Development and application of signal processing methods for crack diagnosis of bar structures

Crack detection methods based on oscillating signals are usually based on two main factors: the dynamical characteristics of the structure and the oscillating signal processing methods. In practice, the change in dynamical characteristics of the structure caused by the crack is very small and difficult to detect directly from the oscillating measurement signal. Therefore, in order to detect these minor changes, modern signal processing methods are given, which is the method of signal processing in time-frequency domains. These methods include the Short-time Fourier Transform (STFT), the Wavelet Transform (WT) v.v. These methods will analyze signals in two time and frequency domains. When using these methods, the signals over time will be represented in the frequency domain while the time information is retained. Therefore, time-frequency methods will be useful for analyzing small or distorted variations in the oscillation signal caused by the crack

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VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY ------o0o------ NGUYEN VAN QUANG DEVELOPMENT AND APPLICATION OF SIGNAL PROCESSING METHODS FOR CRACK DIAGNOSIS OF BAR STRUCTURES Major: Engineering Mechanics Code: 9520101 SUMMARY OF PhD THESIS Hanoi – 2018 The thesis has been completed at: Vietnam Academy of Science and Technology Graduate University of Science and Technology Supervisor: Assoc. Prof. Dr. Nguyen Viet Khoa Reviewer 1: Prof. Dr. Hoang Xuan Luong Reviewer 2: Assoc. Prof. Dr. Luong Xuan Binh Reviewer 3: Assoc. Prof. Dr. Nguyen Phong Dien Thesis is defended before the State level Thesis Assessment Council held at: Graduate University of Science and Technology - Vietnam Academy of Science and Technology. At ... on .. Hardcopy of the thesis can be found at: + Library of Graduate University of Science and Technology + National Library of Vietnam List of the Author’s publications 1. Khoa Viet Nguyen, Quang Van Nguyen. Time-frequency spectrum method for monitoring the sudden crack of a column structure occurred in earthquake shaking duration. Proceeding of the International Symposium Mechanics and Control 2011, p. 158-172. 2. Khoa Viet Nguyen, Quang Van Nguyen. Wavelet based technique for detection of a sudden crack of a beam-like bridge during earthquake excitation. International Conference on Engineering Mechanics and Automation ICEMA August 2012, Hanoi, Vietnam, p. 87-95. 3. Nguyễn Việt Khoa, Nguyễn Văn Quang, Trần Thanh Hải, Cao Văn Mai, Đào Như Mai. Giám sát vết nứt thở của dầm bằng phương pháp phân tích wavelet: nghiên cứu lý thuyết và thực nghiệm. Hội nghị Cơ học toàn quốc lần thứ 9, 2012, p. 539-548. 4. Khoa Viet Nguyen, Hai Thanh Tran, Mai Van Cao, Quang Van Nguyen, Mai Nhu Dao. Experimental study for monitoring a sudden crack of beam under ground excitation. Hội nghị Cơ họ V r n i n ng oàn uốc lần thứ 11, 2013, p. 605-614. 5. Khoa Viet Nguyen, Quang Van Nguyen. Element stiffness index distribution method for multi-cracks detection of a beam- like structure. Advances in Structural Engineering 2016, Vol. 19(7) 1077-1091. 6. Khoa Viet Nguyen, Quang Van Nguyen. Free vibration of a cracked double-beam carrying a concentrated mass. Vietnam Journal of Mechanics, VAST, Vol.38, No.4 (2016), pp. 279- 293. 7. Khoa Viet Nguyen, Quang Van Nguyen, Kien Dinh Nguyen, Mai Van Cao, Thao Thi Bich Dao. Numerical and experimental studies for crack detection of a beam-like structure using element stiffness index distribution method. Vietnam Journal of Mechanics, VAST, Vol.39, No.3 (2017), pp. 203-214. 1 INTRODUCTION Crack detection methods based on oscillating signals are usually based on two main factors: the dynamical characteristics of the structure and the oscillating signal processing methods. In practice, the change in dynamical characteristics of the structure caused by the crack is very small and difficult to detect directly from the oscillating measurement signal. Therefore, in order to detect these minor changes, modern signal processing methods are given, which is the method of signal processing in time-frequency domains. These methods include the Short-time Fourier Transform (STFT), the Wavelet Transform (WT) v.v. These methods will analyze signals in two time and frequency domains. When using these methods, the signals over time will be represented in the frequency domain while the time information is retained. Therefore, time-frequency methods will be useful for analyzing small or distorted variations in the oscillation signal caused by the crack. The objective of the thesis  Study the effect of cracks on the dynamic characteristics of the structure.  Study the applicability of the time-frequency signal processing method in detecting cracks.  Application and development the processing time-frequency oscillation signals methods for cracks detection. Study method  The dynamic characteristics of cracked structures, such as frequencies, mode shapes will be calculated and studied by finite element method. 2  The time-frequency signal processing method will be applied to analyze the simulated vibration signals of the cracked structure.  Develop an oscillating signal processing method to detect changes in element stiffness to detect the crack.  Carry out some experiments to verify the effectiveness of the methods. New findings of the dissertation  The application of wavelet spectral methods for sudden cracks detection.  The application of wavelet analysis for cracks detection based on the effect of cracks and concentrated mass..  Proposed a new method using the "Element stiffness index distribution method" for crack detection of the structure. In this method, the element stiffness index distribution is calculated directly from the oscillation signal. Structure of the thesis The contents of the thesis include the introduction, the conclusion, and 5 chapters: Chapter 1: An overview Presents an overview of the world's research on cracks detection methods based on structural dynamics, signal-processing methods in time-frequency domain for analysis and crack detection. Chapter 2: Theoretical basic Provides a theoretical basic of structural dynamics with cracks. Introduce cracks model of 2-D and 3-D beams. Chapter 3: Theory of oscillating signal processing methods 3 Presentation of theoretical basis of signal processing methods in time-frequency domains and presents an element stiffness index distribution method for crack detection of the structure. Chapter 4: Application of oscillating signal processing methods in some problems Presents the applications of time-frequency methods and an element stiffness index distribution method to detect cracks in different structures. Chapter 5: Experimental verification Presents some experiments to verify the methods developed and applied in the thesis. Conclusion: presents the results of the thesis and some issues that need to be implemented in the future. 4 CHAPTER 1. AN OVERVIEW 1.1. Diagnostic problem We can use direct or indirect methods to detect damage in the structure. Direct methods include visual observation, film shooting, or remove the structural details for inspection. Indirect method is the response singnal analyzing method of the structure under external impact to detect the structural damage. In indirect methods, vibration methods are developed and applied in the world as well as in Vietnam. These methods can be divided into two main groups: the method based on structural dynamics parameter and the method based on oscillation data processing. 1.2. Methods of structural damage detection based on structural dynamics parameter The existence of damage in the structure leads to changes in the frequencies and shape modes. Therefore, the structural characteristics of the damaged structure will contain information about the existence, location and level of damage. In order to detect structural damage, it is essential to study the dynamics of the structure. 1.3. Wavelet analysis method to detect structural damage The change in frequency is the most interest parameter for damage tracking because it is a global parameter of structure. By conventional approach, the natural frequency can be extracted by Fourier transform. However, the information of the time when the frequency changed is lost in this transform. Fortunately, there is another approach which can analyse the frequency change while the information of time is still kept called time-frequency analysis. Recently, some time-frequency based methods have been applied 5 wildly for SHM such as Short Time Fourier transform (STFT), Wigner-Ville Transform (WVT), Auto Regressive (AR), Moving Average (MA), Auto Regressive Moving Average, and Wavelet Transform (WT) [58]. Among these methods, the WT has emerged as an effective method for tracking the change in natural frequency of structures. 6 CHAPTER 2. THEORETICAL BASIC In order to analyze the dynamical characteristics of damaged structures, the thesis will use finite element method because it can analyze complicated structures which analytical method is difficult to perform. So in this chapter, we will present the theoretical basis of finite element method for solving the damaged dynamics problem. 2.2. Finite element models for 2D and 3D beam with crack 2.2.1. 2D beam with crack It is assumed that the cracks only affect the stiffness, not affect the mass and damping coefficient of the beam. An element stiffness matrix of a cracked element can be obtained as following:   2 3 2(0) 2 2 0 1 1 . 2 2 3 l P l W M Pz dz M l MPl EI EI            (2.1) The additional energy due to the crack can be written as:   22 2 III(1) I II 0 1 . a KK K W b da E E          (2.4) The generic component of the flexibility matrix C~ of the intact element can be calculated as: 2 (0) (0) 1 2, , ; , 1,2.ij i j W c P P P M i j P P        (2.6) The additional flexibility coefficient is: 2 (1) (1) 1 2, , ; , 1,2.ij i j W c P P P M i j P P        (2.7) The total flexibility coefficient is: (0) (1).ij ij ijc c c  (2.8) By the principle of virtual work the stiffness matrix of the cracked element can be expressed as: 1 K T C T T c .  (2.11) 7 2.2.2. 3D beam with crack The total compliance C of the cracked element is the sum of the compliance of the intact element and the overall additional compliance due to crack: ( ) (1).oij ij ijc c c  (2.14) The components of the compliance of an intact element can be calculated from Castingliano’s theorem: 2 (0) (0) ; , 1..6,ij i j W c i j P P      (2.15) and the components of the local additional: 2 (1) (1) ; , 1..6.ij i j W c i j P P      (2.16) Where W (0) is the elastic strain energy of the intact element and can be expressed as follows: 2 2 2 2 3 2 22 2 2 3 2 (0) 3 6 2 6 3 5 3 51 2 2 4 0 1 . 2 3 3z z z y y y P l P l P P l P l P l P P lP l P l P l P l W AE GA GA EI EI EI EI EI EI GI                 Where W (1) is the additional strain energy due to crack [116]: 2 2 2 6 6 6 (1) I II III 1 1 1 1 .i i i A W K K K dA E                             (2.19) The stiffness matrix of the cracked element can be obtained as follows: 1 .K T C TTc  (2.36) 2.3. Equation of structural by finite element method In finite element model the governing equation of a beam-like structure can be written as follows [118]: ( ) ; , . e T T T e e e eL t t t f t t f dx f     My( ) Cy( ) Ky( ) N f( ) f N f T (2.37) (0.1) M, C, K are structural mass, damping, and stiffness matrices, respectively; f the excitation force; N T is the transposition of the 8 shape functions at the position x of the interaction force; and y is the nodal displacement of the beam. The displacement of the beam u at the arbitrary position x can be obtained from the shape functions N and the nodal displacement y [119]. Finally, the global stiffness matrix K of the cracked beam is assembled from the element stiffness matrix for intact elements defined in finite element method and matrix K c for cracked elements. Rayleigh damping in the form of C M K   . 2.3. Conclusion This chapter presents cracks models including 2D, 3D beam with crack. In the thesis, these crack models will be applied in 2D beams and frame. This chapter presents the basic equations which used finite element method. This is the basis for calculating the dynamical characteristics of the structure in the thesis. 9 CHAPTER 3. THEORY OF OSCILLATING SIGNAL PROCESSING METHODS In the current oscillating signal processing methods, wavelet analysis, which is a time-frequency method is being developed and applied in many different fields. The natural frequency can be extracted by Fourier transform. However, the information of the time when the frequency changed is lost in this transform. Fortunately, there is another approach which can analyse the frequency change while the information of time is still kept called time-frequency analysis. 3.1. Wavelet analysis method The continuous wavelet transform is defined as follows [76, 85, 120]: * 1 ( , ) ( ) , t b Wf a b f t dt aa            (3.1) where a and b are scale and position, Wf(a,b) are wavelet coefficients at scale a and position b, f(t) is input signal, t b a        is called wavelet function and * t b a        is complex conjugate of t b a        . The wavelet power spectrum: 2 *1( , ) ( ) . t b S a b f t dt a a            (3.12) 3.2. Element stiffness index distribution The ith element stiffness matrix is denoted as: 10 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 ,K i i i i i i i i i e i i i i i i i i k k k k k k k k k k k k k k k k              (3.22) any sub-matrix: 1 1 33 11 34 12 13 14 1 1 43 21 44 22 23 24 1 1 31 32 33 11 34 12 1 1 41 42 43 21 44 22 ( ) ( ) ( ) ( ) , ( ) ( ) ( ) ( ) K i i i i i i i i i i i i i e i i i i i i i i i i i i k k k k k k k k k k k k k k k k k k k k k k k k                           (3.24) in the global stiffness matrix is constructed mainly from the ith element stiffness matrix with some additional components of the (i- 1)th and (i +1)th element stiffness matrices. Therefore, the sub- matrix K i e can be used to characterize the stiffness of the ith element for the crack detection roblem. Indeed, when there is a crack at the ith element, only three submatrices 1,K ie  ,K ie 1 K i e  will be changed as can be seen from equation (3.24). Since the sub-matrix K i e reflects the local stiffness, its norm should serve as a good local indication of its stiffness condition. From this point of view, the change in norm of the submatrix K i e can be used as an indicator of the damage at the ith element. In order to detect the change in norm of the sub-matrix K ie from the global stiffness matrix, we define an element stiffness index distribution as follows:   1 2 1 , ,..., , 1.. , max Q i i i Q         (3.25) where   2 maxK K Ki i T ii e j e ej   is called ith element stiffness index; Q is the number of finite elements. When there is a 11 crack at the ith element, the element stiffness index distribution is expected to have a significant change in the ith element. 3.3. Conclusion This chapter presents the theoretical basis of the wavelet analysis method. The wavelet transform the signal to the frequency domain while the information of time is still kept. The square of the wavelet coefficient module or wavelet spectrum can be expressed as the energy density distribution on the plane time-scale. This chapter presents the theoretical basis for a new method for crack detection based on an element stiffness index distribution. The global stiffness matrix is calculated from the measured frequency response functions instead of mode shapes to avoid limitations of the mode shape-based methods for crack detection. 12 CHAPTER 4. APPLICATION OF OSCILLATING SIGNAL PROCESSING METHODS IN SOME PROBLEMS This chapter will apply this method to solve three diagnostic problems. 4.1. Time-frequency spectrum method for monitoring the sudden crack of a beam structure occurred in earthquake shaking duration 4.1.1. Vibration of the beam-like structure subject to harmonic ground shaking We begin by considering the beam as an Euler–Bernoulli beam subject to the ground excitation. The beam is modeled as Q elements in finite element analysis. The ground excitation is assumed to be a harmonic function dg . Under these assumptions and apply the finite element method the governing equation of motion of the beam can be written as follows [126]: .Md Cd Kd CI KIg gd d    (4.1) 4.1.2. Numerical simulation To match the experimental model, parameters of the beam are: Mass density is 7855 kg/m 3 ; modulus of elasticity E=2.1x10 11 N/m 2 ; L=1.2 m; b=0.06 m; h=0.01 m. Modal damping ratios for all modes are equal to 0.01. During the first half of the excitation structure is modelled as an intact beam and in the second half of the excitation duration, a crack at location Lc=L/2 of the beam is made. the duration of excitement is T=16s. The ground excitation function is chosen as F=0.05sin(35t). Due to this excitation, the beam vibrates mainly with its first natural frequency of 17.8 Hz. 13 a) b) c) d) e) Fig 4.3. IF of beam. a) Crack depth 10%; b) Crack depth 20%; c) Crack depth 30%; d) Crack depth 40%; e) Crack depth 50%. df is the difference between the IFs in the first and the second half of the excitation is investigated. Fig 4.4. Relation between df and crack depth. 4.1.3. Conclusion In this study presents the wavelet power spectrum. The IF can be used to monitor the change in the frequency of the beam for the purpose of crack detection. The existence of the crack is monitored 14 by a decrease in the IF during the excitation. The crack appearance time can be determined by the moment at which the IF starts to decrease 4.2. Free vibration of a cracked double-beam carrying a concentrated mass The finite element model of the double-beam system consisting of two different Euler-Bernoulli beams with rectangular sections connected by a Winkler elastic layer with stiffness modulus mk per unit length is presented in Fig. 4.5. The length of the double-beam is L. Each of the main and auxiliary beams is divided by Q equal elements with length of l. The main beam carries a concentrated mass m at section xm. Fig 4.5. A double-beam element carrying a concentrated mass. The free motion equation of an element of the double-beam system can be derived by using Hamilton’s principle as follows: .MD KD O  (4.16) Where: * ** 11 * * 22 1 1 2 2 , . , , . m m m m                             K K KM M= K= K K KM D OD D= D= O= D OD (4.17) 15 4.2.2. Influence of the concentrated mass on the free vibration of the intact double-beam The influence of the concentrated mass is large when the mass is located at the large amplitude position of the mode shape and vice versa. The MLFs have local minima when the concentrated mass is located at the largest amplitude positions of the mode shapes. While, the MLFs have local maxima when the concentrated mass is located at the nodes of the mode shapes. a1) The 1st mode shape b1) MLF of the 1st frequency a1) The 2nd mode shape b1) MLF of the 2nd frequency a1) The 3rd mode shape b1) MLF of the 3rd frequency Fig 4.7. The first three mode shapes and MLFs. 0 0.2 0.4 0.6 0.8 1 -5 -4 -3 -2 -1 0 N o rm a liz e d a m p lit u d e x/L 0 0.2 0.4 0.6 0.8 1 18.5 19 19.5 20 F re q u e n c y ( H z ) Mass position (x/L) 0 0.2 0.4 0.6 0.8 1 -4 -2 0 N o rm a liz e d a m p lit u d e x/L 0 0.2 0.4 0.6 0.8 1 41 42 43 44 F re q u e n c y ( H z ) Mass position (x/L) 0 0.2 0.4 0.6 0.8 1 -4 -2 0 N o rm a liz e d a m p lit u d e x/L 0 0.2 0.4 0.6 0.8 1 75 76 77 78 79 F re q u e n c y ( H z ) Mass position (x/L) 16 4.2.3. Influence of the