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

Abstract

The invention discloses a beam structure damage identification method based on an exponential optimization modal strain energy method. The method comprises the following steps: (1) respectively obtaining multi-order vibration mode curves before and after beam structure damage through a dynamic test; (2) firstly calculating the vibration mode curvature before and after the beam structure is damaged by using a central difference method, and then adjusting the vibration mode curvature curve after the beam structure is damaged by using an optimal matching method; (3) for the statically determinate structure, the vibration mode curvature before the beam structure is damaged and the adjusted damage vibration mode curvature are subtracted, and damage positioning is carried out on the statically determinate structure through mutation synthesis of a multi-order vibration mode curvature difference curve; and (4) calculating the unit vibration mode curvature through the vibration mode curvature before the beam structure is damaged and the adjusted vibration mode curvature, and quantifying the damage degree by adopting an index optimization modal strain energy method. According to the invention, the damage position and the damage degree of the beam structure can be accurately identified; the method is suitable for different beam structure damage identification; an effective new method is provided for beam structure damage identification, and the method can be applied to bridge damage assessment.

Description

Technical Field

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

[0002] During the service life of a bridge, various factors may cause damage to the bridge structure, and may even cause catastrophic accidents. Therefore, how to predict the actual use of the bridge structure in advance, timely determine the damage location and degree of the bridge structure, and then estimate the remaining life of the bridge has very important research significance and research value. Among them, modal strain energy has been widely used in structural damage identification due to its sensitive characteristics to damage.

[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 in theory. The main problems are: most of the 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 quantitative accuracy of the damage degree 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 exponential optimization modal strain energy method with a simple algorithm, which 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 use the optimal matching method to adjust the modal curvature curve after damage. For statically indeterminate structures, record the zero point position of the modal curvature curve;

[0007] (3) The difference between the modal curvature before damage and the modal curvature after adjustment is used. 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, and quantify the degree of damage according to the unit position. For an over-determined structure, if the damage occurs at the zero position of the modal curvature curve of the first-order modal mode, 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] In the formula, 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 the j-1 measuring point to the j measuring point and the distance from the j measuring point to the j+1 measuring point. , 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] In the formula, represents the modal curvature of the beam structure before damage in the i-th mode at the j-measurement point, It represents the adjusted vibration curvature of the i-th mode of the beam structure at the j-measurement point after damage, represents the undamaged vibration shape curvature of the i-th mode of the j-th element, 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 element of the beam structure, if the rotation angle is constrained, the damage degree calculation formula is as follows:

[0021] ;

[0022] For the edge element of the beam structure, if the rotation angle is unconstrained, the damage degree calculation formula is 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 bi-square 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 start point of the next segment. The optimal matching function f(n) is obtained after fitting.

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

[0031] .

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

[0033] ;

[0034] ;

[0035] Where, DI j is the damage location index of the vibration mode curvature difference at the jth measuring point; is the vibration mode curvature of the beam structure before damage at the jth measuring point, is the adjusted vibration 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 .

[0036] Specifically, in step (3), when the beam structure is an over-determined structure, the vibration mode curve has an inflection point. The inflection point position cannot be used to locate damage and quantify the degree of damage. However, the inflection point positions of vibration mode curves of different orders are different. In this case, multiple vibration modes can be 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 as follows: using modal testing to obtain measured vibration mode curves before and after damage; using the central difference method to calculate 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 at undamaged locations; using the mutation 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 position and degree of damage 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 locating and quantitatively analyzing damage to beam structures, and can be conveniently applied to damage assessment of bridges. 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 It is a single damage working condition positioning diagram of simply supported beams 1, 2, 5, and 10 units of the present invention.

[0045] Figure 8 It is a positioning diagram of multiple damage conditions of simply supported beams 5 and 10 units of the present invention.

[0046] Fig. 9 It is a quantitative error diagram of single damage conditions of simply supported beams 1, 2, 5, and 10 units of the present invention.

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

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

[0049] Fig.12 It is a diagram of the curvature curve of the vibration modes of the three-span continuous beam of the present invention at the 1st and 2nd order.

[0050] Fig.13 It is the vibration mode curvature diagram of the 10 and 18 units of the three-span continuous beam of the present invention before and after damage (damage 60%).

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

[0052] Fig.15 It is the vibration mode curvature diagram of the three-span continuous beam of the present invention before and after damage of 10 and 18 units after optimal matching.

[0053] Fig.16 It 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] Fig.17 It is a positioning diagram of multiple damage conditions after optimal matching of 5 and 10 units of the simply supported beam of the present invention.

[0055] Fig.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] Fig.19 It is a quantitative error diagram of multiple damage conditions after optimal matching of 5 and 10 units of the simply supported beam of the present invention.

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

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

[0059] Fig. 22 It is a cantilever beam structure diagram of the present invention.

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

[0061] Fig.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] Fig.25 It 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] Fig.26 This is a positioning diagram of multiple damage conditions 1 to 5 of a cantilever beam in the second embodiment of the present invention.

[0064] Fig. 27 It is a quantitative error diagram of multiple damage conditions 1 to 5 of the cantilever beam in the second embodiment of the present invention.

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

[0066] Fig.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] Fig.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 is further described below in conjunction with 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 vibration mode curves before and after damage, use the optimal matching method to adjust the vibration mode curvature curve after damage, and record the zero point position of the vibration mode curvature curve for the statically indeterminate structure;

[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 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.

[0073] Step 4: Calculate the unit modal curvature through the modal curvature before damage and the modal curvature after adjustment, and quantify the degree of damage according to the unit position. For the statically indeterminate structure, if the damage occurs at the zero point of the modal curvature curve of the first-order modal vibration mode, use the quantitative results of the damage degree of the modal vibration mode 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 the case of small deflections of a beam subjected to a transverse load and the assumption of rigid cross-section. Figure 2 , taking the perfectly elastic Euler beam as an example to derive the modal strain energy theory.

[0076] When a completely 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 which 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 material mechanics, we know that the differential equation of the deflection curve of a simply supported beam is:

[0081] (3)

[0082] In the formula, express Position section bending moment, express Location section curvature radius.

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

[0084] (4)

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

[0086] (5)

[0087] In the formula, 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 unit, 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 unit modal strain energy expression of the j unit is:

[0089] (6)

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

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

[0092] (7)

[0093] In the formula, c0, c1, c2, c3 are unknown constants.

[0094] When the beam is bent, , the rotation 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 unit. If the above formula is expressed in matrix form, we have:

[0101] (11)

[0102] Solve equation (11) and take the lateral displacement and corners Undetermined coefficients , substituting into formula (7) we can get:

[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] In the formula,

[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] In the formula,

[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

[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 beam structure is damaged, 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] In the formula, 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] Substituting formula (23) into formula (25), we get:

[0133] (26)

[0134] Before damage location, the damage location of the structure has not been determined. Therefore, for any unit j, it is difficult to determine whether the unit is damaged. The traditional solution is to use the stiffness matrix K before damage instead of K d Therefore, the above formula can be simplified to:

[0135] (27)

[0136] From equations (14) and (20), we can know that:

[0137] (28)

[0138] The characteristic equation of the nth-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 normalized mode shape and frequency eigenvalue of the i-th order lumped mass, respectively.

[0141] Multiply the above formula on the left by , simplified to:

[0142] (30)

[0143] Since the mass is normalized, the above formula can be transformed into:

[0144] (31)

[0145] and

[0146] (32)

[0147] in, is the rate of change of the i-th order eigenvalue caused by structural damage, and n-1 is the number of units. Usually, the rate of change of the structural eigenvalue caused by damage is very small and can be ignored, so the above formula can be approximately expressed as:

[0148] (33)

[0149] Right now,

[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 unit modal strain energy in the integral form of the j element after damage is equal to the unit 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 the j-th element is obtained as follows:

[0158] (38)

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

[0160] (39)

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

[0162] (40)

[0163] In the formula, 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 the measuring points, when measuring 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 measuring point numbers). Each unit is 5 cm long and has a rectangular cross section with a cross section width of 3 cm and a cross section 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 actual situations, the damage of bridge structures is diverse, including not only the concrete spalling and steel corrosion of the structure, but also the performance degradation of the material itself, such as the reduction of elastic modulus. These damages will only cause a large change in the stiffness of the structure, but relatively little change in the quality of the structure. Therefore, in the process of finite element software simulation and calculation, the reduction of elastic modulus is usually used to assume that the structure is damaged.

[0167] Considering the different degrees of damage of simply supported beams 1, 2, 5, and 10 units, the measured concentrated mass normalized vibration mode curves before and after damage are extracted for example 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 concentrated mass normalized vibration mode curves before and after damage are extracted. First, the curvature of the measured vibration mode curve 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 normalized vibration mode of the concentrated mass, 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] In the formula, represents the i-th order mode vibration shape of the j measuring point, , l is the unit length.

[0175] 2) Extrapolate two edge measurement points

[0176] (42)

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

[0178] (43)

[0179] In the formula, 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 It can be seen that the unit modal strain energy in integral form after damage is basically the same as the unit modal strain energy in matrix form, but there is a slight error in the unit modal strain energy of two adjacent damaged units, which is caused by the different calculation methods of the two. If the beam structure is divided more densely, the difference between the two can be reduced.

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

[0182] Figure 7 , Figure 8 It is the damage localization diagram of single damage conditions 1 to 4 and multiple damage conditions 5 to 7. It can be seen from the figure that both single damage conditions and multiple damage conditions can accurately locate the damage through the difference in vibration mode curvature. However, the vibration mode curvature before and after the damage at the undamaged unit is too different, that is, the vibration mode curvature before and after the damage not only has a sudden change at the damaged position, but also does not overlap at the undamaged position.

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

[0184] Fig.10is the quantitative error diagram of multiple damage conditions 5~7. 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 of the structure, but the modal curvature difference of the undamaged unit before and after the damage is large; the quantitative effects of equations (38) and (39) are 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 errors are different under different damage conditions, which shows that different positions will have mutual influence after multiple units are damaged.

[0186] Take a three-span continuous beam as an example. Fig.11 As shown in the figure, the span of the three-span continuous beam is arranged as 1000+1500+1000mm, and a unit is divided every 100mm, with a total of 35 units and 36 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 specific parameters are the same as those of the simply supported beam.

[0187] Depend on Fig.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 to locate the damage and quantify the degree of damage. However, the inflection point positions of each order vibration mode curve are different. At this time, 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 Fig.13 It can be seen that the modal curvature curve of the three-span continuous beam after damage has 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 curvatures before and after damage at the undamaged position do not completely overlap, the quantitative results of the damaged unit are also affected.

[0189] From the calculation examples of simply supported beams and three-span continuous beams, the vibration mode curvatures before and after damage change suddenly at the damaged position, and do not completely overlap at the undamaged position, resulting in poor damage identification. Therefore, a new method is proposed here to adjust the vibration mode curvature curve after damage, solve the problem of mutual influence at different positions after multi-unit damage, and more accurately identify damage to the structure.

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

[0191] From the calculation results of the above example, it can be seen that the error of the quantitative results is large due to the obvious difference between the vibration curvature before and after damage in the undamaged position. Therefore, a new data processing method is proposed, named the optimal matching method. This method can make the vibration curvature values ​​before and after damage basically coincide in the undamaged area, so that only the vibration curvature value at the damaged position has a sudden change, while the vibration curvature value at the undamaged position remains consistent, so that the vibration curvature before and after damage basically coincides at the undamaged position, solving the problem of mutual influence at different positions after multi-unit damage. After data processing by this new optimal matching method, it is analyzed again.

[0192] Taking the case where the 10th and 18th units (mid-span units) of a three-span continuous beam are damaged by 60% 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 curvature of the vibration mode after damage Both are calculated by central difference:

[0194] (44)

[0195] In the formula, 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 the measuring points, when measuring 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 Fig.14 The first mutation point of the mutation position is taken as the end point of the previous segment, and the second mutation point is taken 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] In the formula, is the optimal matching function A certain section of the piecewise fitting, It is the piecewise measurement point number of the piecewise function.

[0201] c) Use the optimal matching function The curvature of the vibration mode after the beam structure is damaged Adjust to get the adjusted vibration mode curvature , calculated as follows:

[0202] (47)

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

[0204] 5. Damage location indicators

[0205] Take the curvature of the vibration mode before damage The curvature of the mode after adjustment with the optimal matching function The difference is defined as the damage location index. The damage location index of the vibration mode curvature difference is:

[0206] (48)

[0207] (49)

[0208] In the formula, is the damage location index of the vibration 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. 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. .

[0209] VI. Damage Quantification Method of Exponential Optimization Modal Strain Energy Method

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

[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: Fig.16 , Fig.17 ,It can be seen from the above two figures that the modal curvature processed by the optimal matching method can completely identify the damaged unit. Except for the sudden change in 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: Fig.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 working condition 1 in formula (38) is distributed in the range of -29.08%~-2.07%, the relative error of working condition 2 in formula (38) is distributed in the range of 1.9%~11.5%, the relative error of working condition 3 in formula (38) is distributed in the range of -4.24%~4.84%, and the relative error of working condition 4 in formula (38) is distributed in the range of -17%~-7.49%; the relative error of working condition 1 in formula (39) is distributed in the range of 29.08%~-2.07%, the relative error of working condition 2 in formula (39) is distributed in the range of 2.33% to 12.6%, the relative error of working condition 3 in formula (39) is distributed in the range of 2.49% to 12.7%, and the relative error of working condition 4 in formula (39) is distributed in the range of 3.62% to 12.89%.

[0213] The quantitative results of multiple damage conditions 5 to 7 are as follows: Fig.19 As shown in the figure, it can be seen that 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 5 units is small for small damage and large for large damage, and the quantitative error of 10 units 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 1 unit before and after the optimal matching are unchanged.

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

[0216] (50)

[0217] (51)

[0218] Simplifying it to:

[0219] (52)

[0220] (53)

[0221] In the formula, is the modal curvature of the i-th mode of the j-unit, which is calculated as follows:

[0222] (54)

[0223] It can be seen from the above formula that both are quantitative formulas for the unit mode curvature. The quantitative result of formula (52) is poor, while formula (53) has a good quantitative effect for small damage and a poor quantitative effect 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 that the degree of structural damage is known, the exponent n of the unit curvature is inversely calculated. Taking a simply supported beam as an example, Fig. 20 As shown in the figure, because the simply supported beam is a symmetrical structure, the 1 to 10 units on the left are taken for calculation. Assuming that the 1, 2, 5, and 10 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 index n is as follows Fig.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 is 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) 1 unit:

[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 Fig. 22 , do the same steps as above for the cantilever beam, the fitting result of the cantilever beam is as follows Fig.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 of 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 unit mode shape curvature ratio of the i-th mode of the fixed-end unit;

[0246] 2) The calculation method of 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 unit mode shape curvature ratio 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 represents 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 element of the beam structure, if the rotation angle is constrained, the damage degree calculation formula is as follows:

[0257] (69)

[0258] For the edge element of the beam structure, if the rotation angle is unconstrained, the damage degree calculation formula is as follows:

[0259] (70)

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

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

[0262] Although the formula is fitted with 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 Fig. 20 As shown in the figure, the span of the simply supported beam is 1000mm, and a unit is divided every 50mm, 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 3500kg / m 3 .

[0265] The beam3 beam element of ANSYS software is used to establish a simply supported beam model. Unit 1, unit 2, unit 5 and unit 10 are damaged to different degrees. The damage conditions are the same as those in Table 2. The specific implementation steps are as follows:

[0266] 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 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 between the modal curvature before damage and the modal curvature after adjustment, and use equations (48) and (49) to calculate the modal curvature difference index to locate the damage. The results are shown in Fig.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 an intermediate unit, use equations (60) and (61) to calculate the damage degree of the unit. If the damaged unit is an edge unit, use equations (56), (57) and (58) to calculate the damage degree of the edge unit. The results are shown in Fig.25 As can be seen from the figure, for the single damage condition of the simply supported beam, the quantitative error is 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 Fig. 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 3500kg / m 3 .

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

[0273] Table 3 Damage conditions of cantilever beam multiple units

[0274]

[0275] The specific implementation steps are as follows:

[0276] 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 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 between the modal curvature before damage and the modal curvature after adjustment, and use equations (48) and (49) to calculate the modal curvature difference index to locate the damage. The results are shown in: Fig.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 position. The method for determining the damaged unit position is as described in Example 1. The results are shown in Fig. 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 correctly quantify the damage degree of the damaged part, and the damage degree identification effect is good.

[0280] Example 3: Three-span continuous beam

[0281] like Fig.28 As shown in the figure, the span of the three-span continuous beam is 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 3500kg / m 3 .

[0282] The three-span continuous beam model was established using the beam3 beam element of ANSYS software. The damage of 1 unit, 20 units, 34 units and 47 units of the structure was set. 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 between the modal curvature of the beam structure before damage and the modal curvature after adjustment, and use equations (48) and (49) to calculate the modal curvature difference index to locate the damage. The results are shown in Fig.29, Fig.30 It can be seen that the modal curvature difference index can accurately locate the damage.

[0289] 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 position. The method for determining the damaged unit position is the same as that described in Example 1. The quantitative results are shown in Table 5 (the positions with background color in the table are inflection points):

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

[0291]

[0292] Taking working condition 1 as an example, it can be seen from the table that: unit 47 is the inflection point of the first-order vibration mode of the three-span continuous beam. Its quantitative effect and positioning effect are relatively poor at the first order, with an error of 44.381%, but the quantitative effect is better at the second order, with a quantitative result of 0.185%; unit 34 is at the inflection point of the second-order vibration mode of the three-span continuous beam, so its second-order quantitative effect is poor, with an error of 33.347%, but its first-order quantitative effect is better, with a quantitative result of 1.343%. It can be seen that the inflection point positions of different-order vibration modes of the continuous beam are different. Therefore, when quantifying the inflection point of the first-order modal vibration mode, the modal vibration mode data of other orders can be used to quantify the damage of the inflection point unit of the first-order vibration mode. In summary, the exponential optimization modal strain energy method can correctly quantify the damage of the three-span continuous beam with multiple damage working conditions, and the quantitative error is within 3.5%. The largest error is the quantitative error when the edge unit is damaged by 30%, which is 3.48%, and the damage degree identification effect is good.

[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 exponentially optimized 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 use the optimal matching method to adjust the modal curvature curve after damage. For statically indeterminate structures, record the zero point position of the modal curvature curve; (3) The difference between the modal curvature before damage and the modal curvature after adjustment is used. 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, and quantify the degree of damage according to the unit position. For an over-determined structure, if the damage occurs at the zero position of the modal curvature curve of the first-order modal mode, 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: ； In the formula, 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 the j-1 measuring point to the j measuring point and the distance from the j measuring point to the j+1 measuring point. , 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: ； ； In the formula, represents the modal curvature of the beam structure before damage in the i-th mode at the j-measurement point, It represents the adjusted vibration curvature of the i-th mode of the beam structure at the j-measurement point after damage, represents the undamaged vibration shape curvature of the i-th mode of the j-th element, 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 element of the beam structure, if the rotation angle is constrained, the damage degree calculation formula is as follows: ； For the edge element of the beam structure, if the rotation angle is unconstrained, the damage degree calculation formula is 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 based on the exponential optimization modal strain energy method according to claim 1 is characterized in that: In the 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 based on the exponential optimization modal strain energy method according to claim 1 is characterized in that: 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 bi-square 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 start point of the next segment. The optimal matching function f(n) is obtained after fitting. c) Use the optimal matching function f(n) to calculate the modal curvature of the damaged beam structure Adjust to get the adjusted vibration mode curvature , calculated as follows: 。 4. The beam structure damage identification method based on the exponential optimization modal strain energy method according to claim 1 is characterized in that: In the step (3), the damage location index of the vibration mode curvature difference is: ； ； Where, DI j is the damage location index of the vibration mode curvature difference at the jth measuring point; is the vibration mode curvature of the beam structure before damage at the jth measuring point, is the adjusted vibration 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 .

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

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