Frequency Contour Method for Damage Identification of Beam Structures

Abstract

The present invention discloses a frequency contour method for damage identification of beam-like structures, including the following: (1) obtaining a panoramic damage frequency map; (2) obtaining the measured frequencies of the damaged beam through testing; (3) marking the measured damage frequencies on the panoramic damage frequency map to obtain damage frequency contours, and each order of damage frequency contour represents the combination of all damage positions and damage degrees corresponding to that frequency; (4) projecting the damage frequency contours onto the xOy plane to obtain the intersection points of the contours, and the abscissa and ordinate of these intersection points are the damage position and damage degree. The present invention proposes a panoramic damage frequency representation formula, which is an analytical solution of the theory and has infinite precision; the panoramic damage frequency map drawn based on the panoramic damage frequency representation formula has higher fineness, and in the case of less noise interference, the present invention can accurately locate the damage position and damage degree.

Description

Technical Field

[0001] The present invention relates to damage identification of beam structures, and particularly to a frequency contour method for damage identification of beam structures. Background Art

[0002] Beam structures are the basic elements of complex engineering structures. However, during long-term service, beam structures are affected by various loads and the coupling effect of material degradation, and local damage will inevitably occur. Damage not only endangers the structural safety but also reduces the structural durability. Therefore, studying effective scientific methods and technical means to identify the damage of beam structures is an important prerequisite for avoiding structural failure and carrying out maintenance and replacement in advance. Damage identification is also an important and long-standing research topic in the domestic and international engineering and academic circles.

[0003] In recent decades, the research on damage identification of beam structures based on vibration theory has received much attention. Based on dynamic indexes such as frequency, vibration mode, or frequency response function, a large number of structural damage diagnosis methods have been developed. In contrast, frequency is more suitable for damage characterization because it is easy to obtain and has good robustness to noise. Therefore, damage identification methods based on frequency have received the attention of many scholars. Among them, the frequency contour method is the most typical frequency-based damage identification method. However, the conventional frequency contour maps all use experimental or finite element methods to estimate the relationship between the damage location, degree, and frequency, with poor accuracy and difficulty in being applied to actual engineering. Summary of the Invention

[0004] Object of the Invention: Aiming at the problems existing in the prior art, the present invention provides a frequency contour method for damage identification of beam structures.

[0005] Technical Solution: The frequency contour method for damage identification of beam structures according to the present invention includes the following contents:

[0006] (1) Obtain a panoramic damage frequency map;

[0007] (2) Through testing, obtain the measured frequencies of the damaged beam;

[0008] (3) Mark the measured damage frequencies on the panoramic damage frequency map to obtain damage frequency contours. Each order of damage frequency contour represents the combination of all damage positions and damage degrees corresponding to that frequency;

[0009] (4) Project the damage frequency contours onto the xOy plane to obtain the intersection points of the contours. The abscissa and ordinate of the intersection points are the damage location and damage degree.

[0010] Furthermore, in step (1), a damage frequency panorama is obtained by using an analytical formula of frequency, damage location and damage degree and a calculation and drawing software. The analytical formula of frequency, damage location and damage degree is obtained by the following method:

[0011] The m cracks divide the beam into m+1 sub-beams, and the vibration equation of each sub-beam is listed;

[0012] The cracks affect the stiffness of the beam. The displacement, bending moment and shear force at the connection of each section of the beam are the same, and only the rotation angle is different due to the existence of cracks. The continuity condition equation relationship of the four indicators of displacement, bending moment, shear force and rotation angle is listed. There are 4m continuity condition equations for m cracks, and there are 4 boundary condition equations for the boundary conditions of the entire beam. Therefore, a beam with m cracks has a total of 4m+4 equations.

[0013] By solving the simultaneous equations, we can find the functional relationship between the position and degree of the m cracks and the frequency of the beam, and define this functional relationship as the panoramic characterization formula of the damage frequency of the beam.

[0014] Furthermore, in step (3), the nth-order measured frequency is marked on the damage frequency panorama, where n≥3.

[0015] Compared with the prior art, the present invention has the following significant improvements:

[0016] The prior art uses finite element or experimental methods to obtain the relationship between the damage location and frequency through multiple calculations and measurements, thereby obtaining an approximate panoramic view. The panoramic view obtained by this method is relatively coarse and has a large error. The present invention proposes a damage frequency panoramic characterization formula, which is a theoretical analytical solution with infinite accuracy. The damage frequency panoramic view drawn on the basis of the damage frequency panoramic characterization formula has higher precision. In the case of less noise interference, the present invention can accurately locate the damage location and damage degree. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 A damaged simply supported beam according to an embodiment of the present invention;

[0018] Figure 2 A damaged cantilever beam according to an embodiment of the present invention;

[0019] Figure 3 The first three order damage frequency panoramic diagram and contour lines of the embodiment of the present invention, wherein (a) is the first three order damage frequency panoramic diagram, (b) is the first three order frequency contour lines;

[0020] Figure 4 It is the intersection point of the first three frequency contour line projections of the embodiment of the present invention, wherein (a) is the first three frequency contour line projections, and (b) is a local enlargement of the contour line projections. Detailed implementation manners

[0021] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention.

[0022] A frequency contour method for damage identification of beam-like structures includes the following:

[0023] (1) Obtain a panoramic damage frequency diagram.

[0024] (2) Through testing, obtain the measured frequencies of the damaged beam.

[0025] (3) Mark the damage frequencies obtained from the test on the panoramic damage frequency diagram to obtain damage frequency contours. Each order of damage frequency contour represents the combination of all damage positions and damage degrees corresponding to that frequency.

[0026] (4) Project the damage frequency contours onto the xOy plane to obtain the intersection points of the contours. The abscissa and ordinate of these intersection points are the damage positions and damage degrees.

[0027] In step (1), the panoramic damage frequency diagram can be obtained through conventional multiple experiments or multiple finite models. This method has the defects of low calculation efficiency and low accuracy. Preferably, the panoramic damage frequency diagram can be obtained through the analytical formula of frequency with respect to damage position and damage degree and calculation and drawing software (such as MATLAB software, Python software, etc.). The specific method is as follows:

[0028] m cracks divide the beam into m + 1 sub-beams, and the vibration equations of each sub-beam are listed; the cracks affect the stiffness of the beam. The displacements, bending moments, and shear forces at the joints of each sub-beam are the same, and only the rotation angles are different due to the existence of the cracks; list the equality relationships of the continuity conditions for the four indicators of displacement, bending moment, shear force, and rotation angle. There are 4m continuity condition equalities for m cracks, and there are 4 boundary condition equalities for the boundary conditions of the entire beam. Therefore, a beam with m cracks has a total of 4m + 4 equality equations; solve the equations simultaneously to obtain the functional relationship between the positions and degrees of m cracks and the frequency of the beam. Define this functional relationship as the panoramic damage frequency characterization formula of the beam. The method for obtaining this formula is called the panoramic damage frequency characterization method. This method can obtain the panoramic diagram more quickly and accurately.

[0029] Taking the simply supported beam and the cantilever beam as examples, the panoramic damage frequency characterization method will be introduced in detail.

[0030] 1. Simply supported beam

[0031] For a beam with length, width, and height of L, b, and h respectively, the mass per unit beam length and the bending stiffness are and EI(u), where u is the axial coordinate of the beam and t is the vibration time.

[0032] For a simply supported beam with a single damage, it can be regarded as composed of two sub - beams, as shown in Figure 1 ; they are connected by a rotational spring at the damage position. The stiffness of the spring is related to the degree of damage. Assume the damage position is x = u / L and the damage degree is y, both representing the dimensionless damage position and degree.

[0033] The vibration mode equation of each sub - beam is

[0034] W(v)=A1 sinkLv + A2 cos kLv + A3 sinh kLv + A4 cosh kLv (1)

[0035] where W(v) represents the dimensionless transverse vibration mode function of the beam, v represents the dimensionless coordinate along the length direction of the beam; k is the characteristic parameter, or The coefficients A1, A2, A3, and A4 are determined by the constraints of the beam. For the frequency equation of a beam with infinite degrees of freedom, there are infinitely many frequencies and vibration modes.

[0036] To normalize the length of the beam, let

[0037]

[0038] The partial differential equation of the bending vibration of the beam is simplified to

[0039]

[0040] Using the method of separation of variables to solve, we can assume

[0041] w(u,t)=W(v)T(t) (3)

[0042] where v = u / L, representing the dimensionless coordinate of the beam along the axial direction (any point along the length direction of the beam), then

[0043]

[0044] Equation (3) can be expressed as follows

[0045] T(t)=asin(ωt + v) (5)

[0046] w(u,t)=W(v)sin(ωt + v) (6)

[0047] where a and v are integration constants, and ω is the natural vibration frequency.

[0048] Transform equation (4) into

[0049]

[0050] The vibration mode functions of the left and right sub - beams can be set as

[0051] W1(v) = A1 sinλv + B1 cosλv + C1 sinhλv + D1 coshλv, 0 ≤ v < x (8)

[0052] W2(v) = A2 sinλv + B2 cosλv + C2 sinhλv + D2 coshλv, x ≤ v < 1 (9)

[0053] where, A i 、B i 、C i 、D i (i = 1, 2) are undetermined constants determined by the boundary conditions and continuity conditions. The boundary conditions of the simply - supported beam are

[0054]

[0055] The continuity conditions at the damage position are

[0056]

[0057]

[0058] where, equations (11) and (12) respectively represent the continuity conditions of displacement, bending moment, shear force and rotation angle of the two sub - beams at the damage position. θ is a dimensionless damage flexibility factor, representing a dimensionless coefficient related to the relative crack depth y, and is expressed as follows

[0059] θ = 6πy 2 f(y)(h / L) (13)

[0060] where, f(y) is the correction factor for damage in the damage flexibility. y represents the dimensionless damage degree, that is, the ratio of the damage depth to the beam height. A function with relatively high accuracy for the single - side concentrated crack model is selected as the damage correction factor, as follows

[0061]

[0062] For any damage degree y, the error of equation (14) is less than 0.5%. Therefore, its accuracy can meet the requirements of engineering practice. Substituting equations (8) and (9) into equations (10), (11), (12), it can be expressed in matrix form as

[0063] S Esc ·A = 0 (15)

[0064] where, the coefficient matrix A=(A1 B1 C1 D1 A2 B2 C2 D2) T , SEsc It is expressed as follows:

[0065]

[0066] S Esc The subscript Es in it represents an Euler simply supported beam, and c indicates that the beam contains a crack.

[0067] Let |S Esc | = 0, then the frequency equation can be obtained. Expanding and simplifying this equation gives

[0068] λθ(A + B + C + D + E + F + G) + H = 0 (17)

[0069] where A = sinλsinhλsinλxsinhλx(cosλxsinhλx + sinλxcoshλx), B = -sinλsinλx(sinλxsinhλx + cosλxcoshλx)(sinhλcoshλx + coshλsinhλx), C = sinλsinhλx(sinhλcoshλx - coshλsinhλx), D = sinλcoshλsinhλxsinλx(sinλxsinhλx + cosλxcoshλx), E = -cosλsinhλsinhλxsinλx(sinhλxsinλx - cosλxcoshλx), F = cosλsinλx(coshλxsinλ_x - sinhλxcosλx)(sinhλ_coshλ_x + coshλ_sinhλ_x), G = cosλ_coshλ_sinhλxsinλx(sinhλxcosλx - sinλxcoshλx) and H = -2sinλsinhλ.

[0070] When there is no damage, the damage flexibility factor θ = 0, then Equation (17) can be simplified to H = 0, and further simplified to sinλ = 0, which is consistent with the frequency equation of the undamaged simply supported beam. Equation (17) describes the relationship between the frequency of the damaged simply supported beam and the damage at any position and of any degree.

[0071] 2. Cantilever beam

[0072] For a cantilever beam with a single damage, it can also be regarded as composed of two sub - beams, see Figure 2 ; connected by a rotational spring at the damage position. Assume the damage position is x = u / L and the damage degree is y, both representing the dimensionless damage position and degree.

[0073] Its mode shape equation is the same as that of a simply supported beam, that is, the same as Equation (2), and the mode shape functions of its left and right sub-beams are the same as Equations (8) and (9). One end of the cantilever beam is fixed and the other end is free, and its boundary conditions are

[0074]

[0075] The continuity conditions at the damage position are the same as Equations (11) and (12). Using the same damage correction factor as Equation (14), substituting Equations (8) and (9) into Equations (11), (12) and (18), and expressing it in matrix form, we can get

[0076] S Ecc ·A = 0 (19)

[0077] where the coefficient matrix A = (A1 B1 C1 D1 A2 B2 C2 D2) T , S Ecc represents an Euler cantilever beam with one crack, that is

[0078]

[0079]

[0080] Let |S Ecc | = 0, then the frequency equation can be obtained. Expanding and simplifying this equation gives

[0081] λθ(A + B + C + D + E + F) + G = 0 (21)

[0082] where A = cosλxsinhλx, B = -sinλxcoshλx, C = sinλcoshλxcosh(λ(x - 1)), D = -sinhλcosλxcos(λ(x - 1)), E = -sin(λ(x - 1))cosh(λ(x - 1)), F = cos(λ(x - 1))sinh(λ(x - 1)) and G = -2 - 2cosλcoshλ.

[0083] When there is no damage, the damage flexibility factor θ = 0, then Equation (21) can be simplified to G = 0, that is, 1 + cosλcoshλ = 0, which is the same as the frequency equation of the undamaged cantilever beam.

[0084] G. Bamnios and A. Trochides gave the frequency equation when the damage location is at the fixed end in the literature. In this embodiment, by setting \(x = 0\) in Equation (21), the frequency equation for damage at the fixed end can be obtained. This frequency equation only describes the relationship between the damage degree at the fixed end and the frequency, while the frequency equation in this embodiment describes the relationship between the frequency and damage at any position and of any degree. Therefore, Equation (21) is more general and extensive than the frequency equation in the literature.

[0085] In step (3), the first \(n\) measured frequencies are taken and marked on the panoramic damage frequency diagram, where \(n\) is a positive integer greater than or equal to 3.

[0086] The frequency contour method generally requires at least three frequencies to identify the damage location and degree. More frequencies can also be used to verify and improve the accuracy of damage identification, but the identification principle is the same. Since the lower-order frequencies are convenient to measure and have smaller errors, the first three frequencies are usually used to study the damage identification problem of the beam.

[0087] Taking the cantilever beam model as an example below, the process of identifying damage using the first three frequencies by the frequency contour method is described.

[0088] The length (\(L\)) of the cantilever beam is 1.2 m, the thickness (\(h\)) is 0.02 m, the width (\(b\)) is 0.02 m, the elastic modulus (\(E\)) is \(2.1\times10^{ 11 Pa, and the density (\(\rho\)) is \(7.8\times10^{ 3 kg / m^{ 3 , as Figure 2 shown.

[0089] According to the frequency equation of the cantilever beam and the MATLAB software, the panoramic diagram of the first three damage frequencies of the cantilever beam model can be obtained, as Figure 3 (a) shown. Then, the first three frequencies with the damage location at \(x = 0.33\) and the damage degree at \(y = 0.36\), \(\omega_1 = 11.25\) Hz, \(\omega_2 = 71.57\) Hz, \(\omega_3 = 195.16\) Hz, are marked on the panoramic damage frequency Figure 3 (a), and Figure 3 (b) can be obtained, where curve ① represents the relationship between the damage location and the damage degree when the first-order frequency of the beam is 11.25 Hz, curve ② represents the relationship between the damage location and the damage degree when the second-order frequency of the beam is 71.57 Hz, and curve ③ represents the relationship between the damage location and the damage degree when the third-order frequency of the beam is 195.16 Hz.

[0090] Figure 4 (a) is Figure 3 the projection of the contour lines of the first three damage frequencies in Figure 4 (b) on the \(xOy\) plane, Figure 4(a) Enlarged view of the square local area. It can be seen that three contour lines intersect at a point A. The coordinates of the intersection point A are (0.33, 0.36). Among them, the abscissa 0.33 is the identified damage position, and the ordinate 0.36 is the identified damage degree.

[0091] The above embodiments have described in detail the specific steps and feasibility of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the scope of protection of the present invention.

Claims

1. A frequency contour method for damage identification of beam-like structures, characterized in that, It includes the following: (1) Obtain the analytical formula of frequency, damage location and damage degree, and obtain the damage frequency panorama through the analytical formula of frequency, damage location and damage degree and calculation and drawing software; The specific process of obtaining the analytical formula of frequency, damage location and damage degree is as follows: (1.1) m cracks divide the beam into m+1 sub-beams. List the vibration equations of each sub-beam. (1.2) Cracks affect the stiffness of the beam. The displacement, bending moment and shear force at the connection of each section of the beam are the same, and only the rotation angle is different due to the existence of cracks. The continuity condition equation relationship of the four indicators of displacement, bending moment, shear force and rotation angle is listed. There are 4m continuity condition equations for m cracks, and there are 4 boundary condition equations for the boundary conditions of the entire beam. Therefore, a beam with m cracks has a total of 4m+4 equations; (1.3) Solve the simultaneous equations to find the functional relationship between the position and degree of the m cracks and the frequency of the beam, and define this functional relationship as the panoramic characterization formula of the damage frequency of the beam; (2) Obtain the measured frequency of the damaged beam through testing; (3) Mark the damage frequency obtained by the test on the damage frequency panorama to obtain the damage frequency contour line. Each order of damage frequency contour line represents the combination of all damage locations and damage degrees corresponding to the frequency. (4) Project the damage frequency contour lines onto the xOy plane to obtain the intersection of the contour lines. The horizontal and vertical coordinates of the intersection point are the damage location and damage degree.

2. The frequency contour method for beam structure damage identification according to claim 1, characterized in that In step (3), the nth-order measured frequency is marked on the damage frequency panorama, where n≥3.

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