Beam structure damage identification method based on exponential optimization modal strain energy method

Abstract

The application discloses a beam structure damage identification method of an exponential optimization modal strain energy method, and steps are as follows: (1) a plurality of mode shape curves before and after beam structure damage are obtained through dynamic testing respectively; (2) the mode shape curvature before and after beam structure damage is calculated by using a central difference method, and then the mode shape curvature curve after damage is adjusted by using an optimal matching method; (3) the mode shape curvature before beam structure damage is subtracted from the adjusted mode shape curvature after damage for a statically determinate structure, and damage positioning is carried out by using mode shape curvature difference curve mutation synthesis for a statically indeterminate structure; (4) the mode shape curvature of a unit is calculated by using the mode shape curvature before beam structure damage and the adjusted mode shape curvature, and damage degree quantification is carried out by using the exponential optimization modal strain energy method. The application can accurately identify the damage position and damage degree of a beam structure, is suitable for different beam structure damage identifications, provides an effective new method for beam structure damage identification, and can be applied to bridge damage evaluation.

Description

Technical Field

[0001] The present invention belongs to the technical field of beam structure damage detection, and in particular relates to a method for calculating theoretical damage degree using an exponentially optimized modal strain energy method. Background Art

[0002] During the service life of a bridge, various factors can cause damage to the structure, potentially leading to catastrophic accidents. Therefore, predicting the actual performance of a bridge structure, promptly determining the location and extent of damage, and ultimately estimating the remaining life of the bridge are of vital research significance and value. Modal strain energy, due to its sensitivity to damage, has been widely used in structural damage identification.

[0003] Although a series of studies have been conducted on the damage identification of beam structures using the modal strain energy method and certain achievements have been made, there are still certain problems in practical application and theory. The main problems are: most literature only describes the formula of modal strain energy, but does not explain the origin of the modal strain energy calculation formula, and there is no example to explain the difference between the modal strain energy in integral form and the modal strain energy in matrix form; small damage is taken as the research object, while large damage is rarely studied, and the accuracy of damage degree quantification is not ideal; the modal strain energy of the unit before and after damage not only produces a bulge at the damaged position but also does not overlap at the undamaged position, which not only causes a large error in damage quantification, but also leads to mutual influence between different units when multiple damages occur; therefore, the basic assumptions should be improved and optimized, and a damage identification method that can be applied to different damage degrees should be proposed. Summary of the Invention

[0004] In order to solve the above technical problems, the present invention provides a beam structure damage identification method based on an exponentially optimized modal strain energy method with a simple algorithm. The method is characterized by comprising the following steps:

[0005] (1) Vibration sensors are arranged along the span direction of the beam structure, and modal tests are performed on the beam before and after damage to obtain measured multi-order vibration mode curves;

[0006] (2) Calculate the curvature of the measured modal curves before and after damage, and adjust the modal curvature curve after damage using the optimal matching method. For statically indeterminate structures, record the zero point position of the modal curvature curve;

[0007] (3) Using the difference between the modal curvature before damage and the modal curvature after adjustment, for statically determinate structures, damage location is performed directly through the mutation of the first-order modal curvature difference curve; for statically indeterminate structures, damage location is performed through the mutation of multiple-order modal curvature difference curves;

[0008] (4) Calculate the unit modal curvature by comparing the modal curvature before damage and the modal curvature after adjustment. Quantify the degree of damage based on the unit position. For an indeterminate structure, if damage occurs at the zero point of the modal curvature curve of the first-order modal, use the quantitative results of the damage degree of the modal curves of other orders.

[0009] In step (2), the mode curvature Calculated by the central difference method:

[0010] ;

[0011] Where φ is the vibration mode, subscript j is the measuring point number, i represents the i-th order mode, ε is the average of the distance from measuring point j-1 to measuring point j and the distance from measuring point j to measuring point j+1. , n is the number of measuring points, the first measuring point is arranged at one end of the beam structure, and the nth measuring point is arranged at the other end of the beam structure. The measuring point numbers are continuous, increasing from 1 to n;

[0012] In step (4), the unit mode curvature is calculated as follows:

[0013] ;

[0014] ;

[0015] Where, It represents the modal curvature of the beam structure before damage in the i-th mode at the j-th measuring point. It represents the adjusted modal curvature of the beam structure at the i-th mode at the j-th measuring point after damage. represents the undamaged modal curvature of the i-th mode of element j, represents the damage mode curvature of the i-th mode of the j-th element, and n-1 is the number of elements;

[0016] In step (4), the damage degree is quantitatively calculated as follows:

[0017] For the middle unit of the beam structure, the damage degree calculation formula is as follows:

[0018] ;

[0019] Where n z Denotes the adjustment index of the intermediate unit, De ij is the damage degree of the i-th mode of the j-th element, represents the unit mode shape curvature ratio of the i-th mode of the j-th unit, ;

[0020] For the edge elements of beam structures, if the rotation is constrained, the damage degree is calculated as follows:

[0021] ;

[0022] For the edge elements of beam structures, if the rotation angle is unconstrained, the damage degree is calculated as follows:

[0023] ;

[0024] Where n b Denotes the adjustment index of the edge unit, De ij is the damage degree of the i-th mode of the j-th element, represents the unit mode shape curvature ratio of the i-th mode of the j-th unit, .

[0025] Specifically, in step (1), when testing the vibration mode curve, the vibration mode measurement points before and after the structure is damaged are arranged in the same position, the distance between the measurement points is as consistent as possible, and the number of measurement points is not less than 6.

[0026] Specifically, in step (2), the steps of the optimal matching method are as follows:

[0027] a) Calculate the modal curvature ratio δ, where δ is the modal curvature of the beam structure before damage. Divided by the modal curvature after damage , the calculation formula is as follows:

[0028] ;

[0029] b) Perform bisquare polynomial fitting on the calculated mode curvature ratio δ. The fitting is performed in segments based on the mutation position. The first mutation point of the mutation position is used as the end point of the previous segment, and the second mutation point is used as the starting point of the next segment. The optimal matching function f(n) is obtained through fitting.

[0030] c) Using the optimal matching function f(n) to calculate the modal curvature of the damaged beam structure Adjust to get the adjusted mode curvature , calculated as follows:

[0031] .

[0032] Specifically, in step (3), the damage location index of the mode curvature difference is:

[0033] ;

[0034] ;

[0035] Where, DI j is the damage location index of the mode curvature difference at the jth measuring point; is the modal curvature of the beam structure before damage at the jth measuring point, is the adjusted modal curvature of the beam structure after damage at the jth measuring point, n is the number of measuring points, and the measuring points at both ends of the beam are .

[0036] Specifically, in step (3), when the beam structure is an over-determined structure, there is an inflection point in the mode curve. The inflection point position cannot be used for damage location and damage degree quantification, but the inflection point positions of each order mode curve are different. In this case, multiple order modes are used for damage identification to eliminate the influence of the inflection point.

[0037] Compared with the prior art, the beneficial effects of the present invention are: using modal testing to obtain measured vibration mode curves before and after damage; using the central difference method to calculate the vibration mode curvature, and using the optimal matching method to adjust the vibration mode curvature curves before and after damage to solve the problem of non-overlap in undamaged areas; using the sudden change of the vibration mode curvature difference curve to locate damage; using the vibration mode curvature of the beam structure before damage and the vibration mode curvature after adjustment, respectively calculating the unit vibration mode curvature curve, and using the exponential optimization modal strain energy method to quantify the degree of damage; the present invention can accurately identify the damage location and degree of the beam structure; and can be applied to damage identification of different beam structures (simply supported beams, cantilever beams and three-span continuous beams, etc.), with good adaptability; it provides an effective new method for beam structure damage location and quantitative analysis, which can be conveniently applied to bridge damage assessment. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 It is a flowchart of the method of the present invention.

[0039] Figure 2 It is the Euler beam model and coordinate axis diagram of the present invention.

[0040] Figure 3 It is a schematic diagram of a simply supported beam unit of the present invention.

[0041] Figure 4 It is a diagram of a simply supported beam model of the present invention.

[0042] Figure 5 It is the unit modal strain energy in integral form and matrix form when the simply supported beam 10 units of the present invention are undamaged.

[0043] Figure 6 It is the unit modal strain energy in integral form and matrix form when 10 units of the simply supported beam of the present invention are damaged by 60%.

[0044] Figure 7 This is a single damage working condition positioning diagram of the simply supported beam 1, 2, 5, and 10 units of the present invention.

[0045] Figure 8 This is a multi-damage working condition positioning diagram of the simply supported beam 5 and 10 units of the present invention.

[0046] Figure 9 It is the quantitative error diagram of single damage condition of simply supported beam 1, 2, 5 and 10 units of the present invention.

[0047] Figure 10 It is the quantitative error diagram of multiple damage conditions of simply supported beams with 5 and 10 units in the present invention.

[0048] Figure 11 It is a structural diagram of a three-span continuous beam of the present invention.

[0049] Figure 12 It is a diagram of the curvature curve of the first and second order modes of the three-span continuous beam of the present invention.

[0050] Figure 13 These are the mode curvature diagrams of the 10th and 18th units of the three-span continuous beam of the present invention before and after damage (60% damage).

[0051] Figure 14 It is the optimal matching curve diagram of the three-span continuous beam 10 and 18 units of the present invention.

[0052] Figure 15 These are the modal curvature diagrams of the three-span continuous beam of the present invention before and after damage to 10 and 18 units after optimal matching.

[0053] Figure 16 This is a single damage working condition positioning diagram after optimal matching of 1, 2, 5, and 10 units of the simply supported beam of the present invention.

[0054] Figure 17 This is a positioning diagram of multiple damage working conditions after optimal matching of 5 and 10 units of the simply supported beam of the present invention.

[0055] Figure 18 It is a quantitative error diagram of a single damage condition after optimal matching of 1, 2, 5, and 10 units of the simply supported beam of the present invention.

[0056] Figure 19 It is a quantitative error diagram of multiple damage conditions after optimal matching of 5 and 10 elements of the simply supported beam of the present invention.

[0057] Figure 20 It is a structural diagram of the simply supported beam model of the present invention.

[0058] Figure 21 It is the fitting coefficient diagram of the simply supported beam unit of the present invention.

[0059] Figure 22 It is a structural diagram of the cantilever beam of the present invention.

[0060] Figure 23 It is the cantilever beam unit fitting coefficient diagram of the present invention.

[0061] Figure 24 This is a positioning diagram of single damage conditions 1 to 4 of a simply supported beam in Example 1 of the present invention.

[0062] Figure 25 This is a quantitative error diagram of single damage conditions 1 to 4 of a simply supported beam in Example 1 of the present invention.

[0063] Figure 26 This is a positioning diagram of multiple damage working conditions 1 to 5 of the cantilever beam in Example 2 of the present invention.

[0064] Figure 27 This is a quantitative error diagram of multiple damage conditions 1 to 5 of the cantilever beam in Example 2 of the present invention.

[0065] Figure 28 It is a structural diagram of a three-span continuous beam of the present invention.

[0066] Figure 29 This is a positioning diagram of first-order multiple damage conditions 1 to 3 of a three-span continuous beam in Example 3 of the present invention.

[0067] Figure 30 This is a positioning diagram of second-order multiple damage conditions 1 to 3 of a three-span continuous beam in Example 3 of the present invention. DETAILED DESCRIPTION

[0068] The present invention will be further described below with reference to the accompanying drawings and embodiments. When the following description refers to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements.

[0069] like Figure 1 As shown, the specific steps of the present invention are as follows:

[0070] Step 1: Arrange vibration sensors along the span direction of the beam structure, perform modal tests on the beam before and after damage, and obtain measured multi-order vibration mode curves;

[0071] Step 2: Calculate the curvature of the measured mode shape curves before and after damage, and adjust the mode shape curvature curve after damage using the optimal matching method. For statically indeterminate structures, record the zero point position of the mode shape curvature curve.

[0072] Step 3: Difference the modal curvature before damage and the modal curvature after adjustment. For statically determinate structures, damage location is performed directly through the sudden change of the first-order modal curvature difference curve. For statically indeterminate structures, damage location is performed through the sudden change of multiple-order modal curvature difference curves.

[0073] Step 4: Calculate the unit modal curvature using the modal curvature before damage and the modal curvature after adjustment, and quantify the degree of damage based on the unit position. For statically indeterminate structures, if damage occurs at the zero point of the modal curvature curve of the first-order modal, use the quantitative results of the damage degree of the modal curves of other orders.

[0074] 1. Modal Strain Energy Theory

[0075] The Euler beam is a simplification of the linear elastic theory that is applicable to beams subjected to transverse loads with small deflections and that conforms to the assumption of rigid cross-sections. Figure 2 , taking the perfectly elastic Euler beam as an example to derive the modal strain energy theory.

[0076] When a perfectly elastic Euler beam bends, ignoring the acoustic energy and heat energy generated by the deformation, the work done by the external force is All converted into bending strain energy is stored in the beam. At this point, the work done by the external force is equal to the bending strain energy:

[0077] (1)

[0078] Therefore, the bending strain energy of the beam structure is:

[0079] (2)

[0080] From the mechanics of materials, we know that the differential equation of the deflection curve of a simply supported beam is:

[0081] (3)

[0082] Where, express Position section bending moment, express Position section curvature radius.

[0083] Therefore, by substituting equation (3) into equation (2), the deflection strain energy of the simply supported beam is obtained as:

[0084] (4)

[0085] If we use the i-th order mode shape Instead of deflection , then the modal strain energy of the simply supported beam is:

[0086] (5)

[0087] Where, is the bending stiffness, For Liang Chang, is the modal curvature of the beam.

[0088] Discretize the entire beam and divide it into n units. For any j units, if Figure 3 , the starting position of the j unit on the entire beam is a, the ending position of the j unit on the entire beam is b, and the length is l, then the element modal strain energy expression of the j unit is:

[0089] (6)

[0090] Where, is the bending stiffness of element j when it is undamaged.

[0091] For plane beam elements, based on the displacement boundary conditions at the nodes, the displacement difference function is assumed to be a third-order polynomial function:

[0092] (7)

[0093] Where c0, c1, c2, c3 are unknown constants.

[0094] When the beam is bent, , the angle is:

[0095] (8)

[0096] The known unit boundary conditions are:

[0097] (9)

[0098] Substituting the above formula into formula (7) and formula (8), we have:

[0099] (10)

[0100] In the above formula, l is the length of the beam element. Expressing the above formula in matrix form, we have:

[0101] (11)

[0102] Solve equation (11) and use the lateral displacement and corners Indicates the undetermined coefficient , and put it into formula (7):

[0103] (12)

[0104] In the above formula:

[0105] (13)

[0106] in, Functions are collectively called Hermite functions.

[0107] Substituting formula (12) into formula (6) yields:

[0108] (14)

[0109] Where,

[0110] (15)

[0111] (16)

[0112] In the above formula, B is the geometric matrix, , , is the modal displacement of the jth element in the i-th mode, .

[0113] Therefore, the total modal strain energy of the beam structure is:

[0114] (17)

[0115] Where,

[0116] (18)

[0117] Assuming that the j element is damaged, the modal strain energy after damage and the element modal strain energy of the j element are expressed as follows:

[0118] (19)

[0119] (20)

[0120] 2. Modal Strain Energy Damage Index Theory

[0121] Assuming that the ratio of the modal strain energy of the j element in the a and b segments of the beam to the modal strain energy of the entire beam remains unchanged before and after the damage, we have:

[0122] (twenty one)

[0123] make , , then the above formula can be transformed into:

[0124] (twenty two)

[0125] Simplifying the above formula, we get:

[0126] (twenty three)

[0127] The element stiffness after damage is:

[0128] (twenty four)

[0129] Where, For the degree of damage, .

[0130] definition The damage index De of the unit is expressed by the change of unit stiffness. Therefore, the damage degree of the unit is:

[0131] (25)

[0132] Substitute equation (23) into equation (25) to get:

[0133] (26)

[0134] Before the damage location is determined, the damage location of the structure is not known. Therefore, for any element j, it is difficult to determine whether the element is damaged or not. The traditional solution is to use the stiffness matrix K before damage to replace K d in the calculation. Therefore, the above equation can be simplified as:

[0135] (27)

[0136] And from equations (14) and (20), we have:

[0137] (28)

[0138] And the characteristic equation of the n-order undamped free vibration is:

[0139] (29)

[0140] where K and M are the stiffness matrix and mass matrix of the structure, respectively, and are the i-th order lumped mass normalized modal shape and frequency eigenvalue, respectively.

[0141] Multiply the above equation by on the left side, and simplify to get:

[0142] (30)

[0143] Due to mass normalization, the above equation can be further changed to:

[0144] (31)

[0145] And

[0146] (32)

[0147] where is the rate of change of the i-th order eigenvalue caused by structural damage, and n-1 is the number of elements. Generally, the rate of change of the structural eigenvalue caused by damage is very small and can be ignored. Therefore, the above equation can be approximately represented as:

[0148] (33)

[0149] That is,

[0150] (34)

[0151] Therefore, the damage index of unit j is:

[0152] (35)

[0153] It has been deduced above:

[0154] (36)

[0155] Therefore, it can also be assumed that the element modal strain energy in the integral form of the j element after damage is equal to the element modal strain energy in the matrix form, which can be expressed as:

[0156] (37)

[0157] Substituting equations (36) and (37) into equation (27), the quantitative damage index of the i-th mode of element j is obtained as follows:

[0158] (38)

[0159] Substituting equations (36) and (37) into equation (35), we can obtain another form of quantitative damage index of the i-th mode of unit j:

[0160] (39)

[0161] In the above formula, the curvature of the modal vibration shape of the i-th order at the j-th measuring point is The specific calculation is:

[0162] (40)

[0163] Where, is the vibration mode, subscript j is the measurement point number, for Measuring point to The distance between the measuring points and Measuring point to The average value of the distance between measuring points, when measuring points at both ends , is the number of measuring points.

[0164] 3. Damage modal strain energy analysis of simply supported beams

[0165] Take a simply supported beam with a span of 100 cm as an example. The simply supported beam is divided into 20 units and 21 measuring points. Figure 4(The numbers in the upper circle in the figure are unit numbers, and the numbers in the lower circle are measurement point numbers). Each unit is 5 cm long and has a rectangular cross-section with a width of 3 cm and a height of 5 cm. The material density is 3500 kg / m 3 , the elastic modulus is 2.7×10 3 MPa of material.

[0166] In practice, bridge structures can suffer a variety of damage, including concrete spalling and steel corrosion, as well as degradation of the material's properties, such as a decrease in elastic modulus. These damages typically cause only significant changes in the structure's stiffness, with relatively little impact on its mass. Therefore, in finite element simulations, a decrease in elastic modulus is often used to assume structural damage.

[0167] Considering the different degrees of damage to simply supported beam elements 1, 2, 5, and 10, the measured concentrated mass normalized mode shape curves before and after damage are extracted for calculation. The damage conditions are shown in Table 1:

[0168] Table 1 Damage conditions of simply supported beams

[0169]

[0170] Taking 10 elements with 60% damage as an example, the measured lumped mass normalized mode shape curves before and after damage are extracted. First, the curvature of the measured mode shape curves is calculated using formula (40) to obtain the element modal strain energy in integral form; then the element modal strain energy in matrix form is calculated and the difference between the two is compared.

[0171] Among them, the matrix form must first be used to calculate the modal inclination angle from the measured lumped mass normalized vibration mode, and then calculate the unit modal strain energy. Specifically, the modal inclination angle is calculated as follows:

[0172] 1) First find the first-order derivative of the vibration mode

[0173] (41)

[0174] Where, represents the i-th order mode shape of the j-th measuring point, , l is the unit length.

[0175] 2) Extrapolate two edge measurement points

[0176] (42)

[0177] 3) Average the first-order derivatives of the desired vibration mode to obtain the modal inclination angle

[0178] (43)

[0179] Where, is the desired modal inclination angle, .

[0180] The element modal strain energy in integral form and matrix form before and after damage is as follows: Figure 5 、 Figure 6 As shown, from Figure 5 It can be seen that the element modal strain energy in the integral form before damage is exactly the same as the element modal strain energy in the matrix form; Figure 6 The post-damage integral modal strain energy is essentially the same as the matrix modal strain energy. However, there is a slight discrepancy between the modal strain energies of two adjacent damaged elements. This is due to the different calculation methods. This discrepancy can be minimized by dividing the beam structure more densely.

[0181] Damage identification is performed on the above working conditions, and the results are as follows:

[0182] Figure 7 、 Figure 8 These are the damage localization diagrams for single-damage conditions 1 to 4 and multiple-damage conditions 5 to 7. It can be seen from the figure that both single-damage and multiple-damage conditions can accurately locate damage through the difference in modal curvature. However, the difference between the modal curvatures before and after damage at the undamaged unit is too large, that is, the modal curvatures before and after damage not only change suddenly at the damaged location, but also do not overlap at the undamaged location.

[0183] Figure 9 This is the quantitative error diagram of single damage conditions 1 to 4. It can be seen from the figure that the quantitative relative error of condition 1 in formula (38) is distributed between -29.08% and -2.07%, the quantitative relative error of condition 2 in formula (38) is distributed between 2.2% and 11.7%, the quantitative relative error of condition 3 in formula (38) is distributed between -4.2% and 4.9%, and the quantitative relative error of condition 4 in formula (38) is distributed between -17.48% and -7.48%; the quantitative relative error of condition 1 in formula (39) is distributed between 29.08% and -2.07%, the quantitative relative error of condition 2 in formula (39) is distributed between 2.24% and 12.1%, the quantitative relative error of condition 3 in formula (39) is distributed between -4.07% and 7.29%, and the quantitative relative error of condition 4 in formula (39) is distributed between -16.7% and -1.81%.

[0184] Figure 10This is the quantitative error diagram of multiple damage conditions 5 to 7. It can be seen from the figure that the relative errors of unit 5 quantified by formula (38) are 153.2%, 18.7%, and 1.9%, respectively, and the relative errors of unit 10 quantified by formula (38) are 9.3%, 19.9%, and 80.42%, respectively; the relative errors of unit 5 quantified by formula (39) are 129.4%, 16.63%, and 3.6%, respectively, and the relative errors of unit 10 quantified by formula (39) are 5.99%, 17.82%, and 67.92%, respectively.

[0185] From the above two examples, we can see that the modal curvature difference index can identify the damage location of single damage and multiple damage in the structure, but the modal curvature difference of the undamaged unit before and after the damage is large; the quantitative effects of formula (38) and formula (39) are both poor. In the case of single damage, the quantitative error of units 1 and 10 is large when the damage is small, but small when the damage is large; the quantitative error of units 2 and 5 is small when the damage is small, but large when the damage is large; in the case of multiple damage, the quantitative error of unit 5 is large when the damage is small, but small when the damage is large, and the quantitative result of unit 10 is opposite to that of unit 5; here, units 5 and 10 suffer the same damage in single damage and multiple damage, but the damage quantitative error is different under different damage conditions, which shows that different positions will have mutual influence after the damage of multiple units.

[0186] Take a three-span continuous beam as an example. Figure 11 As shown in the figure, the spans of the three-span continuous beam are arranged as 1000+1500+1000mm, with each unit divided into 100mm intervals, for a total of 35 units and 36 measuring points (the numbers in the circle in the upper row of the figure are the unit numbers, and the numbers in the lower row are the measuring point numbers). The specific parameters are the same as those for the simply supported beam.

[0187] Depend on Figure 12 It can be seen that when the beam structure is an over-determined structure, there is an inflection point in the vibration mode curve. The inflection point position cannot be used for damage location and damage degree quantification. However, the inflection point positions of the vibration mode curves of each order are different. In this case, multi-order vibration modes can be used for damage identification to eliminate the influence of the inflection point.

[0188] Taking the 60% damage of the 10th and 18th units (mid-span units) of the three-span continuous beam as an example, the first-order vibration mode is extracted for calculation, as shown in the following example: Figure 13 It can be seen that the modal curvature curve of the three-span continuous beam after damage produces a sudden change at the damaged position, and the modal curvature curves before and after damage at the undamaged position do not completely overlap; since the modal curvature curves before and after damage at the undamaged position do not completely overlap, it also affects the quantitative results of the damaged unit.

[0189] From the calculation examples of simply supported beams and three-span continuous beams, it can be seen that the modal curvatures before and after damage produce a sudden change at the damaged location and do not completely overlap at the undamaged location, resulting in poor damage identification. Therefore, a new method is proposed here to adjust the modal curvature curve after damage to solve the problem of mutual influence at different locations after multi-unit damage, so as to more accurately identify structural damage.

[0190] 4. Damage Identification Method Based on Optimal Matching Mode Curvature

[0191] From the calculation results of the above example, it can be seen that the significant difference in the modal curvature before and after damage at the undamaged location leads to a large error in the quantitative results. Therefore, a new data processing method is proposed, named the optimal matching method. This method can make the modal curvature values ​​before and after damage basically coincide in the undamaged area, so that the modal curvature values ​​only change suddenly at the damaged location, while the modal curvature values ​​at the undamaged location remain consistent. This makes the modal curvature before and after damage basically coincide at the undamaged location, solving the problem of mutual influence between different locations after multi-unit damage. After data processing using this new optimal matching method, further analysis is carried out.

[0192] Taking the case of 60% damage to units 10 and 18 (mid-span units) of a three-span continuous beam as an example, the specific process is as follows:

[0193] a) First, calculate the modal curvature of the structure before and after damage, where the modal curvature before damage is and the modal curvature after damage Both are calculated by central difference:

[0194] (44)

[0195] Where, is the vibration mode, subscript is the measuring point number, for Measuring point to The distance between the measuring points and Measuring point to The average value of the distance between measuring points, when measuring points at both ends , n is the number of measuring points.

[0196] Then calculate the mode curvature ratio , is the modal curvature of the beam structure before damage Divided by the modal curvature after damage , the calculation formula is as follows:

[0197] (45)

[0198] b) The calculated mode curvature ratio Perform bi-square polynomial fitting, and perform segmented fitting based on the mutation position, such as Figure 14 The first mutation point of the mutation position is used as the end point of the previous segment, and the second mutation point is used as the starting point of the next segment. The optimal matching function is obtained by fitting. , Matlab software can be used to perform bisquare linear fitting.

[0199] (46)

[0200] Where, is the optimal matching function A certain section of the piecewise fitting, The segmented measurement point number of the piecewise function.

[0201] c) Using the optimal matching function Modal curvature of the damaged beam structure Adjust to get the adjusted mode curvature , calculated as follows:

[0202] (47)

[0203] like Figure 15 As shown, the mode curvature after adjustment by the optimal matching method Compared with the mode curvature before damage The undamaged part is basically overlapped, while the damaged part has a larger bulge. The curvature of the vibration mode after adjustment by the optimal matching method is Compared with the mode curvature before damage The degree of damage to the structure is quantified, thus completing the process of matching the modal curvature.

[0204] 5. Damage location indicators

[0205] Take the modal curvature before damage Mode curvature adjusted with the optimal matching function The difference is defined as the damage location index. The damage location index of the mode curvature difference is:

[0206] (48)

[0207] (49)

[0208] Where, is the damage location index of the mode curvature difference at the jth measuring point; For the The modal curvature of the beam structure before damage at the measuring point, For the The modal curvature of the beam structure at the measuring point after damage after adjustment by the optimal matching function, The first measuring point is arranged at one end of the beam structure, and the nth measuring point is arranged at the other end of the beam structure. The measuring point numbers are continuous, increasing from 1 to n. The measuring points at both ends of the beam are arranged at the same time. .

[0209] VI. Damage Quantification Method Using Exponentially Optimized Modal Strain Energy

[0210] The quantitative effect of directly using the modal strain energy index in the previous article is not good and the error is large. Therefore, the optimal matching method is proposed to correct it. The quantitative effect is shown in the figure below:

[0211] Taking the damage of a simply supported beam as an example, the damage conditions are the same as those in Table 1. The damage location results of conditions 1 to 7 are as follows: Figure 16 、 Figure 17 From the two figures above, it can be seen that the modal curvature processed by the optimal matching method can completely identify the damaged unit. Except for the sudden change at the position of the damaged unit, the difference in modal curvature before and after damage at other undamaged units is basically 0.

[0212] The damage quantitative results of single damage conditions 1 to 4 are as follows Figure 18 As shown in the figure, it can be seen that the quantitative effect of modal strain energy processed by the optimal matching method is still not ideal. The relative error of the quantitative method of working condition 1 in formula (38) is distributed between -29.08% and -2.07%, the relative error of the quantitative method of working condition 2 in formula (38) is distributed between 1.9% and 11.5%, the relative error of the quantitative method of working condition 3 in formula (38) is distributed between -4.24% and 4.84%, and the relative error of the quantitative method of working condition 4 in formula (38) is distributed between -17% and -7.49%. The relative error of the quantitative method of working condition 1 in formula (39) is distributed between 29.08% and -2.07%, the relative error of the quantitative method of working condition 2 in formula (39) is distributed between 2.33% and 12.6%, the relative error of the quantitative method of working condition 3 in formula (39) is distributed between 2.49% and 12.7%, and the relative error of the quantitative method of working condition 4 in formula (39) is distributed between 3.62% and 12.89%.

[0213] The quantitative results of multiple damage cases 5 to 7 are as follows Figure 19 As shown in the figure, the relative errors of unit 5 quantified by formula (38) are 156.3%, 18.7%, and 1.9%, respectively, and the relative errors of unit 10 quantified by formula (38) are 9.28%, 19.9%, and 80%, respectively; the relative errors of unit 5 quantified by formula (39) are 5.48%, 9.94%, and 12.65%, respectively, and the relative errors of unit 10 quantified by formula (39) are 12.8%, 10.03%, and 4.91%, respectively.

[0214] The damage location effect of the modal curvature processed by the optimal matching method is better, and the problem of mutual influence between different positions after multi-unit damage is solved; from the quantitative results, the quantitative results of formula (38) before and after the modal curvature processing are basically unchanged, and the quantitative results of formula (39) after the modal curvature processing are improved, but the overall error is still large; among them, the original quantitative error of the 5-unit is small for small damage and large for large damage, and the quantitative error of the 10-unit is large for small damage and small for large damage. Now, after the optimal matching, the quantitative errors of the intermediate units are small for small damage and large for large damage; the quantitative results of the 1-unit remain unchanged before and after the optimal matching.

[0215] Therefore, based on the optimal matching process, the quantitative accuracy of modal strain energy is analyzed. The quantitative index of modal strain energy is known to be:

[0216] (50)

[0217] (51)

[0218] Simplifying it to:

[0219] (52)

[0220] (53)

[0221] Where, is the modal curvature of the i-th mode of element j, which is calculated as follows:

[0222] (54)

[0223] As can be seen from the above formula, both are quantitative formulas for the unit mode curvature. The quantitative result of formula (52) is poor, while the quantitative effect of formula (53) is good for small damage, but poor for large damage. Therefore, it is assumed that the quantitative index is related to the exponent n of the unit curvature, that is,

[0224] (55)

[0225] Assuming the degree of structural damage is known, inversely calculate the exponent n of the unit curvature. Taking a simply supported beam as an example, Figure 20 As shown in the figure, since the simply supported beam is a symmetrical structure, the 1st to 10th units on the left are taken for calculation. Assuming that the 1st, 2nd, 5th, and 10th units of the simply supported beam have different degrees of damage, the specific damage degrees are shown in Table 2:

[0226] Table 2 Single damage condition table of simply supported beam

[0227]

[0228] The inverse exponent n is as follows Figure 21 As shown in the figure above, we can see that the index n is a curve related to the unit curvature ratio. The index n of the middle unit and the edge unit are different. Therefore, the quantitative calculation of the simply supported beam can be divided into edge units and middle units for calculation.

[0229] 1) The damage degree calculation method of the edge element of the beam structure is:

[0230] a) Unit 1:

[0231] (56)

[0232] (57)

[0233] Where n b is the edge element adjustment coefficient, De i1 is the damage level of 1 unit, represents the unit mode shape curvature of the i-th mode of unit 1;

[0234] b) n-1 unit:

[0235] ; (58)

[0236] ; (59)

[0237] 2) The damage degree calculation method of the intermediate unit is:

[0238] ; (60)

[0239] ; (61)

[0240] Where n z Denotes the adjustment coefficient of the intermediate unit, De ij is the damage degree of unit j, , represents the unit mode shape curvature ratio of the i-th mode of the j-th unit, .

[0241] like Figure 22 , do the same steps as above for the cantilever beam, the cantilever beam fitting result is as follows Figure 23 As shown in the figure above, it can be seen that the fitting curves of the fixed end 1 unit and the middle unit are basically the same, the coefficient of the free end 20 unit is different from that of other units, and the fitting curves of the cantilever beam and the simply supported beam are exactly the same, so the quantitative formula of the cantilever beam is:

[0242] 1) The calculation method for the damage degree of the fixed support end of the beam structure is:

[0243] (62)

[0244] (63)

[0245] Where n g is the edge element adjustment coefficient, De ig is the damage degree of the fixed support end, represents the curvature ratio of the unit mode shape of the i-th mode of the fixed end unit;

[0246] 2) The calculation method for the damage degree of the cantilever end of the beam structure is:

[0247] ;(64)

[0248] ; (65)

[0249] Where n x is the edge element adjustment coefficient, De ix is the damage degree of the cantilever end, represents the curvature ratio of the unit mode shape of the i-th mode of the cantilever end unit;

[0250] 3) The damage degree calculation method of the intermediate unit is:

[0251] ; (66)

[0252] ; (67)

[0253] Where n z Denotes the adjustment coefficient of the intermediate unit, De ij Indicates the damage degree of the intermediate unit, .

[0254] Therefore, for the middle unit of the beam structure, the damage degree calculation formula is as follows:

[0255] (68)

[0256] For the edge elements of beam structures, if the rotation is constrained, the damage degree is calculated as follows:

[0257] (69)

[0258] For the edge elements of beam structures, if the rotation angle is unconstrained, the damage degree is calculated as follows:

[0259] (70)

[0260] From the above formula, we can see that for edge elements with no rotational constraints, when the mode curvature ratio is very large, that is, when the damage is very small, the exponent n is infinitely close to 3. Therefore, whether in the quantitative process of the original formula or in the quantitative process after the optimization of the optimal matching method, when the exponent n is 2, the quantitative error of small damage of the edge element is particularly large, while the quantitative effect of large damage is better. Therefore, the exponent n obtained by reverse calculation is related to the mode curvature ratio. The related quadratic function is that when the exponent n increases, the mode curvature ratio increases, and the damage decreases. When the damage is infinitely small, the exponent n is infinitely close to 3 but not equal to 3.

[0261] For the elements in the beam structure, when the mode curvature ratio is very large, that is, when the damage is very small, the exponent n is infinitely close to 2; therefore, in the quantitative formula after the optimal match, when the exponent n is 2, the effect of small damage in the middle element is particularly good, but the effect of large damage is not good. Therefore, the exponent n obtained by inverse calculation is the ratio of the mode curvature ratio. The related linear function is that when the exponent n increases, the mode curvature ratio increases, and the damage decreases. When the damage is infinitely small, the exponent n is infinitely close to 2 but not equal to 2.

[0262] Although the formula is fitted based on statically determinate simply supported beams and cantilever beams, the formula is also applicable to statically indeterminate structures through the optimal matching method. The unit is an edge unit, and other units are calculated as intermediate units.

[0263] Example 1: Simply supported beam

[0264] like Figure 20 As shown in the figure, the span of the simply supported beam is 1000 mm, and each unit is divided into 50 mm intervals, with a total of 20 units and 21 measuring points (the numbers in the upper circle in the figure are the unit numbers, and the numbers in the lower circle are the measuring point numbers). The cross-sectional dimensions of the simply supported beam are b×h=30mm×50mm, and the elastic modulus of the material is 2.7×10 3 MPa, Poisson's ratio is 0.37, and density is 3500 kg / m 3 .

[0265] The simply supported beam model was established using the beam3 beam element in ANSYS software. The damage conditions for 1, 2, 5, and 10 elements were the same as those in Table 2. The specific implementation steps are as follows:

[0266] Step 1: Arrange vibration sensors along the span of the beam structure, perform modal tests on the beam before and after damage, and obtain measured vibration mode curves;

[0267] Step 2: Calculate the modal curvature from the modal shapes before and after damage according to formula (40), and then use formulas (45) to (47) to perform optimal matching on the modal curvature curve after damage;

[0268] Step 3: Difference the modal curvature before damage and the modal curvature after adjustment, and calculate the modal curvature difference index using formulas (48) and (49) to locate the damage. The results are shown in Figure 24 , it can be seen that for the single damage condition of simply supported beam, the modal curvature difference index can correctly locate the damage.

[0269] Step 4: Calculate the damage degree by using the modal curvature of the beam structure before damage and the modal curvature after adjustment. First, determine the damage location of the unit. If the damaged unit is the middle unit, use equations (60) and (61) to calculate the damage degree of the unit. If the damaged unit is the edge unit, use equations (56), (57) and (58) to calculate the damage degree of the edge unit. The results are shown in Figure 25 As can be seen from the figure, for the single damage condition of the simply supported beam, the quantitative errors are all within 1.5%, and the maximum error is -1.44%. Therefore, the damage quantitative indicators of the exponential optimization modal strain energy method can correctly quantify the damage, and the damage degree identification effect is good.

[0270] Example 2: Cantilever beam

[0271] like Figure 22 As shown in the figure, the span of the cantilever beam is 1000mm, and each unit is divided into 50mm intervals, with a total of 20 units and 21 measuring points (the numbers in the upper circle in the figure are the unit numbers, and the numbers in the lower circle are the measuring point numbers). The cross-sectional dimensions of the cantilever beam are b×h=30mm×50mm, and the elastic modulus of the material is 2.7×10 3 MPa, Poisson's ratio is 0.37, and density is 3500 kg / m 3 .

[0272] The cantilever beam model was established using ANSYS software beam3 beam elements. Taking multiple damage conditions as an example, the edge element (1 element) and the mid-span element (10 elements) were simultaneously damaged to varying degrees. The damage conditions are shown in Table 3:

[0273] Table 3 Damage conditions of cantilever beam with multiple elements

[0274]

[0275] The specific implementation steps are as follows:

[0276] Step 1: Arrange vibration sensors along the span of the beam structure, perform modal tests on the beam before and after damage, and obtain measured vibration mode curves;

[0277] Step 2: Calculate the modal curvature from the modal shapes before and after damage according to formula (40), and then use formulas (45) to (47) to perform optimal matching on the modal curvature curve after damage;

[0278] Step 3: Difference the modal curvature before damage and the modal curvature after adjustment, and calculate the modal curvature difference index using formulas (48) and (49) to locate the damage. The results are shown in the table below. Figure 26 It can be seen that for multiple damage conditions of cantilever beams, the modal curvature difference index can correctly locate the damage.

[0279] Step 4: Calculate the damage degree by using the modal curvature of the beam structure before damage and the modal curvature after adjustment. First, determine the damaged unit location. The method for determining the damaged unit location is as described in Example 1. The results are shown in Figure 27 As can be seen from the figure, for the cantilever beam with multiple damage conditions, the absolute value of the quantitative error is basically within 1.3%, and the maximum error is 1.2%. Therefore, the damage quantification index of the exponential optimization modal strain energy method can accurately quantify the damage degree of the damaged parts, and the damage degree identification effect is good.

[0280] Example 3: Three-span continuous beam

[0281] like Figure 28 As shown in the figure, the span of the three-span continuous beam is arranged as 1000+1500+1000mm, and each unit is divided into 50mm intervals, with a total of 70 units and 71 measuring points (the numbers in the upper circle in the figure are the unit numbers, and the numbers in the lower circle are the measuring point numbers). The cross-sectional dimensions are b×h=30mm×50mm, and the elastic modulus of the material is 2.7×10 3 MPa, Poisson's ratio is 0.37, and density is 3500 kg / m 3 .

[0282] A three-span continuous beam model was constructed using ANSYS Beam3 beam elements. Damage conditions were set for 1 element, 20 elements, 34 elements, and 47 elements. The specific damage conditions are shown in Table 4.

[0283] Table 4 Damage conditions of three-span continuous beams

[0284]

[0285] The specific implementation steps are as follows:

[0286] Step 1: Arrange vibration sensors along the span direction of the beam structure, perform modal tests on the beam before and after damage, and obtain measured multi-order vibration mode curves;

[0287] Step 2: Calculate the modal curvature from the modal shapes before and after damage according to formula (40), and then use formulas (45) to (47) to perform optimal matching on the modal curvature curve after damage;

[0288] Step 3: Difference the modal curvature of the beam structure before damage and the modal curvature after adjustment, and calculate the modal curvature difference index using equations (48) and (49) to locate the damage. The results are shown in Figure 29、 Figure 30 It can be seen that the modal curvature difference index can accurately locate damage.

[0289] Step 4: Calculate the damage degree using the modal curvature of the beam structure before damage and the modal curvature after adjustment. First, determine the damaged unit location. The method for determining the damaged unit location is the same as that described in Example 1. The quantitative results are shown in Table 5 (the background color in the table indicates the inflection point location):

[0290] Table 5 Quantitative error of three-span continuous beam

[0291]

[0292] Taking Case 1 as an example, the table shows that element 47 is located at the inflection point of the first-order mode of the three-span continuous beam. Its quantification and location performance at the first-order mode is relatively poor, with an error of 44.381%. However, its quantification performance at the second-order mode is relatively good, with a quantification result of 0.185%. Element 34 is located at the inflection point of the second-order mode of the three-span continuous beam. Therefore, its second-order quantification performance is poor, with an error of 33.347%, but its first-order quantification performance is relatively good, with a quantification result of 1.343%. This indicates that the inflection points of different-order modes of continuous beams are located at different locations. Therefore, when quantifying the inflection point of the first-order mode, data from other-order modes can be used to quantify damage at the inflection point of the first-order mode. In summary, the exponentially optimized modal strain energy method accurately quantifies damage for multiple damage cases of the three-span continuous beam, with a quantification error within 3.5%. The largest error is 3.48% when the edge element is 30% damaged, demonstrating excellent damage identification.

[0293] The above are only three embodiments of the present invention. All equivalent changes and modifications made according to the scope of the patent application of the present invention are within the scope of the present invention.

Claims

1. A beam structure damage identification method based on exponential optimization modal strain energy method, characterized in that: The steps include: (1) Vibration sensors are arranged along the span direction of the beam structure, and modal tests are performed on the beam before and after damage to obtain measured multi-order vibration mode curves; (2) Calculate the curvature of the measured modal curves before and after damage, and adjust the modal curvature curve after damage using the optimal matching method. For statically indeterminate structures, record the zero point position of the modal curvature curve; (3) Using the difference between the modal curvature before damage and the modal curvature after adjustment, for statically determinate structures, damage location is performed directly through the mutation of the first-order modal curvature difference curve; for statically indeterminate structures, damage location is performed through the mutation of multiple-order modal curvature difference curves; (4) Calculate the unit modal curvature by comparing the modal curvature before damage and the modal curvature after adjustment. Quantify the degree of damage based on the unit position. For an indeterminate structure, if damage occurs at the zero point of the modal curvature curve of the first-order modal, use the quantitative results of the damage degree of the modal curves of other orders. In step (2), the mode curvature Calculated by the central difference method: ； Where φ is the vibration mode, subscript j is the measuring point number, i represents the i-th order mode, ε is the average of the distance from measuring point j-1 to measuring point j and the distance from measuring point j to measuring point j+1. , n is the number of measuring points, the first measuring point is arranged at one end of the beam structure, and the nth measuring point is arranged at the other end of the beam structure. The measuring point numbers are continuous, increasing from 1 to n; In step (4), the unit mode curvature is calculated as follows: ； ； Where, It represents the modal curvature of the beam structure before damage in the i-th mode at the j-th measuring point. It represents the adjusted modal curvature of the beam structure at the i-th mode at the j-th measuring point after damage. represents the undamaged modal curvature of the i-th mode of element j, represents the damage mode curvature of the i-th mode of the j-th element, and n-1 is the number of elements; In step (4), the damage degree is quantitatively calculated as follows: For the middle unit of the beam structure, the damage degree calculation formula is as follows: ； Where n z Denotes the adjustment index of the intermediate unit, De ij is the damage degree of the i-th mode of the j-th element, represents the unit mode shape curvature ratio of the i-th mode of the j-th unit, ; For the edge elements of beam structures, if the rotation is constrained, the damage degree is calculated as follows: ； For the edge elements of beam structures, if the rotation angle is unconstrained, the damage degree is calculated as follows: ； Where n b Denotes the adjustment index of the edge unit, De ij is the damage degree of the i-th mode of the j-th element, represents the unit mode shape curvature ratio of the i-th mode of the j-th unit, .

2. The beam structure damage identification method using the exponentially optimized modal strain energy method according to claim 1 is characterized in that: In step (1), when testing the vibration mode curve, the vibration mode measurement points before and after the structure is damaged are arranged in the same position, the distance between the measurement points is as consistent as possible, and the number of measurement points is not less than 6.

3. The beam structure damage identification method using the exponentially optimized modal strain energy method according to claim 1 is characterized by: In step (2), the steps of the optimal matching method are as follows: a) Calculate the modal curvature ratio δ, where δ is the modal curvature of the beam structure before damage. divided by the modal curvature after damage , the calculation formula is as follows: ； b) Perform bisquare polynomial fitting on the calculated mode curvature ratio δ. The fitting is performed in segments based on the mutation position. The first mutation point of the mutation position is used as the end point of the previous segment, and the second mutation point is used as the starting point of the next segment. The optimal matching function f(n) is obtained through fitting. c) Using the optimal matching function f(n) to calculate the modal curvature of the damaged beam structure Adjust to get the adjusted mode curvature , calculated as follows: 。 4. The beam structure damage identification method using the exponentially optimized modal strain energy method according to claim 1 is characterized in that: In step (3), the damage location index of the mode curvature difference is: ； ； Where, DI j is the damage location index of the mode curvature difference at the jth measuring point; is the modal curvature of the beam structure before damage at the jth measuring point, is the adjusted modal curvature of the beam structure after damage at the jth measuring point, n is the number of measuring points, and the measuring points at both ends of the beam are .

5. The beam structure damage identification method using the exponential optimization modal strain energy method according to claim 1 is characterized in that: In step (3), when the beam structure is an over-determined structure, there is an inflection point in the mode curve. The inflection point position cannot be used for damage location and damage degree quantification. However, the inflection point positions of the mode curves of each order are different. In this case, multiple-order mode curves are used for damage identification to eliminate the influence of the inflection point.

Images