8) 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,          ' 0I EI f Au (9) that is,            ''' 0EI Au f I (10) where we can introduce   ' '' ' .u From this,               ' '' ' '' 0EI u Au f I (11) in which EI and A are constant. Thus,              ¨ ¨ ¨ ''' '' ' 0IVEIu EI A u f I u (12) Recall that,     'Au kGA f (13) J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25 353 where   ' f u kG kGA . In this way, the equation of motion can be expressed as follows                   2 ''IV EI I EI I EIu I u Au u f f f kG kG kGA kGA (14) Considering this is a problem of free vibration with mass addition as discontinuity dm at dL , we have                    2 '' 0IV d d EI I EIu I u Au u m u x L kG kG (15) Assuming  xu u , x Lx and   n T t t t f in which u , x , and t are dimensionless parameters, Eqn. (15) is given by                            2 2 2 4 2 4 2 '' 1 0IVx x n x n x n d x n d EI I x EI u I f u A f u f u m f u kG kG LL L (16) Dividing everything by   2 x nA f , Eqn. (16) is described as                        42 4 2 2 2 1 1 '' 1 0IV n d dn n f mEI E I I x u u u u u kG A kG A LAL f L Af (17) Furthermore, according to Blevins [59]:     2 4 2 4 1 4 n L Af EI (18) Thus, the equation of motion can be given by                        22 4 2 4 1 1 '' 0IV n d df I m LE I u u u u u x kG A kGA AL LL (19) 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:                             2 22 2 '' 4 2 4 1 1 0IV n d df R m L u R u u u u x kG AL LL (20) where    1/ L . 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.              2 4 4 0IV d dm L u u u x AL L (21) J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25 354 Therefore, it is noted that the behavior of the beam is a function of its boundary condition  , the intensity of the mass discontinuity dm and its position dL . METHODOLOGY 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                1 *2, | | ,f x W a a f x dx a (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   x , we have      0x dx (23) The corresponding function for wavelet transform is given by            1 , *2| |a x x a a (24) 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 2a and the translation 2a . This way, the wavelet functions are given by                    2 , ,2 2 a a a aTDW f x x dx f x x dx (25) 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. T J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25 355 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 dS and the base signal bS :         1020 Sd WDR dB log Sb (26) in which dS 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, bS 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. (a) (b) (c) (d) Figure 1: WDR index determination using the wavelet functions (a) db5; (b) coif3; (c) sym6; (d) bior6.8. Fig. 1 refers to the free-free beam. By the way, the same procedure was carried out in a simply supported beam. In both situations, the bior6.8 function presented high values of WDR, showing more useful for damage detection in the Timoshenko beam by this technique. 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 presents the investigated cases. N J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25 356 Characteristics of the beam Mass [g] 385.33 Length [mm] 395.0 Base of the cross-section [mm] 19.0 Height of the cross-section [mm] 19.0 Cross section area [mm²] 361.00 Moment of inertia [mm4] 1.086.10-4 Modulus of elasticity [GPa] 66.66 Shear modulus [GPa] 24.18 Density [kg/m³] 2702.27 Poisson’s ratio 0.33 Table 1: Geometric and material properties of the aluminum beam. Case 1 Case 2 Case 3 Case 4 Figure 2: Beam cases with different boundary conditions and damage positions. Frequency-Shift The frequency-shift technique was used for the first three vibration modes of the cases. An additional roving mass am equal to 2% of the total beam mass was applied. The damage was modeled as an additional mass dm varied from 1% to 5% of the beam total mass tm . 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 L/2 m d L/4 m d L/2 m d L/4 m d J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25 357 intensity damage ( ) 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]. Case 1 Case 2 (a1) (a2) (b1) (b2) (c1) (c2) Figure 3: Frequency variation fn(x/L)/max (fn(x/L)) depending on the position of ma in free-free beams (cases C1 and C2): (a) first frequency, (b) second frequency and (c) third frequency.    5% d tm m    2% d tm m J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25 358 Case 3 Case 4 (a1) (a2) (b1) (b2) (c1) (c2) Figure 4: Frequency variation fn(x/L)/max (fn(x/L)) depending on the position of ma in simple supported beams (cases C3 and C4): (a) first frequency, (b) second frequency and (c) third frequency. 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. am J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25 359 (a) (b) Figure 5: Element convergence analysis of DWT of first frequency-shift curve in cases C1 (a) and C2 (b). (a) (b) Figure 6: Frequency-shift curve of C1 beam as function of roving mass am for damage mass dm of 1% (a) and 10% (b). (a) (b) Figure 7: Frequency-shift curve of C3 beam as function of roving mass am for damage mass dm of 1% (a) and 10% (b). 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 am , damage mass ratio dm 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 Tm 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. C oe f. W av el et A J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25 360 (a) (b) Figure 8: Response map of WDR (dB) of free-free (C1) beam as function of damage mass ratio /d Tm m and position ratio /x L for 1st (a) and 2nd (b) frequencies and roving mass am of 1%. (a) (b) Figure 9: Response map of WDR (dB) of simply supported (C3) beam as function of damage mass ratio /d Tm m and position ratio /x L for 1st (a) and 2nd (b) frequencies and roving mass am of 1%. WDR response maps show the influence of damage mass (for a specific roving mass) and damage position for damage localization’s frequency-shift technique for Timoshenko beams in free-free and simply support boundary conditions. Fig. 8a shows a WDR reduction at  2 / 8   0.250L L , i.e., closer to a nodal point for first modal shape of free-free beam. The similar reduction for second modal shape for free-free beam is observed at  1/8   0.125L L and  4 /8   0.500L L . 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 J. C. Santos et alii, Frattura ed Integrità Strutturale, 62 (2022) 349-363; DOI: 10.3221/IGF-ESIS.62.25 361 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. REFERENCES [1] Rytter, A. (1993).Vibrational Based Inspection of Civil Engineering Structures. Aalborg Universitet. [2] Khedmatgozar Dolati, S.S., Caluk, N., Mehrabi, A., Khedmatgozar Dolati, S.S. (2021). Non-Destructive Testing Applications for Steel Bridges, Appl. Sci., 11(20), pp. 9757, DOI: 10.3390/app11209757. [3] Uesaka, M., Mitsuya, Y., Dobashi, K., Kusano, J., Yoshida, E., Oshima, Y., Ishida, M. (2018). On-Site Bridge Inspection by 950 keV/3.95 MeV Portable X-Band Linac X-Ray Sources, Bridg. Optim. - Insp. Cond. Monit., DOI: 10.5772/INTECHOPEN.82275. [4] Salgado, R. (2008). 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