A method for identifying damage probability of a beam structure

Abstract

The application discloses a beam structure damage probability identification method, comprising the following steps: (1) obtaining a damage frequency panorama; (2) marking the damage frequency on the damage frequency panorama to obtain damage frequency contour bands / contour lines, which represent all combinations of damage positions and degrees corresponding to the frequency; (3) projecting the contour bands / contour lines to an xOy plane to form an intersection area / intersection points, and the points in the intersection area or the intersection points are used to indicate the damage positions and degrees; (4) statistically processing the horizontal and vertical coordinates of the boundary vertices of the intersection area or the intersection points to obtain mathematical expectation values representing the damage positions and degrees; (5) calculating the variances of the damage positions and degrees; (6) calculating the probability density functions of the damage positions and degrees according to the central limit theorem; and (7) calculating the possible distribution areas of the damage positions and the probability sizes thereof and the possible intervals of the damage degrees and the probability sizes thereof according to the probability density functions.

Description

Technical Field

[0001] The present invention relates to a method for identifying damage to a beam structure, and in particular to a method for identifying the probability of damage to a beam structure. Background Art

[0002] Beam structures are fundamental elements of complex engineering structures. However, over long periods of service, they are subject to the coupling effects of various loads and material degradation, inevitably causing localized damage. This damage not only compromises structural safety but also reduces durability. Therefore, developing effective scientific methods and technical means to identify damage in beam structures is a crucial prerequisite for preventing structural failure and enabling proactive maintenance and replacement. Damage identification remains a vital and enduring research topic within the engineering and academic communities both domestically and internationally.

[0003] In recent decades, damage identification of beam structures based on vibration theory has attracted considerable attention. Numerous structural damage diagnosis methods have been developed based on dynamic indices such as frequency, mode shape, or frequency response function. In contrast, frequency, due to its ease of acquisition and robustness to noise, is more suitable for damage characterization. Consequently, frequency-based damage identification methods have attracted considerable attention.

[0004] In addition, from a probabilistic perspective, for the inverse problem of damage identification, the probability of identifying the exact damage location and damage extent is 0, and only approximate damage locations and extents can be obtained. How to objectively and accurately describe the possible distribution of damage locations and extents, and at the same time clearly give the probability of damage within a certain specific range, will provide a more scientific basis for damage detection. Summary of the Invention

[0005] Purpose of the invention: In order to solve the problems existing in the prior art, the present invention provides a method for identifying the probability of damage to a beam structure. The method can use probability to describe the location and extent of the damage, and the contour band method has good noise immunity.

[0006] Technical solution: A beam structure damage probability identification method includes the following steps:

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

[0008] (2) Mark multiple damage frequencies obtained through testing or estimation on the damage frequency panorama to obtain contour bands / contour lines of damage frequencies of different orders. Each contour band / contour line of damage frequency represents the combination of all damage locations and damage degrees corresponding to the frequency.

[0009] (3) Projecting the damage frequency contour bands / contour lines onto the xOy plane to form intersection regions of different-order damage frequency contour bands / multiple intersection points of different-order damage frequency contour lines. The points or multiple intersection points within the intersection region are used to indicate the damage location and damage extent.

[0010] Projecting different order damage frequency contour bands onto the xOy plane to form an intersection region of multiple damage frequency contour bands; or projecting different order damage frequency contour lines onto the xOy plane to form a plurality of intersection points of multiple damage frequency contour lines; the point or points within the intersection region are used to indicate the damage location and damage extent;

[0011] (4) Statistically calculate the horizontal and vertical coordinates of the boundary vertices of the intersection area or multiple intersection points to obtain the mathematical expectation value μ representing the damage location and damage degree x 、μ y ;

[0012] (5) Calculate the variance of damage location and damage degree

[0013] (6) According to the central limit theorem, it is assumed that the damage location follows a normal distribution. The degree of damage follows a normal distribution Then the probability density function of the damage location f x (x) and the probability density function of the damage degree f y (y) are

[0014]

[0015]

[0016] When x<0 or x≥1 and y<0 or y≥1, the beam is considered to have no damage;

[0017] (7) Calculate the possible distribution area of ​​the damage location and its probability size based on the probability density function, and calculate the possible range of the damage degree and its probability size.

[0018] Preferably, in step (1), a damage frequency panorama is obtained by using an analytical formula of frequency, damage location, and damage degree and calculation and drawing software; the analytical formula of frequency, damage location, and damage degree is obtained by the following method:

[0019] m cracks divide the beam into m+1 sub-beams. List the vibration equations for each sub-beam.

[0020] Cracks affect the stiffness of the beam. The displacement, bending moment, and shear force at the connections of each sub-beam segment are the same; only the rotation angle varies due to the presence of cracks. The continuity equations for displacement, bending moment, shear force, and rotation are listed. For m cracks, there are 4m continuity equations, and for the entire beam, there are 4 boundary condition equations. Therefore, a beam with m cracks has a total of 4m+4 equations.

[0021] The simultaneous equations are used to solve the functional relationship between the position and degree of the m cracks and the frequency of the beam, and this functional relationship is defined as the panoramic characterization formula of the damage frequency of the beam.

[0022] Optionally, damage frequency identification of beam structures can be achieved based on contour lines:

[0023] In step (2), the measured frequencies of the multi-order damaged beam are obtained through multiple tests, where:

[0024] The first-order frequency of multiple tests is Where k1 is the number of first-order frequency tests;

[0025] The second-order frequency of multiple tests is Where k2 is the number of second-order frequency tests;

[0026] …

[0027] The nth-order frequency of multiple tests is where k n The number of n-th order frequency tests;

[0028] The n-order damage frequency obtained by the test is marked on the panoramic diagram, and each order frequency has multiple contour lines; a total of k1+k2+…+k n damage frequency contour lines;

[0029] In step (3), k1+k2+…+k n Projecting the damage frequency contour bands onto the xOy plane yields a clustered area where the contour lines intersect. All intersection points within this clustered area indicate the possible damage location and extent. The mathematical expectation of the damage location and extent can be calculated using the centroid method and the weighted mean method.

[0030] Optionally, the beam structure damage frequency can be identified based on the contour bands:

[0031] In step (2), considering the influence of noise and error, the measured frequency interval covering the true frequency of the damaged beam is obtained; the n-order measured frequency interval is taken, and this n-order frequency interval is marked on the damage frequency panorama to obtain n damage frequency contour bands, and each order damage frequency contour band represents the combination of all damage positions and damage degrees corresponding to the frequency; specifically, the measured frequency interval covering the true frequency of the damaged beam can be obtained through multiple tests; or the frequency interval covering the true frequency of the damaged beam can be obtained through a single test after estimating the influence of noise and error.

[0032] It is preferred to use the noise hypercube to transform the n-order true frequencies of the beam (f1, f2, ..., f n ) is transformed into the measured frequency with noise and error Right now:

[0033]

[0034]

[0035] …

[0036]

[0037] Among them, ε is the unit of the noise impact, k1, k2, ..., k n is the noise influence coefficient in n directions of the noise hypercube; the value in each direction indicates the degree to which the frequency in this direction is affected by the noise; assuming that the measured frequency is affected by 100% ε level noise, then:

[0038]

[0039]

[0040] …

[0041]

[0042] f i (1-ε), i = 1, 2, ..., n, represents the lower limit of the measured frequency under the influence of noise, or the minimum value of the frequency in multiple tests; f i (1+ε), i=1,2,……,n, which indicates the upper limit of the measured frequency under the influence of noise, or the maximum value of the frequency in multiple tests; thus, n frequency intervals (f1(1-ε),f1(1+ε)), (f2(1-ε),f2(1+ε)),……, (f n (1-ε),f n(1+ε)), all possible n-order frequencies are included in these n frequency intervals; the frequency intervals of these n-order frequencies are marked on the damage frequency panorama to obtain n damage frequency contour bands.

[0043] In step (3), n damage frequency contour bands are projected onto the xOy plane to obtain the intersection area of ​​the n contour bands, and all points in the intersection area are used to indicate the damage location and damage degree;

[0044] In step (4), the intersection area is approximated as a polygon, and the vertex coordinates of the polygon are read. The damage location and damage degree are within these coordinates. The center point of the intersection area is calculated, and the horizontal and vertical coordinates of the center point (μ x , μ y ) represents the mathematical expectation value of damage location and damage extent;

[0045] In step (5), the coordinates of each vertex of the polygon and the horizontal and vertical coordinates of the center point (μ x , μ y ) Calculate the variance of the damage location and the variance of injury severity The horizontal and vertical coordinates of the center point (μ x , μ y ).

[0046] Furthermore, corresponding confidence intervals are obtained based on the confidence levels of different damage locations and damage degrees; or corresponding confidence levels are calculated based on different confidence intervals.

[0047] Furthermore, the method further includes the step of calculating the joint probability density function of the damage location and the damage extent:

[0048] According to the probability statistics method, the covariance Cov(X,Y) and correlation coefficient ρ of the damage location and damage degree are obtained as follows:

[0049] Cov(X,Y)=E((X-EX)(Y-EY))

[0050]

[0051] Where X represents the horizontal coordinate sequence of the polygon vertices, and Y represents the vertical coordinate sequence of the polygon vertices. The two-dimensional joint probability density function of the damage condition is:

[0052]

[0053] 0≤x<1,0≤y<1

[0054] When x < 0 or x ≥ 1 and y < 0 or y ≥ 1, it means there is no damage on the beam;

[0055] According to the two-dimensional joint probability density function of the damage situation, its two-dimensional joint probability density distribution map is obtained; the distribution of the damage location and damage degree is obtained from the two-dimensional joint probability density distribution map.

[0056] Beneficial effects

[0057] Compared with the prior art, the present invention has the following significant improvements:

[0058] (1) The intersections of multiple projection lines / projection bands contain numerous high-precision feature points that indicate the location and extent of damage. The presence of multiple intersections / intersections provides a computational basis for the probabilistic interpretation of damage identification results. In particular, the damage probability identification method based on contour bands takes into account the influence of environmental noise and test errors, making the damage identification method more robust and noise-immune, thus meeting the needs of engineering applications.

[0059] (2) The intersection points of multiple projection lines / the intersection areas of projection bands provide the probability distribution of the damage location and damage extent on the beam, and the probability confidence level and confidence interval are used to describe the location and extent distribution of the damage, providing more scientific damage detection results in the form of probability. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 Schematic diagram of the ε-noise cube (n=3) according to Example 1 of the present invention;

[0061] Figure 2 Schematic diagram of damage frequency contour bands according to Example 1 of the present invention;

[0062] Figure 3 The damage frequency contour band projection diagram of Example 1 of the present invention, wherein (a) is the polygonal intersection area formed by the projection of three frequency contour bands, and (b) is a local magnified view of the contour band intersection area;

[0063] Figure 4 : The damage probability density distribution diagram of Example 1 of the present invention, wherein (a) is the damage location distribution probability diagram, and (b) is the damage degree distribution probability diagram;

[0064] Figure 5 A two-dimensional joint probability distribution diagram of the damage condition in Example 1 of the present invention;

[0065] Figure 6 Schematic diagram of damage frequency contours of Example 2 of the present invention (n=3);

[0066] Figure 7This is a projection diagram of the damage frequency contour line of Example 2 of the present invention, wherein (a) is a projection diagram of three frequency contour lines to form multiple intersection points, and (b) is a local enlarged diagram of the intersection points of the contour lines. DETAILED DESCRIPTION

[0067] A method for identifying beam structure damage probability comprises the following steps:

[0068] (1) Obtain a panoramic view of damage frequency.

[0069] (2) Mark multiple damage frequencies obtained through testing or estimation on the damage frequency panorama to obtain contour bands / contour lines of damage frequencies of different orders. Each contour band / contour line of damage frequency represents the combination of all damage locations and damage degrees corresponding to the frequency.

[0070] (3) Projecting different order damage frequency contour bands onto the xOy plane to form the intersection area of ​​multiple damage frequency contour bands; or projecting different order damage frequency contour lines onto the xOy plane to form the intersection points of multiple damage frequency contour lines, and the number of intersection points is multiple; the points within the intersection area formed by the contour band projection or the multiple intersection points formed by the contour line projection are used to indicate the damage location and damage degree.

[0071] (4) Statistically calculate the horizontal and vertical coordinates of the boundary vertices of the intersection area or multiple intersection points to obtain the mathematical expectation value μ representing the damage location and damage degree x 、μ y .

[0072] (5) Calculate the variance of damage location and damage degree

[0073] (6) According to the central limit theorem, it is assumed that the damage location follows a normal distribution. The degree of damage follows a normal distribution The probability density functions of damage location and damage extent are calculated separately.

[0074] (7) Calculate the possible distribution area of ​​the damage location and its probability size based on the probability density function, and calculate the possible range of the damage degree and its probability size.

[0075] In step (1), preferably, a damage frequency panorama is obtained by using an analytical formula of frequency, damage location, and damage degree and computational drawing software, wherein the computational drawing software includes but is not limited to MATLAB software and Python software. The analytical formula of frequency, damage location, and damage degree is obtained as follows:

[0076] m cracks divide the beam into m+1 sub-beams. List the vibration equations for each sub-beam.

[0077] Cracks affect the stiffness of the beam. The displacement, bending moment, and shear force at the connections of each sub-beam segment are the same; only the rotation angle varies due to the presence of cracks. The continuity equations for displacement, bending moment, shear force, and rotation are listed. For m cracks, there are 4m continuity equations, and for the entire beam, there are 4 boundary condition equations. Therefore, a beam with m cracks has a total of 4m+4 equations.

[0078] The simultaneous equations are used to solve the functional relationship between the position and degree of the m cracks and the frequency of the beam, and this functional relationship is defined as the panoramic characterization formula of the damage frequency of the beam.

[0079] In addition, a comprehensive view of damage frequencies can be obtained through conventional multiple experiments or multiple finite models.

[0080] The technical solutions of steps (2) to (6) of the present invention will be described clearly and completely below in conjunction with two specific embodiments and corresponding drawings.

[0081] Example 1 (Contour Band)

[0082] A beam structure damage probability identification method based on contour strips includes the following steps:

[0083] (1) Obtain a panoramic view of damage frequency.

[0084] (2) Considering the influence of noise and error, the measured frequency range covering the true frequency of the damaged beam is obtained. Specifically, the measured frequency range covering the true frequency of the damaged beam can be obtained through multiple tests; or the frequency range covering the true frequency of the damaged beam can be obtained after estimating the influence of noise and error through a single test.

[0085] Preferably, the noise hypercube is used to simulate the effect of noise on frequency. The nth order true frequency of the beam is defined as (f1, f2, ..., f n ), the measured frequency with noise and error is (f1 * ,f2 * ,……,f n * ), use the noise hypercube to transform the n-order real frequencies of the beam (f1, f2, ..., f n ) is transformed into the measured frequency (f1 * ,f2 * ,……,f n * ), the specific process is as follows:

[0086]

[0087] Among them, ε is the unit of the noise impact, k1, k2, ..., k nis the noise influence coefficient in n directions of the noise hypercube; the value in each direction indicates the degree to which the frequency in this direction is affected by the noise; assuming that the measured frequency is affected by 100% ε level noise, then:

[0088]

[0089] f i (1-ε), i = 1, 2, ..., n, represents the lower limit of the measured frequency under the influence of noise, or the minimum value of the frequency in multiple tests; f i (1+ε), i=1,2,……,n, which indicates the upper limit of the measured frequency under the influence of noise, or the maximum value of the frequency in multiple tests; thus, n frequency intervals (f1(1-ε),f1(1+ε)), (f2(1-ε),f2(1+ε)),……, (f n (1-ε),f n (1+ε)), all possible measured n-order frequencies are contained in this n frequency interval.

[0090] Take the nth-order measured frequency interval and mark it on the damage frequency panorama to obtain n damage frequency contour bands. Each damage frequency contour band represents the combination of all damage locations and damage degrees corresponding to the frequency. Usually, n is a positive integer greater than or equal to 3, that is, at least 3 frequencies are required to more accurately identify the damage location and damage degree.

[0091] (3) Project the n damage frequency contour bands onto the xOy plane to obtain the intersection of the n contour bands. All points within this intersection are used to indicate the damage location and damage extent. Approximate the intersection area as a polygon, and read the coordinates of the polygon vertices. The damage location and damage extent are within these coordinates.

[0092] (4) Calculate the center point of the intersection area, and use the horizontal and vertical coordinates of the center point (μ x , μ y ) represents the mathematical expectation value of the damage location and damage degree; the methods for calculating the center point include but are not limited to the centroid method, weighted mean, and boundary point mean method.

[0093] (5) The coordinates of each vertex of the polygon and the horizontal and vertical coordinates of the center point (μ x , μ y ) Calculate the variance of the damage location and the variance of injury severity

[0094] (6) According to the central limit theorem, it is assumed that the damage location follows a normal distribution. The degree of damage follows a normal distribution Then the probability density function of the damage location f x (x) and the probability density function of the damage degree f y (y) are

[0095]

[0096]

[0097] When x<0 or x≥1 and y<0 or y≥1, it is considered that there is no damage on the beam.

[0098] (7) Calculate the possible distribution area of ​​the damage location and its probability size based on the probability density function, and calculate the possible range of the damage degree and its probability size.

[0099] Based on the obtained probability density functions of damage location and damage degree, a damage probability density distribution diagram can be further obtained. The probability distribution of different damage locations and damage degrees on the beam can be obtained from the damage probability density distribution diagram.

[0100] In addition, the corresponding confidence interval can be obtained according to the confidence levels of different damage locations and damage degrees; or the corresponding confidence level can be calculated according to different confidence intervals.

[0101] Furthermore, the importance of the beams can be graded according to the stress conditions of the beam structure, and the importance of the beams can be used as the confidence level to identify damage.

[0102] Considering the correlation between damage location and damage severity, the following steps are also included to calculate the joint probability density function of damage location and damage severity:

[0103] According to the probability statistics method, the covariance Cov(X,Y) and correlation coefficient ρ of the damage location and damage degree are obtained as follows:

[0104] Cov(X,Y)=E((X-EX)(Y-EY)) (5)

[0105]

[0106] Where X represents the horizontal coordinate sequence of the polygon vertices, and Y represents the vertical coordinate sequence of the polygon vertices. The two-dimensional joint probability density function of the damage condition is:

[0107]

[0108] 0≤x<1,0≤y<1 (7)

[0109] When x<0 or x≥1 and y<0 or y≥1, it means there is no damage on the beam.

[0110] According to the two-dimensional joint probability density function of the damage situation, its two-dimensional joint probability density distribution map can be obtained; the distribution of the damage location and damage degree can be obtained from the two-dimensional joint probability density distribution map.

[0111] Taking n=3 as an example, the first three frequencies of the beam are used to identify damage. Considering that each frequency order may be affected by noise, and the degree of influence varies, in order to simulate the influence of noise on the third frequency order, Figure 1 The ε-noise cube is shown as a noise hypercube. Figure 1 In the figure, the third-order frequency is regarded as a point in the rectangular coordinate system when placed as the horizontal, vertical and vertical coordinates. Due to the influence of test accuracy and environmental noise, the measured frequency value is often not the true value, but a set of approximate values ​​that are very close to the true value. In order to make the approximate value form an interval so that this interval contains the true value, such pseudo-random noise is constructed.

[0112] The ε-noise cube is used as pseudo-random noise to simulate the effect of noise on frequency and transform the true frequency of the beam into the measured frequency with noise and error. The specific process is as follows:

[0113] Transform (f1,f2,f3) into (f1 * ,f2 * ,f3 * ),in

[0114]

[0115] i, j, and k are the unit vectors in the three directions of the ε-noise cube. The size of the unit in each direction represents the degree to which the frequency in that direction is affected by the noise. Assuming that the measured frequency is affected by 100% ε-level noise, then:

[0116]

[0117] f n (1-ε), n=1, 2, 3 means that the measured frequency is affected by noise, which makes it smaller than the natural frequency of the real beam; and f n (1+ε), n=1,2,3 means that due to the influence of noise, the measured frequency is greater than the natural frequency of the real beam. Thus, three frequency intervals are obtained (f1(1-ε), f1(1+ε)), (f2(1-ε), f2(1+ε)), (f3(1-ε), f3(1+ε)), and all possible measured first three frequencies are included in these three intervals. Assume that after multiple measurements, multiple sets of frequency values ​​are obtained, where the maximum and minimum values ​​of each frequency order are f i (1-ε) and f i(1+ε), the actual measured frequency intervals are (f1(1-ε), f1(1+ε)), (f2(1-ε), f2(1+ε)), (f3(1-ε), f3(1+ε)).

[0118] The frequency intervals of these three-order frequencies are marked on the damage frequency panorama to obtain three damage frequency contour bands, which appear as strip-shaped surfaces on the damage frequency panorama.

[0119] Project the three strip-shaped surfaces onto the xOy plane to obtain the intersection area of ​​the three projection surfaces, and approximate the intersection area as a polygon; read the vertex coordinates of the polygon, and the damage location and damage degree are within the range of these coordinates.

[0120] Calculate the center point of the intersection area, and use the horizontal and vertical coordinates of the center point to represent the location and extent of the damage.

[0121] The specific methods and steps are illustrated as follows:

[0122] When the damage position is 0.35 and the damage degree is 0.4 (both are normalized values), the first three frequencies of the beam are f1 = 11.23 Hz, f2 = 71.46 Hz, and f3 = 195.36 Hz. Assuming that the level of pseudo-random noise is ε = 2%, the first three frequency intervals obtained by pseudo-random noise are f1 * ∈(11.01Hz,11.45Hz),f2 * ∈(70.03Hz,72.89Hz),f3 * ∈(191.45Hz,199.27Hz), assume these three frequency intervals to be the frequency intervals obtained by multiple measurements, and mark these three-order frequencies on the damage frequency panorama to obtain three damage frequency contour bands, such as Figure 2 The strip surface ① represents the first-order frequency interval (11.01Hz, 11.45Hz), the strip surface ② represents the second-order frequency interval (70.03Hz, 72.89Hz), and the strip surface ③ represents the third-order frequency interval (191.45Hz, 199.27Hz).

[0123] Project the three strip surfaces onto the xOy plane and we get Figure 3 (a) The intersection of the three projection surfaces can be approximately viewed as a polygon. By enlarging the polygon, we can obtain Figure 3 (b), the vertex coordinates of polygon ABCDE can be read as: 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, and the second value represents the degree of damage. Their average values ​​are

[0124]

[0125]

[0126] The error is

[0127]

[0128]

[0129] It can be seen that the error of the results of this embodiment is small and acceptable in actual engineering. Therefore, 0.343 can be used as the damage position and 0.404 as the damage degree.

[0130] In theory, a certain damage will cause a certain frequency change, and the certain damage can also be identified through multi-order determined frequency changes. However, due to the influence of environmental noise and test errors, it is often impossible to obtain the true value of each order of frequency, that is, under different test conditions, different test equipment may obtain different frequency values, that is, each measured frequency value is a random value with a certain probability distribution around the true value, so the identified damage position and degree will also have a certain probability distribution characteristic. In fact, damage identification based on dynamics is an inverse problem of the dynamic problem. Under normal circumstances, the solution of the inverse problem is often affected by initial conditions, environmental noise, etc., and a certain deviation will appear, and this deviation will show a certain random distribution law with the randomness of the interference conditions. The beam structure damage probability identification method proposed in the present invention belongs to such an inverse problem, so the identification result will also have a certain random distribution characteristic.

[0131] The central limit theorem in probability theory shows that many random variables in reality are formed by the combined influence of a large number of independent random factors, each of which plays a small role in the overall effect. Such random variables often approximately follow a normal distribution. The location and extent of structural damage are random variables, which are affected by many factors. Assuming that the damage location and extent follow a normal distribution, regardless of whether the damage location and extent are correlated, they each follow a normal distribution with their own parameters.

[0132] From equations (8) and (9), we can see that the mathematical expectation value of the damage location is 0.343, and the mathematical expectation value of the damage degree is 0.404. Then, the variance of the damage location and damage degree is calculated from the five points ABCDE respectively.

[0133]

[0134]

[0135] The damage location follows the normal distribution N(0.343,1.27×10 -3 ), the damage degree follows the normal distribution N(0.404,3.80×10 -3 ), then the probability density function of the damage location f x (x) and the probability density function of the damage degree f y (y) are shown in formula (3) and formula (4) respectively.

[0136] Special note: The domain of the normal distribution is the entire real number domain, but when x < 0 or x ≥ 1 and y < 0 or y ≥ 1, the damage probability obtained from the probability density function can be ignored (this is because in a single sampling test, low-probability events will not occur). It can be considered that when x < 0 or x ≥ 1 and y < 0 or y ≥ 1, there is no damage on the beam.

[0137] According to the probability density function of damage location and damage degree, we can get Figure 4 The damage probability density distribution diagram shows the approximate distribution of the damage location and damage degree. This is the damage distribution diagram of the beam obtained by inverse deduction based on the first three natural frequencies of the beam; for the inverse problem, it is difficult to infer an accurate value. From Equations (14) and (15), it can be seen that the probability of any determined damage location and damage degree is zero, but the probability distribution of the damage location and damage degree on the beam can be given. It is more scientific to give the damage detection results in the form of probability. Table 1 shows the probability of damage in a certain interval and the probability of damage degree in a certain interval.

[0138]

[0139]

[0140] Table 1 Interval probability of damage location and damage degree

[0141]

[0142] As shown in Table 1, the probability of damage occurring between 0.3 and 0.4 is the highest, at 83.13%. The probability of damage occurring between 0.2 and 0.3 exceeds 10%, which is not negligible. However, after calculation, the probability of damage occurring between 0.25 and 0.4 is 94.06%, which basically confirms that the damage occurred between 0.25 and 0.4. The probability of damage occurring elsewhere is only 5.94%, and low-probability events will not occur in a single test. The probability of damage between 0.3 and 0.5 is 89.45%, so the probability of damage at other values ​​is even lower. When the damage is between 0.29 and 0.51, the calculated probability is 92.50%, while the probability of damage at other values ​​is only 7.50%.

[0143] Table 2 Calculation of damage interval based on damage confidence

[0144]

[0145]

[0146] As shown in Table 2, corresponding confidence intervals can be obtained based on different confidence levels for damage location and damage severity. For example, for unimportant beam structures, when detecting damage, the damage location and damage severity can be calculated based on a confidence level of 80% or 85%; for general beam structures, the damage location and damage severity can be calculated based on a confidence level of 90% or 95%; and for highly important structures, the damage location and severity can be calculated based on a confidence level of 99% or higher. This approach, in line with structural reliability theory, allows the reliability of bridge structures to be calculated based on probabilistic damage detection.

[0147] In actual engineering, sometimes the damage location and damage degree have a certain correlation. For example, for the same structure, when subjected to the same force, the damage location and damage degree are roughly the same. At this time, the damage location and damage degree have a strong correlation. In order to reflect the damage of the beam more scientifically and reasonably, the following two-dimensional joint probability density is used to study the joint probability density function of the damage location and damage degree. According to the probability statistics method, the covariance Cov(X,Y) and correlation coefficient ρ of the damage location and damage degree are obtained respectively.

[0148] Cov(X,Y)=E((X-EX)(Y-EY))=1.966×10 -4 (5.1)

[0149]

[0150] Then the two-dimensional joint probability density function of the damage situation is

[0151]

[0152] 0≤x<1,0≤y<1 (16)

[0153] Similarly, when x < 0 or x ≥ 1 and y < 0 or y ≥ 1, it means that there is no damage on the beam. According to the two-dimensional joint probability density function of the damage situation, its two-dimensional joint probability density distribution diagram can be obtained, as shown in Figure 5 shown.

[0154] Depend on Figure 5 The distribution of damage location and degree can be seen. According to Table 1, the probability that the damage location is between 0.3 and 0.4 and the damage degree is between 0.3 and 0.5 is 74.41% using the joint probability density, as shown in Equation (17).

[0155]

[0156] When the damage location and damage severity have a strong correlation, the distribution probability of the damage location and damage severity can be calculated using the two-dimensional joint probability.

[0157] As can be seen from Equation (16), using a two-dimensional joint probability density to calculate the probability of damage location and damage extent is relatively complex and not very convenient for engineering applications. However, using a one-dimensional probability density to calculate the damage location and extent separately is relatively simple. In this calculation case, as can be seen from Equation (6.1), the correlation between the damage location and damage extent is very low (ρ = 0.089), almost close to zero, indicating that the damage location and damage extent are almost unrelated. Therefore, the probability distribution of the two can be calculated separately using a one-dimensional probability. That is, in engineering, if it is known that the damage location and damage extent are unrelated, or if their correlation is very low after calculation, the probabilities of the two can be calculated separately using a one-dimensional probability. A simple proof is given below.

[0158] Assuming that the damage degree of the beam is known, the probability density function of the damage location can be obtained through the two-dimensional joint probability density:

[0159]

[0160] Assuming that the damage location of the beam is known, the probability density function of the damage degree can be obtained through the two-dimensional joint probability density:

[0161]

[0162] f X (x) and f Y (y) are both marginal probability density functions of the two-dimensional joint probability density function, so the following relationship exists:

[0163] f x (x) = f X (x) (20)

[0164] f y (y) = f Y (y) (21)

[0165] And the error formula is

[0166]

[0167] From Table 1 we can see that:

[0168]

[0169]

[0170] Thus we can get:

[0171]

[0172] The error is

[0173]

[0174] From this, we can see that when the correlation between the damage location and the damage degree is very small, the one-dimensional probability density function can be used to solve their distribution on the beam. At this time, the error is very small and can be ignored.

[0175] Frequency-based damage identification is an inverse problem of dynamics. Accurately determining the location and extent of damage is difficult, especially under the influence of environmental noise and test errors. Therefore, this embodiment first considers the impact of noise on frequency and proposes a noise-immune damage identification method. Secondly, probability is introduced. Building on the noise-immune contour zone damage identification method, probability density is used to describe the distribution of damage location and extent. Confidence intervals and confidence levels are used to describe the range and probability of damage location and extent, making the results of this inverse problem of damage identification more scientific.

[0176] Example 2 (contour lines)

[0177] A method for identifying beam structure damage probability based on contour lines includes the following steps:

[0178] (1) Obtain a panoramic view of damage frequency.

[0179] (2) Through multiple tests, multiple measured frequencies of the damaged beam's multi-order frequencies are obtained, where:

[0180] The first-order frequency of multiple tests is Where k1 is the number of first-order frequency tests;

[0181] The second-order frequency of multiple tests is Where k2 is the number of second-order frequency tests;

[0182] …

[0183] The nth-order frequency of multiple tests is where k n The number of n-th order frequency tests;

[0184] Mark all the frequencies of the above tests into the damage frequency panorama to obtain k1+k2+…+k n Damage frequency contour lines.

[0185] (3) k1+k2+…+k n Projecting the damage frequency contour bands onto the xOy plane yields a clustered area where the contour lines intersect. All intersection points within this area indicate possible damage locations and damage extents. The coordinates of each intersection point are read from the projection.

[0186] (4) Calculate the center point of all intersections, and use the horizontal and vertical coordinates of the center point (μ x , μ y ) represents the mathematical expectation value of the damage location and damage degree; the methods for calculating the center point include but are not limited to the center of gravity method and weighted mean.

[0187] (5) The coordinates of all intersection points and the horizontal and vertical coordinates of the center point (μ x , μ y ) Calculate the variance of the damage location and the variance of injury severity

[0188] (6) According to the central limit theorem, it is assumed that the damage location follows a normal distribution. The degree of damage follows a normal distribution According to formula (3) and formula (4), the probability density function f of the damage location is obtained x (x) and the probability density function of the damage degree f y (y), when x<0 or x≥1 and y<0 or y≥1, it is considered that there is no damage on the beam.

[0189] (7) Calculate the possible distribution area of ​​the damage location and its probability size based on the probability density function, and calculate the possible range of the damage degree and its probability size.

[0190] Based on the obtained probability density functions of damage location and damage degree, a damage probability density distribution diagram can be further obtained. The probability distribution of different damage locations and damage degrees on the beam can be obtained from the damage probability density distribution diagram.

[0191] like Figure 6 、 Figure 7The figure shows the damage frequency contour diagram and projection diagram when n = 3 and the number of tests is 2 (K1, K2, K3 = 2). By marking all the tested frequencies on the damage frequency panorama, six damage frequency contour lines are obtained, of which ④ indicates two first-order frequencies, ⑤ indicates two second-order frequencies, and ⑥ indicates two third-order frequencies.

[0192] Project the six damage frequency contour bands onto the xOy plane to obtain the clustered area where the contour lines intersect. All intersection points in this area are used to indicate the possible damage location and damage degree. Figure 7 Read out the coordinates of each intersection point.

[0193] Calculate the center point of all intersections, and use the horizontal and vertical coordinates of the center point (μ x , μ y ) represents the mathematical expectation value of the damage location and damage degree; the center point is preferably calculated by the centroid method or the weighted mean method.

[0194] The coordinates of all intersection points and the horizontal and vertical coordinates of the center point (μ x , μ y ) Calculate the variance of the damage location and the variance of injury severity

[0195] Obtain the probability density function of the damage location and damage degree, calculate the possible distribution area of ​​the damage location and its probability size based on the probability density function, and calculate the possible range of the damage degree and its probability size. The specific process of this step is the same as that of Example 1 and will not be repeated here.

[0196] The above embodiment uses n=3 as an example to illustrate the implementation steps, feasibility, and beneficial effects of the present invention. Obviously, the described embodiment is only a part of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without inventive work are within the scope of protection of the present invention.

Claims

1. A beam structure damage probability identification method, characterized in that: The steps include: (1) Obtain a panoramic view of damage frequency; (2) Mark multiple damage frequencies obtained through testing or estimation on the damage frequency panorama to obtain contour bands / contour lines of damage frequencies of different orders. Each contour band / contour line of damage frequency represents the combination of all damage locations and damage degrees corresponding to the frequency. (3) Projecting the damage frequency contour bands / contour lines onto the xOy plane to form intersection regions of different-order damage frequency contour bands / multiple intersection points of different-order damage frequency contour lines. The points or multiple intersection points within the intersection region are used to indicate the damage location and damage extent. Projecting different order damage frequency contour bands onto the xOy plane to form an intersection region of multiple damage frequency contour bands; or projecting different order damage frequency contour lines onto the xOy plane to form a plurality of intersection points of multiple damage frequency contour lines; the point or points within the intersection region are used to indicate the damage location and damage extent; (4) Statistically calculate the horizontal and vertical coordinates of the boundary vertices of the intersection area or multiple intersection points to obtain the mathematical expectation value μ representing the damage location and damage degree x 、μ y ; (5) Calculate the variance of damage location and damage degree (6) According to the central limit theorem, it is assumed that the damage location follows a normal distribution. The degree of damage follows a normal distribution Then the probability density function of the damage location f x (x) and the probability density function of the damage degree f y (y) are When x<0 or x≥1 and y<0 or y≥1, the beam is considered to have no damage; (7) Calculate the possible distribution area of ​​the damage location and its probability size based on the probability density function, and calculate the possible range of the damage degree and its probability size.

2. The beam structure damage probability identification method according to claim 1, characterized in that: In step (1), a damage frequency panorama is obtained by using an analytical formula of frequency, damage location, and damage degree and calculation and drawing software; The analytical formula of the frequency, damage location and damage degree is obtained as follows: m cracks divide the beam into m+1 sub-beams. List the vibration equations for each sub-beam. Cracks affect the stiffness of the beam. The displacement, bending moment, and shear force at the connections of each sub-beam segment are the same; only the rotation angle varies due to the presence of cracks. The continuity equations for displacement, bending moment, shear force, and rotation are listed. For m cracks, there are 4m continuity equations, and for the entire beam, there are 4 boundary condition equations. Therefore, a beam with m cracks has a total of 4m+4 equations. The simultaneous equations are used to solve the functional relationship between the position and degree of the m cracks and the frequency of the beam, and this functional relationship is defined as the panoramic characterization formula of the damage frequency of the beam.

3. The beam structure damage probability identification method according to claim 1, characterized in that: In step (2), the measured frequencies of the multi-order damaged beam are obtained through multiple tests, where: The first-order frequency of multiple tests is Where k1 is the number of first-order frequency tests; The second-order frequency of multiple tests is Where k2 is the number of second-order frequency tests; …… The nth-order frequency of multiple tests is where k n The number of n-th order frequency tests; The n-order damage frequency obtained by the test is marked on the panoramic diagram, and each order frequency has multiple contour lines; a total of k1+k2+…+k n damage frequency contour lines; In step (3), k1+k2+…+k n The damage frequency contour bands are projected onto the xOy plane to obtain the clustered area where the contour lines intersect. All intersection points in the clustered area are used to indicate the possible damage location and damage degree.

4. The beam structure damage probability identification method according to claim 3, characterized in that: The mathematical expectation values ​​of damage location and damage degree are calculated by the center of gravity method and weighted mean method.

5. The beam structure damage probability identification method according to claim 1, characterized in that: In step (2), considering the influence of noise and error, the measured frequency interval covering the true frequency of the damaged beam is obtained; the n-order measured frequency interval is taken and marked on the damage frequency panorama to obtain n damage frequency contour bands. Each order damage frequency contour band represents the combination of all damage locations and damage degrees corresponding to the frequency. In step (3), n damage frequency contour bands are projected onto the xOy plane to obtain the intersection area of ​​the n contour bands, and all points in the intersection area are used to indicate the damage location and damage degree; In step (4), the intersection area is approximated as a polygon, and the vertex coordinates of the polygon are read. The damage location and damage degree are within these coordinates. The center point of the intersection area is calculated, and the horizontal and vertical coordinates of the center point (μ x , μ y ) represents the mathematical expectation value of damage location and damage extent; In step (5), the coordinates of each vertex of the polygon and the horizontal and vertical coordinates of the center point (μ x , μ y ) Calculate the variance of the damage location and the variance of injury severity 6. The beam structure damage probability identification method according to claim 5, characterized in that: In step (2), the measured frequency interval covering the true frequency of the damaged beam is obtained through multiple tests; or the frequency interval covering the true frequency of the damaged beam is obtained after estimating the influence of noise and error through a single test.

7. The beam structure damage probability identification method according to claim 5, characterized in that: In step (2), the noise hypercube is used to transform the n-order true frequencies of the beam (f1, f2, ..., f n ) is transformed into the measured frequency with noise and error Right now: Among them, ε is the unit of the noise impact, k1, k2, ..., k n is the noise influence coefficient in n directions of the noise hypercube; the value in each direction indicates the degree to which the frequency in this direction is affected by the noise; assuming that the measured frequency is affected by 100% ε level noise, then: f i (1-ε), i = 1, 2, ..., n, represents the lower limit of the measured frequency under the influence of noise, or the minimum value of the frequency in multiple tests; f i (1+ε), i=1,2,……,n, which indicates the upper limit of the measured frequency under the influence of noise, or the maximum value of the frequency in multiple tests; thus, n frequency intervals (f1(1-ε),f1(1+ε)), (f2)1-ε),f2(1+ε)),……, (f n (1-ε),f n (1+ε)), all possible measured n-order frequencies are included in this n frequency interval; The frequency interval of the n-order frequency is marked on the damage frequency panorama to obtain n damage frequency contour bands.

8. The beam structure damage probability identification method according to claim 5, characterized in that: In step (5), the horizontal and vertical coordinates of the center point (μ x , μ y ).

9. The beam structure damage probability identification method according to claim 1, characterized in that: According to the confidence levels of different damage locations and damage degrees, the corresponding confidence intervals are obtained; or according to different confidence intervals, the corresponding confidence levels are calculated.

10. The beam structure damage probability identification method according to claim 1, characterized in that: It also includes the steps of calculating the joint probability density function of damage location and damage extent: According to the probability statistics method, the covariance Cov(X,Y) and correlation coefficient ρ of the damage location and damage degree are obtained as follows: Cov(X,Y)=E((X-EX)(Y-EY)) Where X represents the horizontal coordinate sequence of the polygon vertices, and Y represents the vertical coordinate sequence of the polygon vertices. The two-dimensional joint probability density function of the damage condition is: 0≤x<1,0≤y<1 When x < 0 or x ≥ 1 and y < 0 or y ≥ 1, it means there is no damage on the beam; According to the two-dimensional joint probability density function of the damage situation, its two-dimensional joint probability density distribution map is obtained; the distribution of the damage location and damage degree is obtained from the two-dimensional joint probability density distribution map.

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