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Wood Beam Damage Identification Based on the Curvature Mode and Wavelet Transform

Wood Beam Damage Identification Based on the Curvature Mode and Wavelet Transform Hindawi Advances in Civil Engineering Volume 2023, Article ID 2238021, 18 pages https://doi.org/10.1155/2023/2238021 Research Article Wood Beam Damage Identification Based on the Curvature Mode and Wavelet Transform 1 1 1 2 1 1 Zhaobo Meng, Xiancai Ren, Shanqing Chai , Xin Wang, Tengfei Zhao , Feifei Gao, 1 1 Shanwei Wang, and Yufa Liu School of Architecture and Civil Engineering, Liaocheng University, Liaocheng 252000, Shandong, China School of Civil Engineering, Tianshui Normal University, Tianshui 741001, Gansu, China Correspondence should be addressed to Tengfei Zhao; zhaotengfei@lcu.edu.cn Received 24 October 2022; Revised 17 April 2023; Accepted 21 April 2023; Published 3 May 2023 Academic Editor: Dimitrios G. Pavlou Copyright © 2023 Zhaobo Meng et al. Tis 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. In the paper, a simple-supported wooden beam is used as the research object to identify the damage of the wood beam by fnite element analysis and experimental research. First, ANSYS was used to establish the solid fnite element model of the wood beam before and after the damage, and then the discrete wavelet transform was performed on the curvature mode of the wood beam before and after the damage, and the wavelet coefcient diference index was obtained after obtaining the high frequency wavelet coefcients. Ten, the damage location of the wood beam was judged according to the sudden peak of the wavelet coefcient diference index, and the damage degree of the wood beam was estimated by ftting the relationship between the wavelet coefcient diference index and the degree of damage. Finally, the index was verifed by the wooden beam test. Te results show that the wavelet coefcient diference index can accurately identify the damaged location of the wood beams. Te degree of damage to the wood beams at the damage location can be quantitatively estimated by ftting the relationship between the wavelet coefcient diference index and the degree of damage at the damage location. Te research results provide a theoretical basis to identify wooden beam damage. parameters before and after the damage may be used to 1. Introduction determine structural damage [3]. Some scholars have done Wood beams are the main load bearing elements of the much work on the early state damage detection of beams, historic timber structure; under the infuence of environ- and some research results have been achieved [4–6]. He et al. mental conditions, sudden disasters, human damage, and [7] showed that the index of the curvature modal diference other factors, wood beams are prone to damage, afecting the could indicate the location of the damage and the degree of safety, applicability, and durability of the wood structure of a cantilever beam. Xiang et al. [8] suggested the curvature- ancient buildings. Tey can seriously lead to the collapse of based modal utility information entropy as the damage the entire wood structure. Terefore, the study of the detection index, and numerical modeling and testing on identifcation of structural damage to wood beams is supported beams confrmed its validity. Ren et al. [9] studied signifcant. the sensitivity of the natural frequency, the curvature mode, Te three difculties of time, spatial location, and and the diference in the curvature mode to the damage of damage level should be addressed in structural damage wooden beams and conducted a numerical simulation identifcation, and damage identifcation methods have been analysis on the wooden beams. Based on the frst-order created to address the problems mentioned earlier [1, 2]. curvature mode diference index, the infuence of noise Structural damage can cause changes in the physical char- pollution and mesh density on recognition accuracy was acteristics of the structure and in the modal parameters of studied. However, some researchers have found that cur- the structure. As a result, the change in the modal vature modes are inefective in identifying slight damage 2 Advances in Civil Engineering [10]. Terefore, we need to propose a new method to solve where q(x) is the curvature, ρ(x) is the curvature radius, these problems. M(x) is the bending moment of the beam cross section, and Te wavelet transform has the property of local am- EI(x) is the fexural rigidity of the beam section. plifcation of the mutation signal in the time and frequency According to the theory of material mechanics, the domains; so, it can determine the location and degree of curvature function in any section of the beam is expressed as minor damage that occurs in the structure. Janeliukstis et al. follows: [11] suggested a damage identifcation algorithm based on the modal structural vibration wavelet transform, and the € ϕ (x)q (t), (2) q(x) � 􏽘 i i efectiveness of the proposed method was confrmed with i�1 fnite element models of aluminum beams in diferent where ϕ (x) is the j order curvature modal, and q (t) is the i i damaged parts. Bao et al. [12] performed the continuous modal coordinates. According to formulas (1) and (2), wavelet transform of the curvature mode of the structure structural damage leads to a reduction in structural stifness, before and after damage. Tey obtained the wavelet co- which results in an increase in the vibration of the curvature efcient diference index, which was shown to accurately mode. Terefore, sudden change in vibration of the cur- discriminate the damage site and the degree of damage to the vature mode in a certain order can be used to diagnose structure by fnite element simulation and testing of the structural damage. supported beam. Machorrolopez et al. [13] proposed To obtain the curvature mode of the structure before and a method of acoustic emission signal combined with con- after damage, the calculation must be performed with the tinuous wavelet transform, and the proposed method was central diference method based on the displacement mode validated by concrete subjected to bending tests. Te results [14, 15]. Under the premise that the displacement mode and showed that the method could identify the damaged state of the vibration mode of the equally spaced discrete unit nodes concrete beams by the transformed wavelet energy index. are known, the curvature mode is defned as follows: Te abovemrntioned research results carry out relevant u u u damage identifcation studies on diferent structures as re- ϕ (j − 1) − 2ϕ (j) + ϕ (j + 1) i i i (3) ϕ (j) � , search objects but relatively few damage identifcation iu 2 studies on individual wood beam structures. Tis paper proposes a damage identifcation method based on the d d d ϕ (j − 1) − 2ϕ (j) + ϕ (j + 1) i i i (4) curvature mode and wavelet transform for simple-supported ϕ (j) � , id wooden beams. Compared to other methods, this method can better refect the location of the damage to the wooden where ϕ(j) represents the i − th order of curvature modal at beam structure and has a particular sensitivity to the degree j points, u and d represent lossless and lossy states, and l is of damage. Te degree of damage to the wooden beam at the the distance between adjacent nodes. damage location is quantitatively estimated by ftting the relationship equation between the diference index of the 2.2. Principles of Wavelet Transform Damage Identifcation. wavelet coefcient diference index at the damage location and the degree of damage. Specifcally, frst, ANSYS is used For any function f(t) ∈ L (R), the continuous wavelet transform is to build a solid fnite element model of the wooden beam before and after damage and perform modal analysis. Ten, +∞ 1 x − b √�� � the curvature mode of the wooden beam before and after the W (a, b) � 􏽚 f(x)ψ dt, (5) 􏼠 􏼡 |a| a −∞ damage is subjected to a discrete wavelet transform to obtain high frequency wavelet coefcients and obtain the wavelet where W (a, b) is the wavelet coefcient, ψ(x) is the wavelet coefcient diference index. Ten, the damage location of the function, ψ (x) is the complex conjugate of ψ(x), and a and wooden beam is judged according to the sudden change b are the translation and scale factors. peak of the wavelet coefcient diference index, and the In an application, it is necessary to discretize the con- degree of the wooden beam is estimated by ftting the re- tinuous wavelet. Its binary wavelet transform can be lationship between the wavelet coefcient diference index expressed as follows: and the degree of damage. Second, the index was verifed by +∞ 1 x − b wooden beam tests; fnally, the limitations and challenges of j ∗ 􏽱��� W 􏼐2 , b􏼑 � 􏼌 􏼌 􏽚 f(x)ψ 􏼠 􏼡dx. f (6) 􏼌 􏼌 the method are given in the conclusion. j 􏼌 􏼌 a −∞ 􏼌 􏼌 􏼌 􏼌 2. Materials and Methods When b � k2 , the above equation is the discrete wavelet transform. 2.1.PrincipleofDamageIdentifcationintheCurvatureMode. According to the material mechanics theory, the static bending force of the beam is expressed as follows: 2.3. Selection of Wavelet Functions. In signal analysis, choosing and building the appropriate wavelet function is 1 M(x) q(x) � � , (1) essential. Te results of diferent wavelet functions for the ρ(x) EI(x) same problem difer in practice. Terefore, when using wavelet analysis to detect mutation signals, it is necessary to Advances in Civil Engineering 3 combine the analyzed signal’s characteristics with the re- divided, as shown in Figure 3. Te model x-axis is the beam search purpose and select the appropriate wavelet function length direction, the y-axis is the beam section height di- rection, and the z-axis is the beam section width direction. for local mutation detection. In this paper, we refer to the process and principles of wavelet selection in the literature Tat means that the left end of the model constrains UX, UY, [16, 17], combine the analyzed signal’s characteristics, and UZ, ROTX, and ROTY, the right end constrains UY, UZ, fnally select the bior6.8 wavelet function. ROTX, and ROTY, and UZ is constrained at the neutral axis to prevent the wood beam from moving in the z direction. It is assumed that the reduction of the cross-sectional 2.4. Structural Damage Identifcation Index. Tis paper uses height causes damage to the structure, so the damage is the curvature mode indicator as the input signal. Te bior6.8 simulated in the fnite element by reducing the cell cross- wavelet is selected to perform the three-layer discrete sectional size; the specifc operation steps are as follows: at wavelet transform to obtain the high-frequency wavelet the bottom of the wooden beam along the beam width coefcients and then make the diference to get direction, create a rectangular slot body with length 160 mm, d u width 40 mm, depth 2 mm, 5 mm, 10 mm, 20 mm, 30 mm, j j j j (7) DW � W 􏼐2 , k2 􏼑 − W 􏼐2 , k2 􏼑 , fij f f ij ij and 40 mm and then carry out the Boolean operation. Te FEM model of the wooden beam under diferent damage where DW is the diference of wavelet coefcients at i fij conditions is obtained by the command of subtracting the node under the j − th order curvature mode before and after body from the body in the Boolean process. Figure 4(a) d u j j j j structural damage. W (2 , k2 ) and W (2 , k2 ) are the f ij f ij shows the fnite element model of the wood beam for wavelet coefcients at i node in the j − th curvature mode working condition 6, and Figure 4(b) shows the enlarged before and after damage. view of the damage of the wood beam fnite element model for working condition 6. Te degree of damage S is defned as the ratio of the depth of the rectangular slots he with the 2.5. Damage Identifcation Steps height of the beam h, that is, S � he/h, and the degree of (1) First, a modal analysis was performed on the damage is 1%, 2.5%, 5%, 10%, 15%, and 20%. Te damage members of the wooden beam to obtain their frst- conditions are shown in Table 2. order curvature modal data before and after damage. (2) Te bior6.8 wavelet is used to perform the three-layer 3. Results and Discussion discrete wavelet transform in the frst-order curva- ture mode before and after the damage and obtain 3.1. Comparison of the Inherent Frequency under Diferent the high-frequency wavelet coefcients and then WorkingConditions. Based on the abovementioned damage make the diference to get the wavelet coefcient conditions, the intrinsic frequencies of the wood beam diference index. before and after damage are shown in Table 3. Table 3 shows that it can be seen that the overall frequency change before (3) Te damage location was localized on the basis of the and after damage to the wooden beam is small; the maxi- sudden peak of the wavelet coefcient diference mum relative change in frequency before and after damage is index. Te degree of damage to the wooden beam 9.42%. It shows that the frequency change is not sensitive to was estimated by ftting the relationship equation the slight local damage of the wood beam; working con- between the wavelet coefcient diference index and ditions 7–12 compared with the second-order frequency the degree of damage. relative change of working conditions 1–6 showed that its amplitude is small, analysis of the reason, the location of the 2.6. Numerical Simulations. Tis paper uses a simple- damage occurred in the vicinity of the vibration node, supported timber beam as the research object for the lo- resulting in the relative change of frequency amplitude not calization of damage and the study of the degree of damage. apparent, and through this indicator, we can determine the Te wood beam (width 160 mm and height 200 mm) with occurrence of damage to the wood beam. However, the a span of 2400 mm and a net span of 2200 mm, and the location and extent of damage to the wood beam cannot be specifc dimensional arrangement of the wood beam is judged. shown in Figure 1. To obtain the parameters related to the wood timber property in the fnite element model of the simple-supported wood beam, the authors conducted wood 3.2. Comparison of Displacement Vibration Patterns under timber property tests on the same batch of poplar wood Diferent Working Conditions. In order to verify whether beams. All tests were collected, fabricated, and tested displacement vibration is sensitive to damage, the dis- according to national standards [18, 19], part of the test placement vibration data values corresponding to 56 nodes procedure is shown in Figure 2, and the results of the timber on a line on the middle top surface at the level of the center property tests of the wood beam are shown in Table 1. axis of the wooden beam in diferent working conditions Te three-dimensional solid model of the wood beam is were extracted in the article, considering that the test simulated by the Solid185 unit with fnite element ANSYS. measurement points were arranged on the middle axis of the Te mesh cell size is 40 mm, the wood beam model is meshed top surface of the wooden beam (Figure 5), and the spacing by sweep command, and a total of 1960 solid cells are between each node in the model was 0.04 m. Displacement 4 Advances in Civil Engineering Figure 1: Size drawing of the wooden beam (mm). Table 1: Material parameters of wood beams. Elastic modulus (MPa) Poisson’s ratio Modulus of shear (MPa) Densities (kg/m ) E E E μ μ μ G G G L R T LR LT RT LR LT RT 9752.17 2058.57 452.90 0.490 0.410 0.353 585.13 731.41 175.54 Electronic universal testing machine Electronic balance Data Acquisition System Vernier calipers Test pieces Ovens (a) (b) Electronic universal testing machine Electronic universal testing machine Data Acquisition System Data Acquisition System Test pieces Test pieces (c) Figure 2: Material quality test process. (a) Moisture content and density test. (b) Method for determination of the modulus of elasticity in compression parallel to the grain of wood. (c) Method for determining the modulus of elasticity in compression perpendicular to the grain of wood. vibration patterns were converted according to the co- and Figure 8 is the frst two orders of the displacement vibration pattern of two damage in working conditions ordinate relationship between the nodes and the wooden beam model. Te displacement vibration pattern is con- 13–18, where the subpicture part is the magnifcation of the verted to the two-dimensional plane for analysis. In this vibration pattern at the local damage location. section, the extracted displacement vibration patterns are As can be seen in Figures 6–8, in single damage or in two normalized to the maximum amplitude of the vibration damage conditions, the diference between the frst two pattern to facilitate the subsequent analysis. orders of displacement vibration curves of the wooden beam Figures 6 and 7 are the frst two orders of the dis- structure is minimal and difcult to distinguish through the placement vibration pattern of single damage in working naked eye; so, it is difcult to identify the damage to the conditions 1–6 and working conditions 7–12, respectively, wooden beam structure using displacement vibration. 200 Advances in Civil Engineering 5 (a) (b) Figure 3: Finite element model of a wood beam. (a) Finite element model. (b) Boundary conditions. (a) (b) Figure 4: FEM and damage magnifcation of the wooden beam for work conditions 6. (a) Finite element model of the wood beam. (b) Enlarged view of the damage. Table 2: Condition of damage to the wood beam. Degrees Working conditions Damage types Damage locations (m) of damage (%) 0 Undamaged — — 1 520–560 1 2 520–560 2.5 3 520–560 5 4 520–560 10 5 520–560 15 6 520–560 20 One location damage 7 1040–1080 1 8 1040–1080 2.5 9 1040–1080 5 10 1040–1080 10 11 1040–1080 15 12 1040–1080 20 13 520–560, 1040–1080 1 14 520–560, 1040–1080 2.5 15 520–560, 1040–1080 5 Two location damage 16 520–560, 1040–1080 10 17 520–560, 1040–1080 15 18 520–560, 1040–1080 20 3.3. Single Location Damage Identifcation Analysis. In the displacement array of the wooden beam is used as the measured measurements, the higher-order modes of the original data for subsequent damage identifcation studies in structure are not easily measured. At the same time, research this article [20]. In this article, the extracted frst-order scholars have found that lower-order methods play a dom- displacement vibrations are normalized to the maximum inant role in structural vibration; the frst-order magnitude of the pulses to facilitate subsequent analysis. 6 Advances in Civil Engineering Table 3: Changes in intrinsic frequency before and after damage. 1st order frequency 2nd order frequency Working conditions Relative Relative Frequencies (Hz) Frequencies (Hz) frequency change (%) frequency change (%) 0 76.55 — 264.14 — 1 76.55 0.00 264.05 0.03 2 76.52 0.04 263.80 0.13 3 76.42 0.17 263.02 0.42 4 75.95 0.78 260.28 1.46 5 75.14 1.84 256.22 3.00 6 73.90 3.46 250.81 5.05 7 76.53 0.03 264.15 0.00 8 76.46 0.12 264.16 0.01 9 76.19 0.47 264.19 0.02 10 75.20 1.76 264.23 0.03 11 73.65 3.79 264.20 0.02 12 71.45 6.66 263.74 0.15 13 76.51 0.05 264.05 0.03 14 76.39 0.21 263.80 0.13 15 75.99 0.73 263.01 0.43 16 74.51 2.67 260.25 1.47 17 72.29 5.57 256.14 3.03 18 69.34 9.42 250.64 5.11 (a) (b) Figure 5: Wood beam model extraction layer and the extraction line. 1.1 1.25 1.0 1.0 0.8 0.5 0.6 0.0 0.4 -0.5 0.2 -1.0 0 -1.25 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) Intact 10% Intact 10% 1% 15% 1% 15% 2.5% 20% 2.5% 20% 5% 5% (a) (b) Figure 6: Working conditions 1–6 frst- and second-order displacement vibration. (a) First-order vibration. (b) Second-order vibration. Displacement vibration Displacement vibration Advances in Civil Engineering 7 1.1 1.25 1.0 1.0 0.8 0.5 0.6 0.0 0.4 -0.5 0.2 -1.0 0 -1.25 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) Intact 10% Intact 10% 1% 15% 1% 15% 2.5% 20% 2.5% 20% 5% 5% (a) (b) Figure 7: Working conditions 7–12 frst- and second-order displacement vibration. (a) First-order vibration. (b) Second-order vibration. 1.1 1.25 1.0 1.0 0.8 0.5 0.6 0.0 0.4 -0.5 0.2 -1.0 0.0 -1.25 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) Intact 10% Intact 10% 1% 15% 1% 15% 2.5% 20% 2.5% 20% 5% 5% (a) (b) Figure 8: Working conditions 13–18 frst- and second-order displacement vibration. (a) First-order vibration. (b) Second-order vibration. Te normalized frst-order displacement mode calculates difcult to identify its damage location, indicating that the the frst-order curvature modes of each working condition indicator is not sensitive in the case of minor damage to the according to equations (3) and (4). Figures 9 and 10 show the structure, cannot remember the site of structural damage, frst-order curvature modes of the single damage for coupled with the interference of external environmental working conditions 1–6 and working conditions 7–12, noise, and may lead to the indicator not identifying the respectively. location of minor damage to the structure. Also, the wavelet As can be seen from Figures 9 and 10, when the damage transform has the feature of local amplifcation of abrupt degree is greater than the 5% case, the curvature mode at the signals in the time and the frequency domain, which can damage location produces a sudden change, which can be identify the location and the degree of minor structural identifed by the peak of the sudden change to identify the damage. Terefore, in this paper, the frst-order curvature modal data of wooden beams are subjected to a three-layer location of wood beam damage; when the damage degree is less than the 5% case, the curvature mode at the damage discrete wavelet transform using bior6.8 wavelets, and the location of the sudden change is not signifcant, and it is high-frequency wavelet coefcients are derived and then Displacement vibration Displacement vibration Displacement vibration Displacement vibration 8 Advances in Civil Engineering 0.5 efectively reduce the infuence of adjacent units, making the recognition efect more signifcant. In the case of slight 0.0 damage, the wavelet coefcient diference index also has ups -0.5 and downs at the undamaged position but its transformation amplitude is smaller than that of the damaged position, -1.0 which has little infuence on the determination of the -1.5 damaged position; with the increasing damage degree, the wavelet coefcient diference at the undamaged position is -2.0 almost close to the level, and the sudden change peak of the -2.5 wavelet coefcient diference at the damaged position is -3.0 pronounced, and the peak of the wavelet coefcient dif- ference index increases with the increasing damage degree. -3.5 0 0.4 0.8 1.2 1.6 2 2.2 x (m) 3.4. Two Location Damage Identifcation Analysis. To verify Intact 10% the sensitivity of the wavelet coefcient diference index for 1% 15% the identifcation of the two location damage, the working 2.5% 20% conditions (conditions 13–18) were established for the si- 5% multaneous presence of damage at positions 520–560 mm Figure 9: First-order curvature modes of working conditions 1–6. and 1040–1080 mm of the wooden beam. Figure 13 shows the frst-order curvature modal diagrams of the two damage locations for cases 13–18. As shown in Figure 13, consistent 0.5 with the single damage results, the indicator is not sensitive 0.0 in the case of minor damage to the structure. It cannot -0.5 identify the location of the damage to the wood beam and coupled with the interference of external environmental -1.0 noise, which may cause the indicator to fail to identify the -1.5 location of minor damage to the structure. Terefore, in this -2.0 paper, the frst-order curvature modal data of the wood -2.5 beam are subjected to a three-layer discrete wavelet trans- -3.0 form using bior6.8 wavelets, and the high-frequency wavelet -3.5 coefcients are derived and then diferenced to obtain the -4.0 diference maps of wavelet coefcients under each working condition, as shown in Figure 14. -4.5 0 0.4 0.8 1.2 1.6 2 2.2 As seen in Figure 14, the wavelet coefcient diference x (m) index produces a signifcant abrupt change at two pre- determined damage locations of the wood beam. Te lo- Intact 10% cation of damage to the structure of the wood beam can still 1% 15% be accurately determined in the case of minor damage, 2.5% 20% 5% which indicates that the wavelet coefcient diference index is more efective than the curvature mode index in locating Figure 10: First-order curvature modes of working conditions minor damage to the wood beam, and the maximum value of 7–12. the wavelet coefcient diference index increases with the increasing damage degree. divided to obtain the diference maps of the wavelet co- efcients under each working condition, as shown in Fig- 3.5. Identifcation of the Degree of Damage. Te wavelet coefcient diference index was found to increase with the ures 11 and 12. It should be noted that since the damage location is located in the cell between two nodes, the wavelet increase in the degree of damage when a single damage and two damage localizations were performed on the wooden coefcient diference indicator is taken as the extreme value of one of the two mutation nodes; so, the peak of the beam, which indicated that there was a close correlation between the wavelet coefcient diference index and the mutation point in the wavelet coefcient diference map is located in one of the two nodes of the damaged cell, and it degree of damage with high sensitivity; so, the degree of the can indicate the existence of damage near this point. wooden beam could be estimated by ftting the relationship As can be seen in Figures 11 and 12, compared to the equation between the wavelet coefcient diference index problem that the curvature mode index is not apparent to and the degree of damage. recognize minor damages to wood beams, the problem is For single damage, this paper analyzes the working efectively solved using the wavelet coefcient diference conditions 1–6 as an example and adds the simulation of the damage depth of 60 mm at the location of 520 mm–560 mm index. Tis indicator can amplify the signal of small abrupt changes in the modal curvature index and, at the same time, for single damage of the wooden beam by ANSYS, that is, the Curvature modes Curvature modes Advances in Civil Engineering 9 0.008 0.03 0.08 0.006 0.06 0.02 0.004 0.04 0.01 0.002 0.02 0.00 0.000 0.00 -0.002 -0.02 -0.01 -0.004 -0.04 Location of the damage Location of the damage Location of the damage -0.02 x=0.56 m -0.006 -0.06 x=0.56 m x=0.56 m -0.03 -0.008 -0.08 -0.010 -0.04 -0.10 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) x (m) (a) (b) (c) 0.3 0.4 0.6 0.3 0.4 0.2 0.2 0.2 0.1 0.1 0.0 0.0 0.0 -0.1 -0.2 -0.1 -0.2 Location of the damage Location of the damage Location of the damage -0.4 x=0.56 m x=0.56 m -0.3 x=0.56 m -0.2 -0.6 -0.4 -0.3 -0.5 -0.8 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) x (m) (d) (e) (f) Figure 11: Working conditions 1–6 diference of wavelet coefcients. (a) Working condition 1. (b) Working condition 2. (c) Working condition 3. (d) Working condition 4. (e) Working condition 5. (f) Working condition 6. 0.010 0.04 0.15 0.03 0.10 0.005 0.02 0.01 0.05 0.000 0.00 0.00 -0.01 -0.005 -0.02 -0.05 Location of the damage Location of the damage Location of the damage -0.03 x=1.04 m -0.010 x=1.04 m x=1.04 m -0.10 -0.04 -0.015 -0.05 -0.15 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) x (m) (a) (b) (c) 0.3 0.6 0.8 0.6 0.2 0.4 0.4 0.1 0.2 0.2 0.0 0.0 0.0 -0.2 -0.1 -0.2 Location of the damage -0.4 Location of the damage -0.2 -0.4 Location of the damage x=1.04 m -0.6 x=1.04 m x=1.04 m -0.3 -0.6 -0.8 -0.4 -0.8 -1.0 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) x (m) (d) (e) (f) Figure 12: Working conditions 7–12 diference of wavelet coefcients. (a) Working condition 7. (b) Working condition 8. (c) Working condition 9. (d) Working condition 10. (e) Working condition 11. (f) Working condition 12. Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference 10 Advances in Civil Engineering 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5 -4.0 0 0.4 0.8 1.2 1.6 2 2.2 x (m) Intact 10% 1% 15% 2.5% 20% 5% Figure 13: First-order curvature modes of working conditions 9–12. 0.010 0.05 0.10 0.04 0.005 0.03 0.05 0.02 0.000 0.01 0.00 0.00 -0.005 -0.01 -0.05 -0.02 Location of the damage Location of the damage Location of the damage Location of the x=1.04 m -0.010 -0.03 -0.10 Location of the x=1.04 m x=1.04 m damage x=0.56 m Location of the damage x=0.56 m -0.04 damage x=0.56 m -0.015 -0.05 -0.15 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) x (m) (a) (b) (c) 0.3 0.6 0.8 0.6 0.2 0.4 0.4 0.1 0.2 0.2 0.0 0.0 0.0 -0.1 -0.2 -0.2 -0.2 Location of the -0.4 Location of the Location of the damage x=1.04 m -0.4 Location of the -0.3 Location of the Location of the damage x=1.04 m -0.6 damage x=0.56 m damage x=1.04 m damage x=0.56 m damage x=0.56 m -0.6 -0.4 -0.8 0.0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) x (m) (d) (e) (f) Figure 14: Working conditions 13–18 diference of wavelet coefcients. (a) Working condition 13. (b) Working condition 14. (c) Working condition 15. (d) Working condition 16. (e) Working condition 17. (f) Working condition 18. damage degree is 30%. Te maximum value of the diference In order to further obtain the relationship between the in the wavelet coefcient is shown in Table 4 (because the wavelet coefcient diference index and the degree of diference in the wavelet coefcient value is negative, and to damage, this paper is based on the data in Table 4 and on the facilitate analysis, the absolute value is taken). principle of least squares and used Origin software to ft the Wavelet coefficient difference Wavelet coefficient difference Curvature modes Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Advances in Civil Engineering 11 Table 4: Diference of wavelet coefcients for diferent degrees of 1.2 damage at 520 mm–560 mm. 1.0 Degrees Diferences of wavelet Damage locations (mm) of damage (%) coefcients 0.8 1 0.00797 0.6 2.5 0.03038 5 0.0844 0.4 520–560 10 0.23195 15 0.41909 0.2 20 0.61348 30 1.12284 0.0 0 1 2.5 5 10 15 20 30 32 Degree of damage (%) relationship curve between the wavelet coefcient diference Figure 15: 520 mm–560 mm location damage degree and the and the degree of damage under diferent working condi- damage index ftting graph. tions, where the number of polynomials ftted is taken as 3, and the damage ftting curve is shown in Figure 15. Te relationship between the wavelet coefcient difer- 0.8 ence index and the degree of damage in a single damage 0.6 location (520 mm–560 mm) is obtained from Figure 15 as 0.4 follows: 0.2 − 6 3 − 4 2 y � −4.8 × 10 x + 8.6 × 10 x + 0.016x − 0.014, (8) 0.0 -0.2 where x is the level of damage, and y is the wavelet co- efcient diference damage index. -0.4 Location of the damage Te applicability of the ftted formula was verifed in -0.6 x=0.56 m ANSYS by simulating a damage depth of 50 mm (a damage -0.8 level of 25%) at the location of 520 mm–560 mm for a single -1.0 damage to the wood beam. 0 0.4 0.8 1.2 1.6 2 2.2 Figure 16 gives the wavelet coefcient diference plot of x (m) 25% of the degree of damage in the single damage position of Figure 16: Diference of wavelet coefcients for 25% of damage at the wooden beam of 520 mm–560 mm. From Figure 13, the 520−560 mm position. absolute value of the diference in the wavelet coefcient in the position of 520 mm–560 mm is 0.83026. Taking the Table 5: Diference of wavelet coefcients for diferent degrees of absolute value of the diference of the wavelet coefcient into damage at two locations. equation (8), the degree of damage at the position of 520 mm–560 mm is obtained as 24.63% using the MATLAB Degrees Diferences of wavelet Damage locations (mm) solution, which is only a 1.76% error. Terefore, the degree of damage (%) coefcients of the wooden beam can be estimated using equation (8). 1 0.00861 For the two injuries, ANSYS increased the simulation of 2.5 0.03021 the depth of damage of 60 mm at 520 mm–560 mm and 5 0.07732 1040 mm–1080 mm for the two injuries of the timber beam, 520–560 10 0.20317 15 0.3495 that is, the degree of damage was 30%. Te peaks of the 20 0.47594 wavelet coefcient diference for the two damage locations, 30 0.7406 520 mm–560 mm and 1040 mm–1080 mm, are shown in 1 0.01257 Table 5 (because the peaks of the wavelet coefcient dif- 2.5 0.04034 ference is negative, it is taken as an absolute value for the 5 0.10942 convenience of analysis). 1040–1080 10 0.29704 According to the data in Table 5, based on the principle 15 0.49628 of least squares, Origin software was used to ft the re- 20 0.67856 lationship curves between the diference of the wavelet 30 1.02204 coefcients and the degree of damage under diferent working conditions, where the number of ftted polynomials − 5 3 − 4 2 y � −1.4 × 10 x + 7.5 × 10 x + 0.015x − 0.011, was taken as 3. Te damage ftting curves for the two damage locations are shown in Figure 17. (9) Te relationship between the wavelet coefcient difer- − 5 3 − 3 2 ence index and the degree of damage at the two damage y � −2.4 × 10 x + 1.1 × 10 x + 0.022x − 0.018, locations was obtained from Figure 17 as follows: (10) Wavelet coefficient difference Wavelet coefficient difference 12 Advances in Civil Engineering 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 Location of the -0.6 Location of the damage x=1.04 m -0.8 damage x=0.56 m -1.0 0 0.4 0.8 1.2 1.6 2 2.2 x (m) Figure 18: Diference of wavelet coefcients for 25% of damage at two positions. Figure 17: Two location damage degree and the damage index ftting graph. rectangular slot depth he to the beam heighth, that is,S � he/ h. Te specifc rectangular slot locations are shown in Table 6. where x is the level of damage, y is the wavelet coefcient Te test adopted a single-point excitation and a multipoint diference damage indicator at 520 mm–560 mm position, output measurement method to conduct modal tests on the and y is the wavelet coefcient diference damage indicator wooden beam. A force hammer was used to hammer the at the position 1040 mm–1080 mm. wooden beam at a preset reference point. Te structure’s re- Te applicability of the ftted formula was verifed in sponse was collected in batches by multiple INV9822 accel- ANSYS by simulating the damage depth of 50 mm (the erometers from the Beijing Oriental Vibration and Noise damage was 25%) at the locations of 520 mm–560 mm and Technology Research Institute. An average of 19 acceleration 1040 mm–1080 mm for the two damage locations of the sensors must be placed within the clear span of the wooden wood beam. beam. Te magnetic holders are glued to the top surface of the Te wavelet coefcient diference plots of the 25% wooden shaft with 502 glue and the acceleration sensors are damage level for the two damage locations, installed on the magnetic holders. Data were collected and 520 mm–560 mm and 1040 mm–1080 mm, for the wooden analyzed with a multifunctional data acquisition analyzer beams are given in Figure 18. (model INV306N2) and DASP (V11) signal analysis software In Figure 18, the absolute values of the diferences in the using an impact hammer to strike the eight # acceleration wavelet coefcient at the positions 520 mm–560 mm and sensor measurement points three times. Te feld test and the 1040 mm–1080 mm are 0.59787 and 0.83205, respectively. schematic diagram are shown in Figure 20. Te absolute values of the wavelet coefcient diferences at the two damage positions are brought into equations (9) and (10), and the degree of damage at the 520 mm–560 mm 3.6.2. Analysis of Experimental Results. Te inherent fre- position is obtained by using MATLAB to solve for 24.39%, quencies and vibration patterns of the wooden beam were calculated for diferent operating conditions by referring to with only 2.44% error; at the 1040 mm–1080 mm position, the damage degree is 24.61%, with only 1.56% error. the EMA modal test procedure in the DASP&INV product operation and using the guide. Table 7 compares the frst Terefore, the degree of the wooden beam can be estimated using equations (9) and (10). second-order inherent frequencies of the intact wood beam fnite element model with the frst second-order inherent frequencies of the test. From Table 7, it can be seen that the 3.6. Experimental Verifcation error between the frst-order inherent frequency of the fnite element model and the frst-order inherent frequency of the 3.6.1. Experimental Program. To further verify the feasibility tested beam is 1.74%, increases with the increase of the of the proposed method, wooden beams with the same modal order, and the maximum value is 1.41%, which is material and dimensions as the numerical simulation were within the acceptable range. Figure 21 compares the frst two used as the research object. Te wooden beams were divided orders of displacement modal vibration patterns of the fnite into 20 parts of 110 mm each, and the damage was simulated element model of the intact wood beam and the intact wood by changing the cross-sectional dimensions of the wooden beam test. In Figure 21, it can be seen that the modal vi- beams. Te specifc implementation was as follows: rect- bration pattern of the structure is consistent with the modal angular slots of length 160 mm, width 40 mm, and depth vibration pattern characteristics of the supported beam. 2 mm, 5 mm, 10 mm, 20 mm, 30 mm, and 40 mm were Trough the abovementioned analysis, the fnite element manually cut by an electric planer at the bottom of the model and the test results of the wooden beam can be wooden beams along the beam width direction (Figure 19). verifed with each other. Te degree of damage S is defned as the ratio of the 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 1 2.5 5 10 15 20 30 32 Degree of damage (%) 1040 mm-1080 mm 520 mm-560 mm Wavelet coefficient difference Wavelet coefficient difference Advances in Civil Engineering 13 Table 6: Test wood beam damage conditions. Degrees Working conditions Damage types Damage locations (m) Damage units of damage (%) 0 Undamaged — — — 1 480–520 5 1 2 480–520 5 2.5 3 480–520 5 5 One location damage 4 480–520 5 10 5 480–520 5 15 6 480–520 5 20 7 480–520, 1040–1080 5, 10 1 8 480–520, 1040–1080 5, 10 2.5 9 480–520, 1040–1080 5, 10 5 Two location damage 10 480–520, 1040–1080 5, 10 10 11 480–520, 1040–1080 5, 10 15 12 480–520, 1040–1080 5, 10 20 40 mm 30 mm 20 mm 10 mm 5 mm 2 mm (a) (b) (c) (d) Figure 19: Wooden beam grooving site. (a) Cutting. (b) Sanding. (c) Slotting. (d) Diferent depths of slots. wood beam Sensor Computer Acquisition instrument Power Hammer Figure 20: Wooden beam feld test chart. (1) Single Location Damage Identifcation Analysis. Tis measurement process, the wavelet threshold method is used paper uses DASP (V11) signal analysis software to extract to denoise the measured displacement vibration data. Ten, frst-order displacement vibration data under diferent the denoised displacement vibration is normalized to the working conditions. Due to noise interference in the maximum magnitude. Te wavelet coefcient diference 14 Advances in Civil Engineering Table 7: Comparison of frequency of the test wooden beam and the fnite element model. Calculated values of Calculated values of Orders the fnite element test wood beams Errors (%) model (Hz) (Hz) 1 76.55 75.24 1.74 2 264.14 260.48 1.41 COINV_DASP 0 .175608 .351216 .526824 .702432 .087804 .263412 .43902 .614628 .790236 (a) (b) COINV_DASP 0 .116923 116923 .233846 233846 .350769 30 69 .467692 467692 .058462 .175385 .292308 .409231 .526154 (c) (d) Figure 21: Comparison of the frst- and second-order displacement pattern of intact wooden beam and the fnite element model. (a) Vibration pattern in the frst-order displacement mode of fnite elements. (b) Test the frst-order displacement mode of fnite elements. (c) Finite element second-order displacement mode of fnite elements. (d) Test the second-order displacement mode of fnite elements. diagrams under working conditions 1–6 are obtained by index increases with the increasing damage degree. For processing the displacement vibration data according to working conditions 1–6, it should be noted that the location equations (3) to (7), as shown in Figure 22. It should be of the damage to the wooden beam in the test occurred in unit 5, and the nodes at both ends of unit 5 are 0.44 m and noted that since the damage location is located in the cell between two nodes, the wavelet coefcient diference in- 0.55 m away from the end of the beam, respectively, and the dicator is taken as the maximum value of the two mutation horizontal coordinates in the fgure are the beam length, nodes; so, the peak value of the mutation point in the wavelet while the wavelet coefcient diference indicators all peak at coefcient diference diagram is located in one of the two 0.44 m. Tis indicates that the wood beam was damaged near nodes of the damaged cell, and it can indicate the existence this location, and when combined with the actual location of of damage near this point. the damage, it is clear that the wood beam was damaged in In Figure 22, it can be seen that the use of the wavelet unit 5. coefcient diference index can efectively overcome the problem that the curvature mode index is not apparent for (2) Two Location Damage Identifcation Analysis. Te small damage identifcation and can accurately determine wavelet coefcient diference plots for working conditions 7–12 are obtained according to the same processing method the location of the damage to the wood beam structure, and the maximum value of the wavelet coefcient diference for single-site damage, as shown in Figure 23. Advances in Civil Engineering 15 10 mm 5 mm 2 mm 0.00004 0.00020 0.0003 0.00003 0.00015 0.0002 0.00002 0.00010 0.0001 0.00001 0.00005 0.0000 0.00000 0.00000 -0.0001 -0.00001 Location of the damage Location of the damage -0.00005 Location of the damage x=0.44 m x=0.44 m -0.00002 -0.0002 x=0.44 m -0.00010 -0.00003 -0.0003 -0.00015 -0.00004 -0.00020 -0.0004 0 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98 2.2 0 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98 2.2 0 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98 2.2 x (m) x (m) x (m) (a) (b) (c) 40 mm 20 mm 30 mm 0.0006 0.0010 0.0010 0.0004 0.0005 0.0005 0.0002 0.0000 0.0000 0.0000 -0.0005 -0.0005 -0.0002 Location of the damage -0.0004 Location of the damage x=0.44 m -0.0010 -0.0010 Location of the damage x=0.44 m x=0.44 m -0.0006 -0.0015 -0.0015 -0.0008 -0.0010 -0.0020 -0.0020 0 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98 2.2 0 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98 2.2 0 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98 2.2 x (m) x (m) x (m) (d) (e) (f) Figure 22: Working conditions 1–6 diference of wavelet coefcients. (a) Working condition 1. (b) Working condition 2. (c) Working condition 3. (d) Working condition 4. (e) Working condition 5. (f) Working condition 6. 10 mm 5 mm 10 mm 2 mm 2 mm 5 mm 0.00008 0.00020 0.0005 0.0004 0.00006 0.00015 0.0003 0.00004 0.00010 0.0002 0.00002 0.00005 0.0001 0.00000 0.00000 0.0000 -0.00002 -0.0001 -0.00005 -0.00004 -0.0002 Location of the damage -0.00010 -0.00006 x=1.1 m Location of the damage Location of the damage -0.0003 x=1.1 m -0.00015 x=1.1 m -0.00008 Location of the damage -0.0004 x=0.44 m -0.00020 Location of the damage Location of the damage -0.00010 -0.0005 x=0.44 m x=0.44 m -0.00012 -0.00025 -0.0006 02 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98.2 02 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98.2 02 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98.2 x (m) x (m) x (m) (a) (b) (c) Figure 23: Continued. Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference 16 Advances in Civil Engineering 40 mm 30 mm 30 mm 20 mm 40 mm 20 mm 0.0008 0.0015 0.0020 0.0006 0.0015 0.0010 0.0004 0.0010 0.0005 0.0002 0.0005 0.0000 0.0000 0.0000 -0.0002 -0.0005 -0.0005 -0.0004 -0.0010 -0.0010 -0.0006 Location of the damage Location of the damage Location of the damage -0.0015 x=1.1 m x=1.1 m -0.0008 x=1.1 m -0.0015 -0.0020 -0.0010 Location of the damage Location of the damage Location of the damage -0.0020 x=0.44 m x=0.44 m -0.0025 -0.0012 x=0.44 m -0.0014 -0.0025 -0.0030 0 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98 2.2 02 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98.2 02 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98.2 x (m) x (m) x (m) (d) (e) (f) Figure 23: Working conditions 7–12 diference of wavelet coefcients. (a) Working condition 7. (b) Working condition 8. (c) Working condition 9. (d) Working condition 10. (e) Working condition 11. (f) Working condition 12. As shown in Figure 23, the wavelet coefcient diference indicator can accurately locate the damage occurring in units Table 8: Diference of wavelet coefcients for diferent degrees of damage at 480 mm–520 mm. 5 and 10 in the case of two damages. Even in the case of minor damage, the damage location can be correctly de- Degrees Diferences of wavelet Damage location (mm) termined, indicating that the use of the wavelet coefcient of damage (%) coefcients diference index is better than the curvature mode index in 1 0.00003868 locating minor damage, and the maximum value of the 2.5 0.0001406 wavelet coefcient diference index increases with the degree 5 0.0003567 480–520 of damage. For cases 7–12, it should be noted that damage to 10 0.0008155 the wooden beam occurred in units 5 and 10, and the nodes 15 0.00145 20 0.00177 of unit 5 were 0.44 and 0.55 m from the end of the beam, and the nodes of unit 10 were 0.99 and 1.1 m from the end of the beam. Te peak value at 0.44 m and 1.1 m indicates that the diference and the degree of damage under diferent con- beam is damaged near this location, and when combined ditions, where the number of polynomials ftted is taken as 3, with the actual damage location, it is known that the beam is the damage ftting curve at the position of 480 mm–520 mm damaged in unit 5 and unit 10. (unit 5) is shown in Figure 24. It should be noted that only the wavelet coefcient diference corresponding to the (3) Analysis of Damage Level Identifcation. Te wavelet damage degree of 1%, 2.5%, 5%, 10%, and 20% was ftted coefcient diference index increases with the increase in under this section, the wavelet coefcient diference under damage degree when single damage and two damage lo- 15% damage degree was not involved in the ftting, and the calization are performed on wooden beams, so the damage purpose is to let the wavelet coefcient diference of 15% degree of wooden beams can be estimated by ftting the damage degree as a verifcation of the applicability of the relationship equation between the wavelet coefcient dif- ftting formula. ference index and the damage degree. Te relationship between the wavelet coefcient difer- Due to space limitation, this article analyzes the example ence index and the degree of damage in a single damage is of working conditions 1–6 in single damage. Table 8 shows obtained from Figure 24 as follows: the diference in the peak wavelet coefcient for the simple- − 8 3 − 6 2 − 5 − 5 supported wooden beam at the single damage location y � −7.0 × 10 x + 2.6 × 10 x + 6.6 × 10 x − 3.3 × 10 , 480 mm–520 mm (unit 5) with diferent damage levels (11) (because the wavelet coefcient diference value is negative, it is taken as the absolute value for the convenience of wherex is the level of damage, and y is the wavelet coefcient analysis). diference damage indicator at position 480 mm–520 mm In order to further obtain the relationship between the (unit 5). wavelet coefcient diference index and the degree of To verify the applicability of the ftting formula, the damage, this section based on the data in Table 8, based on absolute value of the diference in the wavelet coefcient at the principle of least squares, and using Origin software to ft the position of 480 mm–520 mm (cell #5) was entered into the relationship curve between the wavelet coefcient equation (11) and solved using MATLAB to obtain a damage Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Advances in Civil Engineering 17 0.0020 difcult for the identifcation of the damage to the wooden frame; with the future development of modal test technol- ogy, this problem will gradually be improved. 0.0015 Data Availability 0.0010 Te data used to support the fndings of this study are in- cluded within the article. 0.0005 Conflicts of Interest Te authors declare that they have no conficts of interest. 0.0000 0 1 2.5 5 10 20 Degree of damage (%) Acknowledgments Figure 24: 480 mm–520 mm location damage degree and the Te work was supported by the National Natural Science damage index ftting graph. Foundation of China (52068063), Shandong Province Graduate Natural Science Foundation (ZR2020ME240), level of 14.51% at the position of 480 mm–520 mm (cell #5), Gansu Province Natural Science Foundation Research with an error of only 3.27%; therefore, the damage level of Program (21JR1RE286), Gansu Province Higher Education the wooden beam can be estimated using equation (11). Innovation Fund Project (2020B-173), Fuxi Scientifc Re- search Innovation Team Project (FXD2020-13), and the Maijishan Grottoes Art Research Project of Tianshui Normal 4. Conclusion University (MJS2021-06). Tis paper takes a simple-supported wooden beam as the research object, performs a three-layer discrete wavelet References transform on the frst-order curvature mode before and after the damage, proposes the method of using the wavelet [1] R. Hou and Y. 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Ren, “Structural damages identifcation based on curvature mode and wavelet trans- form,” Journal of Vibration, Measurement & Diagnosis, vol. 316, no. 4, pp. 263–267, 2005. [17] A. Q. Li and Y. L. Ding, Early Warning Teory of Engineering Structure Damage and its Application, Science Press, Beijing, China, 2007. [18] Gb/T 1927-2009, Wood Physical andMechanical TestMaterial Collection methods, China Standard Press, Beijing, Chnia, [19] Gb/T 1929-2009, Wood Physical and Mechanical Specimens Sawing and Specimen Interception methods, China Standard Press, Beijing, China, 2009. [20] B. R. Miao, S. W. Yang, and M. Y. Wang, “Comparison of multiple structural damage identifcation methods using vi- bration response,” Journal of Vibration Engineering, vol. 33, no. 04, pp. 724–733, 2020. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Civil Engineering Hindawi Publishing Corporation

Wood Beam Damage Identification Based on the Curvature Mode and Wavelet Transform

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10.1155/2023/2238021
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Hindawi Advances in Civil Engineering Volume 2023, Article ID 2238021, 18 pages https://doi.org/10.1155/2023/2238021 Research Article Wood Beam Damage Identification Based on the Curvature Mode and Wavelet Transform 1 1 1 2 1 1 Zhaobo Meng, Xiancai Ren, Shanqing Chai , Xin Wang, Tengfei Zhao , Feifei Gao, 1 1 Shanwei Wang, and Yufa Liu School of Architecture and Civil Engineering, Liaocheng University, Liaocheng 252000, Shandong, China School of Civil Engineering, Tianshui Normal University, Tianshui 741001, Gansu, China Correspondence should be addressed to Tengfei Zhao; zhaotengfei@lcu.edu.cn Received 24 October 2022; Revised 17 April 2023; Accepted 21 April 2023; Published 3 May 2023 Academic Editor: Dimitrios G. Pavlou Copyright © 2023 Zhaobo Meng et al. Tis 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. In the paper, a simple-supported wooden beam is used as the research object to identify the damage of the wood beam by fnite element analysis and experimental research. First, ANSYS was used to establish the solid fnite element model of the wood beam before and after the damage, and then the discrete wavelet transform was performed on the curvature mode of the wood beam before and after the damage, and the wavelet coefcient diference index was obtained after obtaining the high frequency wavelet coefcients. Ten, the damage location of the wood beam was judged according to the sudden peak of the wavelet coefcient diference index, and the damage degree of the wood beam was estimated by ftting the relationship between the wavelet coefcient diference index and the degree of damage. Finally, the index was verifed by the wooden beam test. Te results show that the wavelet coefcient diference index can accurately identify the damaged location of the wood beams. Te degree of damage to the wood beams at the damage location can be quantitatively estimated by ftting the relationship between the wavelet coefcient diference index and the degree of damage at the damage location. Te research results provide a theoretical basis to identify wooden beam damage. parameters before and after the damage may be used to 1. Introduction determine structural damage [3]. Some scholars have done Wood beams are the main load bearing elements of the much work on the early state damage detection of beams, historic timber structure; under the infuence of environ- and some research results have been achieved [4–6]. He et al. mental conditions, sudden disasters, human damage, and [7] showed that the index of the curvature modal diference other factors, wood beams are prone to damage, afecting the could indicate the location of the damage and the degree of safety, applicability, and durability of the wood structure of a cantilever beam. Xiang et al. [8] suggested the curvature- ancient buildings. Tey can seriously lead to the collapse of based modal utility information entropy as the damage the entire wood structure. Terefore, the study of the detection index, and numerical modeling and testing on identifcation of structural damage to wood beams is supported beams confrmed its validity. Ren et al. [9] studied signifcant. the sensitivity of the natural frequency, the curvature mode, Te three difculties of time, spatial location, and and the diference in the curvature mode to the damage of damage level should be addressed in structural damage wooden beams and conducted a numerical simulation identifcation, and damage identifcation methods have been analysis on the wooden beams. Based on the frst-order created to address the problems mentioned earlier [1, 2]. curvature mode diference index, the infuence of noise Structural damage can cause changes in the physical char- pollution and mesh density on recognition accuracy was acteristics of the structure and in the modal parameters of studied. However, some researchers have found that cur- the structure. As a result, the change in the modal vature modes are inefective in identifying slight damage 2 Advances in Civil Engineering [10]. Terefore, we need to propose a new method to solve where q(x) is the curvature, ρ(x) is the curvature radius, these problems. M(x) is the bending moment of the beam cross section, and Te wavelet transform has the property of local am- EI(x) is the fexural rigidity of the beam section. plifcation of the mutation signal in the time and frequency According to the theory of material mechanics, the domains; so, it can determine the location and degree of curvature function in any section of the beam is expressed as minor damage that occurs in the structure. Janeliukstis et al. follows: [11] suggested a damage identifcation algorithm based on the modal structural vibration wavelet transform, and the € ϕ (x)q (t), (2) q(x) � 􏽘 i i efectiveness of the proposed method was confrmed with i�1 fnite element models of aluminum beams in diferent where ϕ (x) is the j order curvature modal, and q (t) is the i i damaged parts. Bao et al. [12] performed the continuous modal coordinates. According to formulas (1) and (2), wavelet transform of the curvature mode of the structure structural damage leads to a reduction in structural stifness, before and after damage. Tey obtained the wavelet co- which results in an increase in the vibration of the curvature efcient diference index, which was shown to accurately mode. Terefore, sudden change in vibration of the cur- discriminate the damage site and the degree of damage to the vature mode in a certain order can be used to diagnose structure by fnite element simulation and testing of the structural damage. supported beam. Machorrolopez et al. [13] proposed To obtain the curvature mode of the structure before and a method of acoustic emission signal combined with con- after damage, the calculation must be performed with the tinuous wavelet transform, and the proposed method was central diference method based on the displacement mode validated by concrete subjected to bending tests. Te results [14, 15]. Under the premise that the displacement mode and showed that the method could identify the damaged state of the vibration mode of the equally spaced discrete unit nodes concrete beams by the transformed wavelet energy index. are known, the curvature mode is defned as follows: Te abovemrntioned research results carry out relevant u u u damage identifcation studies on diferent structures as re- ϕ (j − 1) − 2ϕ (j) + ϕ (j + 1) i i i (3) ϕ (j) � , search objects but relatively few damage identifcation iu 2 studies on individual wood beam structures. Tis paper proposes a damage identifcation method based on the d d d ϕ (j − 1) − 2ϕ (j) + ϕ (j + 1) i i i (4) curvature mode and wavelet transform for simple-supported ϕ (j) � , id wooden beams. Compared to other methods, this method can better refect the location of the damage to the wooden where ϕ(j) represents the i − th order of curvature modal at beam structure and has a particular sensitivity to the degree j points, u and d represent lossless and lossy states, and l is of damage. Te degree of damage to the wooden beam at the the distance between adjacent nodes. damage location is quantitatively estimated by ftting the relationship equation between the diference index of the 2.2. Principles of Wavelet Transform Damage Identifcation. wavelet coefcient diference index at the damage location and the degree of damage. Specifcally, frst, ANSYS is used For any function f(t) ∈ L (R), the continuous wavelet transform is to build a solid fnite element model of the wooden beam before and after damage and perform modal analysis. Ten, +∞ 1 x − b √�� � the curvature mode of the wooden beam before and after the W (a, b) � 􏽚 f(x)ψ dt, (5) 􏼠 􏼡 |a| a −∞ damage is subjected to a discrete wavelet transform to obtain high frequency wavelet coefcients and obtain the wavelet where W (a, b) is the wavelet coefcient, ψ(x) is the wavelet coefcient diference index. Ten, the damage location of the function, ψ (x) is the complex conjugate of ψ(x), and a and wooden beam is judged according to the sudden change b are the translation and scale factors. peak of the wavelet coefcient diference index, and the In an application, it is necessary to discretize the con- degree of the wooden beam is estimated by ftting the re- tinuous wavelet. Its binary wavelet transform can be lationship between the wavelet coefcient diference index expressed as follows: and the degree of damage. Second, the index was verifed by +∞ 1 x − b wooden beam tests; fnally, the limitations and challenges of j ∗ 􏽱��� W 􏼐2 , b􏼑 � 􏼌 􏼌 􏽚 f(x)ψ 􏼠 􏼡dx. f (6) 􏼌 􏼌 the method are given in the conclusion. j 􏼌 􏼌 a −∞ 􏼌 􏼌 􏼌 􏼌 2. Materials and Methods When b � k2 , the above equation is the discrete wavelet transform. 2.1.PrincipleofDamageIdentifcationintheCurvatureMode. According to the material mechanics theory, the static bending force of the beam is expressed as follows: 2.3. Selection of Wavelet Functions. In signal analysis, choosing and building the appropriate wavelet function is 1 M(x) q(x) � � , (1) essential. Te results of diferent wavelet functions for the ρ(x) EI(x) same problem difer in practice. Terefore, when using wavelet analysis to detect mutation signals, it is necessary to Advances in Civil Engineering 3 combine the analyzed signal’s characteristics with the re- divided, as shown in Figure 3. Te model x-axis is the beam search purpose and select the appropriate wavelet function length direction, the y-axis is the beam section height di- rection, and the z-axis is the beam section width direction. for local mutation detection. In this paper, we refer to the process and principles of wavelet selection in the literature Tat means that the left end of the model constrains UX, UY, [16, 17], combine the analyzed signal’s characteristics, and UZ, ROTX, and ROTY, the right end constrains UY, UZ, fnally select the bior6.8 wavelet function. ROTX, and ROTY, and UZ is constrained at the neutral axis to prevent the wood beam from moving in the z direction. It is assumed that the reduction of the cross-sectional 2.4. Structural Damage Identifcation Index. Tis paper uses height causes damage to the structure, so the damage is the curvature mode indicator as the input signal. Te bior6.8 simulated in the fnite element by reducing the cell cross- wavelet is selected to perform the three-layer discrete sectional size; the specifc operation steps are as follows: at wavelet transform to obtain the high-frequency wavelet the bottom of the wooden beam along the beam width coefcients and then make the diference to get direction, create a rectangular slot body with length 160 mm, d u width 40 mm, depth 2 mm, 5 mm, 10 mm, 20 mm, 30 mm, j j j j (7) DW � W 􏼐2 , k2 􏼑 − W 􏼐2 , k2 􏼑 , fij f f ij ij and 40 mm and then carry out the Boolean operation. Te FEM model of the wooden beam under diferent damage where DW is the diference of wavelet coefcients at i fij conditions is obtained by the command of subtracting the node under the j − th order curvature mode before and after body from the body in the Boolean process. Figure 4(a) d u j j j j structural damage. W (2 , k2 ) and W (2 , k2 ) are the f ij f ij shows the fnite element model of the wood beam for wavelet coefcients at i node in the j − th curvature mode working condition 6, and Figure 4(b) shows the enlarged before and after damage. view of the damage of the wood beam fnite element model for working condition 6. Te degree of damage S is defned as the ratio of the depth of the rectangular slots he with the 2.5. Damage Identifcation Steps height of the beam h, that is, S � he/h, and the degree of (1) First, a modal analysis was performed on the damage is 1%, 2.5%, 5%, 10%, 15%, and 20%. Te damage members of the wooden beam to obtain their frst- conditions are shown in Table 2. order curvature modal data before and after damage. (2) Te bior6.8 wavelet is used to perform the three-layer 3. Results and Discussion discrete wavelet transform in the frst-order curva- ture mode before and after the damage and obtain 3.1. Comparison of the Inherent Frequency under Diferent the high-frequency wavelet coefcients and then WorkingConditions. Based on the abovementioned damage make the diference to get the wavelet coefcient conditions, the intrinsic frequencies of the wood beam diference index. before and after damage are shown in Table 3. Table 3 shows that it can be seen that the overall frequency change before (3) Te damage location was localized on the basis of the and after damage to the wooden beam is small; the maxi- sudden peak of the wavelet coefcient diference mum relative change in frequency before and after damage is index. Te degree of damage to the wooden beam 9.42%. It shows that the frequency change is not sensitive to was estimated by ftting the relationship equation the slight local damage of the wood beam; working con- between the wavelet coefcient diference index and ditions 7–12 compared with the second-order frequency the degree of damage. relative change of working conditions 1–6 showed that its amplitude is small, analysis of the reason, the location of the 2.6. Numerical Simulations. Tis paper uses a simple- damage occurred in the vicinity of the vibration node, supported timber beam as the research object for the lo- resulting in the relative change of frequency amplitude not calization of damage and the study of the degree of damage. apparent, and through this indicator, we can determine the Te wood beam (width 160 mm and height 200 mm) with occurrence of damage to the wood beam. However, the a span of 2400 mm and a net span of 2200 mm, and the location and extent of damage to the wood beam cannot be specifc dimensional arrangement of the wood beam is judged. shown in Figure 1. To obtain the parameters related to the wood timber property in the fnite element model of the simple-supported wood beam, the authors conducted wood 3.2. Comparison of Displacement Vibration Patterns under timber property tests on the same batch of poplar wood Diferent Working Conditions. In order to verify whether beams. All tests were collected, fabricated, and tested displacement vibration is sensitive to damage, the dis- according to national standards [18, 19], part of the test placement vibration data values corresponding to 56 nodes procedure is shown in Figure 2, and the results of the timber on a line on the middle top surface at the level of the center property tests of the wood beam are shown in Table 1. axis of the wooden beam in diferent working conditions Te three-dimensional solid model of the wood beam is were extracted in the article, considering that the test simulated by the Solid185 unit with fnite element ANSYS. measurement points were arranged on the middle axis of the Te mesh cell size is 40 mm, the wood beam model is meshed top surface of the wooden beam (Figure 5), and the spacing by sweep command, and a total of 1960 solid cells are between each node in the model was 0.04 m. Displacement 4 Advances in Civil Engineering Figure 1: Size drawing of the wooden beam (mm). Table 1: Material parameters of wood beams. Elastic modulus (MPa) Poisson’s ratio Modulus of shear (MPa) Densities (kg/m ) E E E μ μ μ G G G L R T LR LT RT LR LT RT 9752.17 2058.57 452.90 0.490 0.410 0.353 585.13 731.41 175.54 Electronic universal testing machine Electronic balance Data Acquisition System Vernier calipers Test pieces Ovens (a) (b) Electronic universal testing machine Electronic universal testing machine Data Acquisition System Data Acquisition System Test pieces Test pieces (c) Figure 2: Material quality test process. (a) Moisture content and density test. (b) Method for determination of the modulus of elasticity in compression parallel to the grain of wood. (c) Method for determining the modulus of elasticity in compression perpendicular to the grain of wood. vibration patterns were converted according to the co- and Figure 8 is the frst two orders of the displacement vibration pattern of two damage in working conditions ordinate relationship between the nodes and the wooden beam model. Te displacement vibration pattern is con- 13–18, where the subpicture part is the magnifcation of the verted to the two-dimensional plane for analysis. In this vibration pattern at the local damage location. section, the extracted displacement vibration patterns are As can be seen in Figures 6–8, in single damage or in two normalized to the maximum amplitude of the vibration damage conditions, the diference between the frst two pattern to facilitate the subsequent analysis. orders of displacement vibration curves of the wooden beam Figures 6 and 7 are the frst two orders of the dis- structure is minimal and difcult to distinguish through the placement vibration pattern of single damage in working naked eye; so, it is difcult to identify the damage to the conditions 1–6 and working conditions 7–12, respectively, wooden beam structure using displacement vibration. 200 Advances in Civil Engineering 5 (a) (b) Figure 3: Finite element model of a wood beam. (a) Finite element model. (b) Boundary conditions. (a) (b) Figure 4: FEM and damage magnifcation of the wooden beam for work conditions 6. (a) Finite element model of the wood beam. (b) Enlarged view of the damage. Table 2: Condition of damage to the wood beam. Degrees Working conditions Damage types Damage locations (m) of damage (%) 0 Undamaged — — 1 520–560 1 2 520–560 2.5 3 520–560 5 4 520–560 10 5 520–560 15 6 520–560 20 One location damage 7 1040–1080 1 8 1040–1080 2.5 9 1040–1080 5 10 1040–1080 10 11 1040–1080 15 12 1040–1080 20 13 520–560, 1040–1080 1 14 520–560, 1040–1080 2.5 15 520–560, 1040–1080 5 Two location damage 16 520–560, 1040–1080 10 17 520–560, 1040–1080 15 18 520–560, 1040–1080 20 3.3. Single Location Damage Identifcation Analysis. In the displacement array of the wooden beam is used as the measured measurements, the higher-order modes of the original data for subsequent damage identifcation studies in structure are not easily measured. At the same time, research this article [20]. In this article, the extracted frst-order scholars have found that lower-order methods play a dom- displacement vibrations are normalized to the maximum inant role in structural vibration; the frst-order magnitude of the pulses to facilitate subsequent analysis. 6 Advances in Civil Engineering Table 3: Changes in intrinsic frequency before and after damage. 1st order frequency 2nd order frequency Working conditions Relative Relative Frequencies (Hz) Frequencies (Hz) frequency change (%) frequency change (%) 0 76.55 — 264.14 — 1 76.55 0.00 264.05 0.03 2 76.52 0.04 263.80 0.13 3 76.42 0.17 263.02 0.42 4 75.95 0.78 260.28 1.46 5 75.14 1.84 256.22 3.00 6 73.90 3.46 250.81 5.05 7 76.53 0.03 264.15 0.00 8 76.46 0.12 264.16 0.01 9 76.19 0.47 264.19 0.02 10 75.20 1.76 264.23 0.03 11 73.65 3.79 264.20 0.02 12 71.45 6.66 263.74 0.15 13 76.51 0.05 264.05 0.03 14 76.39 0.21 263.80 0.13 15 75.99 0.73 263.01 0.43 16 74.51 2.67 260.25 1.47 17 72.29 5.57 256.14 3.03 18 69.34 9.42 250.64 5.11 (a) (b) Figure 5: Wood beam model extraction layer and the extraction line. 1.1 1.25 1.0 1.0 0.8 0.5 0.6 0.0 0.4 -0.5 0.2 -1.0 0 -1.25 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) Intact 10% Intact 10% 1% 15% 1% 15% 2.5% 20% 2.5% 20% 5% 5% (a) (b) Figure 6: Working conditions 1–6 frst- and second-order displacement vibration. (a) First-order vibration. (b) Second-order vibration. Displacement vibration Displacement vibration Advances in Civil Engineering 7 1.1 1.25 1.0 1.0 0.8 0.5 0.6 0.0 0.4 -0.5 0.2 -1.0 0 -1.25 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) Intact 10% Intact 10% 1% 15% 1% 15% 2.5% 20% 2.5% 20% 5% 5% (a) (b) Figure 7: Working conditions 7–12 frst- and second-order displacement vibration. (a) First-order vibration. (b) Second-order vibration. 1.1 1.25 1.0 1.0 0.8 0.5 0.6 0.0 0.4 -0.5 0.2 -1.0 0.0 -1.25 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) Intact 10% Intact 10% 1% 15% 1% 15% 2.5% 20% 2.5% 20% 5% 5% (a) (b) Figure 8: Working conditions 13–18 frst- and second-order displacement vibration. (a) First-order vibration. (b) Second-order vibration. Te normalized frst-order displacement mode calculates difcult to identify its damage location, indicating that the the frst-order curvature modes of each working condition indicator is not sensitive in the case of minor damage to the according to equations (3) and (4). Figures 9 and 10 show the structure, cannot remember the site of structural damage, frst-order curvature modes of the single damage for coupled with the interference of external environmental working conditions 1–6 and working conditions 7–12, noise, and may lead to the indicator not identifying the respectively. location of minor damage to the structure. Also, the wavelet As can be seen from Figures 9 and 10, when the damage transform has the feature of local amplifcation of abrupt degree is greater than the 5% case, the curvature mode at the signals in the time and the frequency domain, which can damage location produces a sudden change, which can be identify the location and the degree of minor structural identifed by the peak of the sudden change to identify the damage. Terefore, in this paper, the frst-order curvature modal data of wooden beams are subjected to a three-layer location of wood beam damage; when the damage degree is less than the 5% case, the curvature mode at the damage discrete wavelet transform using bior6.8 wavelets, and the location of the sudden change is not signifcant, and it is high-frequency wavelet coefcients are derived and then Displacement vibration Displacement vibration Displacement vibration Displacement vibration 8 Advances in Civil Engineering 0.5 efectively reduce the infuence of adjacent units, making the recognition efect more signifcant. In the case of slight 0.0 damage, the wavelet coefcient diference index also has ups -0.5 and downs at the undamaged position but its transformation amplitude is smaller than that of the damaged position, -1.0 which has little infuence on the determination of the -1.5 damaged position; with the increasing damage degree, the wavelet coefcient diference at the undamaged position is -2.0 almost close to the level, and the sudden change peak of the -2.5 wavelet coefcient diference at the damaged position is -3.0 pronounced, and the peak of the wavelet coefcient dif- ference index increases with the increasing damage degree. -3.5 0 0.4 0.8 1.2 1.6 2 2.2 x (m) 3.4. Two Location Damage Identifcation Analysis. To verify Intact 10% the sensitivity of the wavelet coefcient diference index for 1% 15% the identifcation of the two location damage, the working 2.5% 20% conditions (conditions 13–18) were established for the si- 5% multaneous presence of damage at positions 520–560 mm Figure 9: First-order curvature modes of working conditions 1–6. and 1040–1080 mm of the wooden beam. Figure 13 shows the frst-order curvature modal diagrams of the two damage locations for cases 13–18. As shown in Figure 13, consistent 0.5 with the single damage results, the indicator is not sensitive 0.0 in the case of minor damage to the structure. It cannot -0.5 identify the location of the damage to the wood beam and coupled with the interference of external environmental -1.0 noise, which may cause the indicator to fail to identify the -1.5 location of minor damage to the structure. Terefore, in this -2.0 paper, the frst-order curvature modal data of the wood -2.5 beam are subjected to a three-layer discrete wavelet trans- -3.0 form using bior6.8 wavelets, and the high-frequency wavelet -3.5 coefcients are derived and then diferenced to obtain the -4.0 diference maps of wavelet coefcients under each working condition, as shown in Figure 14. -4.5 0 0.4 0.8 1.2 1.6 2 2.2 As seen in Figure 14, the wavelet coefcient diference x (m) index produces a signifcant abrupt change at two pre- determined damage locations of the wood beam. Te lo- Intact 10% cation of damage to the structure of the wood beam can still 1% 15% be accurately determined in the case of minor damage, 2.5% 20% 5% which indicates that the wavelet coefcient diference index is more efective than the curvature mode index in locating Figure 10: First-order curvature modes of working conditions minor damage to the wood beam, and the maximum value of 7–12. the wavelet coefcient diference index increases with the increasing damage degree. divided to obtain the diference maps of the wavelet co- efcients under each working condition, as shown in Fig- 3.5. Identifcation of the Degree of Damage. Te wavelet coefcient diference index was found to increase with the ures 11 and 12. It should be noted that since the damage location is located in the cell between two nodes, the wavelet increase in the degree of damage when a single damage and two damage localizations were performed on the wooden coefcient diference indicator is taken as the extreme value of one of the two mutation nodes; so, the peak of the beam, which indicated that there was a close correlation between the wavelet coefcient diference index and the mutation point in the wavelet coefcient diference map is located in one of the two nodes of the damaged cell, and it degree of damage with high sensitivity; so, the degree of the can indicate the existence of damage near this point. wooden beam could be estimated by ftting the relationship As can be seen in Figures 11 and 12, compared to the equation between the wavelet coefcient diference index problem that the curvature mode index is not apparent to and the degree of damage. recognize minor damages to wood beams, the problem is For single damage, this paper analyzes the working efectively solved using the wavelet coefcient diference conditions 1–6 as an example and adds the simulation of the damage depth of 60 mm at the location of 520 mm–560 mm index. Tis indicator can amplify the signal of small abrupt changes in the modal curvature index and, at the same time, for single damage of the wooden beam by ANSYS, that is, the Curvature modes Curvature modes Advances in Civil Engineering 9 0.008 0.03 0.08 0.006 0.06 0.02 0.004 0.04 0.01 0.002 0.02 0.00 0.000 0.00 -0.002 -0.02 -0.01 -0.004 -0.04 Location of the damage Location of the damage Location of the damage -0.02 x=0.56 m -0.006 -0.06 x=0.56 m x=0.56 m -0.03 -0.008 -0.08 -0.010 -0.04 -0.10 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) x (m) (a) (b) (c) 0.3 0.4 0.6 0.3 0.4 0.2 0.2 0.2 0.1 0.1 0.0 0.0 0.0 -0.1 -0.2 -0.1 -0.2 Location of the damage Location of the damage Location of the damage -0.4 x=0.56 m x=0.56 m -0.3 x=0.56 m -0.2 -0.6 -0.4 -0.3 -0.5 -0.8 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) x (m) (d) (e) (f) Figure 11: Working conditions 1–6 diference of wavelet coefcients. (a) Working condition 1. (b) Working condition 2. (c) Working condition 3. (d) Working condition 4. (e) Working condition 5. (f) Working condition 6. 0.010 0.04 0.15 0.03 0.10 0.005 0.02 0.01 0.05 0.000 0.00 0.00 -0.01 -0.005 -0.02 -0.05 Location of the damage Location of the damage Location of the damage -0.03 x=1.04 m -0.010 x=1.04 m x=1.04 m -0.10 -0.04 -0.015 -0.05 -0.15 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) x (m) (a) (b) (c) 0.3 0.6 0.8 0.6 0.2 0.4 0.4 0.1 0.2 0.2 0.0 0.0 0.0 -0.2 -0.1 -0.2 Location of the damage -0.4 Location of the damage -0.2 -0.4 Location of the damage x=1.04 m -0.6 x=1.04 m x=1.04 m -0.3 -0.6 -0.8 -0.4 -0.8 -1.0 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) x (m) (d) (e) (f) Figure 12: Working conditions 7–12 diference of wavelet coefcients. (a) Working condition 7. (b) Working condition 8. (c) Working condition 9. (d) Working condition 10. (e) Working condition 11. (f) Working condition 12. Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference 10 Advances in Civil Engineering 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -3.5 -4.0 0 0.4 0.8 1.2 1.6 2 2.2 x (m) Intact 10% 1% 15% 2.5% 20% 5% Figure 13: First-order curvature modes of working conditions 9–12. 0.010 0.05 0.10 0.04 0.005 0.03 0.05 0.02 0.000 0.01 0.00 0.00 -0.005 -0.01 -0.05 -0.02 Location of the damage Location of the damage Location of the damage Location of the x=1.04 m -0.010 -0.03 -0.10 Location of the x=1.04 m x=1.04 m damage x=0.56 m Location of the damage x=0.56 m -0.04 damage x=0.56 m -0.015 -0.05 -0.15 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) x (m) (a) (b) (c) 0.3 0.6 0.8 0.6 0.2 0.4 0.4 0.1 0.2 0.2 0.0 0.0 0.0 -0.1 -0.2 -0.2 -0.2 Location of the -0.4 Location of the Location of the damage x=1.04 m -0.4 Location of the -0.3 Location of the Location of the damage x=1.04 m -0.6 damage x=0.56 m damage x=1.04 m damage x=0.56 m damage x=0.56 m -0.6 -0.4 -0.8 0.0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2 1.6 2 2.2 0 0.4 0.8 1.2 1.6 2 2.2 x (m) x (m) x (m) (d) (e) (f) Figure 14: Working conditions 13–18 diference of wavelet coefcients. (a) Working condition 13. (b) Working condition 14. (c) Working condition 15. (d) Working condition 16. (e) Working condition 17. (f) Working condition 18. damage degree is 30%. Te maximum value of the diference In order to further obtain the relationship between the in the wavelet coefcient is shown in Table 4 (because the wavelet coefcient diference index and the degree of diference in the wavelet coefcient value is negative, and to damage, this paper is based on the data in Table 4 and on the facilitate analysis, the absolute value is taken). principle of least squares and used Origin software to ft the Wavelet coefficient difference Wavelet coefficient difference Curvature modes Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Advances in Civil Engineering 11 Table 4: Diference of wavelet coefcients for diferent degrees of 1.2 damage at 520 mm–560 mm. 1.0 Degrees Diferences of wavelet Damage locations (mm) of damage (%) coefcients 0.8 1 0.00797 0.6 2.5 0.03038 5 0.0844 0.4 520–560 10 0.23195 15 0.41909 0.2 20 0.61348 30 1.12284 0.0 0 1 2.5 5 10 15 20 30 32 Degree of damage (%) relationship curve between the wavelet coefcient diference Figure 15: 520 mm–560 mm location damage degree and the and the degree of damage under diferent working condi- damage index ftting graph. tions, where the number of polynomials ftted is taken as 3, and the damage ftting curve is shown in Figure 15. Te relationship between the wavelet coefcient difer- 0.8 ence index and the degree of damage in a single damage 0.6 location (520 mm–560 mm) is obtained from Figure 15 as 0.4 follows: 0.2 − 6 3 − 4 2 y � −4.8 × 10 x + 8.6 × 10 x + 0.016x − 0.014, (8) 0.0 -0.2 where x is the level of damage, and y is the wavelet co- efcient diference damage index. -0.4 Location of the damage Te applicability of the ftted formula was verifed in -0.6 x=0.56 m ANSYS by simulating a damage depth of 50 mm (a damage -0.8 level of 25%) at the location of 520 mm–560 mm for a single -1.0 damage to the wood beam. 0 0.4 0.8 1.2 1.6 2 2.2 Figure 16 gives the wavelet coefcient diference plot of x (m) 25% of the degree of damage in the single damage position of Figure 16: Diference of wavelet coefcients for 25% of damage at the wooden beam of 520 mm–560 mm. From Figure 13, the 520−560 mm position. absolute value of the diference in the wavelet coefcient in the position of 520 mm–560 mm is 0.83026. Taking the Table 5: Diference of wavelet coefcients for diferent degrees of absolute value of the diference of the wavelet coefcient into damage at two locations. equation (8), the degree of damage at the position of 520 mm–560 mm is obtained as 24.63% using the MATLAB Degrees Diferences of wavelet Damage locations (mm) solution, which is only a 1.76% error. Terefore, the degree of damage (%) coefcients of the wooden beam can be estimated using equation (8). 1 0.00861 For the two injuries, ANSYS increased the simulation of 2.5 0.03021 the depth of damage of 60 mm at 520 mm–560 mm and 5 0.07732 1040 mm–1080 mm for the two injuries of the timber beam, 520–560 10 0.20317 15 0.3495 that is, the degree of damage was 30%. Te peaks of the 20 0.47594 wavelet coefcient diference for the two damage locations, 30 0.7406 520 mm–560 mm and 1040 mm–1080 mm, are shown in 1 0.01257 Table 5 (because the peaks of the wavelet coefcient dif- 2.5 0.04034 ference is negative, it is taken as an absolute value for the 5 0.10942 convenience of analysis). 1040–1080 10 0.29704 According to the data in Table 5, based on the principle 15 0.49628 of least squares, Origin software was used to ft the re- 20 0.67856 lationship curves between the diference of the wavelet 30 1.02204 coefcients and the degree of damage under diferent working conditions, where the number of ftted polynomials − 5 3 − 4 2 y � −1.4 × 10 x + 7.5 × 10 x + 0.015x − 0.011, was taken as 3. Te damage ftting curves for the two damage locations are shown in Figure 17. (9) Te relationship between the wavelet coefcient difer- − 5 3 − 3 2 ence index and the degree of damage at the two damage y � −2.4 × 10 x + 1.1 × 10 x + 0.022x − 0.018, locations was obtained from Figure 17 as follows: (10) Wavelet coefficient difference Wavelet coefficient difference 12 Advances in Civil Engineering 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 Location of the -0.6 Location of the damage x=1.04 m -0.8 damage x=0.56 m -1.0 0 0.4 0.8 1.2 1.6 2 2.2 x (m) Figure 18: Diference of wavelet coefcients for 25% of damage at two positions. Figure 17: Two location damage degree and the damage index ftting graph. rectangular slot depth he to the beam heighth, that is,S � he/ h. Te specifc rectangular slot locations are shown in Table 6. where x is the level of damage, y is the wavelet coefcient Te test adopted a single-point excitation and a multipoint diference damage indicator at 520 mm–560 mm position, output measurement method to conduct modal tests on the and y is the wavelet coefcient diference damage indicator wooden beam. A force hammer was used to hammer the at the position 1040 mm–1080 mm. wooden beam at a preset reference point. Te structure’s re- Te applicability of the ftted formula was verifed in sponse was collected in batches by multiple INV9822 accel- ANSYS by simulating the damage depth of 50 mm (the erometers from the Beijing Oriental Vibration and Noise damage was 25%) at the locations of 520 mm–560 mm and Technology Research Institute. An average of 19 acceleration 1040 mm–1080 mm for the two damage locations of the sensors must be placed within the clear span of the wooden wood beam. beam. Te magnetic holders are glued to the top surface of the Te wavelet coefcient diference plots of the 25% wooden shaft with 502 glue and the acceleration sensors are damage level for the two damage locations, installed on the magnetic holders. Data were collected and 520 mm–560 mm and 1040 mm–1080 mm, for the wooden analyzed with a multifunctional data acquisition analyzer beams are given in Figure 18. (model INV306N2) and DASP (V11) signal analysis software In Figure 18, the absolute values of the diferences in the using an impact hammer to strike the eight # acceleration wavelet coefcient at the positions 520 mm–560 mm and sensor measurement points three times. Te feld test and the 1040 mm–1080 mm are 0.59787 and 0.83205, respectively. schematic diagram are shown in Figure 20. Te absolute values of the wavelet coefcient diferences at the two damage positions are brought into equations (9) and (10), and the degree of damage at the 520 mm–560 mm 3.6.2. Analysis of Experimental Results. Te inherent fre- position is obtained by using MATLAB to solve for 24.39%, quencies and vibration patterns of the wooden beam were calculated for diferent operating conditions by referring to with only 2.44% error; at the 1040 mm–1080 mm position, the damage degree is 24.61%, with only 1.56% error. the EMA modal test procedure in the DASP&INV product operation and using the guide. Table 7 compares the frst Terefore, the degree of the wooden beam can be estimated using equations (9) and (10). second-order inherent frequencies of the intact wood beam fnite element model with the frst second-order inherent frequencies of the test. From Table 7, it can be seen that the 3.6. Experimental Verifcation error between the frst-order inherent frequency of the fnite element model and the frst-order inherent frequency of the 3.6.1. Experimental Program. To further verify the feasibility tested beam is 1.74%, increases with the increase of the of the proposed method, wooden beams with the same modal order, and the maximum value is 1.41%, which is material and dimensions as the numerical simulation were within the acceptable range. Figure 21 compares the frst two used as the research object. Te wooden beams were divided orders of displacement modal vibration patterns of the fnite into 20 parts of 110 mm each, and the damage was simulated element model of the intact wood beam and the intact wood by changing the cross-sectional dimensions of the wooden beam test. In Figure 21, it can be seen that the modal vi- beams. Te specifc implementation was as follows: rect- bration pattern of the structure is consistent with the modal angular slots of length 160 mm, width 40 mm, and depth vibration pattern characteristics of the supported beam. 2 mm, 5 mm, 10 mm, 20 mm, 30 mm, and 40 mm were Trough the abovementioned analysis, the fnite element manually cut by an electric planer at the bottom of the model and the test results of the wooden beam can be wooden beams along the beam width direction (Figure 19). verifed with each other. Te degree of damage S is defned as the ratio of the 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 1 2.5 5 10 15 20 30 32 Degree of damage (%) 1040 mm-1080 mm 520 mm-560 mm Wavelet coefficient difference Wavelet coefficient difference Advances in Civil Engineering 13 Table 6: Test wood beam damage conditions. Degrees Working conditions Damage types Damage locations (m) Damage units of damage (%) 0 Undamaged — — — 1 480–520 5 1 2 480–520 5 2.5 3 480–520 5 5 One location damage 4 480–520 5 10 5 480–520 5 15 6 480–520 5 20 7 480–520, 1040–1080 5, 10 1 8 480–520, 1040–1080 5, 10 2.5 9 480–520, 1040–1080 5, 10 5 Two location damage 10 480–520, 1040–1080 5, 10 10 11 480–520, 1040–1080 5, 10 15 12 480–520, 1040–1080 5, 10 20 40 mm 30 mm 20 mm 10 mm 5 mm 2 mm (a) (b) (c) (d) Figure 19: Wooden beam grooving site. (a) Cutting. (b) Sanding. (c) Slotting. (d) Diferent depths of slots. wood beam Sensor Computer Acquisition instrument Power Hammer Figure 20: Wooden beam feld test chart. (1) Single Location Damage Identifcation Analysis. Tis measurement process, the wavelet threshold method is used paper uses DASP (V11) signal analysis software to extract to denoise the measured displacement vibration data. Ten, frst-order displacement vibration data under diferent the denoised displacement vibration is normalized to the working conditions. Due to noise interference in the maximum magnitude. Te wavelet coefcient diference 14 Advances in Civil Engineering Table 7: Comparison of frequency of the test wooden beam and the fnite element model. Calculated values of Calculated values of Orders the fnite element test wood beams Errors (%) model (Hz) (Hz) 1 76.55 75.24 1.74 2 264.14 260.48 1.41 COINV_DASP 0 .175608 .351216 .526824 .702432 .087804 .263412 .43902 .614628 .790236 (a) (b) COINV_DASP 0 .116923 116923 .233846 233846 .350769 30 69 .467692 467692 .058462 .175385 .292308 .409231 .526154 (c) (d) Figure 21: Comparison of the frst- and second-order displacement pattern of intact wooden beam and the fnite element model. (a) Vibration pattern in the frst-order displacement mode of fnite elements. (b) Test the frst-order displacement mode of fnite elements. (c) Finite element second-order displacement mode of fnite elements. (d) Test the second-order displacement mode of fnite elements. diagrams under working conditions 1–6 are obtained by index increases with the increasing damage degree. For processing the displacement vibration data according to working conditions 1–6, it should be noted that the location equations (3) to (7), as shown in Figure 22. It should be of the damage to the wooden beam in the test occurred in unit 5, and the nodes at both ends of unit 5 are 0.44 m and noted that since the damage location is located in the cell between two nodes, the wavelet coefcient diference in- 0.55 m away from the end of the beam, respectively, and the dicator is taken as the maximum value of the two mutation horizontal coordinates in the fgure are the beam length, nodes; so, the peak value of the mutation point in the wavelet while the wavelet coefcient diference indicators all peak at coefcient diference diagram is located in one of the two 0.44 m. Tis indicates that the wood beam was damaged near nodes of the damaged cell, and it can indicate the existence this location, and when combined with the actual location of of damage near this point. the damage, it is clear that the wood beam was damaged in In Figure 22, it can be seen that the use of the wavelet unit 5. coefcient diference index can efectively overcome the problem that the curvature mode index is not apparent for (2) Two Location Damage Identifcation Analysis. Te small damage identifcation and can accurately determine wavelet coefcient diference plots for working conditions 7–12 are obtained according to the same processing method the location of the damage to the wood beam structure, and the maximum value of the wavelet coefcient diference for single-site damage, as shown in Figure 23. Advances in Civil Engineering 15 10 mm 5 mm 2 mm 0.00004 0.00020 0.0003 0.00003 0.00015 0.0002 0.00002 0.00010 0.0001 0.00001 0.00005 0.0000 0.00000 0.00000 -0.0001 -0.00001 Location of the damage Location of the damage -0.00005 Location of the damage x=0.44 m x=0.44 m -0.00002 -0.0002 x=0.44 m -0.00010 -0.00003 -0.0003 -0.00015 -0.00004 -0.00020 -0.0004 0 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98 2.2 0 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98 2.2 0 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98 2.2 x (m) x (m) x (m) (a) (b) (c) 40 mm 20 mm 30 mm 0.0006 0.0010 0.0010 0.0004 0.0005 0.0005 0.0002 0.0000 0.0000 0.0000 -0.0005 -0.0005 -0.0002 Location of the damage -0.0004 Location of the damage x=0.44 m -0.0010 -0.0010 Location of the damage x=0.44 m x=0.44 m -0.0006 -0.0015 -0.0015 -0.0008 -0.0010 -0.0020 -0.0020 0 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98 2.2 0 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98 2.2 0 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98 2.2 x (m) x (m) x (m) (d) (e) (f) Figure 22: Working conditions 1–6 diference of wavelet coefcients. (a) Working condition 1. (b) Working condition 2. (c) Working condition 3. (d) Working condition 4. (e) Working condition 5. (f) Working condition 6. 10 mm 5 mm 10 mm 2 mm 2 mm 5 mm 0.00008 0.00020 0.0005 0.0004 0.00006 0.00015 0.0003 0.00004 0.00010 0.0002 0.00002 0.00005 0.0001 0.00000 0.00000 0.0000 -0.00002 -0.0001 -0.00005 -0.00004 -0.0002 Location of the damage -0.00010 -0.00006 x=1.1 m Location of the damage Location of the damage -0.0003 x=1.1 m -0.00015 x=1.1 m -0.00008 Location of the damage -0.0004 x=0.44 m -0.00020 Location of the damage Location of the damage -0.00010 -0.0005 x=0.44 m x=0.44 m -0.00012 -0.00025 -0.0006 02 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98.2 02 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98.2 02 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98.2 x (m) x (m) x (m) (a) (b) (c) Figure 23: Continued. Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference 16 Advances in Civil Engineering 40 mm 30 mm 30 mm 20 mm 40 mm 20 mm 0.0008 0.0015 0.0020 0.0006 0.0015 0.0010 0.0004 0.0010 0.0005 0.0002 0.0005 0.0000 0.0000 0.0000 -0.0002 -0.0005 -0.0005 -0.0004 -0.0010 -0.0010 -0.0006 Location of the damage Location of the damage Location of the damage -0.0015 x=1.1 m x=1.1 m -0.0008 x=1.1 m -0.0015 -0.0020 -0.0010 Location of the damage Location of the damage Location of the damage -0.0020 x=0.44 m x=0.44 m -0.0025 -0.0012 x=0.44 m -0.0014 -0.0025 -0.0030 0 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98 2.2 02 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98.2 02 0.22 0.44 0.66 0.88 1.1 1.32 1.54 1.76 1.98.2 x (m) x (m) x (m) (d) (e) (f) Figure 23: Working conditions 7–12 diference of wavelet coefcients. (a) Working condition 7. (b) Working condition 8. (c) Working condition 9. (d) Working condition 10. (e) Working condition 11. (f) Working condition 12. As shown in Figure 23, the wavelet coefcient diference indicator can accurately locate the damage occurring in units Table 8: Diference of wavelet coefcients for diferent degrees of damage at 480 mm–520 mm. 5 and 10 in the case of two damages. Even in the case of minor damage, the damage location can be correctly de- Degrees Diferences of wavelet Damage location (mm) termined, indicating that the use of the wavelet coefcient of damage (%) coefcients diference index is better than the curvature mode index in 1 0.00003868 locating minor damage, and the maximum value of the 2.5 0.0001406 wavelet coefcient diference index increases with the degree 5 0.0003567 480–520 of damage. For cases 7–12, it should be noted that damage to 10 0.0008155 the wooden beam occurred in units 5 and 10, and the nodes 15 0.00145 20 0.00177 of unit 5 were 0.44 and 0.55 m from the end of the beam, and the nodes of unit 10 were 0.99 and 1.1 m from the end of the beam. Te peak value at 0.44 m and 1.1 m indicates that the diference and the degree of damage under diferent con- beam is damaged near this location, and when combined ditions, where the number of polynomials ftted is taken as 3, with the actual damage location, it is known that the beam is the damage ftting curve at the position of 480 mm–520 mm damaged in unit 5 and unit 10. (unit 5) is shown in Figure 24. It should be noted that only the wavelet coefcient diference corresponding to the (3) Analysis of Damage Level Identifcation. Te wavelet damage degree of 1%, 2.5%, 5%, 10%, and 20% was ftted coefcient diference index increases with the increase in under this section, the wavelet coefcient diference under damage degree when single damage and two damage lo- 15% damage degree was not involved in the ftting, and the calization are performed on wooden beams, so the damage purpose is to let the wavelet coefcient diference of 15% degree of wooden beams can be estimated by ftting the damage degree as a verifcation of the applicability of the relationship equation between the wavelet coefcient dif- ftting formula. ference index and the damage degree. Te relationship between the wavelet coefcient difer- Due to space limitation, this article analyzes the example ence index and the degree of damage in a single damage is of working conditions 1–6 in single damage. Table 8 shows obtained from Figure 24 as follows: the diference in the peak wavelet coefcient for the simple- − 8 3 − 6 2 − 5 − 5 supported wooden beam at the single damage location y � −7.0 × 10 x + 2.6 × 10 x + 6.6 × 10 x − 3.3 × 10 , 480 mm–520 mm (unit 5) with diferent damage levels (11) (because the wavelet coefcient diference value is negative, it is taken as the absolute value for the convenience of wherex is the level of damage, and y is the wavelet coefcient analysis). diference damage indicator at position 480 mm–520 mm In order to further obtain the relationship between the (unit 5). wavelet coefcient diference index and the degree of To verify the applicability of the ftting formula, the damage, this section based on the data in Table 8, based on absolute value of the diference in the wavelet coefcient at the principle of least squares, and using Origin software to ft the position of 480 mm–520 mm (cell #5) was entered into the relationship curve between the wavelet coefcient equation (11) and solved using MATLAB to obtain a damage Wavelet coefficient difference Wavelet coefficient difference Wavelet coefficient difference Advances in Civil Engineering 17 0.0020 difcult for the identifcation of the damage to the wooden frame; with the future development of modal test technol- ogy, this problem will gradually be improved. 0.0015 Data Availability 0.0010 Te data used to support the fndings of this study are in- cluded within the article. 0.0005 Conflicts of Interest Te authors declare that they have no conficts of interest. 0.0000 0 1 2.5 5 10 20 Degree of damage (%) Acknowledgments Figure 24: 480 mm–520 mm location damage degree and the Te work was supported by the National Natural Science damage index ftting graph. Foundation of China (52068063), Shandong Province Graduate Natural Science Foundation (ZR2020ME240), level of 14.51% at the position of 480 mm–520 mm (cell #5), Gansu Province Natural Science Foundation Research with an error of only 3.27%; therefore, the damage level of Program (21JR1RE286), Gansu Province Higher Education the wooden beam can be estimated using equation (11). Innovation Fund Project (2020B-173), Fuxi Scientifc Re- search Innovation Team Project (FXD2020-13), and the Maijishan Grottoes Art Research Project of Tianshui Normal 4. Conclusion University (MJS2021-06). Tis paper takes a simple-supported wooden beam as the research object, performs a three-layer discrete wavelet References transform on the frst-order curvature mode before and after the damage, proposes the method of using the wavelet [1] R. Hou and Y. 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Advances in Civil Engineering – Hindawi Publishing Corporation

Published: May 3, 2023

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