Crack detection in bars, beams by measurement of frequency response function

damage. 2. Aims of the study The aims of this study are to develop and apply the method of using frequency response functions to diagnose cracks in elastic bar and beams structure. The contents of the study includes: building models of cracked elastic bar and 3 beams structure; studying the changes of oscillation parameters, mainly frequency response functions due to cracks; conducting experimental studies to measure the oscillation parameters of cracked elastic bar and beams structure in the laboratory and proposing some algorithms to diagnose cracks in the structure based on the built model and experimental measurements. 3. The main contents of the study (1) Study the change of axial oscillation nodes in the bar and the bending oscillation of the beam due to the appearance of cracks in order to diagnose cracks. (2) Experimental study of elastic bar and beam structure containing many cracks by measuring frequency response functions. Since then, we analyze and process the measurement data of frequency response functions to find specific experimental frequencies. (3) Construct an explicit frequency response function in the bar structure, thereby combining with experimental measurement data and using CSM (Crack Scanning Method) to solve the diagnostic problem in the cracked bar. (4) Using Rayleigh formula and CSM to set up and solve the problem of crack diagnosis from experimental frequencies. This thesis includes an introduction, 5 chapters and a conclusion, in which chapter 1 presents an overview of literature; chapter 2 presents the theory of multi-cracked bars and beams; chapter 3 presents the changes in the oscillation nodes; chapter 4 presents the experimental study and chapter 5 presents the algorithms and results of bar and beam structural diagnosis based on the frequency response function and particular frequency. 4 CHAPTER 1: LITERATURE OVERVIEW 1.1. Questions of damage diagnosis With a technical object, there are always two questions: the forward question, studying behavior of the structure; the diagnostic question, in fact is an inverse question, which aims to detect damage in the structure from the measurement data based on the analysis of the forward problem. Specifically for a mechanical system, it is often described by a diagram: Figure 1.1. Mechanical diagram of mechanics with: X: input, external impact, ∑: modeling, describing the structure and characteristics of the system, Y: output, the response of the system. Mechanical systems can be represented by a mathematical equation:   XYL  A crack is a typical form of failure in the structure of buildings and machinery. The crack is generally described by its position and size in the structure. The appearance of cracks in the structure degrades the hardness of the structure in the vicinity of the crack. The diagnosis of cracks in the composition of the structure has attracted many researchers over the past two decades as indicated in the general reports of Doebling et al in 1996, Salawu in 1997 and Sohn et al in 2004. In the diagnosis of damage of the structure in general as well as the cracks in particular, people often use dynamic characteristics. The specific oscillation frequencies, specific X ∑ Y 5 oscillation patterns and frequency response functions (and related characteristics such as the hardness and the softness) are often used. The diagnosis of damage in general and the cracks in particular of the structure based on the frequency change usually only detects the appearance of the crack without determining the crack position. Meanwhile, the cracks affect locally. Therefore the crack information is based on the specific patterns considered in the diagnostic problem. From domestic and foreign studies, it has been shown that specific forms can be used to determine the position of cracks. However, if only the specific form is used for this purpose, it is necessary to have accurate measurement data, which is not always practical in practice. Meanwhile, the response function contains information of both frequency and specific patterns that can be used to analyze the effect of cracks on structural response. Measuring frequency response functions is simple and gives accurate results. Therefore, the development of methods of application of frequency response functions in crack diagnosis is very necessary due to its superiority. 1.2. Frequency response functions in diagnosing structural damage In the measurement data of oscillation characteristics, it was found that using the frequency response functions, which is usually measured directly as input for the diagnosis of damage is better than using frequency and specific patterns. This is due to the remarkable advantages of measured frequency response function data: • The external frequency response function provides information about the specific frequency (resonant frequency), 6 which can also provide additional information about the response of the structure at distant resonant frequencies. • Using the frequency response function will avoid the error of processing the measurement data for frequency separation and the specific form of the measured data (the frequency response function is the input in the separate format analysis). • In addition, important information such as the position of the measurement point and of the force set can be found in the frequency response functions. In recent years the use of a frequency response function to diagnose the damage in structures can be mentioned as in 2005 proposal of Araujo dos Santos et al - a method of determining damage based on the sensitivity of frequency response functions. They pointed out that the damage detection results would be better if we measured low frequencies and stimulus nodes, not cracked nodes. Therefore, there is a wide range of the possibilities of exploiting more information from the frequency response functions. In 2012 Huang et al identified the damage of the five-storey house structure in the structural control problem based on the change of the frequency response functions and the dampers. Here they have shown that with greater noise than 10% it is impossible to determine the damage. 1.3. Reviews and research questions The method of measuring the oscillation characteristics of structures to diagnose the damage is currently the most effective method. However, no matter how we directly analyze the measurement signal or use the model to diagnose the damage, the following two problems still exist. One is that the easy-to-measure characteristics are less sensitive to damage and 7 the second is the measurement error may be greater than the effect of the damage. Therefore, finding other oscillation characteristics which is not sensitive to measurement errors, but is sensitive to the damage to diagnose the damage in the construction is still an unsolved problem. In the oscillation characteristics: frequency and specific oscillation patterns, the drag coefficient and the frequency response function, the frequency and the frequency response function are easily measured and the most accurate. However, the frequency response function is an aggregate feature, including all three previous features (frequency, specific patterns and drag coefficient) and describes the spectral structure of the system. Therefore, the interaction between the vibrational forms and their sensitivity makes the sensitivity of the frequency response function to failure very complex and difficult to identify. This is an obstacle to the use of a frequency response function in diagnosing structural failure. The majority of published works in the world for crack diagnosis by impulse response function are based on finite element method, which does not allow determining the exact position of the crack. Therefore, it is necessary to develop methods aimed at utilizing the precise measurement of the frequency and the frequency response functions in the diagnosis of the damage, which is finding its representations through damage parameters. This allows us to study the frequency sensitivity and frequency response functions for damage and therefore can apply to the structural damage diagnosis. The questions of this thesis are as follows: Study the change of axial oscillation nodes in the bar, bending oscillation of the beam due to the appearance of cracks in order to diagnose the cracks. 8 Experimental study of elastic bar and beam structure containing many cracks by measuring the frequency response function. Since then, we can analyze and process measurement data of frequency response functions in order to find experimental specific frequencies. Constructing the explicit frequency response function in the bar structure, thereby combining with experimental measurement data and using CSM (Crack Scanning Method) to solve the diagnostic problem in the bar containing cracks. Using Rayleigh formula and CSM to set up and solve the problem of crack diagnosis from experimental frequency. CHAPTER 2. THE OSCILLATION OF CRACKED BAR AND BEAM STRUCTURES 2.1. Model of cracks in elastic bar and beam structures The crack, generally understood as an interface in a solid object, causes the state of deformation stress at that interface to be interrupted. The appearance of cracks in the structure changes the dynamic characteristics. Usually cracks are characterized by parameters: position, size and shape. For elastic bars and beams, cracks are considered as changes in the cross section in a segment of very small length b with the depth a. It is precisely the crack pattern opened in the form of a saw which is called The V-shaped crack. The concept of the crack depth and the beginning of the crack is clearly described. Furthermore, it is calculated that the decrease in hardness (or increase in softness) of the bar - beam at the crack- containing cross-section has led to the idea of modeling the crack with a spring which is equivalent to the hardness K at the section containing the crack. Thus, it is possible to describe 9 cracks in elastic beams with a spring that links the two sides of the crack with the hardness determined by experiment and destructive mechanical theory. Figure 2.1. Crack pattern and replacement springs (bending - pulling compressors). 2.2. Axial oscillation of cracked elastic bars The specific oscillation patterns of the elastic bar has the parameters (E, ρ, A, L), which is defined from the equation: ./,/),1,0(,0)()( 00 2  EccLxxx   Suppose that there are n cracks in the bars at the positions e1,..., en with the corresponding depths of a1, ..., and is described by the springs along the axial hardness Kj - a function of the crack depth aj. Then the compatibility condition at the crack is: 2.1.1. Transmission matrix method As the functions nmxm ,...,1),(  are continuous solutions of oscillation equations, they can be expressed in the form of: nmxxx mmm ...,,1,sinBcosA)(   Constants are defined from:          0 0 B A B A m m m H 10 ]2,1,),,...,,,...,,([... 1111   jieehH mmij m ijmmm TTTH nj eee eee e jjjjj jjjjj jjj ,...,1,cossin1sin coscossin1 ),,( 2 2               TT Then, we can build the typical equation (also called the frequency equation) in the axial oscillation of the bar with n cracks by the transmission matrix method. 0)( 1022102110121011  qpnqpnqpnqpn n SCHSSHCCHCSHD  with .)(sin)(;)(cos)( ;)(sin)(;)(cos)( 111111 000000           xq q qq xq q qq xp p pp xp p pp x xSS x xCC x xSS x xCC   2.1.2. The frequency response functions of the axial oscillation of the bar Consider the forced oscillion of the cracked bar described by the equation: .10,/),()()()( 00000 2  xEFLaxxaQxx  It is easy to see that the general solution of the above equation presented in the form: .)()()(),(),( 0 0000   x dsxssxhaQxx  11 Doing the integral on the right and applying the last boundary condition, we get the general expression of the frequency response function: . )1()1( )]()()[1( )(),,(),,( 1 )()( 0 1 00 )( 0000                    n j j p j p n j jj p eKL exKxLxh xxhaxxHxxFRF    Then we analyze the number of the frequency response function in the axial oscillation of the cracked bar. Consider the frequency response function in the vicinity of the first and second specific frequencies, denoted by FRF1 and FRF2. The effect of crack position on the frequency response functions mentioned above is shown in Figure 2.2, Figure 2.3 for the two- ends free bar. The graph shows the change of the modulus of response functions by the position and the depth of one or more cracks. Figure 2.2. Influence of crack position on FRF1 frequency response function of the two-ends free rod (30% crack depth). 12 Figure 2.3. Effect of crack position on FRF2 frequency response function of the two-ends free bar (the crack depth of 30%). • The change in the frequency response function is similar to the change of resonance frequency due to the crack. However, because the frequency response function is a frequency-dependent function, the changes of the the frequency response function due to the crack provides more information than the resonant frequency which is only a fixed numerical value; • A small crack in the bar is difficult to be detected by frequency response functions and it is likely that the question of crack diagnosis does not have a unique answer, especially with a bar with symmetrical boundary conditions; However, the appearance of new peaks in the frequency response function diagram is also a suggestion to diagnose two, three or more cracks. The distance between the new peaks are also a sign to diagnose multi-cracks in a bar; 13 2.2. Bending oscillation of cracked elastic beams 2.2.1. Explicity general test To solve this question, we divide the beam into 1n sections 1,0,1,...,1),,( 101   njj eenjee and consider the equation in each segment. It is easy to see that the general solution of the equation (2.53) has the following form: ).()()()()( 44332211 xLCxLCxLCxLCx jjjjj  Satisfying the condition at the crack and applying boundary conditions gives us a typical equation: .0)(),(),()( 1, 1221 1 12 1 210    n kj kj n j jj n j jj DeDeDD  2.2.2. Rayleigh's formula to calculate the specific oscillation frequency of multi-cracked elastic beams The approximate formula for calculating the frequency of cracked beams is the expression of crack parameters. ).()(),()( 00 1 1, 2 0 4 0 4 jkik n j n ji jiijjikjkjkk eeeee         Chapter 2 conclusion The basic formulas in axial oscillation of cracked elastic bars have been developed, especially the expression of the frequency response functions. The frequency response functions around the first two frequencies (FRF1 and FRF2) depending on the position, the depth and the number of cracks are diagnosed in details. The results of the numerical analysis show that the effect of cracks on the frequency response functions is clearly expressed in the vicinity of the natural frequency and the qualitative change which is similar to the change of the natural 14 frequency due to the cracks. However, large deep cracks can cause new resonant peaks to appear near the initial resonance peak (of the uncracked beam) and the distance between these two resonant peaks depends on both position and depth crack; Basic equations have been established to calculate the frequency and the specific oscillation pattern of multi-cracked elastic beams. These equations are the main tool for studying the effect of cracks on typical oscillation of of beams. In particular, the Rayleigh formula has been established, a manifested expression of the specific frequency for crack parameters. This is the main tool to diagnose cracks by specific frequency using the scanning method by GS. Nguyen Tien Khiem proposed. The new feature of this formula compared to the published results is that it is possible to calculate the second-order component of crack magnitude. This is an important factor to solve some difficulties in diagnosing cracks using the first approximation published in the documents. CHAPTER 3. AXIAL OSCILLATION NODES IN THE CRACKED BAR, BENDING OSCILLATION OF CRACKED BEAMS 3.1. Concept of oscillation nodes in elastic bars and beams One of the oscillation characteristics, which is very similar to the frequency in terms of both the nature and the measurement method, is the oscillation nodes. By definition, the oscillation node is the position in the structure where some patterns of oscillation are suppressed (equal to 0). 3.2. The axial oscillation nodes in the cracked elastic bar )()( )()(tan 021022 012011   pmpm pmpm m SHCH CHSHx   15 Figure 3.1. The effect of the third crack on the displacement of the second node (the second oscillation pattern). The non- displacement contour lines of the second node depend on the first and second crack positions corresponding to the position and the depth of the third crack. 3.3. The oscillation nodes of the cracked elastic beams 0 D C B A )()()()( 0 0 0 0 4321 4 1 4 1 1 4 1 3 1 1 4 1 2 1 1 4 1 1 1 1 0 24 0 23 0 22 0 21 0 14 0 13 0 12 0 11                               mmmmmmmm j n jj j n jj j n jj j n jj xFxFxFxF TETETETE EEEE EEEE 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0 0,05 0,1 0,15 0,2 e2 e1 e3=0.8,a=10% e3=0.85,a=25% e3=0.9025222 e3=0.84,a=20% e3=0.85,a=23,2905% e3=0.8,a=20% e3=0.8,a=25% e3=0.85,a=30% 16 Figure 3.2. The change of the position of the first node (1/3) of the third pattern according to the crack position and the depth varying from 0% to 50% in the two-end single-girder beam. Figure 3.3. The change of the position of the second pattern according to the position of the two cracks on both sides of the node, the two-end single-girder beam. Chapter 3 conclusion Thus, the illustrated numerical results obtained for the bar have been described more clearly and in detail than the results of Delina and Morassi, in which the “push-pull” areas are not clearly defined. The pictures above allow us to determine the direction of movement of the nodes due to the effect of cracks. These are 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 Vi tri vet nut Su th ay d oi d ie m n ut th u nh at 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.1 0.2 0.3 0.4 0.5 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 Vi tri vet nut thu haiVi tri vet nut thu nhat Su th ay d oi d ie m n ut , m od e 2 17 the charts which allow us to partition, even determine the exact position of the crack when the position of the nodes is measured. CHAPTER 4. EXPERIMENTAL MEASUREMENT FOR THE MODEL IN THE LAB 4.1. Measuring frequency response function for the cracked elastic bar model A free two-ends concrete bar model, with a circular section of 0.2m in diameter, 1.5m in length (Figure 4.4) has been made for the experimental measurement of frequency response functions and crack diagnosis by frequency response functions. Two cracks are the saws with the depth of 12% and 5%, which were created at the positions of 0.49m and 1.02m. Figure 4.1. Experimental model and measuring equipment system Figure 4.2. Input signal and FRF of the elastic concrete bar 18 4.2. Measurement of frequency response functions for 006D cracked elastic model Figure 4.3. Experimental model of two-ends elastic beams. Figure 4.6. Frequency response function (FRF) on logarithmic scale and linearly-stimulated scale at p = 0.28m. Chapter 4 conclusion In this chapter, a brief theoretical overview of the measurement of the frequency response function and the use of the frequency response function in determining specific frequencies are presented; The frequency response function of the bar in the first and second separate frequencies vicinity was measured. Only the measurement results in the vicinity of specific frequencies 19 were selected, without the far-fetched frequencies. The result shows the graph of the frequency response function not only at the resonant peaks (used to determine the corresponding frequencies and oscillation patterns) but also shows the behavior of the frequency response function at the frequencies near the resonance. Our understanding of the frequency response function is widened in order to diagnose cracks by frequency response functions. In addition, the measured frequency data of cracked beams were compared with the calculated results. This comparison allows us to simultaneously verify the correctness of both: calculation and measurement. CHAPTER 5. CRACK DIAGNOSIS IN THE ELASTIC BEAMS 5.1. Scanning method in diagnosing cracks with oscillation (1) Selecting a split grid )1....0( 21  neee including the positions of possible cracks with unknown depths ),.....,( 1 naa ; (2) Building the model of beams with above cracks and using this model to establish the equations of crack diagnosis from measurement data; (3) Using the above diagnostic equations together with the given measurement data, determine the vector of magnitude parameters of unknown cracks ),...,( 1 n ; (4) Removing the positions corresponding to the crack magnitude with 0 or negative in the split grid, we get a new split grid with the number of smaller nodes )ˆ,...,ˆ( 1 cnee corresponding to the positive magnitude values )ˆ,...,ˆ( 1 cn ; 20 (5) Using the new split grid of the crack position )ˆ,...,ˆ( 1 cnee to repeat steps 2-3-4 until the new grid is not received, then stop; (6) The final split grid obtained in step 5 is the position of possible cracks ),...,( 1 ree , corresponding to the the positive crack magnitudes ),...,( 1 r ; (7) From the magnitude of the crack found ),...,( 1 r using the T.G. Chondros formula to calculate the crack depth; (8) Finally, we found the position ),...,( 1 ree and the depth ),.....,( 1 raa together with the number of r-cracks, and then the question was solved. 5.2. Crack diagnose for the bar by the frequency response function Developing the following diagnostic equations ).(),,( 1  bγ n j jj   μe Using the frequency response function measurement data for the above concrete bar model, the result of crack position diagnosis for the following figure. Figure 5.1. Results of the crack diagnosis in the concrete bar by FRF1 frequency response function. 21 5.2. Diagnose cracks in beams by the measurement frequency Figure 5.2. Diagnosis results of 03 cracks at the position 0.2, 0.45 and 0.7 with 10% depth (a. diagnosis result is similar to the first, linear; b, c - intermediate nonlinear iterations; d. the final iteration). Chapter 5 conclusion In chapter 5, the following results has been presented: A general crack scanning method was proposed by Prof. Nguyen Tien Khiem was presented and will be applied for two questions of cracking diagnosis in bars from frequency response function and in beams from measured frequencies; The diagnosis algorithm with scanning method has been applied to the bar using the measurement of the frequency response function in the first frequency vicinity. Since the measurement of the frequency response function is sufficiently large (at frequencies in the resonant region), the number of chosen cracks assumed by the scanning method is always equal to the number of frequency division points in the adjacent frequency range. Therefore, the Tikhonov adjustment does not need to be applied. Moreover, the iteration performed in the algorithm is according to the damaged index variable () rather than repeating the crack magnitude. The diagnosis results show that the diagnosis of crack in the bar by the frequency response 22 function only gives accurate results when the crack depth is gr