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MINISTRY OF EDUCATION AND TRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY - CAO VAN MAI DAMAGE DETECTION OF BEAM USING RECEPTANCE FUNCTION Major: Engineering Mechanics Code: 9.52.01.01 SUMMARY OF DOCTORAL THESIS IN MECHANICAL ENGINEERING AND ENGINEERING MECHANICS Hanoi – 2022 The thesis has been completed at: Graduate University Science and Technology – Vietnam Academy of Science and Technology Supervisors: Assoc Prof Dr Nguyen Viet Khoa Reviewer 1: Prof Dr Hoang Xuan Luong Reviewer 2: Prof Dr Tran Ich Thinh Reviewer 3: Assoc Prof Dr Tran Minh Tu Thesis is defended at Graduate University Science and TechnologyVietnam Academy of Science and Technology at : , on , 2022 Hardcopy of the thesis be found at: - Library of Graduate University Science and Technology - Vietnam national library PREFACE The necessity of the thesis Damage detection of structures plays an important role in practice order to assess the damage level, to repair the damage, and to ensure the safety in operation and use, to reduce unwanted risks that may cause damage to people and property Moreover, the detection of location and serverity of damage damage also is very important to aid the reinforcement and repair of the structures more efficient and costeffective Thesis objective The objective of the thesis is to establish the exact receptance function and curvature function of damaged Euler - Bernoulli beam structures for the purpose of damage detection The considered damage types are crack and concentrated mass The influences of damage on the receptance function and curvature function are used for the damage detection of beam structures Content of the thesis The thesis consists of chapters, Chapter presents an overview of the methods of detecting structural damage and the research situation on the effects of crack-type damage and concentrated mass on the natural frequency, mode shape, function receptance and curvature receptance of isotropic homogeneous beam functionally graded beam structures to detect structural damage Chapter develops the exact formulas of the receptance function and its curvature for cracked beam The numerical simulation results using the developed expressions have been used to determine the influence of the crack on the receptance function and receptance curvature, these results can be applied to crack detection Chapter builds the exact formulas of the receptance functions of homogeneous beam and AFG beam without and with concentrated masses and applies the receptance function to determine the location of the concentrated masses Chapter presents experiments to verify the correctness of the developed formulas and the applicability of the proposed method for damage detection Chapter OVERVIEW This chapter presents an overview of the structural damage detections and the study of the effects of crack and concentrated massse on dynamic characteristics such as natural frequency, mode shape, etc., receptance curvature structures to detect structural damage From that, the issues of the thesis is addressed Chapter EXACT RECEPTANCE FUNCTION OF CRACKED BEAMS AND ITS APPLICATION FOR CRACK DETECTION The establishment of the exact formulas for the receptance function and receptance curvature for cracked beam is very important to reduce calculation time in simulating these functions From there, we can study the effect of cracks on them to develop methods for detecting cracks The numerical simulation results using the developed expressions have determined the influence of the crack on the receptance function and receptance curvature, these results can be applied to crack detection 2.1 Receptance function of the intact beam The receptance function of the intact beam is built starting from the Clough equation of motion of the beam [89] Then, using the solution form, orthogonal properties, boundary conditions and normal transformations, we can determine the receptance at due to the force at f can be obtained as: i ( f ) i ( ) ( , f , ) = i =1 i − i ( ) m ( ) d (2.21) “Receptance curvature” defined as the second derivative of the receptance with respect to variable as follows: 2 ( , f , ) n ( f ) d 2n ( ) 2 d n =1 n − m ( ) d n = (2.24) 2.2 The exact formulas of receptance and its curvature of cracked beam In order to derive the exact receptance curvature of the cracked beam, the exact closed form of mode shape is adopted from [42] as follows: 2 k k ( ) = C1 n sin ( − ) + sinh ( − ) U ( − ) + sin i =1 + C2 2 k +C3 2 k i i k 0i k 0i 0i k sin k ( − 0i ) + sinh k ( − 0i ) U ( − 0i ) + cos k n i i sin k ( − 0i ) + sinh k ( − 0i ) U ( − 0i ) + sinh k i =1 n + C4 sin k ( − 0i ) + sinh k ( − 0i ) U ( − 0i ) + cosh k i i i = k n i =1 i i (2.22) Applying the following properties of Heaviside function and Dirac delta, the second derivative and the integral of the square of the mode shape can be derived as follows: n n ( ) d = 4 A k k i =1 j =1 i j 1 1 cos k ( 0i − j ) − sin k ( − 0i − j ) − cosh k ( 0i − j ) k 1 + sin k (1 − 0i ) cosh k (1 − j ) + sin k (1 − j ) cosh k (1 − 0i ) 2 k 2 k − + 1 cos k (1 − 0i ) sinh k (1 − j ) − cos k (1 − j ) sinh k (1 − 0i ) 2 k 2 k 2 k cosh k (1 − j ) sinh k (1 − 0i ) 1 − 0i cos k ( 0i − j ) + sin k ( 0i − j ) 2 2 k − 0i cosh k ( 0i − j ) − sinh k ( 0i − j ) H ( 0i − j ) k 1 − 0i cos k ( j − 0i ) + sin k ( j − 0i ) 2 2 k − 0i cosh k ( j − 0i ) H ( 0i − j ) + k n A i i =1 1 1 sin k 0i − sin k ( − 0i ) (1 − 0i ) cos k 0i − k k + + k + k k 2 k sin k cosh k (1 − 0i ) − n A − (1 − ) sin i i =1 0i + + n A i i =1 k i 4 k cos k 0i − 4 k cos k ( − 0i ) 1 sinh k 0i + sinh k ( − 0i ) − (1 − 0i ) cosh k 0i + k k 1 + sin k (1 − 0i ) cosh k − cos k (1 − 0i ) sinh k 2 k 2 k n − 1 sin k sinh k (1 − 0i ) + cos k cosh k (1 − 0i ) 2 k 2 k A − (1 − ) sinh i =1 0i cos k sinh k (1 − 0i ) 2 k 0i + k 0i + 4 k cosh k 0i + cosh k ( − 0i ) 4ak 1 sin k (1 − 0i ) sinh k − cos k (1 − 0i ) cosh k 2 k 2 k sin k sinh k sin 2 k sinh 2 k + C1C2 + C3C4 + ( C22 − C12 ) + ( C32 + C42 ) 4 k k k k sin k cosh k cos k sinh k + ( C1C3 + C2 C4 ) + ( C2 C4 − C1C3 ) k + ( C1C4 + C2 C3 ) + k + ( C2 C3 − C1C4 ) cos k cosh k k C1 + C22 − C32 + C42 ) + ( C1C4 − C2 C3 ) ( k d 2k ( ) d sin k sinh k k (2.31) n = C1i i k sinh k ( − 0i ) − sin k ( − 0i ) H ( − 0i ) i =1 + cos k ( − 0i ) + cosh k ( − 0i ) ( − 0i ) − C1 k sin k + n C2 i i k sinh k ( − 0i ) − sin k ( − 0i ) H ( − 0i ) i =1 + cos k ( − 0i ) + cosh k ( − 0i ) ( − 0i ) − C2 k cos k + n C3ii k sinh k ( − 0i ) − sin k ( − 0i ) H ( − 0i ) i =1 + cos k ( − 0i ) + cosh k ( − 0i ) ( − 0i ) + C3 k sinh k + n C4 i i k sinh k ( − 0i ) − sin k ( − 0i ) H ( − 0i ) i =1 + cos k ( − 0i ) + cosh k ( − 0i ) ( − 0i ) + C4 k cosh k (2.37) Substituting Eqs (2.31), and (2.37) into (2.24), the exact receptance curvature will be determined 2.3 The receptance curvature of beams with multiple cracks by the finite element method The receptance frequency response function matrix of beams in the finite element method can be written as [88]: α ( ) = Φ −1 ( r −2 ) T Φ (2.46) The receptance curvature of beam is defined as the second derivative of the receptance with respect to the coordinate along the length of beam and is expressed as follows: χ ( ) = α ( ) x (2.47) To determine the receptance curvature, we need to determine the eigenfrequency and eigenform values of the cracked beam by solving the cracked beam eigenvalue equation: My(t ) + Ky(t ) = (2.48) The global mass matrix and global stiffness matrix are are received by assembling the element mass matrix Me , the stiffness matrix of intact element stiffness matrix K e K c is the stiffness matrix of the cracked element: 6l 12 6l 4l EI Ke = l −12 −6l 6l 2l −12 6l 156 2 −6l 2l ml 22l ; Me = 12 −6l 420 54 2 −6l 4l −13l K c = TT c−1T 22l 4l 13l −3l 54 13l 156 −22l −13l −3l −22l 4l (2.49) (2.60) 2.4 Compare with previous publications Table 2.1 Compare and verify the cracked beam calculation program used in the thesis with the article [24] Natural frequency Ref [24] (rad/s) ω1 ω2 ω3 ω4 417.644 2619.704 7337.863 14370.040 Using the finite element method Value Error (rad/s) (%) 417.642 0,0004 2621.721 0,077 7339.719 0,0253 14362.841 0,0501 Using the exact formulas Value Error (rad/s) (%) 417.638 0.0015 2619.929 0.0086 7336.263 0.0218 14377.785 0.0539 Based on the comparison table, we can see that the frequency values calculated by the mathematical method and the exact formulas used by the thesis are similar and very good compared with the results Lee published in the article [24] Therefore, both calculation programs used in the thesis are reliable to calculate the receptance function of cracked beams 2.5 Numerical simulation results Numerical simulations of a simply supported beam with two cracks is presented in this section Parameters of the beam are: Mass density =7800 kg/m3; modulus of elasticity E=2.0x1011 N/m2; L=1 m; b=0.02 m; h=0.01 m., Two cracks with the same depths are made at positions of 0.42L and 0.76L Four levels of the crack depth ranging from 0% to 30% have been applied 2.5.1 Beams intact The numerical simulation results for the intact beam when the forcing frequency is approximately equal to the first three natural frequencies, (Fig 2.6), the receptance curves are smooth except the minimum position The positions of maxima and minima of these receptance graphs coincide with the positions of nodes of the corresponding mode shapes These results imply that when the beam is excited at natural frequencies the distribution of response amplitude along the beam can be predicted easily by using the receptance matrices as well as using the mode shapes (a) ωω1 (b) ωω2 (c) ωω3 (d) ωω1, f = 0.58 (e) ωω2, f = 0.58 (f) ωω3, f = 0.58 Figure 2.6 Receptance matrices of the intact beam However, when the forcing frequency is in between any two natural frequencies, the distribution of response amplitude is more complicated (a) ω=400 Rad/s (b) ω=950 Rad/s Figure 2.7 Receptance matrices of the intact beam As can be seen in figure 2.7, the forcing frequency is in between any two natural frequencies, the response amplitude is complicated at arbitrary frequency and cannot be predicted using mode shapes, but it can be predicted easily by using the 3D graph of receptance matrix 2.5.2 Cracked beam using exact formulas As can be seen in figure 2.8 , when there is a crack, the shape of graph will have changes at the crack positions leading to the changes in the receptance at the crack positions However, the changes in the receptance are small when the crack depth is small and they only become significant when the crack depth is large In our simulation, when the length-to-height ratio is fixed, the changes in the receptance is very difficult to be detected visually when the crack depth is smaller than 40% of the beam height (a) Receptance matrix (b) Receptance, f = 0.58 Figure 2.8 Receptance of the cracked beam, crack depth is 50%, ωω1 As can be seen from Figs 2.9, 2.10, 2.11, there are sharp peaks in the receptance curvature matrices at crack positions When the crack depth increases from 10% to 30%, the height of sharp peaks in receptance curvature increase and becomes clearer Therefore, the height of sharp peaks can be considered as an intensity factor, which relates the change in receptance curvature to the crack depth From these figures, establishing a graph of the first sharp peak height of the receptance curvature corresponding to the first natural frequency versus crack depth, a relationship between the height of sharp peak and crack depth is obtained as shown in Fig 2.12 It can be seen that this relationship is a second-degree polynomial function This relationship can be used for the estimation of crack depth 12 As can be observed from this figs 2.13-2.14, the numerical results obtained for the finite element method are similar to the numerical results when calculated by the exact formulas Therefore, we can once again assert that the receptance curvature can be used to detect cracks The location of the sharp peaks is an indication of the possible location of the crack The height of the given spike can be used to assess the extent of the crack Conclusion of chapter In chapter 2, exact formulas of the receptance function of a cracked beam is presented The influence of the crack on the receptance curvature is investigated The results show that when there are cracks, the receptance curvature is changed significantly at the crack positions and it can be used for crack detection Only receptance curvatures corresponding to low forcing frequencies can be applied efficiently for crack detection while the receptance curvatures corresponding to the high frequencies are not recommended This result has not been reported yet since no detail investigation of receptance curvature graphs at different forcing frequencies has been carried out The results in this chapter are published in 01 ISI Journal article, 01 domestic journal article, 02 international conference papers; 02 papers presented at the national conference: [CT-2], [CT-4], [CT-6], [CT-9], [CT-10], [CT-11] in the List of published works Chapter THE RECEPTANCE OF BEAMS CARRYING CONCENTRATED MASSES In this chapter 3, the general form of receptance functions of the isotropic homogeneous and axially functionally graded (AFG) beams carrying concentrated masses is presented The influences of the concentrated masses and the varying of the material properties along the beam on the receptance matrix are investigated The numerical calculations show that when there are concentrated masses, the receptance matrices of beams are changed Especially, when masses are attached at peak positions of the receptance matrices, these peaks will decrease significantly 13 3.1 Isotropic homogeneous beams carrying concentrated masses Starting from the equation of the beam flexural vibration equation according to Wu et al in [64] and using some basic transformations, using orthogonal properties, we obtain the receptance function of the receptance at due to the force at f obtained: α ( , f , ) = ΦT ( ) q f = ΦT ( ) ( K − M ) Φ ( f −1 ) (3.23) Where: n 2 0 1 d + mk1 ( mk ) k =1 n m ( ) ( ) M = k =1 k mk mk n mkN ( mk ) 1 ( mk ) k =1 n m ( ) ( ) k =1 k mk mk n 2 d + mk22 ( mk ) k =1 n m ( ) ( ) k =1 k N mk mk k =1 n m k ( mk ) N ( mk ) k =1 n 2 N d + mkN ( mk ) k =1 n m ( ) ( ) k mk N mk (3.16) EI 2 d 0 K = L EI22 d EIN2 d (3.17) Substituting Eqs (3.16), (3.17) into Eq (3.23) the formula of the receptance of the simply supported beam carrying concentrated masses will be determined 3.2 AFG beam carrying concentrated masses In this section, let us consider the AFG Euler-Bernoulli beam with uniform cross section carrying concentrated masses Elasticity modulus E(x) and the mass density μ(x) of the AFG Euler-Bernoulli beam is varied as follows [88], [92]: E ( ) = E0 1 − 1 v ; ( ) = 0 1 − 2 v (3.27) Using Adomian decomposition method and some conventional transformations, yields: 14 m v ( v − 1) 1 iv ( i + 1)( i + ) Ck −iv + k =0 k =0 ( k + v − 1)( k + v )( k + v + 1)( k + v + ) i = Ck k + k + v + i =0 m 2n1 iv ( i + 1)( i + )( i + 3) Ck − iv + k =0 ( k + v )( k + v + 1)( k + v + )( k + v + 3) + k +v +3 i =0 + k + k =0 m ( m −i )v i =0 (3.42) Civ + p ( k + 1)( k + )( k + 3)( k + ) m 2 ( m − i ) v Civ + p k =0 ( k + v + 1)( k + v + )( k + v + 3)( k + v + ) − k + v + i =0 The coefficients Ck where k
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