Modal identification method based on statistical moment theory

Abstract

The present application belongs to the technical field of civil engineering structure identification, and particularly relates to a mode shape identification method based on statistical moment theory, comprising the following steps: step one, uniformly arranging measuring points on the surface of a structure along the same direction; step two, obtaining the acceleration response of each measuring point; step three, calculating the second-order statistical moment value of the measuring point by using the acceleration response; step four, calculating the second-order statistical moment ratio of two adjacent measuring points; and step five, constructing the first-order mode shape of the structure by the relationship between the second-order statistical moment and the first-order mode shape of the structure. The mode shape extracted by the method has great advantages in precision, efficiency, convenience and the like, and is not affected by human factors, so that the mode shape of the civil engineering structure can be quickly identified.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of civil engineering structure identification, and particularly relates to a mode shape identification method based on statistical moment theory. BACKGROUND

[0002] Nowadays, civil engineering structures in China are complex and large in size, among which large high-rise buildings and bridge space structures are complex in structure and are greatly affected by loads and environment, and some local key parts are difficult to detect when damaged, which has always puzzled engineering construction personnel. How to identify internal damage of building structures during service is a problem worth thinking and researching. At present, damage identification diagnosis based on modal parameters has been widely applied in the field of civil engineering and has achieved remarkable results.

[0003] Mode shape and related derived indicators are important characteristic parameters of building structures, and how to efficiently extract accurate structural modal shapes from collected signals is an important part of structural damage identification technology. The structural mode shape identification method based on dynamic test data can be classified according to frequency domain method, time domain method and time-frequency domain method.

[0004] 1. Frequency domain method

[0005] The frequency domain method is usually a method for identifying structural modal parameters according to the structural transfer function or frequency response function, which has clear physical meaning and is based on fast Fourier spectrum, so the frequency domain method has been rapidly developed and improved. The frequency domain method has certain limitations in theory, which inevitably brings errors to the identification results. The mode shape identification methods based on frequency domain mainly include peak value method, frequency domain decomposition method, least square complex frequency domain method and transfer rate method, and fast Fourier transform is one of the common time domain to frequency domain methods.

[0006] The frequency domain method is widely used due to its intuitive, clear physical meaning, high signal-to-noise ratio and other advantages, but it has poor identification ability for dense modes and low calculation efficiency.

[0007] 2. Time domain method

[0008] The time domain method is a method for directly establishing a model using the actual response signal of the structure and identifying parameters, which can usually identify modal damping, mode shape and other parameters well and make up for the shortcomings of the frequency domain method. Domestic and foreign scholars have done a lot of research in the field of time domain method of operational modal analysis, and there are many mature theories at present. The mode shape identification methods based on time domain mainly include time series analysis method, stochastic subspace method, least square complex exponential method and characteristic system realization method.

[0009] The time domain method directly identifies the modal parameters by using the response signal, and thus has high identification efficiency, and can identify the parameters of the equipment in operation, reflects the real working mode of the structure, and has advantages for identifying the structure with dense modes. However, the time domain method is sensitive to noise, has poor noise resistance, and is prone to false modes, and thus needs to be further studied to solve the problems.

[0010] 3. Time-frequency domain method

[0011] Most frequency domain and time domain identification methods require that the environmental excitation be white noise or non-white noise stationary excitation, however, the actual engineering cannot always meet the requirement, and the time-frequency analysis method can analyze the changes of the signal in the time domain and the frequency domain, and study the local time-frequency characteristics of the response signal, and thus is suitable for stationary and non-stationary signals. The mode shape identification method based on the time-frequency domain mainly includes: short-time Fourier transform, Hilbert-Huang transform, and wavelet transform.

[0012] The time-frequency method has the advantages of the time domain method and the frequency domain method, has advantages for identifying the structure vibration under non-stationary excitation, and has certain effect for identifying the parameters of the time-varying structure and the nonlinear system, and is a promising identification method. However, the main problems of the time-frequency method at present are: low identification efficiency, and not suitable for engineering practice. SUMMARY

[0013] In view of the above problems in the prior art, the mode shape identification method based on the statistical moment theory is provided, the structural mode shape extracted by the method has great advantages in precision, efficiency, convenience and the like, and is not affected by human factors, and the first-order mode shape of the structure can be quickly identified.

[0014] In order to solve the above technical problems, the technical scheme is adopted as follows:

[0015] The mode shape identification method based on the statistical moment theory comprises the following steps:

[0016] Step 1: uniformly arranging measuring points in the same direction on the surface of the structure;

[0017] Step 2: obtaining the acceleration response of each measuring point;

[0018] Step 3: calculating the second-order statistical moment value of the measuring point by using the acceleration response;

[0019] Step 4: calculating the second-order statistical moment ratio of two adjacent measuring points;

[0020] Step 5: constructing the first-order mode shape of the structure by the relationship between the second-order statistical moment and the first-order mode shape of the structure.

[0021] Furthermore, in steps one and two, the equations of motion for the structure can be expressed as:

[0022]

[0023] In the formula: M, C, and K are the mass, damping, and stiffness matrices of the structure, respectively. U(t) represents the acceleration, velocity, and displacement response vectors of the structural measuring points, respectively, and P(t) represents the external load column vector.

[0024] Under the Rayleigh damping assumption, the acceleration response of the i-th measuring point of the structure can be obtained by using the orthogonality of the mode shapes and decoupling calculations in equation (1):

[0025]

[0026] Furthermore, in steps three and four, the ratio of the second-order statistical moments of the axle contact point response at d1 and d2 is... It can be represented as:

[0027]

[0028] Given that the inherent parameters of the bridge are fixed, the first-order mode shape at any measuring point on the bridge is constant. Therefore, the ratio of the second-order statistical moments of acceleration at d1 and d2 in the bridge response is... It should also be a constant and unaffected by other factors. Similarly, for high-rise building structures with measuring points set on each floor, the vibration mode value at any measuring point should also be a constant, and formula (30) still applies.

[0029] Furthermore, in step five, assuming the structure has L measurement points, the second-order acceleration statistical moments are calculated for each group of simultaneously acquired responses. The statistical moment ratio vector can be obtained using equation (30):

[0030] {R 1,2 ,R 2,3 ,R 3,4 ,...,R i,i+1 ,...,R L-1,L}(31)

[0031] In equation (31), R i-1,i This represents the second-order statistical moment value calculated at the i-th measurement point. The second-order statistical moment value calculated at the (i-1)th measurement point The ratio, and other related symbols are similar;

[0032] Assuming the mode shape at measurement point 1 is the reference point, the first-order mode shape vector of the structure can be constructed by combining equations (30) and (31):

[0033]

[0034] In equation (32), Since it is a constant, normalizing the first-order mode vector yields the normalized first-order mode of the bridge.

[0035] Compared with the prior art, the present invention has the following beneficial effects:

[0036] (1) Through theoretical derivation, numerical simulation and actual bridge test research and analysis, it has been fully demonstrated that the method proposed in this invention can identify the first mode of vibration of the structure well;

[0037] (2) In the numerical simulation part, the present invention considers the effects of external excitation changes, bridge damping ratio and environmental noise respectively. The research results show that compared with the transmissivity method and the random subspace method, the method proposed in this invention can adapt well to the changes of external excitation and bridge damping ratio, and has strong anti-noise ability. The actual bridge test further verifies that the method of this invention has higher recognition accuracy.

[0038] (3) The method of the present invention can be used for mode calculation and analysis under direct measurement test, as well as for mode extraction under new indirect measurement method. Compared with the traditional mode analysis method, the mode extracted by the method of the present invention has great advantages in terms of accuracy, efficiency and convenience, and is not affected by human factors, and can quickly identify the mode of civil engineering structure.

[0039] (4) The method of the present invention is not only applicable to the identification of vibration modes of bridges, but also to the identification of vibration modes of high-rise building structures, and has a wide range of applications. Attached Figure Description

[0040] Figure 1 This is a simplified model diagram of the vehicle-bridge coupling in an embodiment of the mode shape identification method based on statistical moment theory of the present invention.

[0041] Figure 2 This is a flowchart illustrating the first-order vibration mode identification process for bridges, as described in an embodiment of the vibration mode identification method based on statistical moment theory of the present invention.

[0042] Figure 3 This is a schematic diagram of the Qingshui Bridge in an embodiment of the mode shape identification method based on statistical moment theory of the present invention;

[0043] Figure 4 This is a schematic diagram of the numerical simulation unit division of an embodiment of the mode shape identification method based on statistical moment theory of the present invention;

[0044] Figure 5 A comparison of the first-order vibration modes identified by different methods under random traffic flow (Case 1);

[0045] Figure 6 Comparison of the first-order vibration modes identified by different methods under random traffic flow (Case 2);

[0046] Figure 7 Comparison of first-order mode shapes identified by different methods (ζ) n =0.00);

[0047] Figure 8 Comparison of first-order mode shapes identified by different methods (ζ) n =0.01);

[0048] Figure 9 Comparison of first-order mode shapes identified by different methods (ζ) n =0.02);

[0049] Figure 10 Comparison of first-order mode shapes identified by different methods (ζ) n =0.03);

[0050] Figure 11 Comparison of first-order mode shapes identified by different methods (S) NR =10dB);

[0051] Figure 12 Comparison of first-order mode shapes identified by different methods (S) NR =20dB);

[0052] Figure 13 Comparison of first-order mode shapes identified by different methods (S) NR =30dB);

[0053] Figure 14 Comparison of first-order mode shapes identified by different methods (S) NR =40dB);

[0054] Figure 15 This is the fundamental frequency diagram of the Qingshui Bridge in an embodiment of the mode shape identification method based on statistical moment theory of the present invention;

[0055] Figure 16 This is a signal diagram of the test vehicle ① in an embodiment of the mode shape identification method based on statistical moment theory of the present invention;

[0056] Figure 17 This is a signal diagram of the test vehicle ② in an embodiment of the mode shape identification method based on statistical moment theory of the present invention;

[0057] Figure 18 This is a frequency diagram of the test vehicle ① in an embodiment of the mode shape identification method based on statistical moment theory of the present invention;

[0058] Figure 19 This is a frequency diagram of the test vehicle ② in an embodiment of the mode shape identification method based on statistical moment theory of the present invention;

[0059] Figure 20 Comparison of different methods for identifying the first-order vibration mode. Detailed Implementation

[0060] To enable those skilled in the art to better understand this invention, it primarily focuses on the theoretical derivation, numerical simulation, and real-bridge experiments for identifying vibration modes in bridge structures; the same principles apply to high-rise building structures. The technical solution of this invention is further described below based on bridge structures, in conjunction with accompanying drawings and embodiments.

[0061] First, the vehicle-bridge coupling and multi-degree-of-freedom statistical moment theories were derived theoretically, and the equivalence relationship between statistical moments and mode shapes was explained, demonstrating the feasibility of the proposed method. Then, numerical simulation was used to study the mode shape identification effect under factors such as external excitation changes, bridge damping ratio, and environmental noise, and the results were compared with those of the transmissibility method and the random subspace method (SSI). Finally, the method was preliminarily verified through a real bridge test.

[0062] 1. Theoretical Foundation

[0063] 1.1 Vehicle-Axle Coupling Theory

[0064] The simplified model of the vehicle-bridge coupling system in the new indirect measurement method is as follows: Figure 1 As shown. In this model, the moving excitation vehicle Test vehicles ① and ② are simplified to mass blocks supported by springs. The mass of the moving excitation vehicle is m. v The stiffness coefficient of the supporting spring is k v The mass of test vehicle ① is m v1 The stiffness coefficient of the supporting spring is k v1 The damping of the supporting spring is c. v1 The mass of test vehicle ② is m v2 The stiffness coefficient of the supporting spring is k v2 The damping of the supporting spring is c. v2 The bridge surface roughness is r(x), and the bridge is simplified as a uniform and constant-section Eular-Bernoulli beam with a length of L; the flexural stiffness of the bridge section is EI; due to the moving excitation vehicle As the test vehicle is used only as an external excitation for the bridge, in order to focus more on its physical characteristics without losing the generality of the problem, the following assumptions are proposed: (1) The mass of test vehicle ① and test vehicle ② is negligible compared with the mass of the bridge. (2) To facilitate theoretical derivation, the damping ratio of the moving excitation vehicle and the bridge is ignored, while it is taken into consideration in numerical simulation and experiment. (3) In the theoretical derivation, the moving excitation vehicle travels at a constant speed v on the bridge.

[0065] The structural mode identification method based on statistical moment theory of the present invention includes the following steps:

[0066] Step 1: Distribute measuring points evenly along the same direction on the surface of the structure;

[0067] Step 2: Obtain the acceleration response at each measuring point;

[0068] Step 3: Calculate the second-order statistical moment value of the measuring point using the acceleration response;

[0069] Step 4: Calculate the ratio of the second-order statistical moments between two adjacent measurement points;

[0070] Step 5: Construct the first-order mode shape of the structure by using the relationship between the second-order statistical moments and the first-order mode shape of the structure.

[0071] Furthermore, in steps one and two, measuring points are uniformly spaced along the same direction on the structure surface, and the equation of motion of the structure can be expressed as:

[0072]

[0073] In the formula: M, C, and K are the mass, damping, and stiffness matrices of the structure, respectively. U(t) represents the acceleration, velocity, and displacement response vectors of the structural measuring points, respectively, and P(t) represents the external load column vector.

[0074] Under the Rayleigh damping assumption, equation (1) can be used to obtain the equation of motion for the structural modal response by utilizing the modal orthogonality and decoupling:

[0075]

[0076] In equations (2) to (3), Y n (t) represents the generalized coordinates corresponding to the nth mode shape; M n ξ n ω n φ n These represent the nth-order generalized mass, damping ratio, natural circular frequency, and standard mode of the structure, respectively; P n (t) represents the generalized force corresponding to the nth mode, which can be obtained by solving the equation of the nth uncoupled mode shape:

[0077]

[0078] For a low-critical-damping structural system, in equation (4):

[0079]

[0080] The acceleration response at the i-th measuring point of the structure can be expressed as:

[0081]

[0082] The motion equations for the moving excitation vehicle and the bridge are expressed as follows:

[0083]

[0084] Among them, f c (t) can be expressed as:

[0085]

[0086] In the formula: q v , These represent the absolute vertical displacement, velocity, and acceleration of the moving excitation vehicle, u. b This represents the absolute vertical displacement of the bridge. The second derivative of displacement with respect to time; u b "" represents the fourth derivative of the displacement with respect to the position x of the moving excitation vehicle; f c (t) represents the external excitation of the random traffic flow; δ represents the Diclave function; and t represents time.

[0087] This invention employs a method of collecting the vehicle body acceleration response from two stationary test vehicles. This method is unaffected by the roughness of the bridge surface; therefore, the motion equations of the two test vehicles can be expressed as:

[0088]

[0089] In the formula: q v1 , These represent the absolute vertical displacement, velocity, and acceleration of test vehicle ①, respectively. v2 , These represent the absolute vertical displacement, velocity, and acceleration of test vehicle ②, respectively.

[0090] Based on the literature review and analysis, during the process of collecting the vehicle-bridge response while the test vehicle is stationary, the inverted vehicle-bridge contact point response can extract bridge information better than the vehicle body response. Therefore, this invention inverts the collected vehicle body response to the vehicle-bridge contact point response under stationary conditions. Based on the transformation of equation (10), we can obtain:

[0091]

[0092] q v1 , Acceleration signals that can be collected by test vehicle ① Integrating, we obtain the result. Similarly, based on the transformation of equation (11), we can obtain the contact point response of test vehicle ②:

[0093]

[0094] 1.2 Analysis based on the theory of statistical moments with multiple degrees of freedom

[0095] The literature performs Fourier transforms on the response of the moving excitation vehicle and the axle contact point in Section 1.1, and the following relationship can be derived in the frequency domain:

[0096] Db (x,w)=H bv D v (w) (14)

[0097] D v1 (w)=H bv1 D b (x,w) (15)

[0098] D v2 (w)=H bv2 D b (x,w) (16)

[0099] In the above formula: D b (x,t),D v (w),D v1 (w),D v2 (w) represents the frequency domain expression of the vertical displacement of the vehicle-bridge contact point, the moving excitation vehicle, test vehicle ①, and test vehicle ②, respectively.

[0100] Where H bv H bv1 H bv2 The expression is:

[0101]

[0102] In the formula, w, w1, and w2 represent the frequencies of the mobile excitation vehicle, test vehicle ①, and test vehicle ②, respectively; φ j (d) represents the mode shape value of the j-th mode of the bridge at x = d; H j (w) is the frequency response function; Ω j The expression is:

[0103]

[0104] From equations (14), (15), and (16), we can deduce that:

[0105] D v1 (w)=H bv1 H bv D v (w) (21)

[0106] D v2 (w)=H bv2 H bv D v (w) (22)

[0107] The power spectral density function of the displacement response of test vehicle ① has the following relationship with the frequency domain expression of the displacement response:

[0108]

[0109] In the formula S v1 Let T be the power spectral density function of the displacement response of test vehicle ①; T is the length of the response signal truncation.

[0110] Within the linear elastic range, the second-order central moment of displacement of the test vehicle ① at x = d on the bridge (hereinafter referred to as the second-order displacement moment) can be obtained from the relationship between the power spectrum and variance of the structural response:

[0111]

[0112] In the formula: Let x and d represent the variance and second moment of displacement of test vehicle ① at x = d on the bridge, respectively.

[0113] Substituting equations (18), (21), and (22) into equation (24), we get:

[0114]

[0115] The second moment of displacement of test vehicle ② Similarly, we can obtain the following.

[0116] Will Compare have to:

[0117]

[0118] The literature has shown that equation (26) still applies to the vehicle-bridge contact point response at different bridge locations, when w1 = w2,m v1 =m v2 At that time, the first-order fundamental frequency signal of the bridge structure is extracted, and equation (26) is rewritten as:

[0119]

[0120] In the formula φ represents the second-order statistical moments of displacement at the vehicle-bridge contact points at x = d1 and x = d2 on the bridge, respectively; j (d1),φ j (d2) represents the first mode shape values ​​at x = d1 and x = d2 on the bridge, respectively.

[0121] Vanmarcke (1976) proposed an approximate result for narrowband response, namely, that under low damping conditions, the variance of the acceleration response and the contrast of the displacement response should have the following relationship:

[0122]

[0123] In the formula and These represent the variance of the acceleration response and the second-order center distance of acceleration (or simply the second-order moment of acceleration) at the vehicle-bridge contact point at x = d on the bridge, respectively.

[0124] Substituting equation (28) into equation (27), we get:

[0125]

[0126] In equation (29), These represent the second-order statistical moments of acceleration at the vehicle-bridge contact points at x = d1 and x = d2 on the bridge, respectively.

[0127] According to equation (29), the ratio of the second-order statistical moments of acceleration at the axle contact point at d1 and d2 is... It can be represented as:

[0128]

[0129] Given that the inherent parameters of the bridge are fixed, the first-order mode shape at any measuring point on the bridge is constant. Therefore, the ratio of the second-order statistical moments of acceleration at d1 and d2 in the bridge response is... It should also be a constant and unaffected by other factors. Similarly, for high-rise building structures with measuring points set on each floor, the vibration mode value at any measuring point should also be a constant, and formula (30) still applies.

[0130] Assuming the bridge has L measuring points, the second-order acceleration statistical moments are calculated for each group of simultaneously collected responses. The statistical moment ratio vector can be obtained using equation (30):

[0131] {R 1,2 ,R 2,3 ,R 3,4 ,...,R i,i+1 ,...,R L-1,L}(31)

[0132] In equation (31), R i-1,i This represents the second-order statistical moment value calculated at the i-th measurement point. The second-order statistical moment value calculated at the (i-1)th measurement point The ratio of , and other related symbols are similar.

[0133] Assuming the mode shape at measurement point 1 is the reference point, the first-order mode shape vector of the bridge can be constructed by combining equations (30) and (31):

[0134]

[0135] In equation (32), Since it is a constant, normalizing the first-order mode vector yields the normalized first-order mode of the bridge.

[0136] The above theory still allows for the extraction of bridge vibration modes using statistical moment ratios for directly measured acceleration data. In order to promote the indirect measurement method to practical applications and facilitate its application in actual engineering, the theoretical derivation uses the acceleration signal collected by the test vehicle to invert the acceleration signal at the contact point to extract the first-order vibration mode of the bridge.

[0137] 1.3 Bridge Structure Vibration Mode Identification Process

[0138] Considering that the identification of the first-order vibration mode is the easiest and relatively accurate method for testing civil engineering structures, the process for identifying the first-order vibration mode of a bridge using the method of this invention is as follows: Figure 2 As shown.

[0139] 2 Numerical Simulation

[0140] The Qingshui Bridge on the first-class highway section from Mawu Town to Longtan Town in Fuling District, Chongqing, was used as the model for numerical simulation. A schematic diagram of the bridge is shown below. Figure 3 As shown, the bridge has a span of 30m, uses C50 concrete, and has an elastic modulus of E = 3.45 × 10⁻⁶. 10 N / m 2 The moment of inertia of the cross section is 1.57m. 4 .

[0141] To maintain consistency with the parameters of the test vehicle used on the actual bridge, the distance between the two test vehicles used in the numerical simulation was 2 meters. Therefore, the bridge was divided into 15 equally spaced units, as shown in the schematic diagram of the unit division. Figure 4 As shown, each rectangular block represents a different unit of the bridge and is represented by a circled number. For example, the third unit is represented by "③". The numbers 0 to 15 are the unit node numbers, which are the test point numbers where the test vehicle stops.

[0142] During the data acquisition process, while the test vehicles are stationary, a continuous flow of random traffic will act as an excitation, causing random vibrations in the bridge to approximate the actual data acquisition process. Two test vehicles will first be stationary at two measurement points in Unit ①, with the front vehicle being test vehicle ② and the rear vehicle being test vehicle ①. They will simultaneously acquire signals for 30 seconds while stationary. After completion, the two test vehicles will move to the next measurement point at a fixed interval of 2 meters. Due to space limitations, the specific data acquisition methods and steps for the two test vehicles can be found in the references. After the node data acquisition is completed, according to... Figure 2 The flowchart shown calculates the first vibration mode of the bridge.

[0143] In the process of mode shape identification, the influence of factors such as changes in external excitation, bridge damping ratio, and environmental noise cannot be ignored. The numerical simulation section will employ the method of this invention, the transmissibility method, and the random subspace method for mode shape identification, respectively, to discuss the impact of the aforementioned factors on the accuracy of mode shape identification. Finally, by comparing the errors of the three methods, the characteristics of the method of this invention will be analyzed.

[0144] 2.1 Research on Changes in External Incentives

[0145] According to the parameter settings of the actual bridge test, the masses of the two test vehicles are m respectively. v1 =m v2 =1470kg, vehicle damping is c v1 =c v2 =1000 N·s·m -1 The stiffness is k v1 =k v2 =524076 N·m -1 The test vehicle's body frequency is w v1 =w v2 =3Hz, bridge damping ratio is taken as ζ n =0.03, random traffic flow considers four moving vehicles with different speeds, masses, and bridge entry times to approximate random excitation, as shown in Table 1. The four moving vehicles continuously cycle on the bridge according to Table 1 until the test vehicle completes signal acquisition.

[0146] The Modal Guarantee Criterion (MAC) is used to compare and analyze the differences in the first-order vibration modes proposed by the method in this invention, the transitivity method, and the random subspace method. The MAC is defined as follows:

[0147]

[0148] Where, φ a and φ b These represent the vectors of the calculated simulated mode shape and the standard mode shape, respectively. The closer the MAC value is to 1, the closer the simulated mode shape is to the standard mode shape; conversely, the closer the MAC value is to 1, the closer the simulated mode shape is to the standard mode shape.

[0149] Table 1 Changes in External Incentives

[0150]

[0151] Using the external excitation variations shown in Table 1, the first-order vibration modes of the bridge were identified using the method of this invention, the transmissibility method, and the random subspace method, respectively. MAC was used as the standard for evaluating the accuracy of the vibration modes, and φ in formula (29) was used as the standard. a φ represents the model mode vectors calculated by different methods. b The standard mode shape vectors, mode shape curves, and comparison results obtained from the bridge model analysis using ABAQUS finite element method are shown below. Figure 5 , Figure 6 As shown in Table 2.

[0152] Table 2. MAC values ​​for mode identification under different methods

[0153]

[0154] Depend onFigure 5 , Figure 6 As shown in Table 2, under both types of random traffic flow excitation, the mode shape MAC extracted using the method of this invention is above 99.9998%. The maximum values ​​of mode shape MAC extracted by the transmissivity method and the random subspace method are 99.9996% and 99.9994%, respectively.

[0155] 2.2 Research on Bridge Damping Ratio

[0156] The test vehicle parameters are the same as in Section 2.1. The external excitation is set according to Case 1 in Table 1. Four damping ratio conditions are set for the bridge: 0.00 (undamped), 0.01, 0.02, and 0.03. The three methods described in Section 2.1 are still used to identify the bridge vibration modes. The vibration mode curves and comparison results are as follows: Figure 7 , Figure 8 , Figure 9 , Figure 10 As shown in Table 3.

[0157] Table 3. MAC values ​​for mode identification under different methods

[0158]

[0159] Depend on Figure 7 , Figure 8 , Figure 9 , Figure 10 As shown in Table 3, the mode shapes with MAC values ​​extracted by the method of the present invention and the random subspace method are both above 99.9990%, while the mode shape with MAC values ​​extracted by the transmissivity method is the lowest at 99.9949%. Therefore, it can be concluded that both the method of the present invention and the random subspace method can better identify the first-order mode shapes of bridge structures under the influence of the bridge damping ratio.

[0160] 2.3 Impact of Environmental Noise

[0161] In actual data acquisition environments, the impact of environmental noise on signals is unavoidable.

[0162] This section simulates contaminated signals by adding white noise to the signals collected by the test vehicle. The signal-to-noise ratio (SNR) is used as an indicator to judge the intensity of the added noise. The SNR is defined as:

[0163]

[0164] Where N j y represents the total number of data points collected by the test vehicle at the j-th node. ij δ represents the data corresponding to the i-th sampling interval collected by the test vehicle at the j-th node. ij This represents the noisy data corresponding to the i-th sampling interval collected by the test vehicle at the j-th node. From formula (28), it is known that the greater the noise intensity, the higher the signal-to-noise ratio S. NRThe smaller it is.

[0165] The test vehicle parameters are the same as in Section 2.1. The external excitation is set according to Case 1 in Table 1, and the bridge damping ratio is taken as ζ. n =0.03, set S respectively NR =10, 20, 30, 40 dB, four groups of noise-involved damage conditions, the three methods described in Section 2.1 are still used to identify bridge vibration modes, the vibration mode curves and comparison results are as follows. Figure 11 , Figure 12 , Figure 13 , Figure 14 As shown in Table 4.

[0166] Table 4. MAC values ​​for mode identification under different methods

[0167]

[0168] Depend on Figure 11 , Figure 12 , Figure 13 , Figure 14 As shown in Table 4, environmental noise affects the accuracy of mode shape identification. The transmissibility method has relatively poor accuracy, especially under 10dB noise, where the MAC is only 99.8913%. The method of this invention and the random subspace method achieve better results, both exceeding 99.9786%. It is worth noting that although both the method of this invention and the random subspace method can identify mode shapes well, the random subspace-based method has a more complex identification process and lower computational efficiency. In contrast, the mode shape identification method of this invention, based on statistical moment theory, has a simpler process and higher computational efficiency. Table 5 shows the computational time efficiency of the method of this invention and the random subspace method for extracting mode shapes under different influencing factors.

[0169] Table 5. Mode identification efficiency under the influence of different factors.

[0170]

[0171] 3. Real Bridge Test

[0172] The bridge selected for the actual bridge test in this invention is the Qingshui Bridge on the first-class highway section from Mawu Town to Longtan Town in Fuling District, Chongqing. This bridge is a single-span simply supported beam bridge with a span length of 30m, and this span was chosen as the test span. According to the on-site design data, the main beam of the bridge is a 1×30m ​​prestressed concrete simply supported T-beam, the total length of the bridge is 42.08m, and the total width of the bridge deck is 20m. The superstructure span arrangement is a 1×30m ​​prestressed concrete simply supported T-beam with a beam height of 2.0m; the substructure abutments are gravity-type U-shaped abutments with open-cut spread foundations; the bridge deck pavement is asphalt concrete. The concrete grade is C50, and the bridge's elastic modulus is E = 3.45 × 10⁻⁶. 10 N / m 2The moment of inertia of the cross section is 1.57m. 4 .

[0173] This experiment consisted of three parts. The first part was a forced vibration test, in which the test vehicle was forced to vibrate, and its own frequency was determined to be 3.00 Hz. The second part was a direct measurement test, in which the fundamental frequency of the bridge was quickly extracted by placing the sensor directly on the bridge surface. The value was 5.37 Hz, and the spectrum is shown below. Figure 15 As shown; the third part is the transfer test of the test vehicle, and the Shimizu Bridge unit division and measuring point layout are still as shown. Figure 4 As shown, two test vehicles, from left to right at a fixed interval of 2 meters, first synchronously and statically collected signals for 30 seconds at measurement points 1 and 0. After completion, they moved to the next measurement point. The first vehicle is test vehicle ②, and the second vehicle is test vehicle ①. Due to space limitations, the specific acquisition process can be found in the references. The vertical acceleration signals and corresponding spectrum diagrams collected by the two test vehicles at measurement points 1 and 2 are shown below. Figure 16 , Figure 17 , Figure 18 , Figure 19 As shown, the peak values ​​on the spectrum represent the vehicle frequency of 3.00Hz and the bridge fundamental frequency of 5.37Hz, respectively. This indicates that the acceleration signal collected by the test vehicle contains both vehicle frequency information and bridge frequency signal, and the vibration transmission of the test vehicle is good. Figure 20 Table 6 shows a comparison of the first mode shape of the bridge and the corresponding MAC value identified by different methods. In the figure, "Direct Measurement - Random Subspace Method" means that the first mode shape of the bridge is extracted from the acceleration signal collected by the direct measurement method using the random subspace method, and "Indirect Measurement - Method of the Invention" means that the first mode shape of the bridge is extracted from the contact point acceleration signal inverted by indirect measurement using the method of the present invention. The other names are similar.

[0174] To highlight the contrast effect of mode shape identification, the direct measurement method of the second part of the experiment was used to collect the vertical acceleration of the bridge. The specific operation steps are as follows: the acceleration sensor is directly placed on each measuring point on the bridge surface, and at the same time, the vehicle-bridge contact point position corresponding to the indirect measurement method is used to extract the first mode shape of the bridge using the random subspace method, and this is used as the standard mode shape vector in formula (27).

[0175] Table 6. MAC values ​​for mode identification under different methods

[0176]

[0177] Depend on Figure 20As shown in Table 6, the MAC value identified by the method of the present invention is 99.9921%, which is higher than the MAC value identified by the random subspace method (99.9903%) and the MAC value identified by the transmissivity method (99.9881%). Furthermore, the nodal error of the first mode shape identified by the direct measurement-method of the present invention and the indirect measurement-method of the present invention is only 3.58%, further highlighting the advantages of the mode shape identification method of the present invention.

[0178] 4. Conclusion

[0179] The feasibility of the method of the present invention was further analyzed through numerical simulation and actual bridge test, and the following conclusions were drawn:

[0180] (1) Through theoretical derivation, numerical simulation and actual bridge test research and analysis, it has been fully demonstrated that the method proposed in this invention can identify the first-order vibration mode of the bridge well;

[0181] (2) In the numerical simulation, this invention considers the effects of external excitation changes, bridge damping ratio, and environmental noise. The results show that compared with the transmissivity method and the random subspace method, the method proposed in this invention can adapt well to changes in external excitation and bridge damping ratio, and has stronger noise resistance. The actual bridge test further verifies that the method of this invention has higher recognition accuracy.

[0182] (3) The method of the present invention can be used for mode shape calculation and analysis under direct measurement test, as well as for mode shape extraction under new indirect measurement method. Compared with the traditional mode shape analysis method, the mode shape extracted by the method of the present invention has great advantages in terms of accuracy, efficiency and convenience, and is not affected by human factors. It can provide a new method for rapid identification of bridge mode shape.

[0183] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A mode shape identification method based on statistical moment theory, characterized in that, Includes the following steps: Step 1: Distribute measuring points evenly along the same direction on the surface of the structure; Step 2: Obtain the acceleration response at each measuring point; Step 3: Calculate the second-order statistical moment value of the measuring point using the acceleration response; Step 4: Calculate the ratio of the second-order statistical moments between two adjacent measurement points; Step 5: Construct the first-order mode shape of the structure by using the relationship between the second-order statistical moments and the first-order mode shape of the structure, specifically as follows: Assuming the bridge has L measuring points, calculate the second-order acceleration statistical moment for each group of simultaneously collected responses, using the formula... The statistical moment ratio vector can be obtained: {R 1,2 ,R 2,3 ,R 3,4 ,...,R i,i+1 ,...,R L-1,L }(31) In equation (31), R i-1,i This represents the second-order statistical moment value calculated at the i-th measurement point. The second-order statistical moment value calculated at the (i-1)th measurement point The ratio, and other related symbols are similar. Assuming the mode shape at measurement point 1 is the reference point, the first-order mode shape vector of the bridge can be constructed by combining equations (30) and (31): In equation (32), Since it is a constant, normalizing the first-order mode vector yields the normalized first-order mode of the bridge.

2. The mode shape identification method based on statistical moment theory according to claim 1, characterized in that: In steps one and two, the equations of motion for the structure can be expressed as: In the formula: M, C, and K are the mass, damping, and stiffness matrices of the structure, respectively. U(t) represents the acceleration, velocity, and displacement response vectors of the structural measuring points, respectively, and P(t) represents the external load column vector. Under the Rayleigh damping assumption, the acceleration response of the i-th measuring point of the structure can be obtained by using the orthogonality of the mode shapes and decoupling calculations in equation (1): In the formula: Let i be the acceleration of the measurement point on structure i. Let n be the nth mode shape value at measurement point i of the structure. Let be the second derivative of the generalized coordinates with respect to time corresponding to the nth mode shape.

3. The mode shape identification method based on statistical moment theory according to claim 2, characterized in that: In steps three and four, the ratio of the second-order acceleration statistical moments at adjacent measuring points d1 and d2 is... It can be represented as: In the formula: and φ1(d1) and φ1(d2) represent the second-order statistical moments of acceleration at the vehicle-bridge contact points at x = d1 and x = d2 on the bridge, respectively, and represent the first-order mode values ​​at x = d1 and x = d2 on the bridge, respectively. Given that the inherent parameters of the structure are fixed, the first-order mode shape at any measuring point of the structure is constant. Therefore, the ratio of the statistical moments of the second-order acceleration at d1 and d2 of the structural response is... It should also be a constant and unaffected by other factors.

4. The mode shape identification method based on statistical moment theory according to claim 3, characterized in that: In step five, assuming the structure has L measurement points, the second-order acceleration statistical moments are calculated for each group of simultaneously acquired responses. The statistical moment ratio vector can be obtained using equation (30): {R 1,2 ,R 2,3 ,R 3,4 ,...,R i,i+1 ,...,R L-1,L }(31) In equation (31), R i-1,i This represents the second-order statistical moment value calculated at the i-th measurement point. The second-order statistical moment value calculated at the (i-1)th measurement point The ratio, and other related symbols are similar; Assuming the mode shape at measurement point 1 is the reference point, the first-order mode shape vector of the structure can be constructed by combining equations (30) and (31): In equation (32), φ1 is the first mode shape of the structure. The value of the first-order mode shape at the i-th measurement point of the structure. Since it is a constant, normalizing the first-order mode vector yields the normalized first-order mode of the structure.

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