Mechanical structure response reconstruction method and system based on optimal modal response estimation

Abstract

The present application relates to a kind of mechanical structure response reconstruction method and system based on optimal modal response estimation, belong to structural response reconstruction field, method includes according to free vibration modal response form construction response estimation signal;Residual between response estimation signal and real measurement signal is calculated;According to the spectral characteristics of residual signal, optimization objective function is constructed;Optimization function is solved using gradient descent algorithm, and the modal response of each order estimated is obtained;According to modal transfer ratio equation, structural response reconstruction is completed.The method of the present application is based on the modal superposition method of linear system as theoretical basis, by constructing response estimation signal and optimizing the spectral characteristics of residual signal, the optimal modal response estimation signal is obtained, and finally the response reconstruction is carried out through modal transfer ratio equation, this method can realize the response reconstruction of structure with dense modal characteristics based on single sensor data and ensure the accuracy of response reconstruction.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of structural response reconstruction, and particularly relates to a mechanical structural response reconstruction method and system based on optimal modal response estimation. BACKGROUND

[0002] Structural Health Monitoring (SHM) is a process of collecting detection information through a series of sensors deployed on a target system to infer the structural integrity of the system and quantify its performance degradation. However, in practical applications, due to economic constraints, it is impossible to arrange sensors at all positions; at the same time, due to the complexity of the geometric shape of the system structure, it is difficult to arrange sensors in some key structural areas. Therefore, when the system response of the key structural area cannot be obtained, the dynamic response reconstruction technology needs to be used to reconstruct the system response of these areas using the measured system response data.

[0003] At present, response reconstruction methods are mainly divided into two categories: frequency domain reconstruction methods and time domain reconstruction methods. The frequency domain method mainly includes a reconstruction method based on a generalized transfer rate matrix. The method reconstructs the response by constructing a generalized transfer rate matrix between the measured points and the points to be reconstructed. Although the frequency domain method uses data in the frequency response domain, and converts the reconstructed frequency domain response back to the time domain through inverse Fourier transform, this will introduce additional computational cost and numerical truncation error. The time domain reconstruction method mainly includes a reconstruction method based on a modal transfer matrix and a reconstruction method based on empirical mode decomposition. The former reconstructs the response by calculating the modal shape of the structure and constructing the transfer matrix between the measured points and the points to be reconstructed, but usually requires the number of sensors to be greater than the number of truncated modes; the latter extracts the modal response of each order through a band-pass filter and an empirical mode decomposition algorithm, and reconstructs the response through the mode shape ratio between the points to be reconstructed and the measured points. Compared with the frequency domain method, the time domain method has more advantages in computational efficiency, but has challenges in dealing with structures with dense modal characteristics.

[0004] Dense modes have a greater impact on structural response reconstruction based on empirical mode decomposition. The reason is that if the two natural frequencies of the vibrating structure are very close to each other and the respective damping ratios are also large, they will appear as two overlapping peaks in the frequency domain, making it difficult to accurately extract the modal response of each order through a band-pass filter, thereby reducing the accuracy of structural response reconstruction. Therefore, it is of great academic significance and application prospect to study the method for reconstructing the response of dense modal structures. SUMMARY

[0005] In view of the deficiencies of the prior art, the mechanical structure response reconstruction method and system based on optimal modal response estimation are provided, which can be used for response reconstruction of mechanical structures with two very close inherent frequencies, guaranteeing the reconstruction accuracy, and can be widely applied to dense modal structure response reconstruction in multiple fields, and has high calculation efficiency and high accuracy.

[0006] To achieve the above object, the technical scheme is as follows:

[0007] In one aspect, the mechanical structure response reconstruction method based on optimal modal response estimation is provided, which comprises the following steps:

[0008] S1, simulating the actual working condition of the mechanical structure, placing an acceleration sensor on any known point on the mechanical structure, and collecting original acceleration data by using the acceleration sensor and then performing band-pass filtering and denoising;

[0009] S2, constructing a calculation model of the response estimation signal, and constructing the response estimation signal of the known point according to the free vibration modal response form as follows:

[0010]

[0011] wherein, ξ i and ω i respectively represent the i-th modal damping ratio and inherent frequency, respectively represent the amplitude and phase of the estimated modal response, represents the known point response estimation signal; t represents the time length of the signal, and n represents the number of structure modes;

[0012] S3, calculating the residual error between the known point response estimation signal and the true measurement signal:

[0013]

[0014] wherein, x(t) represents the true measurement signal of the known point, and r(t) represents the residual error signal between the known point response estimation signal and the true measurement signal;

[0015] S4, constructing an optimization objective function according to the spectral characteristics of the residual error signal:

[0016]

[0017] wherein, r(ω) is the frequency domain form of the residual error signal r(t), ω upp is the upper limit value of the signal spectrum in the frequency domain;

[0018] S5, using a gradient descent algorithm to solve the optimization objective function, and obtaining u i and θ iaccurately estimate values of u i and θ i and then obtain unknown point modal responses of each order from the estimated values of u

[0019]

[0020] S6, superimpose unknown point modal responses of each order in step S5 according to the modal transfer ratio equation to complete structure response reconstruction:

[0021]

[0022] wherein subscript k represents a known point, u represents an unknown point, i represents the i-th order mode, φ iu represents the u-th component of the i-th order mode, q ik represents the i-th order mode of the k-th known point, x u (t) is the structure response of the unknown point.

[0023] Preferably, the specific steps of accurately estimating values of u i and θ i are as follows:

[0024] S51, obtain partial derivatives of the optimization objective function with respect to the estimated values of u and θ and obtain the gradient of the objective function at this point

[0025] S52, set the learning rate eta;

[0026] S53, update the parameter X, wherein the parameter X is a column vector composed of u and θ and accurate estimated values of u and θ

[0027] Preferably, the parameter X is updated according to the formula in step S53.

[0028] Preferably, step S1 specifically comprises the following sub-steps:

[0029] S11, select an excitation mode, apply an external excitation, and collect the dynamic response of the structure under the excitation by arranging acceleration sensors;

[0030] S12, use a filter to remove high-frequency noise, and perform Fourier transform on the time-domain signal to convert the signal to the frequency domain;

[0031] S13, use a modal identification method to extract modal parameters.

[0032] Preferably, the collected raw acceleration data is denoised by the FIR band-pass filter in step S12.

[0033] Preferably, the sampling frequency in step S1 is 5-10 kHz.

[0034] In another aspect, the present application also provides a response reconstruction system for the mechanical structure response reconstruction method based on optimal modal response estimation described above, the system comprising a collection unit, a response estimation signal construction unit, a residual signal calculation unit, an optimization objective function construction unit, an optimization objective function solving unit and a structure response reconstruction unit.

[0035] The collection unit is used to collect raw acceleration data by using an acceleration sensor and then perform band-pass filtering and denoising.

[0036] The response estimation signal construction unit is used to construct a calculation model of the response estimation signal.

[0037] The residual signal calculation unit is used to calculate the residual between the known point response estimation signal and the true measurement signal.

[0038] The optimization objective function construction unit is used to construct an optimization objective function according to the spectral characteristics of the residual signal.

[0039] The optimization objective function solving unit is used to solve the optimization objective function using a gradient descent algorithm to obtain the estimated modal responses of unknown points.

[0040] The structure response reconstruction unit is used to superimpose the modal responses of unknown points according to the modal transfer ratio equation to complete the structure response reconstruction.

[0041] Compared with the prior art, the present application has the following beneficial effects:

[0042] (1) The present application provides a mechanical structure response reconstruction method based on optimal modal response estimation, which can be used for response reconstruction of mechanical structures with two very close natural frequencies, ensuring reconstruction accuracy, and can be widely applied to dense modal structure response reconstruction in multiple fields, with high calculation efficiency and high accuracy. Secondly, this method only uses a single sensor to obtain modal responses and can reconstruct the response at any position in the structure. It is efficient and can reduce economic cost.

[0043] (2) The present application provides a mechanical structure response reconstruction system based on optimal modal response estimation, which can reconstruct multiple unknown points by collecting signals of any known point, complete the reconstruction of dense modal of mechanical structure, and is convenient to use, fast in reconstruction speed and high in reconstruction accuracy. BRIEF DESCRIPTION OF DRAWINGS

[0044] Figure 1A flowchart of the present application;

[0045] Figure 2 A schematic block diagram of the system of the present application;

[0046] Figure 3 An experimental diagram for acquiring turbine blade vibration response data in embodiment 1 of the present application;

[0047] Figure 4 A comparison diagram of the measurement results and reconstruction results of point P2 in embodiment 1 of the present application;

[0048] Figure 5 A comparison diagram of the measurement results and reconstruction results of point P3 in embodiment 1 of the present application;

[0049] Figure 6 A six-degree-of-freedom model diagram in embodiment 2 of the present application;

[0050] Figure 7a A displacement response diagram of x1 in embodiment 2 of the present application; Figure 7b A Fourier spectrum diagram of x1 in embodiment 2 of the present application;

[0051] Figure 8a A reconstruction effect diagram of x2, Figure 8b A reconstruction effect diagram of x3, Figure 8c A reconstruction effect diagram of x4. DETAILED DESCRIPTION

[0052] The exemplary embodiments, features and aspects of the present application will be described in detail below with reference to the accompanying drawings. The same reference numbers in the drawings represent functionally identical or similar elements. Although various aspects of the embodiments are shown in the drawings, the drawings are not necessarily drawn to scale unless specifically indicated.

[0053] The present application proposes a mechanical structure response reconstruction method based on optimal modal response estimation, as shown in Figure 1 The method comprises the following steps:

[0054] S1, simulate the actual working condition of the mechanical structure, place an acceleration sensor on any known point on the mechanical structure, and perform band-pass filtering and denoising after collecting the original acceleration data by the acceleration sensor.

[0055] S2, construct a calculation model of the response estimation signal, and construct the response estimation signal of the known point according to the free vibration modal response form as follows:

[0056]

[0057] wherein, ξ i , ω irespectively represent the damping ratio and the natural frequency of the i-th mode, respectively represent the amplitude and the phase of the estimated modal response, represents the known point response estimation signal; t represents the time length of the signal, and n represents the number of modes of the structure.

[0058] S3, calculate the residual error between the known point response estimation signal and the true measurement signal:

[0059]

[0060] wherein x(t) represents the known point true measurement signal, and r(t) represents the residual error signal between the known point response estimation signal and the true measurement signal.

[0061] S4, construct an optimization objective function according to the spectral characteristics of the residual error signal:

[0062]

[0063] wherein r(ω) is the frequency domain form of the residual error signal r(t), ω upp is the upper limit value of the frequency domain of the signal spectrum.

[0064] S5, use a gradient descent algorithm to solve the optimization objective function, to obtain accurate estimates of u i and θ i , and obtain the modal responses of the unknown points according to the estimates of u i and θ i .

[0065]

[0066] wherein the specific steps of obtaining accurate estimates of u i and θ i are as follows:

[0067] S51, for the optimization objective function, obtain the partial derivatives of the objective function with respect to the estimates u and θ , and obtain the gradient of the objective function at this point

[0068] S52, set the learning rate eta.

[0069] S53, update the parameters X according to the formula , wherein the parameter X is a column vector composed of u and θ , and after multiple iterations, accurate estimates of u and θ are obtained, and the estimated modal responses q i (t) are obtained.

[0070] S6, according to the modal transfer equation, superimpose the unknown point modal response of each order in step S5 to complete the reconstruction of the structure response:

[0071]

[0072] wherein subscript k represents a known point, u represents an unknown point, i represents the i-th order modal, φ iu represents the u-th component of the i-th order modal, q ik represents the i-th order modal of the k-th known point, x u (t) is the structure response of the unknown point.

[0073] The specific method of the modal transfer equation is as follows:

[0074] The motion equation of a multi-degree-of-freedom system is as follows:

[0075]

[0076] wherein M is the mass matrix, C is the damping matrix, and K is the stiffness matrix; the column vector x and the force F(t) both have degrees of freedom (DOF), if the number of system degrees of freedom is n, then the matrix size is, and the dimension of the column vector x and the force F(t) is n x 1.

[0077] b. Calculate the system modal parameters:

[0078] The premise of modal superposition is to calculate the characteristic frequency and the corresponding mode shape, and the following eigenvalue equation is used for calculation:

[0079] (K-ω 2 M)φ=0;

[0080] The calculation result is a set of natural frequencies ω i and the corresponding mode shapes φ i , wherein i ranges from 1 to n.

[0081] c. System modal orthogonality:

[0082] The system stiffness matrix has orthogonality:

[0083]

[0084] wherein m qi is called the i-th order modal mass, and similarly for the stiffness matrix:

[0085]

[0086] wherein K qiThe modal stiffness of the i-th order is called, when the system damping satisfies the requirement of proportional damping, C = αM + βK, where α and β are proportional coefficients, and the mode shape also satisfies the orthogonality with respect to the damping matrix:

[0087]

[0088] C qi The modal damping of the i-th order is called.

[0089] The modal superposition method of the system is as follows:

[0090] When the external excitation F(t) = 0, the modal response of each order of the structure can be expressed as:

[0091]

[0092] According to the mode shape orthogonality, the mode shapes of each order are linearly independent, that is, the n mode shapes of the n-dimensional system form a group of bases of the n-dimensional vector space. The coordinate system based on the orthogonal bases is called the modal coordinate system. The displacement response x of the system is expressed as:

[0093] x = φq.

[0094] On the other hand, the present application also provides a response reconstruction system for the mechanical structure response reconstruction method based on the optimal modal response estimation, as shown in Figure 2 The system includes a collection unit 1, a response estimation signal construction unit 2, a residual signal calculation unit 3, an optimization objective function construction unit 4, an optimization objective function solving unit 5, and a structure response reconstruction unit 6.

[0095] The collection unit 1 is used to collect original acceleration data by using an acceleration sensor and then perform band-pass filtering and denoising.

[0096] The response estimation signal construction unit 2 is used to construct a calculation model of the response estimation signal.

[0097] The residual signal calculation unit 3 is used to calculate the residual between the known point response estimation signal and the true measurement signal.

[0098] The optimization objective function construction unit 4 is used to construct an optimization objective function according to the spectral characteristics of the residual signal.

[0099] The optimization objective function solving unit 5 is used to solve the optimization objective function by using a gradient descent algorithm to obtain the estimated modal response of each order of the unknown point.

[0100] The structure response reconstruction unit 6 is used to superimpose the modal response of each order of the unknown point according to the modal transfer ratio equation to complete the structure response reconstruction. Specific embodiment 1

[0102] The embodiment provides a turbine blade response reconstruction method based on optimal modal response estimation, as shown in the following formula (1): Figure 1 The method comprises the following steps:

[0103] S1, in the experiment, the turbine blade is hit by a rubber hammer, so as to simulate the actual working condition of the turbine blade. Three points P1, P2 and P3 on the turbine blade are respectively provided with acceleration sensors. The sensors record the acceleration response at a sampling frequency of 5 kHz. The experimental arrangement is as shown in the following figure: Figure 3 The signal data of the acceleration sensor at the P1 point is used for reconstructing the signal data of the P2 point and the P3 point, and the actual data collected at the P2 point and the P3 point is used for comparison with the reconstructed signal data of the P2 point and the P3 point. The actual data and the reconstructed data are compared to prove the feasibility of the application. In actual application, only the signal of one point needs to be collected, and the signal reconstruction of the remaining multiple points can be completed.

[0104] After the acceleration data is collected, the original acceleration data is denoised through an FIR band-pass filter. Meanwhile, modal information such as the damping ratio, the natural frequency and the mode shape matrix of the structure is obtained according to experimental modal analysis (EMA).

[0105] In the embodiment, the specific steps of step S1 are as follows:

[0106] S11, an excitation mode is selected, an external excitation is applied, and the dynamic response of the structure under the action of the excitation is recorded through a sensor arrangement.

[0107] S12, signal processing: high-frequency noise is removed by using a filter, and Fourier transform is performed on the time-domain signal, so that the signal is converted to the frequency domain.

[0108] S13, modal parameters are extracted by using a modal identification method. Common algorithms include the stationary random excitation method, the subspace method and the frequency response function method.

[0109] In the embodiment, in the experimental process, the P1 point response is taken as an original signal, and the P2 and P3 point responses are reconstructed respectively.

[0110] S2, the calculation formula of the response estimation signal is constructed according to the free vibration response form as follows:

[0111]

[0112] Wherein, ξ i and ω i represent the damping ratio and the natural frequency of the i th mode respectively, i i represent the amplitude and the phase of the estimated modal response respectively, represent the amplitude and the phase of the estimated modal response respectively, represent the amplitude and the phase of the estimated modal response respectively.

[0113] ​S3, calculate the residual between the response estimation signal and the real measurement signal:

[0114]

[0115] where x(t) represents the P1 point measurement signal, and r(t) represents the residual signal between the response estimation signal and the real measurement signal.

[0116] S4, construct an optimization objective function according to the spectral characteristics of the residual signal:

[0117]

[0118] where r(ω) is the frequency domain form of the residual signal r(t), ω upp is the upper limit value of the signal spectrum in the frequency domain.

[0119] S5, use a gradient descent algorithm to solve the optimization function to obtain the accurate parameter u i and θ i estimation values as shown in Table 1, thereby obtaining the estimated modal response of each order:

[0120]

[0121]

[0122] Table 1

[0123]

[0124] S6, complete the structure response reconstruction according to the modal transfer ratio equation:

[0125]

[0126] The reconstruction results of P2 and P3 points in this embodiment and the actual measurement results are shown in Figure 4 and Figure 5 . Specific embodiment 2

[0128] This embodiment proposes a six-degree-of-freedom mass-spring-damper response reconstruction method based on optimal modal response estimation, as shown in Figure 1 , which specifically includes the following steps:

[0129] S1, as shown in Figure 6 , a six-degree-of-freedom mass-spring-damper numerical model is established in this embodiment. The displacements of m1, m2, …, m6 are represented as x1, x2, …, x6. The system parameters are set as follows:

[0130]

[0131] ​The theoretical natural frequencies of the system are shown in Table 2 using M6 and K6.

[0132] Table 2

[0133]

[0134] S2, set the initial displacement of the system as x(0) = [0 0 0 0 0 0] T , and the initial velocity as The sampling frequency is set as 1000 Hz. The Newmark-β algorithm is used to obtain the displacement response of the system. Take x1 as the known signal, and take x2, x3, x4 as the signals to be reconstructed. FIG. 8(a) and FIG. 8(b) show the displacement response of x1 and the corresponding Fourier spectrum. From the spectrum, it can be observed that there is a significant close modal problem.

[0135] S3, construct the response estimation signal according to the free vibration response form

[0136]

[0137] wherein, ξ i , ω i represent the i-th modal damping ratio and natural frequency respectively, represent the amplitude and phase of the estimated modal response respectively, represent the response estimation signal.

[0138] S4, calculate the residual error between the response estimation signal and the true measured signal:

[0139]

[0140] wherein, x(t) represents the P1 point measured signal, and r(t) represents the residual error between the response estimation signal and the true measured signal.

[0141] S5, construct the optimization objective function according to the spectral characteristics of the residual signal:

[0142]

[0143] wherein, r(ω) is the frequency domain form of the residual signal r(t), ω upp is the upper limit value of the signal spectrum in the frequency domain.

[0144] S6, use the gradient descent algorithm to solve the optimization function, and the solving result is shown in Table 3, to obtain the estimated modal responses:

[0145]

[0146] Table 3

[0147]

[0148] S7, complete the structure response reconstruction according to the modal transfer ratio equation:

[0149]

[0150] The displacement response schematic diagram of the embodiment is shown in Figure 7a The Fourier spectrum schematic diagram is shown in Figure 7b The reconstruction results are shown in Figure 8a- Figure 8c

[0151] As can be seen from the above, the method can be used for response reconstruction of mechanical structures with two inherent frequencies very close to each other, ensuring the reconstruction accuracy, and can be widely applied to response reconstruction of dense modal structures in multiple fields, with high calculation efficiency and high accuracy, and can be widely applied in multiple fields.

[0152] The above-described embodiments only describe the preferred embodiments of the present application, and do not limit the scope of the present application. Without departing from the design spirit of the present application, various modifications and improvements to the technical solutions of the present application made by those skilled in the art shall fall within the protection scope determined by the claims of the present application.​

Claims

1. A mechanical structure response reconstruction method based on optimal modal response estimation, characterized in that: It includes the following steps: S1. Simulate the actual working conditions of the mechanical structure, place an accelerometer at any known point on the mechanical structure, use the accelerometer to collect raw acceleration data, and then perform bandpass filtering to remove noise; S2. Construct a calculation model for the response estimation signal. The response estimation signal of the known point is constructed according to the free vibration modal response form as follows: ； in, 、 represent the i-th order modal damping ratio and natural frequency respectively, 、 represent the amplitude and phase of the estimated modal response, respectively, Represents the estimated signal of the known point response; = , t represents the time length of the signal, and n represents the number of structural modes; S3. Calculate the residual between the known point response estimation signal and the true measurement signal: ； in, Represents the real measurement signal of the known point, Residual signal representing the difference between the estimated signal of the known point response and the true measured signal; S4. Construct the optimization objective function according to the spectral characteristics of the residual signal: ； in, is the residual signal The frequency domain form of is the frequency domain upper limit of the signal spectrum; S5. Use the gradient descent algorithm to solve the optimization objective function and obtain and estimated value, and based on and The estimated value of the unknown point is the modal response of each order : ； S6. According to the modal transfer ratio equation, the modal responses of each order of the unknown point in step S5 are superimposed to complete the structural response reconstruction: ； Among them, the subscript represents a known point, represents the unknown point, i represents the i-th mode, represents the ith mode A quantity, Representative The i-th mode of a known point, is the structural response of the unknown point.

2. The mechanical structure response reconstruction method based on optimal modal response estimation according to claim 1, characterized in that: In step S5, and The specific steps to accurately estimate the value are as follows: S51. For the optimization objective function, find its estimate and The partial derivative of the objective function at this point is obtained ; S52, set the learning rate eta; S53, update parameters , where the parameters is and After multiple iterations, an accurate estimate is obtained. and .

3. The mechanical structure response reconstruction method based on optimal modal response estimation according to claim 2, characterized in that: In step S53, according to the formula Update the parameter X.

4. The mechanical structure response reconstruction method based on optimal modal response estimation according to claim 1, characterized in that: Step S1 specifically includes the following sub-steps: S11, selecting an excitation method, applying external excitation, and collecting the dynamic response of the structure under the excitation by arranging acceleration sensors; S12, using a filter to remove high-frequency noise, and performing Fourier transform on the time domain signal to convert the signal into the frequency domain; S13. Extract modal parameters using modal identification method.

5. The mechanical structure response reconstruction method based on optimal modal response estimation according to claim 4, characterized in that: In step S12, the collected raw acceleration data is denoised by using an FIR bandpass filter.

6. The mechanical structure response reconstruction method based on optimal modal response estimation according to claim 4, characterized in that: The sampling frequency in step S1 is 5-10 kHz.

7. A response reconstruction system for the mechanical structure response reconstruction method based on optimal modal response estimation according to claim 1, characterized in that: It includes an acquisition unit, a response estimation signal construction unit, a residual signal calculation unit, an optimization objective function construction unit, an optimization objective function solution unit and a structural response reconstruction unit; The acquisition unit is used to collect raw acceleration data using an acceleration sensor and then perform bandpass filtering to remove noise; The response estimation signal construction unit is used to construct a calculation model of the response estimation signal; The residual signal calculation unit is used to calculate the residual between the known point response estimation signal and the real measurement signal; The optimization objective function construction unit is used to construct an optimization objective function according to the spectrum characteristics of the residual signal; The optimization objective function solving unit is used to solve the optimization objective function using a gradient descent algorithm to obtain estimated modal responses of various orders of unknown points; The structural response reconstruction unit is used to superimpose the modal responses of each order of the unknown point according to the modal transfer ratio equation to complete the structural response reconstruction.

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