CN112307932B - Parameterized full-field visual vibration modal decomposition method - Google Patents
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Abstract
The invention discloses a parameterized full-field visual vibration modal decomposition method, which relates to the technical field of vibration modal parameter identification, and comprises the following steps: step 1, parameterizing a time domain modal coordinate; step 2, parameterizing a spatial domain modal shape; step 3, constructing a time-space domain parameterized modal stack model; step 4, performing time-space domain joint optimization on the parameterized modal stack model; and 5, decomposing a parameter matrix and reconstructing a mode. By implementing the method, the difficulty of high-dimensional degree of freedom calculation caused by the advantage of high visual spatial resolution is avoided, the problems of accurate decomposition and parameter identification of high-dimensional and full-field visual vibration modes are solved, the complex time-space domain background noise in the visual vibration data is inhibited, and the calculation capacity and the calculation efficiency are improved.
Description
Technical Field
The invention relates to the technical field of vibration modal parameter identification, in particular to a parameterized full-field visual vibration modal decomposition method.
Background
In a traditional operation modal analysis method, a vibration signal acquired by an accelerometer or a laser vibration meter is generally adopted for modal analysis, a small number of degrees of freedom needs to be calculated, and only dozens to hundreds of degrees of freedom exist under dense sensor arrangement, so that the spatial resolution of modal parameter identification is very low. In contrast, the total number of degrees of freedom required to be calculated for full-field modal parameter identification based on visual vibration measurement exceeds thousands, and both the calculation capability of modal identification and the hardware condition of a computer bring huge challenges. The traditional operation modal analysis methods, such as frequency domain decomposition, time domain decomposition, random subspace method, etc., are only suitable for modal parameter identification calculation with a small number of degrees of freedom, because they mostly rely on algorithms of calculating cross power density, principal component analysis or singular value decomposition, and other eigenvalue decomposition, and in the case of high dimensional degree of freedom brought by visual vibration full field displacement, the matrix to be calculated is high dimensional or even ultra-high dimensional, such as calculating matrix eigenvalue or singular value decomposition of tens of thousands of times tens of thousands of scales, which is hardly realized under the present calculation conditions and the calculation cost is too high. Therefore, for the operation modal analysis based on the visual vibration measurement, it is urgently needed to develop an effective and rapid high-dimensional degree-of-freedom processing method so as to solve the problem of the visual full-field modal analysis, improve the calculation efficiency, and save the calculation cost and the calculation time.
The visual vibration modal analysis method in the prior art almost avoids the computational difficulty of high-dimensional degree of freedom in the visual full-field modal identification without exception, and the method still processes a small number of degrees of freedom. Specifically, in the current visual vibration modal identification method, only a small amount of preset motions of reference pixels are selected for calculating modal parameters such as modal shape. The method greatly weakens the advantage that the visual vibration measurement technology has high spatial resolution, and considers the dynamic characteristics of the structure, such as fatigue damage, cracks, wear marks and the like, so that the method mostly has local characteristics and usually needs a modal mode with higher spatial resolution to detect and position the damage. Therefore, how to fully utilize the advantage of high spatial resolution and solve the problem of identifying the parameters of the visual full-field vibration mode, and improve the calculation capability and the calculation efficiency is an important problem to be solved in the visual vibration analysis.
Therefore, those skilled in the art are dedicated to developing a parameterized full-field visual vibration mode decomposition method, which solves the computational difficulty of high-dimensional degree of freedom due to the advantage of high spatial resolution of vision, solves the problems of accurate decomposition and parameter identification of high-dimensional and full-field visual vibration modes, and suppresses complex time-space domain background noise existing in visual vibration data.
Disclosure of Invention
In view of the above-mentioned drawbacks of the prior art, the technical problems to be solved by the present invention are: how to handle the high-dimensional degree-of-freedom computational problem, and how to suppress the background noise problem in the visual vibration data.
In order to achieve the above object, the present invention provides a parameterized full-field visual vibration modal decomposition method, which comprises the following steps:
step 2, parameterizing a spatial domain modal shape;
step 3, constructing a time-space domain parameterized modal stack model;
step 4, performing time-space domain joint optimization on the parameterized modal stack model;
and 5, decomposing a parameter matrix and reconstructing a mode.
Further, in step 1, the mode coordinate q of the ith mode is i (t) is:
q i (t)＝a i (t)cos(2πf i t+ψ i0 ),t＝1,2,...,N T
wherein, a i (t) is amplitude, f i To have a damped natural frequency, # i0 Is a constant phase, t is time, N T Is the number of sampling points.
Further, the step 1 further comprises: estimation value of damping natural frequency from signal power spectrum density by using peak value selection algorithm
Wherein the content of the first and second substances,
b i (t) and
for two new amplitudes to be estimated, expand them into a Fourier series:
wherein the content of the first and second substances,
and
two amplitude-expanded fourier coefficients, respectively; l is the Fourier order; f 0 Is the frequency resolution, and the calculation formula is F 0 ＝f s 2N T ，f s Is the signal sampling frequency.
Further, the parameterized modal coordinates q i (t) discretizing to obtain the following formula:
q i ＝ρ i T Θ i T
Θ i ＝[A i B C i B]
wherein the content of the first and second substances,
where (·) T is the transposed symbol and diag [ · ] is the diagonal matrix.
Further, the step 2 further comprises: using two-dimensional Fourier series to convert the i-th order mode shape phi i,θ (x, y) is expanded in a space domain, and the expression is as follows:
wherein θ is a horizontal direction or a vertical direction; n and M are two-dimensional Fourier orders along the x-axis and y-axis;
are Fourier coefficients; f w =2 π/Gw and F h ＝2π/Gh
Fundamental frequencies of the x-axis and the y-axis, respectively; h and w are the height and width of the video image, respectively.
Further, the step 2 further comprises: discretizing the parameterized ith-order spatial mode shape to obtain:
φ i ＝Hz i
φ i ＝Hz i
wherein the content of the first and second substances,
further, the step 3 further comprises: the step 3 further comprises: using parameterized time domain modal coordinates q i And parameterized spatial domain mode shape phi i Establishing a time-space domain modal superposition model, wherein the expression is as follows:
wherein Q is the number of modes, and
Ω＝[Ω 1 … Ω Q ]
Θ＝[Θ 1 … Θ Q ]。
further, the step 4 further includes: adopting a target optimization criterion of double regular parameters, wherein an optimization parameter matrix omega is as follows:
wherein | · | purple sweet F An F norm representing a matrix; the last three F norms are regular terms to solve the ill-conditioned problem; lambda [ alpha ] 1 And λ 2 Is twoA regularization parameter.
Further, the parameter matrix
Comprises a plurality of sub-parameter matrixes,
using singular value decomposition method to each sub-matrix
Decomposing to obtain:
wherein
Is the maximum singular value, mu i And upsilon i The corresponding left singular vector and right singular vector.
Further, the final modal shape and modal coordinate are obtained through reconstruction, and the expression is as follows:
wherein, the first and the second end of the pipe are connected with each other,
in order to estimate the mode shape of the mode,
is the estimated modal coordinates; i | · | purple wind ∞ Representing the infinite norm of the vector.
Compared with the prior art, the invention at least has the following beneficial technical effects:
1. the parameterized full-field visual vibration modal decomposition method can convert high-dimensional freedom data into low-dimensional parameter matrix decomposition, and can efficiently solve the problems of insufficient computing capacity and low computing efficiency.
2. The invention provides a time-space domain dual-regular parameter target optimization criterion, which can inhibit time domain and space domain noises simultaneously and improve the estimation precision of modal parameters.
The conception, specific structure and technical effects of the present invention will be further described in conjunction with the accompanying drawings to fully understand the purpose, characteristics and effects of the present invention.
Drawings
FIG. 1 is a basic framework diagram of the parametric full-field visual vibration modal decomposition method of the present invention;
FIG. 2 is the top 4 th order full field modal shape estimated by the parameterized full field visual vibration modal decomposition method of the present invention;
fig. 3 shows the first 4 th order full-field mode shape reconstructed by Time Domain Decomposition (TDD) according to an embodiment of the present invention.
Detailed Description
The technical contents of the preferred embodiments of the present invention will be made clear and easily understood by referring to the drawings attached to the specification. The present invention may be embodied in many different forms of embodiments and the scope of the invention is not limited to the embodiments set forth herein.
As shown in fig. 1, a parameterized full-field visual vibration mode decomposition method includes the following steps:
in the time domain, each order of modal coordinates are subjected to parametric expansion by using a one-dimensional Fourier series to obtain a series of time domain parameters:
step 2, parameterizing the airspace modal shape;
in the airspace, parameterizing and unfolding each level of modal shape by using two-dimensional Fourier series to obtain a series of airspace parameters:
step 3, constructing a time-space domain parameterized modal stack model;
carrying out space-time coupling on the parameterized modal coordinate of the time domain and the parameterized modal shape of the space domain, and establishing a time-space domain parameterized modal superposition model:
the obtained parameter matrix omega comprises a plurality of sub-parameter matrices,
Ω＝[Ω 1 … Ω Q ]
wherein omega i ＝z i ρ i T ，i＝1,…,Q。
Step 4, performing time-space domain joint optimization on the parameterized modal stack model;
constructing a double regular parameter target optimization criterion for optimizing and solving a parameter matrix omega;
the solution is as follows:
step 5, parameter matrix decomposition and modal reconstruction;
performing Singular Value Decomposition (SVD) on the submatrix of the parameter matrix obtained by optimization solution to obtain a left eigenvector mu corresponding to the maximum eigenvalue i And a right feature vector v i ：
And then reconstructing to obtain a final modal shape and a modal coordinate, wherein the expression is as follows:
the modal shape results of the final reconstruction of each order are shown in fig. 2. The reconstructed modal shape is very close to the real modal shape, and the error is very small, so that the method has very high modal parameter identification accuracy.
To verify the superiority of the present invention, the modal shape results extracted by Time Domain Decomposition (TDD) are shown in fig. 3. It can be seen that the conventional TDD method has more noise interference.
Further, the modal confidence of each order of modality reconstructed by the present invention and TDD method is given, as shown in table 1.
Table 1: mode confidence contrast between the invention and TDD method
Table 1 shows that the modal confidence of the present invention is higher than that of the conventional TDD method, which indicates that the modal parameter identification result of the present invention is more accurate.
Finally, to verify the computational efficiency and the computational power of the present invention, the computation times of the present invention and the frequency domain decomposition method (FFD) and the time domain decomposition method (TDD) in different degrees of freedom are compared, as shown in table 2. With the increase of the size of the video image, the number of degrees of freedom required to be calculated is rapidly increased, while the traditional frequency domain decomposition (FFD) method and Time Domain Decomposition (TDD) method are difficult to calculate or have long calculation time under the number of hundreds or thousands of degrees of freedom, but the invention can still calculate under the number of tens of thousands or even higher degrees of freedom, and the calculation time is only a few seconds, which shows that the invention has strong calculation capability and high calculation efficiency.
Table 2: comparison of computing time and computing power of different modal parameter identification methods
In conclusion, the parameterized full-field visual vibration mode decomposition method is adopted, the high-dimensional degree of freedom calculation difficulty caused by the advantage of high visual spatial resolution is avoided, the problems of accurate decomposition and parameter identification of the high-dimensional and full-field visual vibration modes are solved, the complex time-space domain background noise existing in the visual vibration data is restrained, and the calculation capacity and the calculation efficiency are improved.
The foregoing detailed description of the preferred embodiments of the invention has been presented. It should be understood that numerous modifications and variations could be devised by those skilled in the art in light of the present teachings without departing from the inventive concepts. Therefore, the technical solutions that can be obtained by a person skilled in the art through logical analysis, reasoning or limited experiments based on the prior art according to the concepts of the present invention should be within the scope of protection determined by the claims.
Claims (7)
1. An application of a parameterized full-field visual vibration modal decomposition method in processing of a high spatial resolution video image, the high spatial resolution video image having a time domain modal coordinate and a space domain modal shape, the method comprising the steps of:
step 1, parameterizing the time domain modal coordinate;
mode coordinate q of ith mode i (t) is:
q i (t)＝a i (t)cos(2πf i t+ψ i0 ),t＝1,2,...,N T
wherein, a i (t) is amplitude, f i To have a damped natural frequency, # i0 Is a constant phase, t is time, N T Is the number of sampling points;
estimation value of damping natural frequency from signal power spectrum density by using peak value selection algorithm
Wherein the content of the first and second substances,
b i (t) and
for two new amplitudes to be estimated, expand them into a Fourier series:
wherein, the first and the second end of the pipe are connected with each other,
and
two amplitude-expanded fourier coefficients, respectively; l is the Fourier order; f 0 Is the frequency resolution, and the calculation formula is F 0 ＝f s /2N T ，f s Is the signal sampling frequency;
step 2, parameterizing the airspace modal shape;
using two-dimensional Fourier series to convert the i-th order mode shape phi i,θ (x, y) is expanded in a space domain, and the expression is as follows:
wherein θ is a horizontal direction or a vertical direction; n and M are two-dimensional Fourier orders along the x-axis and y-axis;
are the Fourier coefficients; f w =2 π/Gw and
fundamental frequencies of the x-axis and the y-axis, respectively; h and w are the height and width of the video image respectively;
step 3, constructing a time-space domain parameterized modal stack model;
step 4, performing time-space domain joint optimization on the parameterized modal stack model;
and 5, decomposing a parameter matrix and reconstructing a mode.
2. Use of a method of full field visual vibration modality decomposition of parameterisation according to claim 1, in video image processing with high spatial resolution, characterised in that the parameterised modality coordinates q, are adapted to i (t) discretizing to obtain the following formula:
Θ i ＝[A i B C i B]
wherein, the first and the second end of the pipe are connected with each other,
where (·) T is the transposed symbol and diag [. Cndot. ] is the diagonal matrix.
3. Use of a method of parametric full-field visual vibration modality decomposition according to claim 2, characterized in that said step 2 further comprises: discretizing the parameterized ith-order spatial mode shape to obtain:
φ i ＝Hz i
φ i ＝Hz i
wherein, the first and the second end of the pipe are connected with each other,
4. use of a method of parametric full-field visual vibration mode decomposition according to claim 3 in high spatial resolution video image processing, wherein said step 3 further comprises: using parameterized time-domain modal coordinates q i And parameterized spatial domain mode shape phi i Establishing a time-space domain modal superposition model, wherein the expression is as follows:
wherein Q is the number of modes, and
Ω＝[Ω 1 … Ω Q ]
Θ＝[Θ 1 … Θ Q ]。
5. use of a method of parametric full-field visual vibration modality decomposition according to claim 4, in high spatial resolution video image processing, wherein said step 4 further comprises: adopting a target optimization criterion of double regular parameters, wherein an optimization parameter matrix omega is as follows:
wherein | · | charging F An F norm representing a matrix; the last three F norms are regular terms to solve the ill-conditioned problem; lambda 1 And λ 2 Are two canonical parameters.
6. Use of a method of decomposition of a full-field visual vibration mode parametrized according to claim 5 for the processing of high spatial resolution video images, characterized in that the parameter matrix
Comprises a plurality of sub-parameter matrixes,
using singular value decomposition method to each sub-matrix
Decomposing to obtain:
wherein
Is the maximum singular value, mu i And upsilon i The corresponding left singular vector and right singular vector.
7. Use of a parametric full-field visual vibration mode decomposition method according to claim 6 in high spatial resolution video image processing to reconstruct the mode shape and the mode coordinates, expressed as:
wherein the content of the first and second substances,
in order to estimate the mode shape of the mode,
is the estimated modal coordinates; i | · | purple wind ∞ Representing an infinite norm of the vector.
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