DSpace at VNU: A new method for beam-damage-diagnosis using adaptive fuzzy neural structure and wavelet analysis

Mechanical Systems and Signal Processing 39 (2013) 181–194 Contents lists available at SciVerse ScienceDirect Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp A new method for beam-damage-diagnosis using adaptive fuzzy neural structure and wavelet analysis Sy Dzung Nguyen a,c, Kieu Nhi Ngo b, Quang Thinh Tran c, Seung-Bok Choi d,n a Inha University, Republic of Korea Laboratory of Applied Mechanics (LAM) of Ho Chi Minh University of Technology, Vietnam Ho Chi Minh University of Industry, HUI, Vietnam d Department of Mechanical Engineering, Smart Structures and Systems Laboratory, Inha University, Incheon 402-751, Republic of Korea b c a r t i c l e i n f o abstract Article history: Received 18 December 2012 Received in revised form 22 March 2013 Accepted 23 March 2013 Available online 16 April 2013 In this work, we present a new beam-damage-locating (BDL) method based on an algorithm which is a combination of an adaptive fuzzy neural structure (AFNS) and an average quantity solution to wavelet transform coefficient (AQWTC) of beam vibration signal The AFNS is used for remembering undamaged-beam dynamic properties, while the AQWTC is used for signal analysis Firstly, the beam is divided into elements and excited to be vibrated Vibrating signal at each element, which is displacement in this work, is measured, filtered and transformed into wavelet signal with a used-scale-sheet to calculate the corresponding difference of AQWTC between two cases: undamaged status and the status at the checked time Database about this difference is then used for finding out the elements having strange features in wavelet quantitative analysis, which directly represents the beamdamage signs The effectiveness of the proposed approach which combines fuzzy neural structure and wavelet transform methods is demonstrated by experiment on measured data sets in a vibrated beam-type steel frame structure & 2013 Elsevier Ltd All rights reserved Keywords: Fuzzy neural networks Wavelet transform Damage location Damage diagnosis Structure health monitor Introduction It is well known that the health monitoring of structures is one of the serious public issues which is concerned by many researchers The approach for the health monitoring of structures can be classified into two major groups: model-based and data-driven methods [1] In the first method group, the fault growth trend of structure is forecasted based on the understanding of failure model progression Since there are many different structures and many different types of failures, it is difficult to develop accurate models in most practical instances Moreover, in some cases the fault propagation of the structures is quite complex and not fully understood In the second method group, collected condition data is used for building the fault propagation models This group is based on mechanical properties in which some physical characteristics of the structure such as structure stiffness affect to structure vibration characteristics Damage appearing on structure reduces its stiffness, or reduces horizontal section area, or both of these Hence, the changes of the vibration characteristics of the structure, such as the natural frequency, displacement or mode shape are signs to observe the damage in the structure These features are very important for structural health monitoring systems including structural damage detection, location and quantification [2] Many solutions to structure damage identification and prediction based on these signs have n Corresponding author Tel.: +82 32 860 7319 E-mail addresses: nsidung@yahoo.com (S.D Nguyen), seungbok@inha.ac.kr (S.-B Choi) 0888-3270/$ - see front matter & 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.ymssp.2013.03.023 182 S.D Nguyen et al / Mechanical Systems and Signal Processing 39 (2013) 181–194 Nomenclature the ith hyperplane of the NF A(i) a scale parameter in wavelet transform a–a damage degree 3.56% (.) að:Þ j ; j ẳ 0n coefficient of A a ẳ ẵas1 as2 ⋯asm Š scale vector used for surveying process b position index in wavelet transform b–b damage degree 6.9% cdj damage coefficient of the jth element crj relative damage index c–c damage degree 11.16% d–d damage degree 23.4% Er ð:Þ average-square error of the ANN HAS horizontal section area M number of fuzzy rules N number of neurons at the hidden layer of the ANN Ne number of divided elements in the surveyed beam n number of dimensions of the input data space of the data set (or the factors relative to the vibration-exciting statuses) P number of samples in the data set Ptest number of vibration exciting statuses used for damage checking process pHB Rk TΣ T Σcheck W Wf :ị xi ; yi ị xi ẳ ẵxi1 Yi yiị j zji :ị ikj :ị ẵi i ψn pure cluster typed a hyperbox number of k-labeled fuzzy sets set of training data sets corresponding to undamaged time of beam set of checking data sets weight vector of the ANN wavelet transform f ð:Þ the ith input (xi )–output (yi ) data sample of training data set xi2 :::xin Š the ith input vector of training data set (the ith vibration-exciting status) displacement function at the ith position of the surveyed beam output value of the ith data sample corresponding to the jth hyperplane, A(j) average quantity of wavelet transform input-space-data group typed a hyperbox of the NF system membership value of the ith data sample in jth k-labeled-fuzzy set the ith input (ℵi )–output (℘i ) data sample of the ANN mother wavelet complex conjugate of the mother wavelet function ψ been presented [3–8] It is remarked that since the vibration signal of the real structures is the measured signal with noise distortion, the filtering noise and the use of signal-analysis tools are to be carefully treated One of effective approaches to handle this problem is to use wavelet transform [9,10] Since most of real physical systems are nonlinear, ill-defined and uncertain, it is difficult to directly establish models by conventional mathematical means [11] In addition, database for structure-health-monitor system, which is based on data-driven methods, has to be correlatively established at two times: structure-undamaged time and the checking time This is really difficult if this process is based on traditional ways because we are not able to exactly repeat an exciting status at two different times Hence mathematical models such as artificial neural networks technique (ANN) [6,12–14] or fuzzy logic (FL) [15], or combination of these models [1,11] are frequently used to overcome the impending challenges with different degrees In [12], ANN was used for predicting the onset of corrosion in concrete bridge decks taking into account the parameter uncertainty In [6], a solution to bridge-damaged detection was proposed in which the location of the damage was obtained by using modal energy-based damage index values The wavelet transform method analyzes the signal in two dimensions: time-frequency or space-frequency In this use, instead of using only a constant width window such as Fourier transform, wavelet transform uses a window-width-variable parameter called scale a which can play a role similar to frequency By this way, wavelet transform is able to locally analyze signals to find out irregular events in dynamic response signal such as vibration signal of the damaged structure [16] There have been many researches using wavelet transform for damage detection [3,9,10] in both types of damages: fatigue or cracks ANN technique and FL are types of artificial intelligence techniques They have the potential to deliver effectively solutions to problems which are difficult or impossible to be performed by conventional linear methods [11] Since these models can map any complicated functional relationship between independent and dependent variables, they can provide better prediction and identification capabilities than traditional methods [17–19] NF is a fuzzy system built based on ANN Since these mathematical models could combine to get advantages of FL and ANN, this has saliently strong points than ANN or FL only NF has usefully been used for identification and prediction [1,20,21] To build NF, data space was classified to establish data clusters having common features [20,21] This process creates the data clusters having a role as a skeleton to build fuzzy sets of the fuzzy insurance system On the other hand, this classifying process could separate alien data samples from the data set to place them in distinct clusters At these distinct clusters membership values of other data samples in the data set are very small, or even are zero As a result, this could reduce influence of these alien data samples on calculated output of data samples to increase accuracy of NF system Hence in this side, the impact of the above data classifying process on system can be seen as a noise-filtering process for the data set This is one of featuring advantages of the NF built based on this way However, the effectiveness of this issue depends on S.D Nguyen et al / Mechanical Systems and Signal Processing 39 (2013) 181–194 183 suitable degree of the NF-membership function for data set features As usually, to adjust the role of each data cluster in output-value-calculating process of the net, membership function is adjusted using an adjustment parameter, such as γ in [20,21] It is known that the choice of the adjustment parameter depends on features of each data space Actually, it significantly affects the prediction accuracy of the model However it is very difficult to find out appropriate values of this for each real application Consequently, the main contribution of this work is to propose a novel approach for beam-damage-diagnosis which is easily applicable in practice The proposed approach focuses on the combination of fuzzy neural structure and wavelet transform methods In order to achieve this goal, AFNS is used for building NF system with data clusters typed hyperboxes which are created and joined as a part of input data space of the ANN Subsequently, the role of each data cluster in outputvalue-calculating process of the net is adjusted by the training process of the ANN By this way, the influence of unsatisfactory-relationship-value quantity between fuzzy sets on calculating result of the fuzzy influence system is compensated by the ANN when calculating the output value This increases accuracy of the model Furthermore, in order to overcome the difficulty of signal analysis, a wavelet-quantitative analysis method is proposed In this research, the damage positions are investigated using two main steps As a first step, database for the beam health monitor system, which is based on the change of its dynamic property signals, is established at two times: the structure-undamaged time and the checking time under same vibration exciting status (VES) Usually this is really difficult due to that we can not exactly repeat VES at two different times However, the use of the proposed AFNS could exactly interpolate data at any time when the structure is not damaged The second step is to analyze the created database using the wavelet-quantitative analysis based on average wavelet transform of vibration signal with a used-scale-sheet to calculate the corresponding difference of AQWTC between two cases: the undamaged status and the status at the checked time Database about this difference is then used for finding out the elements having strange features in wavelet quantitative analysis, which are beam-damage signs This could overcome the difficulty about finding out the optimal scale in wavelet transform for each application The proposed method is experimentally implemented on the beam-typed steel frame structure Experiment results are analyzed to verify the effectiveness of the proposed method as well as to evaluate the application possibility of the method for actual structures Proposed fuzzy neural system To well interact to a system its mathematical model needs to be built firstly In structure damage location using datadriven methods as mentioned in the introduction, this model is usually established by identification based on measured data sets A given data set consisting of input–output samples is expressed by ðxi ; yi ị; xi ẳ ẵxi1 xi2 :::xin ; i ẳ 1:::P: ð1Þ The above database is used for identifying the dynamic response of structure In the above equation,xi is the input vector,yi is corresponding output of the ith data sample in the set having P data samples There are various ways to identify system based on this database In this paper, the proposed AFNS shown in Fig is used for this work This AFNS contains two substructures: an adaptive neuro-fuzzy system (NF) built by the neuro-fuzzy-building algorithm named HLM1of [20] and cascade-forward neural networks (ANN) The ANN consists of one input layer, one hidden layer and one output layer Number of input signals depends on feature of training data set and structure of the NF; number of neurons at the hidden layer of the net is adaptively established in training process The proposed structure of the AFNS in Fig is established with main steps as follows: (1) Building data clusters of given data set – Using the algorithm Hyperplane Clustering of [21] for building min–max hyperbox clusters Γ ðkÞ in input data space and hyperplane classes AðkÞ ; k ¼ 1:::M; in output data space – Using the algorithm for cutting and separating the hyperbox having largest sample number named CSHL of [20] to build pure data clusters, in which each pure cluster contains common-labeled samples Building NF system (2) – Using the algorithm HLM1 for training process to build NF system Structure of the ANN – Input vector of the ANN is M+n dimensions The number of neuron N in the hidden layer is N ¼N0 at the beginning time of the training process and adaptively adjusted in the AFNS training process Where N0 is default number In this (3) paper, we use N0 ¼ 100 The number of neuron in the output layer is The ‘sum’ function is used for input of all of neurons The purelin function, f sị ẳ s, is used for output of the neuron at output layer Transfer function is used for all of neurons at the hidden layer as follows: f sị ẳ 1 ỵ exp 2sị – The ith input–output sample of data set used for training the ANN is signed and established as follows: ½ℵi i ẳ ẵxi yiị ị; yi : 2ị 184 S.D Nguyen et al / Mechanical Systems and Signal Processing 39 (2013) 181–194 Input layer Hyper-planes of the NF pHB11 The ANN layer μi11 xi = [xi xin ] y1( t ) (i = P ) xi1 xi2 pHB1R1 μi1R1 pHBj1 μij1 Wi(in 2) (i = M ) b (2) y ij pHBjRj Output Wk(lr) (k = N ) μijR j ˆyt W j(in1 ) xin ( j = M + n ) pHBM1 bk(1) μiM (t) yM pHBMRM μi1MR M Wi(in 2) (i = n) Fig Structure of the AFNS based on an adaptive neuro-fuzzy system (NF) and cascade-forward neural networks (ANN) This is combined by the ith input–output sample of the data set (1), ðxi ; yi Þ, and vector of values of corresponding ðiÞ ðiÞ hyperplanesyðiÞ of the NF system trained by this data set (1), yiị ẳ ẵyiị y2 ::: yM These values are calculated as follows: n kị ẳ akị yiị j xij ỵ a0 ; k k ẳ 1:::M; i ẳ 1:::P jẳ1 (k) whereakị created by training process of the NF p ; p ¼ 0…n, are coefficients of A Training the ANN to adjust net parameters Let W be the weight vector of the ANN, W ¼ ½w1 w2 :::wH ŠT The error equation of the ANN can be written as follows: (4) Er ðWÞ ¼ P ^ i ðWÞÞ2 ∑ ð℘ −℘ Pi¼1 i P ẳ e2i Wị iẳ1 ẳ V T WịVWị where VWị ẳ ẵv1 Wị v2 Wị ::: vP WịT ẳ ẵe1 Wị e2 Wị ::: eP WịT Using the algorithm Levenberg–Marquardt, the weight vector W of the ANN at the (k+1)th loop, signed Wk+1, is calculated as follows: W kỵ1 ẳ W k ẵJ T W k ịJW k ị ỵ I1 J T W k ịVW k Þ where, I is unit-square matrix, size H; μ is an adaptive index; and J is the matrix Jacobian as follows: ∂v ∂v1 ∂v1 ⋯ ∂w ∂w ∂w H ∂v ∂v ∂v2 2 ⋯ ∂w H JðW k Þ ¼ ∂w1 ∂w2 ⋮ ⋮ ⋱ ⋮ ∂vP ∂vP ∂vP ⋯ ∂wH ∂w1 ∂w2 ðW k Þ S.D Nguyen et al / Mechanical Systems and Signal Processing 39 (2013) 181–194 185 Wavelet transform 3.1 Theory basis Wavelet analysis provides a powerful tool to characterize local features of a signal Unlike the Fourier transform, where the function used as the basis of decomposition is always a sinusoidal wave, other basis functions can be selected for wavelet shape according as the features of the signal These basis functions named mother wavelet ψ A mother wavelet is a function of zero average: Z ỵ tịdt ¼ 0: −∞ This can be dilated or compressed with a scale parameter a, and translated by a position parameter b as follows:   t−b a 0; bR a;b tị ẳ p a a The wavelet transform of f at the scale a and position b is computed by correlating f(t) with a wavelet atom The continuous wavelet transform of f(t) is defined as follows [16]:   Z ỵ tb Wf a; bị ¼ dt f ðtÞ pffiffiffi ψ n a a −∞ where ψ n is complex conjugate of the mother wavelet functionψ When using the wavelet for signal analysis, if the scale parameter a is small, it results in very narrow windows and is appropriate for high frequency components in the signal f(t) Oppositely if the scale parameter a is large, it results in wide windows and is suitable for the low frequency components in the signal f(t) The choice of scale can be based on actual analysis of the demand signal The greater the scale is, the more details of the frequency division are In fact, the advantages of wavelet transform in signal analysis can be realized by selecting only appropriate wavelet function and wavelet scale 3.2 Average quantity wavelet coefficient 3.2.1 Selection of mother wavelet function In signal analysis using wavelet transform, selection of the most appropriate wavelet mother function is really essential In this work, the following considerations are used for this: – The database used in this work is the displacement signal of the surveyed vibration beam which has a nearly symmetric shape In the database, it can be observed that signal value can usually reach zero at the initial and final points in a period Consequently, the wavelet mother function has to be compactly supported or nearly compactly supported with a finite duration In addition, both of orthogonal and biorthogonal properties are not required in this work since the reconstruction of the original signal is not required – In signal decomposition and reconstruction, orthogonal wavelets such as Daubechies wavelets [22,23] are usually used due to their efficiencies Relation to the orthogonal wavelets, to satisfy the orthogonal property, integral of the wavelet function and integral of the square of the wavelet function must respectively be equal to zero and one However, the above constraint makes the orthogonal wavelets non-differentiable [24] In our research the wavelet NF model is proposed, in which an optimization solution is utilized By this approach, derivatives of the wavelet function are required in order to minimize errors between the model outputs and the actual outputs For this reason, in this use a nonorthogonal differentiable wavelet mother function is suitable Based on issues abovementioned, it can be seen that the Mexican hat function is an appropriate selection for this work In fact the Mexican hat wavelet function has several principal characteristics Firstly, it is a rapidly vanishing function [25] and is a computationally efficient function Secondly, it can be analytically differentiated and can be used conveniently for decomposing multidimensional time series Equation of the selected wavelet is given as follows: "  " 2 #  2 # t−b t−b ψ a;b tị ẳ p 1=4 exp 3ị a a 3.2.2 Sampling frequency Actually, the appropriate sampling solution is usually selected according to the target of each use For example, the relation to digitally recorded noisy data decomposition and reconstruction, sampling at unequally spaced times is usually used In [26], to obtain information from noisy signals using wavelet methods, Hall and Penev suggested an adaptive sampling rule Differently, the relation to signal analysis to investigate crucial characteristics of the signal source without 186 S.D Nguyen et al / Mechanical Systems and Signal Processing 39 (2013) 181–194 reconstruction the initial signal, sampling at equally spaced times can be used In [24], a time-frequency domain wavelet analysis of acceleration signal of earthquake records was accomplished based on this solution In this study, the goal is to investigate signal characteristics to find out beam damage signs, without initial signal reconstruction Consequently, firstly the initial signal f ðtÞ is sampled at equally spaced times, corresponding to a constant sampling frequency A set of translation parameter b is then taken at the same points where the f ðtÞ is sampled A scale vector a ẳ ẵas1 as2 asm , which is being depicted in detail in the next section, is then created to achieve an appropriate range of frequency resolution Subsequently, the created parameters a and b are used to dilate or compress the mother wavelet in order to accomplish a family of wavelet ψ a;b ðtÞ Next f ðtÞ is multiplied by the wavelets ψ a;b ðtÞ at different values of the scale a and translation b Finally, the continuous wavelet transform (CWT) coefficient Wf ða; bÞ is then obtained by summing the created produces Wf ða; bÞ indicates the correlation between the signal and the wavelet functions ψ a;b ðtÞ 3.2.3 Building the scale vector As abovementioned, the selection of the optimal scale for beam-damage diagnosis is very important to achieve best effective result However, the common way presented above is not enough to choose the best value of the scale In addition, due to the analyzed data the measured signals with noise distortion are occurred Consequently, the result of using only one value of the scale which is considered to be suitable for wavelet analysis to obtain information relative to the system is sometimes unsatisfactory Instead of getting the system information we may receive only features relative to noisy To overcome these, in this work the use of the scale vector a is proposed to build the AQWTC for finding out beam damage signs To create the scale vector a ẳ ẵas1 as2 asm , the modified GramSchmidt algorithm of [27] is used, where m is number of wavelets obtained This work is performed as follows: – Building element data sets: The surveyed beam is divided into Ne elements By measuring vibration signal at each element, which is displacement signal in this work, Ne element data sets called subsets are obtained – Building the element scale vector: Using the modified Gram–Schmidt algorithm for subsets, Ne element scale vectors, signed K i ; i ¼ 1:::N e ; are correspondingly established Relation to building K i ; the process can be summarized as follows Firstly, empty wavelets whose supports not contain any data are deleted from the wavelet decomposition Result of this work is a nonzero wavelet coefficient frame to be created Next, from this frame the wavelet which gives best approximates to the measured data is selected Subsequently, this wavelet is combined with the remainder of the nonzero wavelet coefficient frame one at a time to determine the best combination This procedure is repeated for all nonzero wavelet coefficients and finally the element scale vector is created as a result of the accomplished process – Establishing the scale vector: The vector ais structured based on the following set: K ¼ fK ∪K ∪:::∪K Ne g where, ∪ denotes the union operator 3.2.4 Average quantity wavelet coefficient Based on Ne element data sets and the scale vector a created, AQWTC of the jth element at ith vibration status of the beam, signed zji , is calculated as follows: zji ¼ m ∑ Wf ji ðask ; bị; j ẳ 1Ne mkẳ1 4ị where f ji is the function relating to jth element at ith vibration state of the beam In this paper f is displacement function of element Beam-damaged location algorithm (BDLA) Based on the AFNS and using wavelet analysis as presented above, we propose an algorithm for beam-damaged location named BDLA Fig shows flow chart of the BDLA, and Fig presents relationship of parameters used in the algorithm At the beam-undamaged time: Step Building data sets The beam is vibrated by different exciting statuses Displacement is measured and filtered to build Ne data sets The jth data set, corresponding to the jth element, has P data samples ðxi ; zji Þ; in which xi ẳ ẵxi1 :::; xin ; i ¼ 1:::P, expresses the ith vibration-exciting status (VES); n depicts factors relative to the vibration-exciting statuses; zji is the AQWTC calculated based on (4) as follows: zji ¼ Bm m k ẳ bị Wf ji ask ; bị, j¼1…Newhere B is number of sampled points of function f (it is also the size of vector b) Step Identifying elements at the beam-undamaged time Ne the AFNS are trained to identify elements based on Ne corresponding data sets created in Step At the checking time: S.D Nguyen et al / Mechanical Systems and Signal Processing 39 (2013) 181–194 187 START Building jthdata sets ( xi , z ji ) at undamaged time of the beam, j=1 Ne ,i=1 P AFNSj, j=1 Ne , trained Step Undamaged time Step by ( xi z ji ) to identify the beam at undamaged time Set up Ptest ; k=0 k =: k +1 kP= Step test ? Y N Calculating Damage Coefficients - Making vibration of the beam by xk - Using the AFNSj for Checking time cdj (6), crj(7) calculating zˆ jk Determine condition of the beam based on - Calculating z jk based on (4), j = Ne Step c dj, c rj Y Continue ? N STOP Fig Flow chart of the proposed algorithm BDLA The ith vibration exciting status( xi , i = Ptest ) xi1 AFNSj xi2 zˆ ji (j=1…Ne) Calculate crj , cdj j = Ne zji xin - Measure vibration signal - Calculate zji Fig Principle for calculating damage indexes based on the AFNS Step Calculating damage coefficients – Perform Ptest vibration exciting statuses, which could be different from the vibration statuses used for beam-undamaged identification as presented in Step At each exciting status, vibration signal at elements is measured, filtered to calculate AQWTC (zji ; ) based on (4) j ¼ 1:::Ne ; i ¼ 1:::P test Based onzji ;calculate AQWTC at each element for all Ptest vibration exciting statuses as follows: zj ¼ P test m ∑ ∑ Wf ji ask ; bị; j ẳ 1:::N e mBP test i ẳ k ẳ bị 5ị where B is number of sampled points of function f (it is also the size of vector b) – Based on Ptest vibration exciting statuses above, use Ne the AFNS built in Step for establishing AQWTC corresponding to each exciting status at each element, and then calculate AQWTC ðzðnotÞ Þat each element for all Ptest vibration exciting j 188 S.D Nguyen et al / Mechanical Systems and Signal Processing 39 (2013) 181194 statuses as follows: znotị ẳ j P test ∑ z^ ; j ¼ 1:::Ne P test i ¼ ji where ‘not’ expresses value of the index AQWTC corresponding to beam-undamaged time Damage coefficient cdi of the jth element is calculated as follows: ðnotÞ cdj ẳ zj zj ; j ẳ 1Ne 6ị – Relative change of AQWTC on each element at one time is expressed by a relative damage index, signed crj , calculated based on (6) as follows: crj ¼ cdj ; max ẵcdk j ẳ 1:::N e 7ị k ¼ 1:::N e Step Locating damage of beam Element having crj ¼ (7) is the one most decreased in flexural rigidity, and degree of this damage is shown bycdj (6) Experimental investigation The proposed algorithm, BDLA, is used for data sets measured on a vibrated beam-typed steel frame to find out its damaged locations 5.1 Test rig and procedures The experimental model is shown in Fig Motor D carrying a mass M is fixed on a beam-typed iron frame at many different positions The center deviation level of M, Md, is easily varied by varying distance d from position of fixed M to the rotation axle center of D The frequency converter V is used to change angular velocityωof D Hence, the factors relative to the vibration-exciting statuses n is The frame, length L¼3 m, is divided into 12 equal parts by 13 nodes signed Y1,…,Y13 Sensors (1), the vibration signal measuring system LAM_BRIDGE (2), and a computer (3) are used to measure vibration signal at these nodes at different vibration-exciting-statuses (VES) At the frame-undamaged time: By changing the position of D on the frame, changing the Md and angular velocity ω of D, we created P ¼1500 VES to measure corresponding vibration signals of the structure at Y2,…,Y12 By this way, a data set named T Σ consisting of Ne element subsets, each of these subsets having P¼ 1500 (n ¼3) input–output samples was established The T Σ was then used for training the AFNS to identify all elements of the frame at its undamaged time At the checking time: To make frame-damaged conditions, the frame was cut at one or two positions In each case, the reduction of horizontal section area (HSA) of the frame at the cut positions was performed by a group in levels: a–a (3.56% of HSA), b–b (6.9% of HSA), c–c (11.16% of HSA) and d–d (23.4% of HSA) Corresponding to each damage status above, by exciting to produce vibration of the frame in order to measure vibration signal at its nodes we established a checking data set, named T Σcheck used for checking process Based on T Σcheck and the AFNSs trained by T Σ , the algorithm BDLA was then used to find out beam-damaged locations corresponding to these conditions of the frame, upon this the effectiveness of the proposed algorithm was estimated 5.2 Results and discussion 5.2.1 Analysis of sensitivity In order to estimate sensitivity of the signal used for the proposed algorithm, AQWTC zji (4), surveys are performed in this section In this survey, the beam was damaged at Y6 with degrees, a–a, b–b, c–c Analysis of vibration signal at this position (Y6) was performed to calculate AQWTC zji (4) and damage index cd (6) in two cases: (1) using the ANN for identifying undamaged status of the beam and a value a¼ 0.5 of the scale used for wavelet transform vibration signal, and (2) using the proposed method with a used-scale vector a In both of these cases, beam-vibration-exciting statuses were the same Results presented in Fig 5a, b shows that although condition of the beam, beam-vibration-exciting statuses and measuring position are the same, AQWTC zji and damage index cd in the first case is smaller than corresponding parameters in the second case Namely cd is 0.014, 0.02, and 0.028, in the first case; and is 0.0444, 0.0509, and 0.0542 in the second case, corresponding to damage degrees a–a, b–b, and c–c, respectively Besides, high increase of zji in Fig 5b after each cycle more clearly expresses damaged feature appearing in the beam than in Fig 5a These show that the proposed algorithm, BDLA, in this work depicts better real condition of the beam at damaged location, even at small damaged degree 3.56%, a–a S.D Nguyen et al / Mechanical Systems and Signal Processing 39 (2013) 181–194 Y13 Y12 Y11 Y10 Y9 Y8 Y7 Y6 Y5 Y4 Y3 Y2 189 Y1 L= 3m Fig Experimental model: (a) photograph of the model; (b) the dividing nodes 0.8 0.6 Average Wavelet Index, zji Average Wavelet Index, zji 0.8 c-c, cd=0.028 b-b, cd=0.020 a-a, cd=0.014 0.4 0.2 c-c, cd=0.0542 b-b, cd=0.0509 0.6 a-a, cd=0.0444 0.4 0.2 0.0 0.0 200 400 600 800 Time (sampled points) 1000 200 400 600 800 1000 Time (sampled points) Fig The beam was damaged at Y6 with degrees: a–a, b–b, c–c AQWTC zji (4) and damage index cd (6) were calculated based on analysis of vibration signal at this position (Y6) of the beam in two cases: (1) using neural networks for identifying with a value of scale a ¼0.5 used for wavelet transform (a); and (2) using the proposed method (b) The above results can be analyzed based on the use of the scale parameter as follows When using one value of the scale a for wavelet transform (called the one-scale a, such as, a ¼ 0:5 as abovementioned), if the one-scale a does not correspond with any frequency component of the signal, wavelet coefficient values are small for both damaged and undamaged statuses of the structure Hence no strange sign for damage diagnosis appears However, it is different if the proposed scale vector ais used By this way, in the group of scales as1 ; as2 ; :::asm belong to the scale vector a; there is more probability of that at least one ask ∈a corresponds with frequency component of the analyzed signal to create the marked sign for damage diagnosis In addition, it can be seen that when damage appears such as stiffness reduction of the beam, the natural frequency of the vibration beam decreases Consequently, natural frequencies of the beam at two statuses, damaged and undamaged structure, are different In this circumstance, if the scale parameter a is used for both, there may be two cases The first is that neither of these frequencies correspond with the scale a Hence there is no clear sign for considering In the second case, the scale a corresponds with one of these frequencies or with both of them but different degrees As a result, wavelet coefficients corresponding to vibration signal of the beam at damaged and undamaged statuses are different This different feature is the positive sign for damage diagnosis In this side, it can be clearly seen that if the scale vector a is used, probability of appearing the second case is higher than using the one-scale a Besides, if the scale vector a together with the proposed average-quantity solution to wavelet transform coefficient (AQWTC) is used, the above different feature is accumulated to create the more striking sign for damage diagnosis 190 S.D Nguyen et al / Mechanical Systems and Signal Processing 39 (2013) 181–194 Based on the argument and verifying results as abovementioned, it can be observed that the proposed method has advantage compared with the previous method as usually used It is noted that the sensitivity of the signal used for the proposed algorithm, AQWTC zji (4), is higher than the sensitivity of wavelet transform coefficient based on the one-scale a 5.2.2 Analysis of vibration signal at elements Here, the beam was damaged at positions: Y4 and Y8 in degree 11.16%, c–c Analysis of vibration signal is not only performed at damaged positions but also at the other along the beam Vibration signal at positions: Y2, Y4, Y6, Y8, Y10 and Y12, were all measured and filtered to calculate AQWTC zji (4) of each element The results presented in Fig show that when the beam is damaged, AQWTC at all elements in it are increased, even at undamaged elements Y2, Y6, Y10 and Y12 However, at damaged positions the increase is faster Namely at Y4 and Y8 AQWTC, zji, get the largest In graphs ofzji , graphs of z4i and z8i are highest This feature is used for damaged location of the proposed algorithm, BDLA 5.2.3 Single damage and the frame divided into elements In this experiment, the frame was divided into four elements and it was damaged at only one position Four divided elements were as follows: Y1–Y4, Y4–Y7, Y7–Y10, and Y10–Y13 having the equal lengths, L/4 We made single damage positions on the frame, i.e the frame reduced HSA at only one position, as follows: – Or the frame was cut at Y4+ (the middle of Y4 and Y5) with levels a–a, b–b, c–c, d–d; – Or the frame was cut at Y6+ (the middle of Y6 and Y7) with levels a–a, b–b, c–c, d–d Thus, in this test the different cut points all belong to the 2nd element The algorithm BDLA was used in order to determine these defected positions The results are shown in Figs and 8, corresponding to the cut positions to be Y4+ and Y6+, respectively These figures show that the relative damage index gets the maximum value, cr2 ¼1, at the damaged element (the 2nd element), and crð:Þ o at the other elements It means that in cases of the single damage as presented above, the proposed algorithm BDLA rightly determines the positions in which HAS was reduced, even at low defected level (3.56%) 5.2.4 Case of double damage In this case, the beam was divided into elements and it was simultaneously damaged at two positions belonging to 2nd and 4th elements with degree 6.9% (b–b) Damage coefficient cdi (6) of each element was calculated and depicted in Fig Result shows that at two damaged elements, damage coefficient gets the largest values This means that the proposed method exactly locates the damaged positions appearing on the beam, even when damages simultaneously appeared and damage degrees are quite small (6.9%) 5.2.5 Single/double damage and the frame divided into elements The frame was divided into elements as following Y1–Y5, Y5–Y9, and Y9–Y13 having the equal lengths, L/3 BDLA and [6] were used to determine the positions reduced HSA of the frame in two cases, single damage or double damage with three levels b–b, c–c, d–d In the first case, the frame was cut at Y6+ In the second case, the frame was simultaneously cut at two positions Y6+ and Y10+ (the middle of Y10 and Y11) Figs 10 and 11 show corresponding results Fig 10 presents the experiment results in case single damage in the beam at Y6+ corresponding to three damage degrees, b–b (Fig 10a), c–c (Fig 10b) and d–d (Fig 10c) The diagrams show that the BDLA exactly determines the position reduced HSA at all these levels However, using [6], the damaged position is exactly determined only when damage level is higher than the level b–b 30 Average Wavelet Index, zji z 2i z 4i 25 z 6i z 8i 20 z 10i z 12i 15 10 0 100 200 300 400 500 600 Time (sampled points) Fig The beam was damaged at Y4 and Y8, damage degree 11.16% (c–c) AQWTC zji (4) was calculated based on analysis of vibration signal at Y2, Y4, Y6, Y8, Y10 and Y12 1 1.0 0.8 Da level 3.56% at Y4+ (elem 2) 0.670 0.6 0.330 0.4 0.190 0.2 Relative Damage Index, cr Relative Damage Index, cr S.D Nguyen et al / Mechanical Systems and Signal Processing 39 (2013) 181–194 0.0 1.0 0.8 0.6 0.4 0.415 0.2 0.123 0.188 Da level 11.16% at Y4+ (elem 2) 0.8 0.6 0.4 0.147 0.076 0.063 0.0 Relative Damage Index, cr 1.0 Element (1-4) Element (1-4) Relative Damage Index, cr Da level 6.9% at Y4+ (elem 2) 0.0 0.2 191 1.0 Da level 23.4% at Y4+ (elem 2) 0.8 0.6 0.4 0.2 0.090 0.024 0.0 Element (1-4) 0.010 Element (1-4) 1.0 0.8 Da level 3.56% at Y6+ (elem 2) 0.71 0.6 0.4 0.2 0.18 0.002 0.0 Relative Damage Index, cr Relative Damage Index, cr Fig Relative damage index cr (7) was calculated by BDLA in case the beam damaged at Y4+ (the middle of Y4 and Y5) belongs to the 2nd element with different degrees a–a, b–b, c–c, d–d; results presented in a–d, respectively 1.0 0.8 0.595 0.6 0.4 0.2 0.151 0.045 0.0 Da level 11.16% at Y6+ (elem 2) 0.8 0.6 0.4 0.272 0.2 0.151 0.089 Relative Damage Index, cr Relative Damage Index, cr 1.0 Da level 23.4% at Y6+ (elem 2) 0.8 0.6 0.4 0.2 0.120 0.059 0.010 0.0 0.0 Element (1-4) Element (1-4) Element (1-4) 1.0 Da level 6.9% at Y6+ (elem 2) 4 Element (1-4) Fig Relative damage index cr (7) was calculated by BDLA in case the beam damaged at Y6+ (the middle of Y6 and Y7) belonging to the 2nd element with different degrees a–a, b–b, c–c, d–d; results presented in a–d, respectively (6.9%) Fig 10a shows that if it is based on [6], element damaged is the 3rd element due to its relative damage index gets the largest value (cr3 ¼ 1) In fact, damage is at the 2nd element Fig 11 presents next experiment results in case double damage at Y6+ and Y10+, corresponding to three damage degrees, b–b (Fig 11a), c–c (Fig 11b) and d–d (Fig 11c) This figure shows that 192 S.D Nguyen et al / Mechanical Systems and Signal Processing 39 (2013) 181–194 Damage Index, cd 10.0478 9.7681 10 Da level 6.9% at e.2 and e.4 5.2132 4.036 4.299 4.1022 2 Elements (1-6) 1.0 Relative damage index, crj (j=1 3) Relative damage index, crj (j=1 3) Fig Using the proposed algorithm, BDLA, in case the beam simultaneously damaged at two positions with degree b–b, belonging to 2nd and 4th elements BDLA [6] 0.8 0.6 0.38 0.4 0.28 0.2 0.17 0.14 1.0 BDLA [6] 0.8 0.6 0.51 0.4 0.36 0.2 0.11 0.13 0.0 0.0 Relative damage index, crj, (j=1 3) Elements (1-3) Elements (1-3) 1.0 BDLA [6] 0.8 0.6 0.4 0.29 0.21 0.2 0.14 0.10 0.0 Elements (1-3) Fig 10 Damage location results of the BDLA and [6] when the frame was divided into elements, single damage position at Y6+ (belongs to the 2nd element) with different damage levels b–b (a), c–c (b), d–d (c) in this case both of methods, BDLA and [6], could exactly determine the positions reduced HSA at all these levels since the relative damage indexes of them get the largest at damaged elements 2nd and 3rd elements Conclusions Reality shows that the effectiveness of the structure health monitors depends on many issues in which ability of mathematical models and sensitivity of signal analysis tools are frequently treated In order to improve accuracy degree of S.D Nguyen et al / Mechanical Systems and Signal Processing 39 (2013) 181–194 Relative damage index, crj (j=1 3) 0.8 0.66 0.6 0.50 0.4 0.2 1.0 BDLA [6] 0.35 0.15 Relative damage index, crj (j=1 3) 1.0 193 BDLA [6] 0.8 0.77 0.6 0.45 0.4 0.24 0.2 0.15 0.0 0.0 Elements (1-3) 1.0 Relative damage index, crj (j=1 3) Elements (1-3) BDLA [6] 0.8 0.8 0.6 0.6 0.5 0.4 0.42 0.2 0.0 Elements (1-3) Fig 11 Damage location results of BDLA and [6] when the frame was divided into elements, and the double damage was at Y6+, belonging to the 2th element and at Y10+, belonging to the 3rd element with three damage degrees: b–b (a); c–c (b) and d–d (c) structure health monitoring system, in this work we focus on two major issues: modeling method and signal analysis In order to achieve the research goal, two steps were performed based on the proposed beam-damage-locating algorithm named BDLA which was developed based on a new AFNS, and a new solution using wavelet transform for signal analysis As a first step, the database for actual structure health monitor system, which is based on the change of its dynamic property signals, has been established at two times: at structure-undamaged time and the checking time under same VES In fact, this is really difficult due to that we can not exactly repeat VES at two different times In this work, the use of AFNS of algorithm BDLA could exactly interpolate data at any time when the structure is not damaged, and hence it can resolve this difficulty The second step proposed in this work for signal analysis is average wavelet transform of vibration signal with a used-scalesheet to calculate the corresponding difference of AQWTC between two cases: undamaged status and the status at the checked time Database about this difference is then used for finding out the elements having strange features in wavelet quantitative analysis, which are beam-damage signs This could overcome the difficulty about finding out the optimal scale in wavelet transform of vibration signal for each application Based on the analyzed arguments as well as verifying results via experimental investigation as abovementioned, it can be observed that the proposed method has crucial advantages, such as the sensitivity and the ability to interpolate data The proposed algorithm, BDLA associated with AQWTC and the AFNS can predict or estimate the beam-damage location much better than conventional methods It is finally remarked that the proposed method can be extended to automaticstructure-health-monitoring systems Acknowledgment This work was partially supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (No 2010-0015090) 194 S.D Nguyen et al / Mechanical Systems and Signal Processing 39 (2013) 181–194 References [1] Chaochao Chen, Bin Zhang, George Vachtsevanos, 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