CN109685036B - Structural response slow change and transient component separation method - Google Patents

Abstract

The invention relates to a method for separating gradual and transient components of structural response, the method comprising: defining gradual and transient components in the response according to the multi-time scale characteristics of the structural response; calculating a real-valued signal of the measured structural response The complex-valued analytical signal of , and the separation problem of slowly varying and transient components in the structural response is equivalently transformed into a constrained variational optimization problem based on the bandwidth of modal estimation; the alternating direction multiplier method is used to solve it, Enables simultaneous separation of slowly varying and transient components in the structural response. The method provided by the invention has the characteristics of self-adaptation, non-recursion and nonlinearity; the application of the alternate direction multiplier method ensures the nonlinear separation of each response component; the embedded Wiener filter can effectively filter out the potential in the response component noise, thereby ensuring the fidelity of the response separation results. In turn, it provides the required, accurate and independent slow and transient response data for further health monitoring methods such as structural condition assessment.

Description

A Method for Separation of Gradient and Transient Component of Structural Response

technical field

The invention belongs to the fields of civil engineering structure health monitoring and information processing, and in particular relates to a method for separating slow change and transient components of structural response.

Background technique

Civil engineering structures are subjected to various loads during service, which makes the structural response information obtained by sensors contain response components caused by various loads. Taking a large-span space structure as an example, the response of its peripheral structural components is mainly caused by the gravity caused by the structure or the component itself, the temperature response caused by solar radiation (daily temperature difference), sudden cooling and annual temperature difference, and the live load caused by wind load. Response composition. Among the above loads, the time scale of self-weight is the longest; the time scale of annual temperature difference is second; the sudden cooling process usually takes several days to complete, so its time scale is slightly longer than that of daily temperature difference; The wind load has the shortest time scale, and the live load response caused by it has obvious randomness. Taking the frequency component of the response component caused by the daily temperature difference as the limit, the response component lower than the frequency component is defined as the slow component, and the response component higher than the frequency component is the transient component, then the slow component is mainly determined by the temperature. The transient component is mainly caused by the load with short acting time and strong randomness, such as wind load. Response separation is to use the means of signal separation to decompose the measured response information into non-interfering slowly varying temperature response components and transient live load response components.

The most commonly used signal separation method is principal component analysis, which takes the first principal component of the measured response as the subspace where the response component caused by temperature action is located, and realizes the gradual change in the measured response through data reconstruction of the first principal component. Separation of temperature components. However, considering that the principal component analysis is essentially a linear tool, the necessary premise for this method to effectively separate the slow and transient components in the measured response is that the influence of the ambient temperature on it is linear. When dealing with nonlinear problems with complex structural responses, PCA will not be fully effective.

In recent years, some nonlinear signal separation methods for dealing with nonlinear problems have emerged, such as wavelet packet decomposition and empirical mode decomposition. The problem with these methods is that wavelet analysis, as a signal time-frequency analysis method, has a variable time-frequency window, but it is actually a mechanical lattice division of the time-frequency plane, which is not an adaptive signal in nature. Approach. Empirical mode decomposition is a highly adaptive signal separation method of recursive mode type. It separates signals layer by layer by means of cyclic screening, so this method cannot extract different eigenmodes at the same time, which makes its computational efficiency low; , Empirical Mode Decomposition lacks a perfect theoretical basis, and its smoothing of time series is completely based on the time domain characteristics of the signal, which makes it inevitable that there are endpoint effects, the rationality of spline function fitting, and the state aliasing, etc.

By analyzing the existing signal separation methods, it is found that the current methods are still insufficient in realizing the effective separation of the slow and transient components in the response signal, so it is necessary to design an adaptive non-recursive nonlinear nonlinearity with strong anti-noise ability. Signal separation method to achieve synchronous and nonlinear high-efficiency separation of slow and transient components in the measured response information obtained from the sensor, thereby providing the required, accurate, and Separate ramp and transient response component data.

SUMMARY OF THE INVENTION

In view of this, the purpose of the present invention is to overcome the deficiencies of the prior art and provide a method for separating the gradual change of the structural response and the transient components.

In order to achieve the above objects, the present invention adopts the following technical solutions: a method for separating the gradual change of the structural response and the transient component, the method comprising:

According to the multi-time scale characteristics of the measured structural response of the sensor, the gradual and transient components in the structural response are defined;

According to the real-valued signal of the structural response obtained by the sensor, the corresponding complex-valued analytical signal is calculated;

The separation problem of slowly varying and transient components in the structural response is equivalently transformed into a constrained variational optimization problem based on modal estimation bandwidth;

The constrained variational optimization problem is solved by the alternating-direction multiplier method to achieve the simultaneous separation of the slow and transient components of the structural response.

Further, the multi-time-scale characteristics of the structural response are caused by various loads that the civil structure is subjected to during service.

Further, various loads suffered by the civil structure during service include:

Dead loads caused by the mass of the structure itself, temperature effects caused by solar radiation, sudden cooling and annual temperature differences, wind loads caused by air movement, traffic loads caused by vehicles, and live loads caused by human activities.

Further, the response components with different time scale characteristics are simultaneously included in the response signal obtained by the actual measurement of the civil structure by the sensor.

Further, the sensor includes at least one of the following items:

Stress Sensors, Strain Sensors and Displacement Sensors.

Further, the gradual and transient components in the structural response are defined, including:

According to the user's needs, define the frequency limit values of the gradual change and transient components in the structural response to be separated; and define the response components of the structural response whose frequency components are lower than the frequency limit value as the gradual change component, and the response component whose frequency component is higher than or equal to the frequency limit value in the structural response is defined as the transient component.

Further, the complex-valued analytical signal is composed of two parts, a real part and an imaginary part;

Among them, the real part is the real-valued signal of the structural response obtained by the sensor, and the imaginary part is the Hilbert transform of the real-valued signal, and its expression is:

z(t)=x(t)+jH[x(t)] Equation (1)

In formula (1), x(t) is the real-valued signal of the structural response obtained by the sensor; H[x(t)] is the Hilbert transform of x(t); j is the imaginary part unit, j ² =- 1; z(t) is the complex-valued analytical signal of the structural response to the real-valued signal x(t).

Further, the described problem of separating the slowly varying and transient components in the structural response is equivalently converted into a constrained variational optimization problem based on the modal estimation bandwidth, specifically including:

Considering each response component caused by various loads as an eigenmode function u(t) with a specific center frequency ω, the measured response signal x(t) with K response components obtained by the sensor is expressed as:

In formula (2), u _k (t) is the k-th order eigenmode function, and the center frequency ω _k of each order eigenmode function u _k (t) exists such that ω ₁ <ω ₂ <... <ω _K ; K is the total number of eigenmode functions, that is, the number of response components contained in the measured response signal;

The separation of different frequency components in the structural response is achieved by solving a constrained variational problem that minimizes the sum of the estimated bandwidths of each mode;

The constrained variational optimization model based on the modal estimation bandwidth is:

为对时间t求导数；||·||₂为L²范数。In formula (3), {u _k } is the set of eigenmode functions, {u _k }:={u ₁ ,u ₂ ,...,u _K }; {ω _k } is the set of eigenmode functions. Central frequency set, {ω _k }:={ω ₁ ,ω ₂ ,...,ω _K }; δ(t) is the Dirac function;

is the derivative with respect to time t; ||·|| ₂ is the L ² norm.

Further, using the alternating direction multiplier method to solve the constrained variational optimization problem includes:

Iteratively solve the explicit solution of each order eigenmode function u _k and the corresponding center frequency ω _k in the Fourier domain;

Update the Lagrangian multiplier λ(ω) by dual ascending to realize the advancement of the entire iterative update process;

The sum of the residuals of the eigenmode functions of each order obtained by two iterations is used as the convergence condition to separate the response components in the measured response;

Wherein, the convergence condition of the iterative update process is:

和

分别为第k阶本征模态u_k(t)在第n次和n+1次迭代时的傅里叶变换；

为收敛容许残差；In formula (4),

and

are the Fourier transform of the k-th eigenmode u _k (t) at the nth and n+1th iterations, respectively;

is the allowable residual error for convergence;

进行傅里叶逆变换，得K个本征模态函数{u_k}，实现对结构原实测响应中各响应成分分量的分离，并根据所述定义得到实测响应信息中的缓变和瞬变成分。For those that satisfy the convergence condition shown in Eq. (4)

Perform inverse Fourier transform to obtain K eigenmode functions {u _k }, realize the separation of each response component in the original measured response of the structure, and obtain the slow transition and transient in the measured response information according to the definition Element.

The present invention has the beneficial effects that the existing signal separation method is only suitable for dealing with linear problems, the method does not have adaptability, the calculation efficiency is low, and the problems such as end effect and modal aliasing cannot be avoided. The invention aims to take the measured response of civil structures as the research object, and provides an adaptive non-recursive nonlinear signal separation method, which can realize the synchronous separation of the slow and transient components in the measured response information obtained from the sensor; The application of the alternating direction multiplier method ensures the nonlinear separation of each response component; meanwhile, the adaptive Wiener filter embedded in the method can effectively filter out the potential noise in the response component, thus ensuring the fidelity of the response separation result. In turn, it provides the required, accurate and independent slow and transient response component data for further health monitoring methods such as structural condition assessment.

Description of drawings

FIG. 1 is a flow chart of a method for separating the gradual change of the structural response and the transient component of the present invention.

Detailed ways

The technical solutions in the embodiments of the present invention will be described in further detail below with reference to the accompanying drawings in the embodiments of the present invention:

As shown in FIG. 1 , the present invention provides an embodiment of a method for separating gradual and transient components of a structural response, the method comprising the following steps:

S1. According to the multi-time scale characteristics of the measured structural response of the sensor, define the gradual and transient components in the structural response;

S2. Calculate the corresponding complex-valued analytical signal for the structural response real-valued signal obtained by the sensor;

S3. Convert the separation problem of slowly varying and transient components in the structural response equivalently into a constrained variational optimization problem based on modal estimation bandwidth;

S4. Use the alternating direction multiplier method to solve the constrained variational optimization problem to realize the synchronous separation of the slow and transient components in the structural response.

The implementation of step S1 is described in further detail below,

For a large-span space structure in the normal use stage, the response of its peripheral structural components is mainly caused by the gravity caused by the structure or the component itself, the temperature response caused by solar radiation (daily temperature difference), sudden cooling and annual temperature difference, and the activity caused by wind load. load response composition. That is, the real-time structural response obtained by the sensor can be split into:

x(t)=x _G (t)+x _T (t)+x _W (t) Equation (1)

In formula (1), x(t) is the real-time structural response information obtained by the sensor; _xG ( _t ) is the gravitational response component caused by the mass of the structure itself; xT(t) is the daily temperature difference, sudden cooling and annual Temperature response component due to temperature difference; x _W (t) is the live load response component due to wind load.

Considering that among the loads that cause the structural response, the time scale of self-weight is the longest; the time scale of annual temperature difference is the second, which shows a cyclical trend in units of years; the sudden cooling process usually takes several days to complete, so the time scale of The time scale of daily temperature difference is slightly longer than that of daily temperature difference; the time scale of daily temperature difference is shorter, and it shows a periodic trend in days; the time scale of wind load is the shortest, and the live load response caused by it has obvious randomness.

It can be seen that the time scales of various types of loads are different, and the corresponding response components also have different time scale characteristics, that is, each component in the response information obtained by the sensor has different frequency components.

Specifically, the gradual and transient components in the structural response are defined, including:

According to the user's needs, define the frequency limit value of the gradual change and transient components in the structural response; and define the response component of the structural response whose frequency component is lower than the frequency limit value as the gradual change component. Response components of the structural response whose frequency components are higher than or equal to the frequency threshold are defined as transient components.

In this embodiment, the frequency component of the response component caused by the daily temperature difference is used as the frequency limit value, the response component lower than the frequency limit value is defined as the slowly changing component, and the response component higher than or equal to the frequency limit value is defined as the instantaneous component The variable component, that is, the slow component is mainly caused by the temperature effect, and the transient component is mainly caused by the wind load. By applying a signal separation method to the raw response time history obtained from the sensor, the overall structural response can be split into two parts: the slow component due to self-weight and temperature change, and the transient component due to wind load.

The implementation of step S2 is described in further detail below,

The structural response time history x(t) obtained by the sensor is a real-valued signal, which has a conjugate-symmetric spectrum in the frequency domain. However, since the negative spectral part is redundant, it is necessary to convert the real-valued signal into a complex-valued analytical signal with a single-sided spectrum in the frequency domain, thereby removing the negative spectral part of the real-valued signal and retaining its positive spectral part. A complex-valued analytical signal is defined as:

z(t)=x(t)+jH[x(t)] Equation (2)

In formula (2), H[x(t)] is the Hilbert transform of the real-valued signal x(t); j is the imaginary part unit, j ² =-1; z(t) is the structural response real-valued signal Complex-valued analytical signal of x(t).

Specifically, the Hilbert transform for the structural response time history x(t) refers to the output obtained by inputting the real-valued signal x(t) into a linear time-invariant system with an impulse response of 1/(πt), namely The new signal H[x(t)] is obtained by convolving the original signal x(t) with the impulse response 1/(πt). Its mathematical expression is:

In formula (3), h(t) is the impulse response of the linear time-invariant system, h(t)=1/(πt); * is the convolution operator.

The implementation of step S3 is described in further detail below,

The precondition that the structural response separation problem can be equivalently transformed into a constrained variational optimization problem based on modal estimation bandwidth is that the structure is in normal use. Because the stress state of its components is always in the elastic stage during the normal use of the structure, each response component in the total response measured by the sensor satisfies the superposition principle.

Under this premise, according to the multi-time scale characteristics of the measured response signal of the structure, the response components caused by various types of loads are regarded as the eigenmode function u(t) with a specific center frequency ω. In the elastic stage, the response time history x(t) obtained by the sensor can be expressed as the general form shown in equation (4):

In formula (4), u _k (t) is the k-th order eigenmode function, and the center frequency ω _k of each order eigenmode function u _k (t) exists such that ω ₁ <ω ₂ <... <ω _K ; K is the total number of eigenmode functions.

Variational modal decomposition decomposes the original response signal x(t) into K eigenmode function representations by solving K eigenmode functions u _k (t) that minimize the sum of the modal estimated bandwidths. The essence is Convert the signal separation problem into a constrained variational model optimization problem to solve. Since the variational modal decomposition is based on the modal estimation bandwidth expansion, the estimated bandwidth of the modal u _k (t) is first calculated by the following steps:

(1) Convert the real-valued mode u _k (t) into a complex-valued analytical signal through the Hilbert transform, and obtain the unilateral spectrum:

In formula (5), δ(t) is the Dirac function;

的修正，将模态解析信号的中心频率移动至零频：(2) By index

, which shifts the center frequency of the modal-analytical signal to zero frequency:

(3) Calculate the square of the L ² norm of the signal gradient after translation to obtain the estimated bandwidth of the modal function u _k (t):

为对时间t求导数；||·||₂为L²范数。In formula (7),

is the derivative with respect to time t; ||·|| ₂ is the L ² norm.

Based on the estimated bandwidth of each order mode, variational modal decomposition achieves the separation of each response component in the original measured response by solving the following constrained variational problem:

In formula (8), {u _k } is the set of eigenmode functions, {u _k }:={u ₁ ,u ₂ ,...,u _K }; {ω _k } is the set of eigenmode functions. Central frequency set, {ω _k }:={ω ₁ ,ω ₂ ,...,ω _K }.

The quadratic penalty factor α and the Lagrangian multiplier λ(t) are introduced, and the constrained variational problem shown in Eq. (8) is converted into an unconstrained variational problem for solving. The augmented Lagrangian expression is:

The implementation of step S4 is described in further detail below,

达到极小值的最优解。具体的迭代公式为：The alternating direction multiplier method is used to solve the minimization problem shown in Eq. (9). By iteratively updating u _k , ω _k and λ(t), we seek to make the augmented Lagrangian expression

The optimal solution that reaches the minimum value. The specific iteration formula is:

为第n次迭代的第k阶本征模态及其相应的中心频率；

为第n+1次迭代的第k阶本征模态及其相应的中心频率；λⁿ(t)为第n次迭代的更新拉格朗日乘子；τ为噪声容限参数；i为本征模态及其中心频率的阶次。In formulas (10)-(12),

is the k-th eigenmode of the n-th iteration and its corresponding center frequency;

is the k-th eigenmode of the n+1th iteration and its corresponding center frequency; λ ⁿ (t) is the updated Lagrangian multiplier of the n-th iteration; τ is the noise tolerance parameter; i is the The order of the eigenmodes and their center frequencies.

Convert the optimization problems shown in equations (10) and (11) into the integral form in the Fourier domain to solve, and obtain the explicit solutions of the eigenmodes of each order and the corresponding center frequencies in the loop iteration process, while using the dual Updating update Lagrange multipliers.

为第k阶本征模态u_k(t)的傅里叶变换；

为拉格朗日乘子λ(t)的傅里叶变换；二次惩罚因子α控制着维纳滤波器的带宽。In formulas (13)-(15),

is the Fourier transform of the k-th eigenmode u _k (t);

is the Fourier transform of the Lagrange multiplier λ(t); the quadratic penalty factor α controls the bandwidth of the Wiener filter.

The convergence condition of the above iterative update process is:

为收敛容许残差，通常情况下，可取

In formula (16),

To allow the residual error to converge, usually, it is desirable to

进行傅里叶逆变换，得K个本征模态函数{u_k}，实现对结构实测响应信息中各分量的分离，并根据S1中的相关定义，将频率成分低于频率界限值的各响应分量相加构成缓变成分，将频率成分高于或等于频率界限值的各响应分量相加构成瞬变成分。For those satisfying the convergence condition shown in Eq. (16)

Inverse Fourier transform is performed to obtain K eigenmode functions {u _k }, which realizes the separation of each component in the measured response information of the structure, and according to the relevant definition in S1, separates each component whose frequency component is lower than the frequency limit value. The response components are added to form a gradual component, and the response components whose frequency components are higher than or equal to the frequency threshold are added to form an instantaneous component.

It can be understood that, when the separated different frequency components are not superimposed, all the response components contained in the measured response signal can be obtained.

From the perspective of fully mining the useful information in the measured response of civil structures, the present embodiment is only suitable for dealing with linear problems in the existing signal separation method, the method is not adaptive, the calculation efficiency is low, and the end effect and the end point effect cannot be avoided. In order to solve the problems of modal aliasing and other problems, a method of structural response gradual change and transient component separation based on variational modal decomposition is proposed to realize the adaptive and Synchronized non-recursive, non-linear effective separation, which in turn provides the required, accurate and independent slow and transient response component data for further health monitoring methods such as structural condition assessment.

It can be understood that, the same or similar parts in the above embodiments may refer to each other, and the content not described in detail in some embodiments may refer to the same or similar content in other embodiments.

It should be noted that any process or method description in the flowcharts or otherwise described herein may be understood to represent code comprising one or more executable instructions for implementing a particular logical function or step of the process modules, segments, or portions, and the scope of the preferred embodiments of the invention includes alternative implementations, which may be performed out of the order shown or discussed, including in a substantially simultaneous manner or in the reverse order depending on the functionality involved perform functions, which should be understood by those skilled in the art to which embodiments of the present invention pertain.

It should be understood that various parts of the present invention may be implemented in hardware, software, firmware or a combination thereof. In the above-described embodiments, various steps or methods may be implemented in software or firmware stored in memory and executed by a suitable instruction execution system. For example, if implemented in hardware, as in another embodiment, it can be implemented by any one or a combination of the following techniques known in the art: Discrete logic circuits, application specific integrated circuits with suitable combinational logic gates, Programmable Gate Arrays (PGA), Field Programmable Gate Arrays (FPGA), etc.

Those skilled in the art can understand that all or part of the steps carried by the methods of the above embodiments can be completed by instructing the relevant hardware through a program, and the program can be stored in a computer-readable storage medium, and the program can be stored in a computer-readable storage medium. When executed, one or a combination of the steps of the method embodiment is included.

In addition, each functional unit in each embodiment of the present invention may be integrated into one processing module, or each unit may exist physically alone, or two or more units may be integrated into one module. The above-mentioned integrated modules can be implemented in the form of hardware, and can also be implemented in the form of software function modules. If the integrated modules are implemented in the form of software functional modules and sold or used as independent products, they may also be stored in a computer-readable storage medium.

The above-mentioned storage medium may be a read-only memory, a magnetic disk or an optical disk, and the like.

In the description of this specification, description with reference to the terms "one embodiment," "some embodiments," "example," "specific example," or "some examples", etc., mean specific features described in connection with the embodiment or example , structure, material or feature is included in at least one embodiment or example of the present invention. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.

Although the embodiments of the present invention have been shown and described above, it should be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Embodiments are subject to variations, modifications, substitutions and variations.

Claims (7)

1. A method for separating gradual changes in structural response and transient components, characterized in that the method comprises: According to the multi-time scale characteristics of the measured structural response of the sensor, the gradual and transient components in the structural response are defined; According to the real-valued signal of the structural response obtained by the sensor, the corresponding complex-valued analytical signal is calculated; The separation problem of slowly varying and transient components in the structural response is equivalently transformed into a constrained variational optimization problem based on modal estimation bandwidth; The constrained variational optimization problem is solved by the alternating direction multiplier method, and the synchronous separation of the slow and transient components in the structural response is realized; The description defines the gradual and transient components in the structural response, including: According to the user's needs, define the frequency limit values of the gradual change and transient components in the structural response to be separated; and define the response components of the structural response whose frequency components are lower than the frequency limit value as the gradual change component, and define the response component whose frequency component in the structural response is higher than or equal to the frequency limit value as the transient component; The multi-time-scale nature of the structural response is caused by the various loads experienced by civil structures during service. 2 . The method for separating gradual change and transient components of structural response according to claim 1 , wherein the various loads suffered by the civil structure during service include: 2 . Dead loads caused by the mass of the structure itself, temperature effects caused by solar radiation, sudden cooling and annual temperature differences, wind loads caused by air movement, traffic loads caused by vehicles, and live loads caused by human activities. 3. A method for separating gradual change and transient components of structural response according to claim 1, wherein the response components with different time scale characteristics are simultaneously included in the response obtained by actually measuring the civil structure by the sensor in the signal. 4 . The method for separating gradual changes in structural response and transient components according to claim 1 , wherein the sensor comprises at least one of the following items: 5 . Stress Sensors, Strain Sensors and Displacement Sensors. 5 . The method for separating gradual change and transient components of a structural response according to claim 1 , wherein the complex-valued analytical signal is composed of two parts, a real part and an imaginary part; 6 . Among them, the real part is the real-valued signal of the structural response obtained by the sensor, and the imaginary part is the Hilbert transform of the real-valued signal, and its expression is: z(t)=x(t)+jH[x(t)] Equation (1) In formula (1), x(t) is the real-valued signal of the structural response obtained by the sensor; H[x(t)] is the Hilbert transform of x(t); j is the imaginary part unit, j ² =- 1; z(t) is the complex-valued analytical signal of the structural response to the real-valued signal x(t). 6. A method for separating gradual and transient components of structural response according to any one of claims 1 to 4, wherein the problem of separating gradual and transient components in structural response, etc. It can be efficiently transformed into a constrained variational optimization problem based on modal estimation bandwidth, which includes: Considering each response component caused by various loads as an eigenmode function u(t) with a specific center frequency ω, the measured response signal x(t) with K response components obtained by the sensor is expressed as:

In formula (2), u _k (t) is the k-th order eigenmode function, and the center frequency ω _k of each order eigenmode function u _k (t) exists such that ω ₁ <ω ₂ <…<ω The relationship between _K ; K is the total number of eigenmode functions, that is, the number of response components contained in the measured response signal; The separation of different frequency components in the structural response is achieved by solving a constrained variational problem that minimizes the sum of the estimated bandwidths of the eigenmodes; The constrained variational optimization model based on the modal estimation bandwidth is:

In formula (3), {u _k } is the set of eigenmode functions, {u _k }:={u ₁ ,u ₂ ,...,u _K }; {ω _k } is the center frequency of the eigenmode function Set, {ω _k }:={ω ₁ ,ω ₂ ,...,ω _K }; δ(t) is the Dirac function;

is the derivative with respect to time t; ||·|| ₂ is the L2 norm. 7 . The method for separating gradual and transient components of structural response according to claim 6 , wherein the method of solving the constrained variational optimization problem by using the alternating direction multiplier method comprises: 8 . Iteratively solve the explicit solution of each order eigenmode function u _k and the corresponding center frequency ω _k in the Fourier domain; Update the Lagrangian multiplier λ(ω) by dual ascending to realize the advancement of the entire iterative update process; The sum of the residuals of the eigenmode functions of each order obtained by two iterations is used as the convergence condition to separate the response components in the measured response; Wherein, the convergence condition of the iterative update process is:

In formula (4),

and

are the Fourier transform of the k-th eigenmode u _k (t) at the nth and n+1th iterations, respectively;

is the allowable residual error for convergence; For those that satisfy the convergence condition shown in Eq. (4)

Perform inverse Fourier transform to obtain K eigenmode functions {u _k }, realize the separation of each response component in the original measured response of the structure, and obtain the slow transition and transient in the measured response information according to the definition Element.

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