A beam structure damage identification method based on frequency contour band method

Abstract

This invention discloses a damage identification method for beam structures based on the frequency contour band method, comprising the following steps: (1) obtaining a panoramic view of the damage frequency; (2) considering the influence of noise and error, obtaining a measured frequency range covering the true frequency of the damaged beam; (3) taking n-order measured frequency ranges and marking them all on the panoramic view of the damage frequency to obtain damage frequency contour bands, each order of damage frequency contour bands representing all combinations of damage locations and damage degrees corresponding to that frequency; (4) projecting the damage frequency contour bands onto the xOy plane to obtain the intersection area of ​​n contour bands, all points within the intersection area are used to indicate possible damage locations and damage degrees; (5) calculating the center point of the intersection area, using the horizontal and vertical coordinates of the center point to represent the location and degree of damage. This invention considers the influence of environmental noise and test errors, enabling the frequency-based damage identification method to achieve a certain degree of noise immunity and exhibiting good robustness.

Description

Technical Field

[0001] This invention relates to a method for identifying damage in beam structures, and more particularly to a method for identifying damage in beam structures based on the frequency contour band method. Background Technology

[0002] Beam structures are fundamental elements of complex engineering structures. However, during long-term service, beam structures are subjected to various loads and the coupled effects of material degradation, inevitably leading to localized damage. This damage not only jeopardizes structural safety but also reduces structural durability. Therefore, researching effective scientific methods and techniques to identify damage in beam structures is a crucial prerequisite for preventing structural failure and enabling timely maintenance and replacement. Damage identification remains a vital and ever-evolving research topic in the engineering and academic communities both domestically and internationally.

[0003] In recent decades, research on damage identification of beam structures based on vibration theory has received considerable attention. Numerous structural damage diagnosis methods have been developed based on dynamic indicators such as frequency, mode shape, or frequency response function. In contrast, frequency is more suitable for damage characterization due to its ease of acquisition and good robustness to noise. Therefore, frequency-based damage identification methods have attracted much attention from scholars, among which the frequency contour method is the most typical frequency-based damage identification method.

[0004] While the damage identification method based on frequency contour lines is theoretically simple and easy to implement, in actual engineering, frequencies are easily affected by environmental noise and measurement errors, causing frequency contour lines to fail to form common intersections in the xOy plane. Consequently, information indicating the location and extent of damage cannot be obtained, ultimately leading to the failure of the method. In other words, the method fails to achieve noise immunity, has low robustness and effectiveness, and is only suitable for ideal situations. Summary of the Invention

[0005] Purpose of the invention: To address the problem of insufficient damage identification capability of the frequency contour line method under noisy conditions, this invention provides a damage identification method for beam structures based on the frequency contour band method, which has good noise immunity capability.

[0006] Technical solution: A damage identification method for beam structures based on the frequency-equidistant band method, comprising the following:

[0007] (1) Obtain a panoramic view of the damage frequency;

[0008] (2) Considering the effects of noise and error, the measured frequency range of the covered damaged beam is obtained;

[0009] (3) Take the nth order measured frequency interval and mark the nth order frequency interval on the damage frequency panoramic map to obtain the damage frequency contour band. Each order damage frequency contour band represents the combination of all damage locations and damage degrees corresponding to that frequency; n≥3;

[0010] (4) Project the damage frequency contour bands onto the xOy plane to obtain the intersection region of n contour bands. All points within the intersection region are used to indicate the damage location and damage degree.

[0011] (5) Calculate the center point of the intersecting area, and use the horizontal and vertical coordinates of the center point to represent the location and extent of the damage.

[0012] Furthermore, in step (1), a panoramic view of the damage frequency is obtained using analytical formulas relating frequency to damage location and severity, and computational plotting software (such as MATLAB, Python, etc.). The analytical formulas relating frequency to damage location and severity are obtained as follows:

[0013] The beam is divided into m+1 sub-beams by m cracks. The vibration equations for each sub-beam are listed.

[0014] Cracks affect the stiffness of the beam. The displacement, bending moment, and shear force at the connection of each segment of the beam are the same, only the rotation angle is different due to the presence of cracks. List the continuity condition equations for the four indicators of displacement, bending moment, shear force, and rotation angle. There are 4m continuity condition equations for m cracks, and there are 4 boundary condition equations for the entire beam. Therefore, a beam containing m cracks has a total of 4m+4 equations.

[0015] By solving the simultaneous equations, the functional relationship between the location and extent of m cracks and the frequency of the beam is obtained. This functional relationship is defined as the panoramic characterization formula for the damage frequency of the beam.

[0016] Furthermore, in step (2), the measured frequency range covering the true frequency of the damaged beam is obtained through multiple tests; or the frequency range covering the true frequency of the damaged beam is obtained by estimating the effects of noise and error through a single test.

[0017] Furthermore, in step (5), the center point can be calculated using methods such as the centroid method, the weighted average method, and the boundary point average method.

[0018] Furthermore, a noise hypercube is used as pseudo-random noise. This pseudo-random noise is used to simulate the effect of noise on these nth-order frequencies. ε is the level of the pseudo-random noise, i.e., the unit of the magnitude of the noise's influence. The specific value of ε depends mainly on the test accuracy. If the accuracy is 1%, then 1% can be used; if the accuracy is 0.1%, then 0.1% can be used. Usually, ε is less than or equal to 2%, and preferably less than or equal to 1%.

[0019] Specifically, a noise hypercube is used to represent the nth order true frequency (f1, f2, ..., fn) of the beam. n Transformed into measured frequencies that contain noise and errors. The process is as follows:

[0020]

[0021] Where ε is the unit of the magnitude of the noise's influence, k1,k2,……,k n These are the noise influence magnitude coefficients in n directions of the noise hypercube; the value in each direction represents the magnitude of the noise influence on the frequency in that direction; assuming the measured frequency is affected by 100% ε-level noise, then:

[0022]

[0023] f i (1-ε), i=1,2,……,n, represents the lower limit of the measured frequency due to noise, or the minimum value of this frequency order when tested multiple times; f i (1+ε), i=1,2,……,n, represents the upper limit of the measured frequency due to noise, or the maximum value of this frequency order during multiple tests; thus, n frequency intervals can be obtained (f1(1-ε),f1(1+ε)), (f2(1-ε),f2(1+ε)),……,(f n (1-ε),f n (1+ε)), all possible nth order frequencies that can be measured are contained within these n frequency intervals;

[0024] The frequency range of these n-order frequencies is marked on the damage frequency panorama, resulting in n damage frequency contour bands; the damage frequency contour bands appear as banded curved surfaces on the damage frequency panorama.

[0025] Beneficial effects

[0026] Compared with the prior art, the present invention has the following significant advantages:

[0027] (1) The damage frequency panoramic characterization method was used. The relationship between damage location, degree and frequency was expressed analytically through the damage frequency panoramic characterization formula, so that each frequency point in the damage frequency panoramic map can better match the actual situation.

[0028] (2) By drawing a panoramic view of damage frequency using computer drawing software, it is possible to achieve efficient graphical display of any damage condition and any order frequency.

[0029] (3) The frequency of the damaged beam is no longer a single frequency value, but a frequency range considering the influence of environmental noise and test error. This frequency range can be obtained by multiple measurements under different test methods and test conditions, or it can be the calculated range after considering the influence of environmental noise and test error after a single test.

[0030] (4) Use frequency range instead of frequency value to label the damage frequency panorama, and expand the one-dimensional contour lines into two-dimensional contour bands to contain more damage information.

[0031] (5) Projecting the frequency contour bands onto the xOy plane yields multiple projection bands. The intersection or common area of ​​these multiple projection bands can be the range where the damage location and degree may be located. However, the projection curves of the contour lines often do not have a unified intersection point due to the influence of noise and error, making it impossible to determine the damage location and degree, thus rendering the frequency contour method ineffective.

[0032] (6) The location and extent of damage are represented by calculating the center point of the intersection area of ​​multiple projection bands, which makes the method somewhat noise-immune.

[0033] This method takes into account the effects of environmental noise and testing errors, making the frequency-based damage identification method more robust and noise-immune, thus meeting the needs of engineering applications. Furthermore, the intersecting area of ​​multiple projection bands contains numerous feature points indicating the location and extent of damage. More accurate damage location and extent can be calculated using methods such as the centroid method, weighted average method, and boundary point average method. Attached Figure Description

[0034] Figure 1 This is a simply supported beam with damage, as described in an embodiment of the present invention.

[0035] Figure 2 The damaged cantilever beam is an embodiment of the present invention;

[0036] Figure 3 This is a schematic diagram of an ε-noise cube (n=3) according to an embodiment of the present invention;

[0037] Figure 4 The first-order frequency of this invention varies with the location and extent of damage, as shown in the embodiment of the invention, where (a) is a three-dimensional diagram and (b) is a planar diagram;

[0038] Figure 5 The second-order frequency of this invention varies with the location and extent of damage, as shown in embodiment (a) and (b) respectively.

[0039] Figure 6 The third-order frequency of this invention varies with the location and extent of damage, as shown in embodiment (a) and (b) respectively.

[0040] Figure 7 This is a schematic diagram of the damage frequency contour band in an embodiment of the present invention;

[0041] Figure 8 This is a projection diagram of the frequency contour bands in an embodiment of the present invention, wherein (a) is the intersection region of the polygonal shape formed by the projection of the three frequency contour bands, and (b) is a magnified view of a portion of the intersection region of the contour bands. Detailed Implementation

[0042] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0043] A damage identification method for beam structures based on the frequency contour band method includes the following:

[0044] (1) Obtain a panoramic view of damage frequency.

[0045] (2) Obtain the measured frequency range of the true frequency of the damaged beam through multiple tests; or obtain the frequency range of the true frequency of the damaged beam by estimating the noise and error effects through a single test; take the first n frequency ranges. Usually, n is a positive integer greater than or equal to 3, that is, at least 3 frequencies are needed to accurately identify the location and degree of damage.

[0046] (3) Mark the frequency range on the damage frequency panorama to obtain the damage frequency contour band. Each damage frequency contour band represents the combination of all damage locations and damage degrees corresponding to that frequency.

[0047] (4) Project the damage frequency contour bands onto the xOy plane to obtain the intersection region of n contour bands. All points within this intersection region may indicate the location and extent of the damage. It should be noted that due to noise and error, the exact location and extent of the damage cannot be obtained.

[0048] (5) Calculate the center point of the intersecting area, and use the horizontal and vertical coordinates of the center point to represent the location and degree of damage; specifically, the center point can be calculated by the centroid method, weighted average, or boundary point average method.

[0049] In step (1), a panoramic image can be obtained by conventional finite element analysis or experiments, which involves multiple calculations and measurements to obtain an approximate panoramic image by determining the relationship between the damage location and frequency. However, the accuracy of this method is not high. Therefore, it is preferable to obtain a panoramic image of the damage frequency using an analytical formula relating frequency to the damage location and degree. The specific steps are as follows:

[0050] Assuming that m cracks divide the beam into m+1 sub-beams, list the vibration equations for each sub-beam.

[0051] Cracks affect the stiffness of the beam. The displacement, bending moment, and shear force at the connection of each segment of the beam are the same, only the rotation angle is different due to the presence of cracks. List the continuity condition equations for the four indicators of displacement, bending moment, shear force, and rotation angle. There are 4m continuity condition equations for m cracks, and there are 4 boundary condition equations for the entire beam. Therefore, a beam containing m cracks has a total of 4m+4 equations.

[0052] By solving the simultaneous equations, the functional relationship between the location and extent of m cracks and the frequency of the beam is obtained. This functional relationship is defined as the panoramic characterization formula for the damage frequency of the beam.

[0053] Based on the formula for panoramic characterization of damage frequency, a high-precision panoramic image of damage frequency is obtained by combining computational plotting software, including but not limited to MATLAB and Python.

[0054] Taking simply supported beams and cantilever beams as examples, this method for panoramic characterization of damage frequency is described in detail.

[0055] 1. Simply supported beam

[0056] A beam has length L, width b, and height h, respectively. Its mass per unit length and bending stiffness are respectively... And EI(u), where u is the axial coordinate of the beam and t is the vibration time.

[0057] For a simply supported beam with a single damage, it can be considered as consisting of two sub-beams, see... Figure 1 A rotating spring is used to connect the damaged area. The stiffness of the spring is related to the degree of damage. Assuming the damaged location is x = u / L and the degree of damage is y, both represent dimensionless damaged location and degree.

[0058] The mode equation for each segment of the beam is:

[0059] W(v)A1sin kLv+A2cos kLv+A3sinh kLv+A4cosh kLv (1)

[0060] Where W(v) represents the dimensionless transverse mode shape function of the beam, v represents the dimensionless coordinate along the beam length, and k is a characteristic parameter. or The coefficients A1, A2, A3, and A4 are determined by the constraints of the beam. For the frequency equation of an infinitely free beam, there are infinitely many frequencies and mode shapes.

[0061] To normalize the beam length, let...

[0062]

[0063] The partial differential equation for the bending vibration of the beam is simplified to:

[0064]

[0065] To solve this using the method of separation of variables, we can assume...

[0066] w(u,t)=W(v)T(t) (3)

[0067] Where v = u / L represents the dimensionless coordinate of the beam's axial direction (at any point along the beam's length), then...

[0068]

[0069] Equation (3) can be expressed as follows:

[0070] T(t)=a sin(ωt+v) (5)

[0071] w(u,t)=W(v)sin(ωt+v) (6)

[0072] Where a and v are integration constants, and ω is the natural frequency.

[0073] Transform equation (4) into

[0074]

[0075] The mode shape functions of the left and right sub-beams can then be set as follows:

[0076] W1(v)=A1 sinλv+B1 cosλv+C1 sinhλv+D1 coshλv,0≤v≤x (8)

[0077] W2(v)=A2sinλv+B2cosλv+C2sinhλv+D2coshλv,x≤v≤1 (9)

[0078] Among them, 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 for the simply supported beam are...

[0079]

[0080] The continuity condition at the location of the damage is

[0081]

[0082]

[0083] Equations (11) and (12) represent the continuity conditions of displacement, bending moment, shear force, and rotation angle of the two sub-beams at the damage location, respectively. θ is a dimensionless damage compliance factor, representing a dimensionless coefficient related to the relative crack depth y, expressed as follows:

[0084] θ=6πy 2 f(y)(h / L) (13)

[0085] Where f(y) is the damage compliance correction factor, and y represents the dimensionless damage degree, i.e., the ratio of damage depth to beam height. A function with relatively high accuracy for the single-sided concentrated crack model is selected as the damage correction factor, as follows:

[0086]

[0087] Equation (14) has an error of less than 0.5% for any damage level y, thus its accuracy meets the actual engineering requirements. Substituting equations (8) and (9) into equations (10), (11), and (12), and expressing them in matrix form, we can obtain...

[0088] S Esc ·A=0 (15)

[0089] Wherein, the coefficient matrix S Esc It is expressed as follows:

[0090]

[0091] S Esc The subscript Es in the text indicates an Euler simply supported beam, and c indicates that the beam contains a crack.

[0092] Order | S Esc |=0, which gives us the frequency equation. Expanding and simplifying this equation, we get...

[0093] λθ(A+B+C+D+E+F+G)+H=0 (17)

[0094] Among them, A=sinλsinhλsinλxsinhλx(cosλxsinhλx+sinλxcoshλx), B=-sinλsinλx(sinλxsinhλx+cosλxcoshλx)(sinh λcoshλx+coshλsinhλx), C=sinλsinhλx (sinhλcoshλx-coshλsinhλx), D=sinλcoshλsinhλxsinλx (sinλxsinhλx +cosλxcoshλx), E=-cosλsinhλsinhλxsinλx(sinhλxsinλx-cosλxcoshλx), F=cosλsinλx(coshλxsinλx-sinhλx cosλx)(sinhλcoshλx+coshλsinhλx), G=cosλcoshλsinhλxsinλx(sinhλxcosλx-sinλxcoshλx) and H=-2 sinλsinhλ.

[0095] When there is no damage, the damage compliance factor θ = 0, then equation (17) can be simplified to H = 0. Further simplification yields sinλ = 0, which is consistent with the frequency equation of a simply supported beam without damage. Equation (17) describes the relationship between the frequency of a damaged simply supported beam and damage at any location and of any degree.

[0096] 2. Cantilever beam

[0097] For a single-damage cantilever beam, it can also be considered as consisting of two sub-beams, see... Figure 2 A rotating spring is used to connect the damaged area. Assume the damaged area is x = u / L and the damaged degree is y, both representing dimensionless damaged location and degree.

[0098] Its mode shape equation is consistent with that of a simply supported beam, i.e., consistent with equation (2), and the mode shape functions of its left and right sub-beams are consistent with equations (8) and (9). The cantilever beam is fixed at one end and free at the other, and its boundary conditions are...

[0099]

[0100] The continuity condition at the damage location is consistent with equations (11) and (12), and the same damage correction factor as in equation (14) is used. Substituting equations (8) and (9) into equations (11), (12), and (18), the result can be expressed in matrix form.

[0101] S Ecc A = 0 (19)

[0102] Wherein, the coefficient matrix S EccThis refers to an Euler cantilever beam, which includes a crack, i.e.

[0103]

[0104]

[0105] Order | S Ecc |=0, which gives us the frequency equation. Expanding and simplifying this equation, we get...

[0106] λθ(A+B+C+D+E+F)+G=0 (21)

[0107] Among them, 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.

[0108] When there is no damage, the damage compliance factor θ = 0, then equation (21) can be simplified to G = 0, i.e., 1 + cosλcoshλ = 0, which is consistent with the frequency equation of the undamaged cantilever beam.

[0109] G. Bamnios and A. Trochides provided a frequency equation for damage at a fixed end in their literature. In this embodiment, setting x = 0 in equation (21) yields the frequency equation for damage at a fixed end. This frequency equation only describes the relationship between the degree of damage at the fixed end and the frequency, while the frequency equation in this embodiment describes the relationship between the frequency and damage at any location and of any degree. Therefore, equation (21) is more general and extensive than the frequency equation in the literature.

[0110] Furthermore, a noise hypercube is used as pseudo-random noise, which is used to simulate the effect of noise on these nth order frequencies.

[0111] Specifically, a noise hypercube is used to represent the nth order true frequency (f1, f2, ..., fn) of the beam. n Transformed into measured frequencies that contain noise and errors. The steps are as follows:

[0112]

[0113] Wherein, ε is the level of pseudo-random noise, that is, the unit of the magnitude of the noise's influence. The value of ε mainly depends on the accuracy of the test. If the accuracy is 1%, then 1% can be used; if the accuracy is 0.1%, then 0.1% can be used. Generally, ε is less than or equal to 2%, and more preferably less than or equal to 1%.

[0114] k1,k2,……,k n These are the noise influence magnitude coefficients in n directions of the noise hypercube; the value in each direction represents the magnitude of the noise influence on the frequency in that direction; assuming the measured frequency is affected by 100% ε-level noise, then:

[0115]

[0116] f i (1-ε), i=1,2,……,n, represents the lower limit of the measured frequency due to noise, or the minimum value of this frequency order when tested multiple times; f i (1+ε), i=1,2,……,n, represents the upper limit of the measured frequency due to noise, or the maximum value of this frequency order during multiple tests; thus, n frequency intervals can be obtained (f1(1-ε),f1(1+ε)), (f2(1-ε),f2(1+ε)),……,(f n (1-ε),f n (1+ε)), all possible nth order frequencies that can be measured are contained within these n frequency intervals.

[0117] By marking the frequency range of these n-order frequencies on the damage frequency panorama, n damage frequency contour bands are obtained. The damage frequency contour bands appear as banded curved surfaces on the damage frequency panorama.

[0118] Projecting n strip surfaces onto the xOy plane yields the intersection region of the n projection surfaces. The intersection region is approximated as a polygon. The vertex coordinates of the polygon are read, and the damage location and degree are within these coordinate ranges.

[0119] Calculate the center point of the intersecting region, and use the horizontal and vertical coordinates of the center point to represent the location and extent of the damage.

[0120] This embodiment uses the first three frequencies to identify impairments (n=3). Each frequency may be affected by noise, and the degree of influence varies. To simulate the effect of noise on the third-order frequencies, the following method is used: Figure 3 The ε-noise cube shown is the noise hypercube of this embodiment. Figure 3In this context, the third-order frequency is represented as a point on a Cartesian coordinate system, with the horizontal, vertical, and horizontal axes as coordinates. Due to the influence of testing accuracy and environmental noise, the measured frequency value is often not the true value, but a set of approximations that are very close to the true value. In order to form an interval of approximations that includes the true value, such pseudo-random noise is constructed.

[0121] Using an ε-noise cube as pseudo-random noise, the effect of noise on frequency is simulated, transforming the true frequency of the beam into a measured frequency with noise and error. The specific process is as follows:

[0122] Transform (f1,f2,f3) into in

[0123]

[0124] i, j, and k are unit vectors in the three directions of the ε-noise cube. The magnitude of each unit vector in each direction represents the extent to which the frequency in that direction is affected by the noise. Assuming the measured frequency is affected by 100% ε-level noise, then:

[0125]

[0126] f n (1-ε), n=1,2,3 indicates that the measured frequency is less than the natural frequency of the actual beam due to the influence of noise; while f n (1+ε), n=1,2,3 indicates that the measured frequency is greater than the natural frequency of the actual beam due to noise. Therefore, three frequency intervals are obtained: (f1(1-ε), f1(1+ε)), (f2(1-ε), f2(1+ε)), and (f3(1-ε), f3(1+ε)). All possible measured first three frequencies are contained within these three intervals. Assume that multiple measurements yield multiple sets of frequency values, where the maximum and minimum values ​​of each frequency are f1(1-ε), f2(1+ε)), and f3(1+ε) respectively. i (1-ε) and f i (1+ε), then the actual measured frequency intervals are (f1(1-ε),f1(1+ε)), (f2(1-ε),f2(1+ε)), (f3(1-ε),f3(1+ε)).

[0127] By marking the frequency ranges of these three orders on the damage frequency panorama, three damage frequency contour bands are obtained. The damage frequency contour bands appear as banded curved surfaces on the damage frequency panorama.

[0128] The three strip surfaces are projected onto the xOy plane to obtain the intersection region of the three projection surfaces. The intersection region is approximated as a polygon. The vertex coordinates of the polygon are read, and the damage location and damage degree are within these coordinate ranges.

[0129] Calculate the center point of the intersecting region, and use the horizontal and vertical coordinates of the center point to represent the location and extent of the damage.

[0130] In theory, frequency can accurately identify the location and extent of damage. However, due to environmental noise and testing errors, both the overall damage frequency map and the measured frequency may contain errors. Therefore, in practice, noise and errors can lead to deviations in damage identification, or even identification failure. The following section uses pseudo-random noise to simulate the final error caused by environmental noise and testing errors, thereby studying the sensitivity between frequency and damage.

[0131] Because the damage frequency panorama characterization formula based on Timoshenko beam theory is a very complex implicit functional relationship, it is difficult to analyze the sensitivity of frequency to damage from analytical formulas, and it is also difficult to analyze the sensitivity of damage to frequency. Therefore, this embodiment studies and describes the sensitivity between frequency and damage through a damage frequency panorama. Figure 4 , Figure 5 and Figure 6 As shown, where Figure 4 (a), 5(a), and 6(a) are three-dimensional images of the panoramic view of the first three order damage frequencies, respectively. Figure 4 (b), 5(b), and 6(b) are projections of the panoramic image onto the xOy plane, with each color representing a contour band of a different frequency.

[0132] according to Figure 2 Sensitivity analysis was performed at the first-order frequency range (46.93 Hz, 58.18 Hz). Figure 4 As shown in (a), each 1Hz frequency variation is represented by a color, and this is projected onto the xOy plane. Figure 4 As shown in (b), the larger the damage and the closer the damage is to the fixed end, the greater the frequency change gradient, that is, the smaller the width of the frequency equal height band, indicating that the frequency is more sensitive to the damage at this time.

[0133] according to Figure 5 Sensitivity analysis was performed at the second-order frequencies, ranging from 312.20 Hz to 362.40 Hz. Figure 5 As shown in (a), each 5Hz frequency variation is represented by a color, and this is projected onto the xOy plane. Figure 5 As shown in (b), the greater the damage, the closer the damage is to the fixed end, or the closer the damage is to the mode equilibrium point, the greater the frequency change gradient, that is, the smaller the width of the frequency equal height band, indicating that the frequency is more sensitive to the damage at this time.

[0134] according to Figure 6 Sensitivity analysis was performed at the third-order frequency range (897.80 Hz, 1005 Hz). Figure 6As shown in (a), each 10Hz frequency variation is represented by a color, and this is projected onto the xOy plane. Figure 6 As shown in (b), the greater the damage, the closer the damage is to the fixed end, or the closer the damage is to the mode equilibrium point, the greater the frequency change gradient, that is, the smaller the width of the frequency equal height band, indicating that the frequency is more sensitive to the damage at this time.

[0135] because Figure 4 , Figure 5 and Figure 6 It can be seen that when the damage has a greater impact on the frequency, its sensitivity to the frequency also increases. Therefore, when the damage has a smaller impact on the frequency, the error of identifying the damage by frequency will be larger, and when the damage has a greater impact on the frequency, the error of identifying the damage by frequency will be smaller.

[0136] This embodiment obtains the measured frequency range of the first three frequencies through multiple tests, and marks the frequency range on the damage frequency panorama. Specific methods and steps are illustrated below:

[0137] Taking the damage location as 0.35 and the damage degree as 0.4 (both normalized values), the first three frequencies of the beam are f1 = 11.23 Hz, f2 = 71.46 Hz, and f3 = 195.36 Hz. Assuming the pseudo-random noise level is ε = 2%, the first three frequency intervals obtained from the pseudo-random noise are as follows: These three frequency ranges are assumed to be frequency ranges obtained from multiple measurements, and these three frequencies are plotted on the damage frequency panorama, resulting in three damage frequency contour bands, such as... Figure 7 As shown in the figure. Among them, the strip surface ① represents the first frequency range (11.01Hz, 11.45Hz), the strip surface ② represents the second frequency range (70.03Hz, 72.89Hz), and the strip surface ③ represents the third frequency range (191.45Hz, 199.27Hz).

[0138] Projecting the three strip surfaces onto the xOy plane yields... Figure 8 (a) The intersection of the three projection planes can be approximated as a polygon. Enlarging the polygon, we obtain... Figure 8 (b) Read the vertex coordinates of polygon ABCDE, namely A(0.295, 0.452), B(0.348, 0.490), C(0.400, 0.408), D(0.356, 0.332), and E(0.318, 0.340). The first value of these five points represents the location of the damage, the second value represents the degree of damage, and their average values ​​are respectively...

[0139]

[0140]

[0141] Where the error is

[0142]

[0143]

[0144] Therefore, it can be seen that the number of intersection points of contour bands is greater than that of frequency contour lines, and the calculation is based on more data points. Thus, using the damage frequency contour band method can effectively reduce detection errors. According to the error calculation, the error of the result in this embodiment is small and acceptable in practical engineering. Therefore, 0.343 can be taken as the damage location and 0.404 as the damage degree.

[0145] The frequency contour band method for damage identification based on the panoramic representation of damage frequency improves the one-dimensional line into a two-dimensional surface, enabling the frequency contour band method to contain more damage information. Then, the damage feature value is identified by calculating the mean. This method makes the frequency-based damage identification method achieve a certain degree of noise immunity and has good robustness.

[0146] The above embodiments, using n=3 as an example, have provided a detailed description of the implementation steps, feasibility, and beneficial effects of the present invention. Obviously, the described embodiments are merely a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

Claims

1. A method for damage identification of a beam-like structure based on frequency contour band method, characterized in that, Includes the following steps: (1) Obtain a panoramic view of damage frequency; (2) Considering the effects of noise and error, obtain the measured frequency range covering the true frequency of the damaged beam; specifically, use the noise hypercube to measure the nth order true frequency of the beam ( , ,……, ) transformed into the measured frequency with noise and error ( , ,……, The process is as follows: ； in, It is a unit for measuring the magnitude of noise impact. , ,……, It is a noise hypercube The magnitude coefficient of noise influence in each direction; the value in each direction represents the extent to which the frequency in that direction is affected by noise. Assuming the measured frequency is affected by 100% ε-level noise, then: ； This indicates the lower limit of the measured frequency due to noise, or the minimum value of that frequency order when tested multiple times. This indicates the upper limit of the measured frequency due to noise, or the maximum value of that frequency order across multiple tests; thus, n frequency intervals are obtained. ； All possible nth-order frequencies are contained within these n frequency intervals; these n frequency intervals are marked on the damage frequency panorama to obtain n damage frequency contour bands. (3) Take the n-order measured frequency intervals and mark these n-order frequency intervals on the damage frequency panorama to obtain the damage frequency contour bands. Each damage frequency contour band represents the combination of all damage locations and damage degrees corresponding to that frequency; n≥3; (4) Project the damage frequency contour bands onto the xOy plane to obtain the intersection region of n contour bands. All points within the intersection region are used to indicate the damage location and damage degree. (5) Calculate the center point of the intersecting area, and use the horizontal and vertical coordinates of the center point to represent the location and extent of the damage.

2. The method for identifying damage to beam structures according to claim 1, characterized in that, In step (1), a panoramic view of the damage frequency is obtained by using analytical formulas relating frequency to damage location and damage degree, as well as by using calculation and plotting software.

3. The beam-like structure damage identification method according to claim 2, characterized in that, In step (1), the analytical formulas for the frequency, damage location, and damage extent are obtained as follows: The beam is divided into m+1 sub-beams by m cracks. The vibration equations for each sub-beam are listed. Cracks affect the stiffness of the beam. The displacement, bending moment, and shear force at the connection of each segment of the beam are the same, only the rotation angle is different due to the presence of cracks. List the continuity condition equations for the four indicators of displacement, bending moment, shear force, and rotation angle. There are 4m continuity condition equations for m cracks, and there are 4 boundary condition equations for the entire beam. Therefore, a beam containing m cracks has a total of 4m+4 equations. By solving the simultaneous equations, the functional relationship between the location and extent of m cracks and the frequency of the beam is obtained. This functional relationship is defined as the panoramic characterization formula for the damage frequency of the beam.

4. The beam-like structure damage identification method according to claim 1, characterized in that, In step (2), the measured frequency range covering the true frequency of the damaged beam is obtained through multiple tests.

5. The beam-like structure damage identification method according to claim 1, characterized in that, In step (2), after estimating the effects of noise and error through a single test, the frequency range covering the true frequency of the damaged beam is obtained.

6. The method for identifying damage to beam structures according to claim 1, characterized in that, In step (5), the center point is calculated using the centroid method, the weighted average method, and the boundary point average method.

7. The beam-like structure damage identification method of claim 1, wherein The damage frequency contour band appears as a banded curved surface on the damage frequency panoramic view.

8. The beam-like structure damage identification method of claim 1, wherein ε≤2%.

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