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

Abstract

The invention relates to a mechanical structure response reconstruction method and system based on optimal modal response estimation, and belongs to the field of structure response reconstruction, and the method comprises the steps: constructing a response estimation signal according to a free vibration modal response form; calculating a residual error between the response estimation signal and the real measurement signal; constructing an optimization objective function according to the spectral characteristics of the residual signals; solving the optimization function by using a gradient descent algorithm to obtain an estimated modal response of each order; and structural response reconstruction is completed according to the modal transfer ratio equation. According to the method, a modal superposition method of a linear system is taken as a theoretical basis, an optimal modal response estimation signal is obtained by constructing a response estimation signal and optimizing the spectral characteristics of a residual signal, and finally, response reconstruction is performed through a modal transfer ratio equation, so that the optimal modal response estimation signal is obtained. According to the method, response reconstruction of a structure with dense modal characteristics can be realized based on single sensor data, and the accuracy of response reconstruction is ensured.

Description

Technical Field

[0001] The invention belongs to the field of structural response reconstruction, and in particular relates to a mechanical structure response reconstruction method and system based on optimal modal response estimation. Background Art

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

[0003] At present, the response reconstruction methods are mainly divided into two categories: frequency domain reconstruction method and time domain reconstruction method. The frequency domain method mainly includes the reconstruction method based on the generalized transmissibility matrix. This method reconstructs the response by constructing the generalized transmissibility matrix between the measured points and the points to be reconstructed. Although the frequency domain method uses the data in the frequency response domain and converts the reconstructed frequency domain response back to the time domain through the inverse Fourier transform, this will introduce additional computational costs and numerical truncation errors. The time domain reconstruction method mainly includes the reconstruction method based on the modal transfer matrix and the reconstruction method based on the empirical mode decomposition. The former calculates the modal vibration shape of the structure and constructs the transfer matrix between the measured points and the points to be reconstructed for response reconstruction, but it usually requires that the number of sensors is more than the number of truncated modes; the latter extracts the modal responses of each order through the bandpass filter and the empirical mode decomposition algorithm, and reconstructs the response through the vibration 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 there are challenges in dealing with structures with dense modal characteristics.

[0004] Dense modes have a great influence on the reconstruction of structural response based on empirical mode decomposition. The reason is that if the two natural frequencies of the vibrating structure are very close and their damping ratios are also large, then in the frequency domain, two peaks will overlap with each other, making it difficult to accurately extract the modal responses of each order through a bandpass filter, thereby reducing the accuracy of structural response reconstruction. Therefore, studying methods for the reconstruction of dense modal structural responses has important academic significance and application prospects. Summary of the invention

[0005] In view of the shortcomings of the prior art, the present invention provides a method and system for reconstructing the response of a mechanical structure based on optimal modal response estimation, which can be used to reconstruct the responses of two mechanical structures with very close natural frequencies, ensuring the reconstruction accuracy. It can be widely used in the reconstruction of dense modal structure responses in multiple fields, with high computational efficiency and high accuracy.

[0006] To achieve the above object, the present invention discloses the following technical solution:

[0007] On the one hand, the present invention proposes a mechanical structure response reconstruction method based on optimal modal response estimation, which comprises the following steps:

[0008] S1. Simulate the actual working condition of the mechanical structure, place an acceleration sensor at any known point on the mechanical structure, collect the original acceleration data using the acceleration sensor, and then perform bandpass filtering to remove noise;

[0009] S2. Construct a calculation model for the response estimation signal. Construct the response estimation signal of the known point according to the free vibration modal response form as follows:

[0010]

[0011] Among them, ξ i ,ω i 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;

[0012] S3. Calculate the residual between the known point response estimation signal and the true measurement signal:

[0013]

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

[0015] S4. Construct an optimization objective function based on the spectral characteristics of the residual signal:

[0016]

[0017] Among them, r(ω) is the frequency domain form of the residual signal r(t), ω upp is the frequency domain upper limit of the signal spectrum;

[0018] S5. Use the gradient descent algorithm to solve the optimization objective function and get u i and θ iAccurately estimate the value and according to u i and θ i The estimated value of is used to obtain the modal responses of each unknown point:

[0019]

[0020] 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:

[0021]

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

[0023] Preferably, in step S5, u i and θ i The specific steps to accurately estimate the value are as follows:

[0024] S51. For the optimization objective function, find its value for the estimated quantity and The partial derivative of the objective function at this point is obtained

[0025] S52, set the learning rate eta;

[0026] S53, update parameter X, where parameter X is and After multiple iterations, an accurate estimate is obtained. and

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

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

[0029] S11, selecting an excitation method, applying external excitation, and collecting the dynamic response of the structure under the excitation by arranging acceleration sensors;

[0030] 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;

[0031] S13. Extract modal parameters using modal identification method.

[0032] Preferably, in step S12, the collected raw acceleration data is denoised by using a FIR bandpass filter.

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

[0034] On the other hand, the present invention also provides a response reconstruction system for the above-mentioned mechanical structure response reconstruction method based on optimal modal response estimation, the system comprising 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;

[0035] The acquisition unit is used to collect raw acceleration data using an acceleration sensor and then perform bandpass filtering to remove noise;

[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 real measurement signal;

[0038] The optimization objective function construction unit is used to construct an optimization objective function according to the frequency spectrum 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 estimated modal responses of each order of unknown points;

[0040] 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.

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

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

[0043] (2) The present invention provides a mechanical structure response reconstruction system based on optimal modal response estimation, which can reconstruct multiple dense unknown points by collecting signals from any known point, thereby completing the reconstruction of the dense modes of the mechanical structure. It is easy to use, has a fast reconstruction speed and high reconstruction accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1It is a schematic diagram of the process of the present invention;

[0045] Figure 2 is a schematic block diagram of the system of the present invention;

[0046] Figure 3 This is an experimental diagram of obtaining turbine blade vibration response data in specific embodiment 1 of the present invention;

[0047] Figure 4 It is a schematic diagram comparing the measurement result and reconstruction result of point P2 in specific embodiment 1 of the present invention;

[0048] Figure 5 It is a schematic diagram comparing the measurement result and reconstruction result of point P3 in specific embodiment 1 of the present invention;

[0049] Figure 6 This is a schematic diagram of a six-degree-of-freedom model in specific embodiment 2 of the present invention;

[0050] Figure 7a is a schematic diagram of the displacement response of x1 in specific embodiment 2 of the present invention; Figure 7b is a schematic diagram of the Fourier spectrum of x1 in specific embodiment 2 of the present invention;

[0051] Figure 8a This is a schematic diagram of the reconstruction effect of x2. Figure 8b This is a schematic diagram of the reconstruction effect of x3. Figure 8c This is a schematic diagram of the reconstruction effect of x4. DETAILED DESCRIPTION

[0052] The exemplary embodiments, features and aspects of the present invention will be described in detail below with reference to the accompanying drawings. The same reference numerals in the accompanying drawings represent elements with the same or similar functions. Although various aspects of the embodiments are shown in the accompanying drawings, the drawings are not necessarily drawn to scale unless otherwise specified.

[0053] The present invention proposes a mechanical structure response reconstruction method based on optimal modal response estimation, such as Figure 1 As shown, it includes the following steps:

[0054] S1. Simulate the actual working conditions of the mechanical structure, place an acceleration sensor at any known point on the mechanical structure, use the acceleration sensor to collect raw acceleration data, and then perform bandpass filtering to remove noise.

[0055] S2. Construct a calculation model for the response estimation signal. Construct the response estimation signal of the known point according to the free vibration modal response form as follows:

[0056]

[0057] Among them, ξ i ,ω irepresent 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.

[0058] S3. Calculate the residual between the known point response estimation signal and the true measurement signal:

[0059]

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

[0061] S4. Construct an optimization objective function based on the spectrum characteristics of the residual signal:

[0062]

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

[0064] S5. Use the gradient descent algorithm to solve the optimization objective function and get u i and θ i Accurately estimate the value and according to u i and θ i The estimated value of is used to obtain the modal responses of each unknown point:

[0065]

[0066] Among them, we get u i and θ i The specific steps to accurately estimate the value are as follows:

[0067] S51. For the optimization objective function, find its value for the estimated quantity and The partial derivative of the objective function at this point is obtained

[0068] S52. Set the learning rate eta.

[0069] S53, according to the formula Update parameter X, where parameter X is given by and After multiple iterations, an accurate estimate is obtained. and And get the estimated modal responses q i (t).

[0070] 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:

[0071]

[0072] Where, the subscript k represents a known point, u represents an unknown point, i represents the i-th mode, and φ iu represents the uth component of the i-th mode, q ik represents the i-th mode of the k-th known point, x u (t) is the structural response of the unknown point.

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

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

[0075]

[0076] Among them, 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 degrees of freedom of the system is n, the matrix size is, and the column vector x and the force F(t) dimensions are n×1.

[0077] b Calculate the system modal parameters:

[0078] The premise of modal superposition is to calculate the eigenfrequency and the corresponding vibration mode, which are calculated using the following eigenvalue equation:

[0079] (K-ω 2 M)φ=0;

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

[0081] c System modal orthogonality:

[0082] The system stiffness matrix is ​​orthogonal:

[0083]

[0084] Among them, m qi It is called the i-th order modal mass. Similarly, for the stiffness matrix, we have:

[0085]

[0086] Among them, K qiIt is called the i-th order modal stiffness. When the system damping meets the requirements of proportional damping, C = αM + βK, where α and β are proportional coefficients, and the vibration mode also satisfies orthogonality with respect to the damping matrix:

[0087]

[0088] Among them, C qi is called the i-th order modal damping.

[0089] d System modal superposition method:

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

[0091]

[0092] According to the orthogonality of vibration modes, we know that the vibration modes of each order are linearly independent, that is, the n vibration modes of the n-dimensional system constitute a set of bases of the n-dimensional vector space. The coordinate system based on this orthogonal basis is called the modal coordinate system, and the displacement response x of the system is expressed as:

[0093] x=φq.

[0094] On the other hand, the present invention also provides a response reconstruction system for the above-mentioned mechanical structure response reconstruction method based on optimal modal response estimation, such as Figure 2 As shown, 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 solution unit 5 and a structural response reconstruction unit 6.

[0095] The acquisition unit 1 is used to acquire raw acceleration data using an acceleration sensor and then perform bandpass 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 real measurement signal.

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

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

[0100] The structural response reconstruction unit 6 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. Specific embodiment 1

[0102] This embodiment provides a turbine blade response reconstruction method based on optimal modal response estimation, such as Figure 1 As shown, it includes the following steps:

[0103] S1. In the experiment, the turbine blades are hammered with a rubber hammer to simulate the actual working conditions of the turbine blades. Acceleration sensors are placed at points P1, P2 and P3 on the turbine blades. The sensors record the acceleration response at a sampling frequency of 5kHz. The experimental arrangement is as follows: Figure 3 As shown. The signal data of the acceleration sensor at point P1 is used to reconstruct the signal data at points P2 and P3, and the actual data collected at points P2 and P3 are used to compare with the reconstructed signal data of points P2 and P3. The actual data and the reconstructed data are compared to prove the feasibility of the present application. In practical applications, only the signal of one point needs to be collected to complete the reconstruction of the signals of the remaining multiple points.

[0104] After collecting the acceleration data, the original acceleration data is denoised by FIR bandpass filter. At the same time, the modal information such as the damping ratio, natural frequency, vibration mode matrix, etc. of the structure is obtained according to the experimental modal analysis (EMA).

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

[0106] S11. Select an excitation method, apply external excitation, and record the dynamic response of the structure under the excitation through sensor arrangement.

[0107] S12. Signal processing: Use filters to remove high-frequency noise and perform Fourier transform on the time domain signal to convert the signal to the frequency domain.

[0108] S13. Use modal identification methods to extract modal parameters. Common algorithms include stationary random excitation method, subspace method and frequency response function method.

[0109] In this embodiment, during the experiment, the response at point P1 is used as the original signal, and the responses at points P2 and P3 are reconstructed respectively.

[0110] S2. The calculation formula for constructing the response estimation signal according to the free vibration response form is as follows:

[0111]

[0112] Among them, ξ i ,ω i 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 response estimation signal.

[0113] S3. Calculate the residual between the response estimation signal and the true measurement signal:

[0114]

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

[0116] S4. Construct an optimization objective function based on the spectrum characteristics of the residual signal:

[0117]

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

[0119] S5. Use the gradient descent algorithm to solve the optimization function and obtain the parameter u i and θ i An accurate estimate of

[0120] As shown in Table 1, the estimated modal responses of each order are obtained:

[0121]

[0122] Table 1

[0123]

[0124] S6. Complete the structural response reconstruction according to the modal transfer ratio equation:

[0125]

[0126] The reconstruction results and measured results of points P2 and P3 in this embodiment are as follows: Figure 4 and Figure 5 shown. Specific embodiment 2

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

[0129] S1, such as Figure 6 As shown, 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 respectively. The system parameters are set as follows:

[0130]

[0131] Using M6 and K6, the theoretical natural frequencies of the system are shown in Table 2.

[0132] Table 2

[0133]

[0134] S2. Set the initial displacement of the system to x(0) = [0 0 0 0 0 0] T , the initial velocity is The sampling frequency is set to 1000 Hz. The Newmark–β algorithm is used to obtain the displacement response of the system. x1 is taken as the known signal. x2, x3, and x4 are the signals to be reconstructed. Figure 8(a) and Figure 8(b) show the displacement response of x1 and the corresponding Fourier spectrum. Obvious tight modal problems can be observed from the spectrum.

[0135] S3. Constructed according to the free vibration response form

[0136]

[0137] Among them, ξ i ,ω i 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 response estimation signal.

[0138] S4. Calculate the residual between the response estimation signal and the true measurement signal:

[0139]

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

[0141] S5. Construct an optimization objective function based on the spectrum characteristics of the residual signal:

[0142]

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

[0144] S6. Use the gradient descent algorithm to solve the optimization function. The solution results are shown in Table 3, and the estimated modal responses of each order are obtained:

[0145]

[0146] Table 3

[0147]

[0148] S7. Complete the structural response reconstruction according to the modal transfer ratio equation:

[0149]

[0150] The displacement response diagram of this embodiment is as follows Figure 7a As shown, the Fourier spectrum diagram is as follows Figure 7b The reconstruction results are shown in Figure 8a-8c shown.

[0151] In summary, it can be seen that the method of this method can be used for the reconstruction of the response of two very close natural frequencies of mechanical structures to ensure the reconstruction accuracy. It can be widely used in the reconstruction of dense modal structural responses in multiple fields. It has high computational efficiency and high accuracy, and can be promoted and applied in multiple fields.

[0152] The embodiments described above are only descriptions of the preferred implementation modes of the present invention, and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope determined by the claims of the present invention.

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 condition of the mechanical structure, place an acceleration sensor at any known point on the mechanical structure, collect the original acceleration data using the acceleration sensor, and then perform bandpass filtering to remove noise; S2. Construct a calculation model for the response estimation signal. Construct the response estimation signal of the known point according to the free vibration modal response form as follows: Among them, ξ i ,ω i 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: Where x(t) represents the true measurement signal of the known point, and r(t) represents the residual signal between the response estimation signal of the known point and the true measurement signal; S4. Construct an optimization objective function based on the spectral characteristics of the residual signal: Among them, r(ω) is the frequency domain form of the residual signal r(t), ω upp is the frequency domain upper limit of the signal spectrum; S5. Use the gradient descent algorithm to solve the optimization objective function and get u i and θ i The estimated value of u i and θ i The estimated value of the unknown point modal response q i (t): 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: Where, the subscript k represents a known point, u represents an unknown point, i represents the i-th mode, and φ iu represents the uth component of the i-th mode, q ik represents the i-th mode of the k-th known point, x u (t) 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, u is obtained i and θ i The specific steps to accurately estimate the value are as follows: S51. For the optimization objective function, find its value for the estimated quantity and The partial derivative of the objective function at this point is obtained S52, set the learning rate eta; S53, update parameter X, where parameter X 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 1, 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 is characterized in that: In step S12, the collected raw acceleration data is denoised by using a 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 frequency 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 each order 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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