Method for calculating local equivalent stiffness of beam structure with frequency equivalence

Abstract

This invention discloses a method for calculating the local equivalent stiffness of a beam structure based on frequency equivalence. The steps are as follows: Based on the length d of the damaged region of the beam structure, set an appropriate length ε of the locally damaged beam segment; calculate the equivalent stiffness coefficient c. eq Calculate the equivalent stiffness of the locally damaged beam segment, where equivalent stiffness = c. eq × Bending stiffness of the undamaged beam section; c for locally damaged beam segments, excluding those near the zero point of the modal curvature curve. eq It can be calculated using an approximate method; for each location of the beam structure, c eq Based on the modal curvature curves of the beam structure before damage, strain energy methods can be used to calculate the curvature for simply supported beams, cantilever beams, and statically indeterminate beams. For single-span beams with uniform cross-sections, theoretical modal curvature curves are used. For other structures, the modal curvature curves are obtained through finite element model calculations or actual measurements, and the curvature is calculated using the central difference method. This invention proposes a concise method for calculating the equivalent stiffness between local damage measurement points, providing a theoretical basis for calculating the actual damage degree when conducting quantitative damage tests based on dynamic indicators.

Description

Technical Field

[0001] This invention belongs to the field of structural health monitoring and relates to a method for calculating the theoretical damage degree of beam structures. Specifically, it relates to a method for calculating the local equivalent stiffness of beam structures based on frequency equivalence. Background Technology

[0002] In recent years, the number of old bridges in my country has been increasing, and the problems they present have become increasingly prominent. Damage to beam structures typically manifests as localized damage such as cracking and concrete crushing. However, the spacing between measuring points is usually fixed during damage identification. When damage is detected in the structure, it is likely that localized damage occurs between two measuring points. In this case, what is the equivalent damage level between the two measuring points? This question is crucial for the reasonable interpretation of quantitative damage index results. Due to the difficulty in quantifying damage level, there are few reports in the literature on experimental verification. This method proposes a simple method for calculating the equivalent stiffness between measuring points of localized damage based on the frequency equivalence of beam structures, providing a theoretical basis for calculating the actual damage level when conducting quantitative damage tests based on dynamic indices. Summary of the Invention

[0003] To address the problem of calculating the dynamic equivalent stiffness of beam structures under local damage, this invention proposes a frequency-equivalent method for calculating the local equivalent stiffness of beam structures.

[0004] The method for calculating the local equivalent stiffness of a beam structure based on frequency equivalence described in this invention comprises the following steps:

[0005] (1) Set an appropriate length ε for locally damaged beam segments based on the length d of the damaged area of ​​the beam structure;

[0006] (2) Calculate the equivalent stiffness coefficient c eq ;

[0007] (3) Calculate the equivalent stiffness of the locally damaged beam segment, equivalent stiffness = c eq × Bending stiffness of the section of the undamaged beam;

[0008] Specifically, in step (2), the equivalent stiffness coefficient c eq Calculate using the following method:

[0009] 1) For locally damaged beam segments, excluding those near the zero point of the modal curvature curve, the equivalent stiffness coefficient is approximately calculated using the following method:

[0010] The distance between adjacent zero points on the curvature curve of the beam structure is L. c0 , for the middle L c0 The equivalent stiffness calculation method for locally damaged beam segments within a range of / 3 is as follows:

[0011]

[0012] Among them, ceq EI is the equivalent stiffness coefficient. The origin is at the left end of the beam, the x-axis is along the beam length and points to the right end of the beam, z is the distance of the locally damaged beam segment from the origin, and ε is the length of the locally damaged beam segment. The number of locally damaged beam segments can be calculated using ε and the beam span. u (x) represents the flexural stiffness of the section at position x of the undamaged beam, EI d (x) represents the flexural stiffness of the section at position x of the locally damaged beam segment within the range [z, z+ε]; when it is a beam with a uniform cross-section, EI u (x) is a constant EI, and the equivalent stiffness coefficient can be simplified to the following formula:

[0013]

[0014] Among them, EI di To divide a locally damaged beam segment of length ε into equal segments num, the flexural stiffness of the i-th segment is calculated; 2) For each location of the beam structure, the equivalent stiffness coefficient is calculated more accurately using the following method:

[0015]

[0016] Where, φ n "" represents the curvature curve of the nth mode when the beam structure is undamaged, and exp is an exponent related to the structure type; a) Simply supported beam with uniform cross-section

[0017] Frequency equation:

[0018] sin(a n L) = 0;

[0019] Where L is the span of the beam;

[0020] The solution is:

[0021]

[0022] The frequency solution is:

[0023]

[0024] Where m is the linear density of the beam with uniform cross-section:

[0025] Mode shape function:

[0026]

[0027] Modal curvature curve function and exponent exp:

[0028]

[0029] exp = 2;

[0030] b) Cantilever beam with uniform cross-section

[0031] Frequency equation:

[0032]

[0033] The solution is:

[0034]

[0035] The frequency solution is:

[0036]

[0037] Mode shape function:

[0038]

[0039] Modal curvature curve function and exponent exp:

[0040]

[0041] exp = 2;

[0042] c) Fixed beam

[0043] Frequency equation:

[0044] 1+cosh 2 (a n L)-2cos(a n L)cosh(a n L)-sinh 2 (a n L) = 0; the solution is:

[0045]

[0046] The frequency solution is:

[0047]

[0048] Mode shape function:

[0049]

[0050] Modal curvature curve function and exponent exp:

[0051]

[0052] d) Fixed-supported-hinged beams with uniform cross-section

[0053] Frequency equation:

[0054] cos(a n L)sinh(a n L)-cosh(an L)sin(a n L) = 0;

[0055] The solution is:

[0056]

[0057] The frequency solution is:

[0058]

[0059] Mode shape function:

[0060]

[0061] Mode curvature curve function:

[0062]

[0063] e) Fixed-supported-directionally-supported beam with uniform cross-section

[0064] Frequency equation:

[0065] sin(a n L)cosh(a n L)+sinh(a n L)cos(a n L) = 0;

[0066] The solution is:

[0067]

[0068] The frequency solution is:

[0069]

[0070] Mode shape function:

[0071]

[0072] Modal curvature curve function and exponent exp:

[0073]

[0074] exp = 2.5;

[0075] f) For other beam structures, the mode shape curves can be calculated by finite element model or obtained by actual measurement. The mode shape curvature curve is calculated by the central difference method. For the measured mode shape curvature curve, a continuous function is used for fitting to eliminate the influence of damage and noise. The fitted mode shape curvature curve is used for calculation.

[0076] g) For statically indeterminate beam structures, the locally damaged beam segment within 2L / 3 of the fixed support end or directional support end has an exp = 2.5, and other locally damaged beam segments have an exp = 2.

[0077] Specifically, in step (2), for simply supported beams and cantilever beams, when calculating the local equivalent stiffness of the first mode, the number of locally damaged beam segments is not less than 1, and when calculating the nth mode, the number of locally damaged beam segments is not less than 4(n-1); for statically indeterminate beams, when calculating the local equivalent stiffness, the number of locally damaged beam segments in each span is not less than 4n.

[0078] Specifically, in step (2), the integral can be calculated using mathematical software such as MATLAB.

[0079] This invention focuses on the equivalent stiffness of local damage in beam structures and proposes a method for calculating the local equivalent stiffness of beam structures based on frequency equivalence. Through numerical examples of simply supported beams, cantilever beams, fixed beams, fixed-hinged beams, and three-span continuous beams, the effectiveness of the frequency equivalence-based method for calculating the local equivalent stiffness of beam structures is verified, providing a theoretical basis for calculating the actual damage degree when conducting quantitative tests on damage degree based on dynamic indicators. Attached Figure Description

[0080] Figure 1 This is a schematic diagram of the equivalent stiffness of a simply supported beam segment with local damage, as per the present invention.

[0081] Figure 2 This is a model diagram of a simply supported beam according to Embodiment 1 of the present invention.

[0082] Figure 3 This is the first-order vibration mode c of the simply supported beam in Embodiment 1 of the present invention. eq value.

[0083] Figure 4 This is the second-order vibration mode c of the simply supported beam in Embodiment 1 of the present invention. eq value.

[0084] Figure 5 This is the third-order vibration mode c of the simply supported beam in Embodiment 1 of the present invention. eq value.

[0085] Figure 6 This is a graph showing the frequency variation of local damage in a simply supported beam according to Embodiment 1 of the present invention.

[0086] Figure 7 This is a diagram showing the first-order frequency error of the equivalent stiffness of a simply supported beam (20 beam segments) according to Embodiment 1 of the present invention.

[0087] Figure 8 This is a diagram showing the second and third order frequency errors of the equivalent stiffness of a simply supported beam in Embodiment 1 of the present invention (20 beam segments).

[0088] Figure 9This is a diagram showing the 4th and 5th order frequency error of the equivalent stiffness of a simply supported beam in Embodiment 1 of the present invention (20 beam segments).

[0089] Figure 10 This is a diagram showing the first-order frequency error of the equivalent stiffness of a simply supported beam (10 beam segments) according to Embodiment 1 of the present invention.

[0090] Figure 11 This is a diagram showing the second and third order frequency errors of the equivalent stiffness of a simply supported beam in Embodiment 1 of the present invention (10 beam segments).

[0091] Figure 12 This is a frequency error diagram of the equivalent stiffness of a simply supported beam in Embodiment 1 of the present invention, from the first to the third order (4 beam segments).

[0092] Figure 13 This is a frequency error diagram of the equivalent stiffness of a simply supported beam in Embodiment 1 of the present invention, from the first to the third order (one beam segment).

[0093] Figure 14 This is the first-order vibration mode c of the cantilever beam in Embodiment 2 of the present invention. eq value.

[0094] Figure 15 This is the second-order vibration mode c of the cantilever beam in Embodiment 2 of the present invention. eq value.

[0095] Figure 16 This is the third-order vibration mode c of the cantilever beam in Embodiment 2 of the present invention. eq value.

[0096] Figure 17 This is a graph showing the frequency variation of local damage to a cantilever beam in Embodiment 2 of the present invention.

[0097] Figure 18 This is the first-order frequency error diagram of the equivalent stiffness of the cantilever beam in Embodiment 2 of the present invention (20 beam segments).

[0098] Figure 19 This is a diagram showing the second and third order frequency errors of the equivalent stiffness of the cantilever beam in Embodiment 2 of the present invention (20 beam segments).

[0099] Figure 20 This is a diagram showing the 4th and 5th order frequency error of the equivalent stiffness of the cantilever beam in Embodiment 2 of the present invention (20 beam segments).

[0100] Figure 21 This is the first-order frequency error diagram of the equivalent stiffness of the cantilever beam in Embodiment 2 of the present invention (10 beam segments).

[0101] Figure 22 This is a diagram showing the second and third order frequency errors of the equivalent stiffness of the cantilever beam in Embodiment 2 of the present invention (10 beam segments).

[0102] Figure 23This is a frequency error diagram of the equivalent stiffness of the cantilever beam in Embodiment 2 of the present invention, from the first to the third order (4 beam segments).

[0103] Figure 24 This is a frequency error diagram of the first to third order equivalent stiffness of the cantilever beam in Embodiment 2 of the present invention (one beam segment).

[0104] Figure 25 This is the first-order vibration mode c of the fixed beam in Embodiment 3 of the present invention. eq value.

[0105] Figure 26 This is a graph showing the frequency variation of local damage to the fixed beam in Embodiment 3 of the present invention.

[0106] Figure 27 This is a frequency error diagram of the first to third order equivalent stiffness of the fixed beam in Embodiment 3 of the present invention (20 beam segments).

[0107] Figure 28 This is a diagram showing the 4th and 5th order frequency error of the equivalent stiffness of the fixed beam in Embodiment 3 of the present invention (20 beam segments).

[0108] Figure 29 This is a frequency error diagram of the equivalent stiffness of the fixed beam in Embodiment 3 of the present invention, from the first to the fourth order (10 beam segments).

[0109] Figure 30 This is a frequency error diagram of the first to third order equivalent stiffness of the fixed beam in Embodiment 3 of the present invention (4 beam segments).

[0110] Figure 31 This is a frequency error diagram of the first to third order equivalent stiffness of the fixed beam in Embodiment 3 of the present invention (one beam segment).

[0111] Figure 32 This is the first-order vibration mode c of the fixed-hinged beam in Embodiment 4 of the present invention. eq value.

[0112] Figure 33 This is a graph showing the frequency variation of local damage to a fixed-hinged beam according to Embodiment 4 of the present invention.

[0113] Figure 34 This is the first-order frequency error diagram of the equivalent stiffness of the fixed-hinged beam with exp=2 and 2.5 in Embodiment 4 of the present invention (20 beam segments).

[0114] Figure 35 This is the first-order frequency error diagram (20 beam segments) of the fixed-hinged beam with equivalent stiffness exp=2.5 and 3 in Embodiment 4 of the present invention.

[0115] Figure 36 This is a frequency error diagram of the equivalent stiffness of the fixed-hinged beam in Embodiment 4 of the present invention, from the first to the third order (20 beam segments).

[0116] Figure 37This is a diagram showing the 4th and 5th order frequency error of the equivalent stiffness of the fixed-hinged beam in Embodiment 4 of the present invention (20 beam segments).

[0117] Figure 38 This is a frequency error diagram of the equivalent stiffness of the fixed-hinged beam in Embodiment 4 of the present invention, from the first to the fourth order (10 beam segments).

[0118] Figure 39 This is a frequency error diagram of the equivalent stiffness of the fixed-hinged beam in Embodiment 4 of the present invention, from the first to the third order (4 beam segments).

[0119] Figure 40 This is a frequency error diagram of the equivalent stiffness of the fixed-hinged beam in Embodiment 4 of the present invention, from the first to the third order (one beam segment).

[0120] Figure 41 This is a model diagram of a three-span continuous beam according to Embodiment 5 of the present invention.

[0121] Figure 42 This is the damage vibration mode diagram of the left 1 / 3 of the three-span continuous beam segment 5 in Embodiment 5 of the present invention.

[0122] Figure 43 This is the damage mode curvature curve of the left 1 / 3 of the three-span continuous beam segment 5 in Embodiment 5 of the present invention.

[0123] Figure 44 This is a curve fitting diagram of the first-order vibration mode curvature of the left 1 / 3 damaged beam segment 5 of the three-span continuous beam in Embodiment 5 of the present invention.

[0124] Figure 45 This is a curve fitting diagram of the second-order vibration mode curvature of the left 1 / 3 damaged beam segment 5 of the three-span continuous beam according to Embodiment 5 of the present invention.

[0125] Figure 46 This is the damage mode curvature curve of the left 1 / 3 of the three-span continuous beam segment 15 in Embodiment 5 of the present invention.

[0126] Figure 47 This is a curve fitting diagram of the first-order vibration mode curvature of the left 1 / 3 of the three-span continuous beam segment 15 in Embodiment 5 of the present invention.

[0127] Figure 48 This is a curve fitting diagram of the second-order vibration mode curvature of the left 1 / 3 of the three-span continuous beam segment 15 in Embodiment 5 of the present invention.

[0128] Figure 49 This is the damage mode curvature curve of the right 1 / 3 of the three-span continuous beam segment in Embodiment 5 of the present invention.

[0129] Figure 50 This is a curve fitting diagram of the first-order vibration mode curvature of the right 1 / 3 damaged beam segment 30 of the three-span continuous beam in Embodiment 5 of the present invention. Detailed Implementation

[0130] 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.

[0131] Figure 1 This is a schematic diagram illustrating the equivalent stiffness of a locally damaged segment of a simply supported beam according to the present invention. In the diagram, L represents the beam length, the origin is at the left end of the beam, the x-axis is along the beam length and points to the right end of the beam, z is the distance of the locally damaged segment from the origin, ε is the length of the locally damaged segment, d is the actual length of the damaged segment, b is the distance of the actual damaged segment from the left measuring point, and EI... d EI represents the stiffness of the actual damaged beam segment. eq For the equivalent stiffness of the beam segment between measuring points, φ 1d ,φ 2d These are the first and second mode shapes under the damage state, f 1d ,f 2d These are the first and second order frequencies of the damage state, φ. 1eq ,φ 2eq These are the first and second mode shapes of the equivalent model, f 1eq ,f 2eq These are the first and second order frequencies of the equivalent model, f. id ,f ieq These are the frequency of the i-th order damage state and the equivalent model frequency, respectively.

[0132] The method for calculating the local equivalent stiffness of a beam structure based on frequency equivalence as described in this invention comprises the following specific steps:

[0133] Step 1: Based on the length d of the damaged area of ​​the beam structure, set an appropriate length ε for the locally damaged beam segment;

[0134] Step 2: Calculate the equivalent stiffness coefficient c eq ;

[0135] Step 3: Calculate the equivalent stiffness of the locally damaged beam segment. Equivalent stiffness = c eq × Bending stiffness of the section of the undamaged beam;

[0136] Specifically, in step 2, the equivalent stiffness coefficient c eq Calculate using the following method:

[0137] 1) For locally damaged beam segments, excluding those near the zero point of the modal curvature curve, the equivalent stiffness coefficient is approximately calculated using the following method:

[0138] The distance between adjacent zero points on the curvature curve of the beam structure is L. c0 , for the middle L c0 The equivalent stiffness calculation method for locally damaged beam segments within a range of / 3 is as follows:

[0139]

[0140] Among them, c eq c is the equivalent stiffness coefficient. eq EI u (x) represents the equivalent stiffness, with the origin at the left end of the beam. The x-axis runs along the beam's length, pointing towards the right end. z is the distance of the locally damaged beam segment from the origin, and ε is the length of the locally damaged beam segment. EI u (x) represents the flexural stiffness of the section at position x of the undamaged beam, EI d (x) represents the flexural stiffness of the section at position x of the locally damaged beam segment within the range of [z, z+ε].

[0141] When it is a beam with a uniform cross-section, EI u (x) is a constant EI, and the equivalent stiffness EI eq It can be simplified to the following formula for calculation:

[0142]

[0143] Among them, EI di To divide a locally damaged beam segment of length ε into equal segments num, the flexural stiffness of the i-th segment is given.

[0144] 2) The equivalent stiffness coefficient at each location of the beam structure is calculated more accurately using the following method:

[0145]

[0146] Where, φ n "" represents the curvature curve of the nth mode when the beam structure is undamaged, and exp is an exponent related to the structure type.

[0147] a) Simply supported beam with uniform cross-section

[0148] The flexibility matrix of the first-order mode structure is:

[0149]

[0150] Where ω1 is the first-order angular frequency, φ1=[φ 11 φ 12 …φ 1k ] T Let be the first-order mode shape vector, and k be the number of measurement points in the mode shape. The deflection curve w can be obtained by multiplying the compliance matrix by the load column vector, with P = [10…0]. T For example:

[0151]

[0152] For a specific beam structure with uniform cross-section, the frequency is

[0153]

[0154] In the formula, C i It is a constant related to the structure type and order. Substituting into equation (5), we get:

[0155]

[0156] The formula for calculating strain energy is as follows:

[0157]

[0158] Substituting equation (7) into the above equation, we get:

[0159]

[0160] Generalizing the above equation and removing the constants, we get:

[0161]

[0162] Where: φ n "" represents the curvature curve of the nth mode when the beam structure is undamaged, and exp is an exponent related to the structure type.

[0163] The equivalent stiffness can be calculated by using the method of equal strain energy, and equation (3) can be obtained.

[0164] Frequency equation for a simply supported beam:

[0165]

[0166] Where L is the span of the beam.

[0167] The solution is:

[0168]

[0169] The frequency solution is:

[0170]

[0171] Where m is the linear density of a beam with a uniform cross-section.

[0172] Mode shape function:

[0173]

[0174] Modal curvature curve function and exponent exp:

[0175]

[0176] exp=2 (16)

[0177] b) Cantilever beam with uniform cross-section

[0178] Frequency equation:

[0179]

[0180] The solution is:

[0181]

[0182] The frequency solution is:

[0183]

[0184] Mode shape function:

[0185]

[0186] Modal curvature curve function and exponent exp:

[0187]

[0188] exp=2 (22)

[0189] c) Fixed beam

[0190] Frequency equation:

[0191] 1+cosh 2 (a n L)-2cos(a n L)cosh(a n L)-sinh 2 (a n L)=0 (23) The solution is:

[0192]

[0193] The frequency solution is:

[0194]

[0195] Mode shape function:

[0196]

[0197] Modal curvature curve function and exponent exp:

[0198]

[0199] exp = 2.5 (28)

[0200] d) Fixed-supported-hinged beams with uniform cross-section

[0201] Frequency equation:

[0202] cos(a n L)sinh(a n L)-cosh(an L)sin(a n L)=0 (29)

[0203] The solution is:

[0204]

[0205] The frequency solution is:

[0206]

[0207] Mode shape function:

[0208]

[0209] Mode curvature curve function:

[0210]

[0211] e) Fixed-supported-directionally-supported beam with uniform cross-section

[0212] Frequency equation:

[0213] sin(a n L)cosh(a n L)+sinh(a n L)cos(a n L)=0 (34)

[0214] The solution is:

[0215]

[0216] The frequency solution is:

[0217]

[0218] Mode shape function:

[0219]

[0220] Modal curvature curve function and exponent exp:

[0221]

[0222] exp = 2.5 (39)

[0223] f) For other beam structures, when the theoretical mode shape curve cannot be obtained, the mode shape curve can be calculated by the finite element model or obtained by actual measurement. The mode shape curvature curve is calculated by the central difference method. The measured mode shape curvature curve is fitted with a continuous function to eliminate the influence of damage and noise. The fitted mode shape curvature curve is then used for calculation.

[0224] g) For statically indeterminate beam structures, the locally damaged beam segment within 2L / 3 of the fixed support end or directional support end has an exp = 2.5, and other locally damaged beam segments have an exp = 2.

[0225] In step 2, for simply supported beams and cantilever beams, when calculating the local equivalent stiffness of the first mode, the number of locally damaged beam segments is not less than 1, and for the nth mode, the number of locally damaged beam segments is not less than 4(n-1); for statically indeterminate beams, when calculating the local equivalent stiffness, the number of locally damaged beam segments in each span is not less than 4n.

[0226] In step 2, the integral can be calculated using mathematical software such as MATLAB.

[0227] Example 1: A simply supported beam with a span of 10m, cross-sectional dimensions b×h=30cm×50cm, C40 concrete, and an elastic modulus E=3.25×10 4 MPa, Poisson's ratio is 0.2, and density is 25 kN / m³ 3 A model of a locally damaged beam segment, 0.5m long, divided into three equal segments, is shown below. Figure 2 The stiffness of the locally damaged area decreased uniformly to 0.1EI.

[0228] From step 1, the length of the damaged area of ​​the structure is d = 0.5 / 3m, and the length of the locally damaged beam segment is ε = 0.5m. When damaging each segment one by one, 60 equivalent stiffness coefficients c can be obtained. eq .

[0229] c in the approximation method in step 2 eq The calculation is as follows:

[0230]

[0231] The exact calculation method in step 2, c eq The calculation was performed using a programmed method, and the c values ​​for the first three vibration modes were calculated. eq Each as Figures 3-5 The peak positions in the figure represent the zero points of the mode curvature. As can be seen from the figure, at the midpoints of each peak, the c calculated in step 2... eq The values ​​are basically consistent, therefore, for the middle third of the peak value range, the approximate method in step 2 can be used to calculate c. eq value.

[0232] Each localized damage area is damaged one by one, resulting in a decrease in structural frequency, such as Figure 6 The first-order frequency changes the most, and the maximum changes in each order frequency range from -8.8% to -12.3%.

[0233] When the locally damaged beam segment is divided into 20 segments, each segment is 0.5m long (containing 3 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figures 7-9 As can be seen, the first-order frequency error is almost zero, the second-order is within 0.2%, and the third to fifth orders show a slight increase, all within 1%. Table 1 shows the influence of different exponents exp on the frequency error. It is evident that the frequency error is minimized when exp = 2; decreasing or increasing exp increases the frequency error. Therefore, exp = 2 is chosen for the simply supported beam.

[0234] When the locally damaged beam segment is divided into 10 segments, each segment is 1m long (containing 6 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figure 10 , Figure 11 As can be seen, the first-order frequency error is almost zero, the second-order is within 0.8%, and the third-order is slightly greater than 1%. The frequency error is higher than that of 20 beam segments.

[0235] When the locally damaged beam segment is divided into 4 segments, each segment is 2.5m long (containing 15 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figure 12 As can be seen, the first-order frequency error is still almost zero, the second-order error has two slightly greater than 1%, and the third-order error has five greater than 1%, indicating a further increase in frequency error.

[0236] Table 1. Comparison of frequency errors between equivalent stiffness beams with different exponents (exp) and actual damaged beams.

[0237]

[0238] Note: exp = 0 means that c is calculated using the approximate method in step 2. eq Value; a / b, where a represents the maximum frequency error % (reflecting the local maximum frequency error), and b represents the sum of squares of all frequency error % (reflecting the average frequency error).

[0239] When the locally damaged beam segment is divided into one segment, each segment is 10m long (containing 60 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figure 13 As can be seen, the error of the first-order frequency is still almost zero, the error of the second-order frequency is slightly close to 2% in two places, and the error of the third-order frequency is greater than 2.5% in three places. Except for the first-order frequency, the errors of the other orders are significantly larger. Therefore, the number of locally damaged beam segments has a certain impact on the results. It is recommended that the number of beam segments with second-order and higher vibration modes be divided into not less than 4(n-1).

[0240] Step 3: Equivalent stiffness = c eq EI.

[0241] Example 2: A 10m span cantilever beam with the fixed support end on the left and other parameters the same as a simply supported beam.

[0242] c calculated for the first 3 modeseq Each as Figures 14-16 The peak positions in the figure represent the zero points of the mode shape curvature. It can be seen from the figure that near each peak, the c calculated in step 2... eq The differences are significant, therefore, except for the 1 / 3 range near the peak, the approximate method in step 2 can be used to calculate c. eq value.

[0243] Each localized damage area is damaged one by one, resulting in a decrease in structural frequency, such as Figure 17 The first-order frequency changes the most, and the maximum changes in each order frequency range from -10.7% to -20.8%.

[0244] When the locally damaged beam segment is divided into 20 segments, each segment is 0.5m long (containing 3 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figures 18-20 As can be seen, the first-order frequency error is almost zero, the second-order is within 0.7%, and the third to fifth orders show a slight increase, all within 1.2%. Table 2 shows the influence of different exponents exp on the frequency error. It is evident that the frequency error is minimized when exp = 2; decreasing or increasing exp increases the frequency error. Therefore, exp = 2 is chosen for the cantilever beam.

[0245] Table 2 Comparison of frequency errors between equivalent stiffness beams with different exponents (exp) and actual damaged beams

[0246]

[0247] Note: exp = 0 means that c is calculated using the approximate method in step 2. eq Value; a / b, where a represents the maximum frequency error % (reflecting the local maximum frequency error), and b represents the sum of squares of all frequency error % (reflecting the average frequency error).

[0248] When the locally damaged beam segment is divided into 10 segments, each segment is 1m long (containing 6 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figure 21 , Figure 22 It can be seen that the first-order frequency error is almost zero, while the second and third-order errors are not significantly different, and are all within 1% except for a few points. The frequency error is higher than that of 20 beam segments.

[0249] When the locally damaged beam segment is divided into 4 segments, each segment is 2.5m long (containing 15 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figure 23 As can be seen, the first-order frequency error is still almost zero, the second-order error has two points slightly greater than 1%, and the error near the fixed end of the third-order error is significantly larger.

[0250] When the locally damaged beam segment is divided into one segment, each segment is 10m long (containing 60 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figure 24 As can be seen, the error at the first-order frequency is still almost zero, while the errors near the fixed supports at the second and third orders are significantly larger. Therefore, the number of locally damaged beam segments has a certain impact on the results. It is recommended that the number of beam segments with second-order and higher vibration modes be divided into no less than 4(n-1).

[0251] Example 3: A 10m span fixed-support beam with other parameters the same as a simply supported beam.

[0252] c for first-order mode shape calculation eq Each as Figure 25 The peak positions in the figure represent the zero points of the mode shape curvature. It can be seen from the figure that near each peak, the c calculated in step 2... eq The differences are significant, therefore, except for the 1 / 3 range near the peak, the approximate method in step 2 can be used to calculate c. eq value.

[0253] Each localized damage area is damaged one by one, resulting in a decrease in structural frequency, such as Figure 26 The first-order frequency changes the most, and the maximum changes in each order frequency range from -8.3% to -14.5%.

[0254] Table 3 shows the influence of different exponents exp on frequency error. It can be seen that when the exponent exp = 2.5 or 3, the frequency error is significantly reduced. Considering the cases of dividing the beam into 10 segments and 4 segments, exp = 2.5 is taken for the fixed beam.

[0255] When the locally damaged beam segment is divided into 20 segments, each segment is 0.5m long (containing 3 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figure 27 , Figure 28 As can be seen, the frequency errors of the 1st to 3rd orders are slightly larger at the extremes, and are all within 0.5% in the middle. The 4th and 5th orders show a slight increase, but are all within 0.8%.

[0256] When the locally damaged beam segment is divided into 10 segments, each segment is 1m long (containing 6 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figure 29 As can be seen, the frequency errors of the first to third orders are all within 1.5%, while the frequency error of the fourth order is significantly larger near the end.

[0257] Table 3 Comparison of frequency errors between equivalent stiffness beams with different exponents (exp) and actual damaged beams

[0258]

[0259] Note: a / b, where a represents the maximum frequency error % (reflecting the local maximum frequency error), and b represents the sum of squares of all frequency error % (reflecting the average frequency error).

[0260] When the locally damaged beam segment is divided into 4 segments, each segment is 2.5m long (containing 15 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figure 30 As can be seen, the errors of the first-order frequencies are all less than 0.4%, the errors of the second-order frequencies are larger at the ends, and the errors of the third-order frequencies are larger at most positions.

[0261] When the locally damaged beam segment is divided into one segment, each segment is 10m long (containing 60 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figure 31 As can be seen, the error at the first-order frequency end is relatively large, reaching a maximum of 10%, and the errors at the second and third-order frequencies are significantly larger, indicating that even when divided into a single beam segment, the first-order vibration mode cannot be calculated accurately. Therefore, for fixed-support beams, it is recommended that the number of beam segments be no less than 4n.

[0262] Example 4: A 10m span fixed-hinged beam with the fixed end on the right and the hinged end on the left. Other parameters are the same as those of a simply supported beam.

[0263] c for first-order mode shape calculation eq Each as Figure 32 The peak positions in the figure represent the zero points of the mode shape curvature. It can be seen from the figure that near each peak, the c calculated in step 2... eq The differences are significant, therefore, except for the 1 / 3 range near the peak, the approximate method in step 2 can be used to calculate c. eq value.

[0264] Each localized damage area is damaged one by one, resulting in a decrease in structural frequency, such as Figure 33 The first-order frequency changes the most, and the maximum changes in each order frequency range from -8.6% to -15.6%.

[0265] The influence of different exponents (exp) on frequency error is analyzed in Table 4. It can be seen that the frequency error is smallest when exp = 2.5, and also relatively small when exp = 2 and 3. The frequency comparison for each exponent at the first mode shape is shown in Table 4. Figure 34 , Figure 35 It can be seen that the frequency error at each position where exp=3 is generally greater than that at exp=2.5. When exp=2, the frequency error near the hinged end is significantly smaller than that at exp=2.5. Considering that the boundary of the simply supported beam is the hinged end, exp is taken as 2. Therefore, for a fixed-hinged beam, the range of 2L / 3 from the fixed end is taken as exp=2.5, and the range of L / 3 from the hinged end is taken as exp=2.

[0266] Table 4 Comparison of frequency errors between equivalent stiffness beams with different exponents (exp) and actual damaged beams

[0267]

[0268] Note: a / b, where a represents the maximum frequency error % (reflecting the local maximum frequency error), and b represents the sum of squares of all frequency error % (reflecting the average frequency error).

[0269] When the locally damaged beam segment is divided into 20 segments, each segment is 0.5m long (containing 3 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figure 36 , Figure 37 It can be seen that the frequency errors of the 1st to 3rd orders are slightly larger at the extremes, and are all within 0.5% in the middle. The 4th and 5th orders show a slight increase, but are all within 1%.

[0270] When the locally damaged beam segment is divided into 10 segments, each segment is 1m long (containing 6 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figure 38 As can be seen, the frequency errors of the first to third orders are all within 1.5%, while the frequency error of the fourth order is significantly larger near the end.

[0271] When the locally damaged beam segment is divided into 4 segments, each segment is 2.5m long (containing 15 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figure 39 It can be seen that the error of the first-order frequency is less than 0.4% except for a few points at the ends which are around 1%. The error of the second-order frequency is larger at the ends, and the error of the third-order frequency is larger at most positions.

[0272] When the locally damaged beam segment is divided into one segment, each segment is 10m long (containing 60 locally damaged areas). The error between the frequency calculated from the equivalent stiffness and the actual frequency of the locally damaged area is as follows: Figure 40 As can be seen, the errors at each frequency order are quite large, indicating that even when divided into a single beam segment, the first-order vibration mode cannot be calculated accurately. Therefore, for fixed-hinged beams, it is recommended that the number of beam segments be no less than 4n.

[0273] Example 5: A 3×10m three-span continuous beam, divided into segments of 0.5m each, with other parameters the same as a simply supported beam. The model is as follows. Figure 41 .

[0274] The analysis covers three working conditions: damage to the left 1 / 3 area of ​​beam segment 5, damage to the left 1 / 3 area of ​​beam segment 15, and damage to the right 1 / 3 area of ​​beam segment 30.

[0275] When the stiffness of the left 1 / 3 region of beam segment 5 decreases uniformly to 0.1EI, the measured first and second order vibration mode curves are as follows: Figure 42The finite difference method is used to determine the mode curvature curve, such as... Figure 43 Remove the curvature values ​​at the damaged location and perform local fitting on the remaining curve, such as... Figure 44 , Figure 45 The equation for the curvature curve of the undamaged mode shape can be obtained as follows:

[0276] φ1″=4.259×10 -4 x 2 -4.414×10 -3 x + 5.656 × 10 -4 (41)

[0277] φ2″=6.963×10 -4 x 2 -6.100×10 -3 x + 5.267 × 10 -4 (42)

[0278] The equivalent stiffness coefficient is calculated using the following formula:

[0279]

[0280] c can be obtained eq(n＝1) =0.2736, c eq(n＝2) =0.2708, when the exponent is 2.5, the result is c. eq(n＝1) =0.2800, c eq(n＝2) =0.2764, c calculated using the approximation method in step 2. eq(1) =0.25, the frequency errors calculated for each equivalent stiffness coefficient are shown in Table 5. It can be seen that when exp = 2, the errors for the first and second orders of frequency are the smallest, at -0.008% and -0.016% respectively, almost zero. When exp = 2.5, the frequency error increases slightly. eq(1) The frequency error is relatively the largest at this point; therefore, when high precision is not required, c can be used. eq(1) When a precise value is needed, c is calculated using exp=2. eq value.

[0281] Table 5 Comparison of equivalent stiffness coefficient and frequency error in the left 1 / 3 region of beam segment 5

[0282]

[0283] When the stiffness of the left 1 / 3 region of beam segment 15 decreases uniformly to 0.1EI, the curvature curves of the first and second order measured vibration modes are obtained using the finite difference method, as follows: Figure 46 Remove the curvature values ​​at the damaged location and perform local fitting on the remaining curve, such as... Figure 47 , Figure 48 The equation for the curvature curve of the undamaged mode shape can be obtained as follows:

[0284] φ1″=-4.049×10 -4 x 2 +3.630×10 -3 x + 1.835 × 10 -3 (44)

[0285] φ2″=-6.398×10 -4 x 2 +4.157×10 -3 x + 7.966 × 10 -3 (45)

[0286] The equivalent stiffness coefficient is calculated using the following formula:

[0287]

[0288] The calculations of the equivalent stiffness coefficients and frequency errors are shown in Table 6. It can be seen that when exp = 2, the errors for the first and second orders of frequency are the smallest, at 0.009% and 0.008% respectively, almost zero. When exp = 2.5, the frequency error increases slightly. eq(1) =0.25 and the first-order result c eq(n＝1) =0.2312 has little correlation with the second-order result c eq(n＝2) =0.1945 is significantly different because beam segment 15 is close to the zero point of the second mode curvature.

[0289] Table 6 Comparison of equivalent stiffness coefficient and frequency error in the left 1 / 3 region of beam segment 15

[0290]

[0291] When the stiffness of the right 1 / 3 region of beam segment 30 uniformly decreases to 0.1EI, the curvature curves of the first and second order measured vibration modes are obtained using the finite difference method, such as... Figure 49 As can be seen, the damage location is at the zero point of the second-order mode curvature curve, therefore the second-order mode curvature curve shows almost no abrupt change. Removing the curvature value at the damage location and performing local fitting on the remaining curve, as shown... Figure 50 The equation for the curvature curve of the undamaged mode shape can be obtained as follows:

[0292] φ1″=5.101×10 -4 x 2 -5.094×10 -3 x + 3.654 × 10 -3 (47)

[0293] The equivalent stiffness coefficient is calculated using the following formula:

[0294]

[0295] The calculations of the equivalent stiffness coefficients and frequency errors are shown in Table 7. It can be seen that the second-order frequency hardly changes after damage. Therefore, at the zero point of the mode shape curvature curve, the degree of damage does not affect the frequency value. For the first-order mode shape, the values ​​of each c... eq The values ​​are close, and the frequency errors are very small.

[0296] Table 7 Comparison of equivalent stiffness coefficient and frequency error in the right 1 / 3 region of beam segment 30

[0297]

[0298] The above descriptions are merely five embodiments of the present invention. All equivalent changes and modifications made within the scope of the claims of the present invention are within the scope of the present invention.

Claims

1. A method for calculating the local equivalent stiffness of a beam structure based on frequency equivalence, characterized in that... Includes the following steps: (1) Based on the length d of the damaged area of ​​the beam structure, set an appropriate length for the locally damaged beam segment. ; (2) Calculate the equivalent stiffness coefficient ; (3) Calculate the equivalent stiffness of the locally damaged beam segment, equivalent stiffness = Bending stiffness of the section of a non-damaging beam; Specifically, in step (2), the equivalent stiffness coefficient Calculate using the following method: 1) For locally damaged beam segments, excluding those near the zero point of the modal curvature curve, the equivalent stiffness coefficient is approximately calculated using the following method: The distance between adjacent zero points of the beam mode curvature curve is L c0 , the intermediate L c0 / 3 range, the equivalent stiffness calculation method of the local damage beam segment is: ； in, The x-axis represents the equivalent stiffness coefficient, with the origin at the left end of the beam, the x-axis along the beam length pointing to the right end, and z representing the distance of the locally damaged beam segment from the origin. For the length of the locally damaged beam segment, through The number of locally damaged beam segments can be calculated from the span of the beam. The bending stiffness of the section at position x of the undamaged beam. for The flexural stiffness of the section at position x of the locally damaged beam segment within the range; When it is a beam with a uniform cross-section Since EI is a constant, the equivalent stiffness coefficient can be simplified to the following formula: ； in, To make the length of The locally damaged beam segment is divided into num segments, and the cross-sectional bending stiffness of the i-th segment is ; 2) The equivalent stiffness coefficient at each location of the beam structure is calculated more accurately using the following method: ； in, Let be the curvature curve of the nth mode when the beam structure is undamaged, and exp be an exponent related to the structure type; a) Simply supported beam with uniform cross-section Frequency equation: ； Where L is the span of the beam; The solution is: ； The frequency solution is: ； Where m is the linear density of the beam with uniform cross-section: Mode shape function: ； Modal curvature curve function and exponent exp: ； ； b) Cantilever beam with uniform cross-section Frequency equation: ； The solution is: ； The frequency solution is: ； Mode shape function: ； Modal curvature curve function and exponent exp: ； ； c) Fixed beam Frequency equation: ； The solution is: ； The frequency solution is: ； Mode shape function: ； Modal curvature curve function and exponent exp: ； ； d) Fixed-supported-hinged beams with uniform cross-section Frequency equation: ； The solution is: ； The frequency solution is: ； Mode shape function: ； Mode curvature curve function: ； e) Fixed-supported-directionally-supported beam with uniform cross-section Frequency equation: ； The solution is: ； The frequency solution is: ； Mode shape function: ； Modal curvature curve function and exponent exp: ； ； f) For other beam structures, the mode shape curves are calculated by finite element model or obtained by actual measurement. The mode shape curvature curves are calculated by the central difference method. For the measured mode shape curvature curves, a continuous function is used for fitting to eliminate the influence of damage and noise. The fitted mode shape curvature curves are then used for calculation. g) For statically indeterminate beam structures, the locally damaged beam segment within 2L / 3 of the fixed support end or directional support end has an exp=2.5, and other locally damaged beam segments have an exp=2.

2. The method for calculating the local equivalent stiffness of a beam structure based on frequency equivalence according to claim 1, characterized in that: In step (2), for simply supported beams and cantilever beams, when calculating the local equivalent stiffness of the first mode, the number of locally damaged beam segments is not less than 1, and when calculating the nth mode, the number of locally damaged beam segments is not less than 4(n-1); for statically indeterminate beams, when calculating the local equivalent stiffness, the number of locally damaged beam segments in each span is not less than 4n.

3. The method for calculating the local equivalent stiffness of a beam structure based on frequency equivalence according to claim 1, characterized in that: In step (2), the integral is calculated using MATLAB mathematical software.

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