A method for identifying multiple local nonlinear locations

Abstract

A method for identifying multiple local nonlinear locations includes: acquiring a single broadband excitation F and the displacement response x of each degree of freedom i under a normal excitation level that induces nonlinearity. i (t) and velocity response are divided into subsystems s with degree of freedom i as the center. i The invention determines the nonlinearity of each subsystem and, based on the actual connections within the system, achieves preliminary nonlinear location identification. Then, according to the nonlinear basis functions corresponding to the maximum values ​​of the integrity indices for multiple degrees of freedom, it further identifies whether the connections that could not be determined in step 2 exhibit local nonlinearity. This invention achieves accurate location identification of systems with multiple local nonlinearities, requiring only system input and output data under a single broadband excitation force to calculate the location information of local nonlinearities within the system. Furthermore, it is not limited by the type or number of nonlinearities, or the excitation location, and maintains good location identification performance even in noisy environments.

Description

Technical Field

[0001] This invention belongs to the field of nonlinear system identification and relates to a method for identifying multiple local nonlinear locations. Background Technology

[0002] In practical engineering applications, the dynamic behavior of many mechanical and civil structural systems can be described by multi-free system models, expressed in the form of mass-damped-spring equations. All such structural systems are prone to local nonlinearities due to long-term use or harsh working environments. Common local nonlinearities include breathing cracks in structures, radial clearances in rolling bearings of rotor systems, hysteresis at structural connections and boundaries, and Hertzian contact. Therefore, identifying the location of local nonlinearities in a system is of significant practical importance for structural damage localization and health monitoring. Furthermore, as a crucial component of the nonlinear structural modeling process, its accuracy has significant academic and engineering value for subsequent nonlinear model identification and nonlinear parameter identification.

[0003] The identification of local nonlinear locations in structures is an important problem, and many different techniques have been developed to obtain this information. Currently, the main methods include inverse path methods, model update methods, force-state mapping methods, and sparse identification methods. Most of these techniques require multiple measurements to obtain system response data under different excitation levels, or require numerical integration or differentiation of sampled data to obtain displacement, velocity, and acceleration data. This can introduce more errors, especially in noisy environments. Recently, some scholars have proposed identifying the local nonlinear locations of a system based on the difference in the order of the subsystem's state-space model, effectively solving the above problems. However, this method still has certain limitations. For structures with multiple local nonlinearities, it cannot achieve accurate nonlinear location identification. Summary of the Invention

[0004] In view of some or all of the above-mentioned technical problems existing in the prior art, the present invention provides a method for identifying multiple local nonlinear positions, which solves the problem that it is impossible to accurately identify the local nonlinear positions of a system when the system has multiple local nonlinearities.

[0005] To achieve the above objectives, the present invention provides a method for identifying multiple local nonlinear locations, comprising the following steps:

[0006] Step 1: Under the normal excitation level that induces nonlinearity, obtain a single broadband excitation F and the displacement response x of i under this single broadband excitation. i (t) and velocity response

[0007] Step 2: Divide the system into subsystems s with degree of freedom i as the center.i And determine the nonlinearity of each subsystem, and combine the actual connection relationship of the system to achieve preliminary nonlinear position identification;

[0008] Step 3: Based on the nonlinear basis functions corresponding to the maximum values ​​of the integrity indices of multiple degrees of freedom, further identify whether there is local nonlinearity in the connections that could not be determined in Step 2.

[0009] The specific details of step 1 are as follows:

[0010] Obtain a single broadband excitation F under a normal excitation level that induces nonlinearity;

[0011] Obtain the displacement response x of each degree of freedom i of the system under the single broadband excitation force. i (t) and velocity response

[0012] The specific details of step 2 are as follows:

[0013] Based on the system structure, subsystems s are divided. i ;

[0014] Construct a discrete-time state-space model of the subsystem and calculate the logarithmic difference Δ between adjacent singular values ​​σ. j,j+1

[0015] Calculate the linear exponent NLI of each subsystem i ;

[0016] According to the linear exponent NLI i Determine the nonlinearity of each subsystem:

[0017] Based on the nonlinear detection results of each subsystem and the actual connection relationships of the system, a preliminary nonlinear position identification is performed on the nonlinear system, revealing questionable connections that cannot be determined.

[0018] The specific details of step 3 are as follows:

[0019] Establish a dictionary set D of nonlinear basis functions at questionable connections;

[0020] The nonlinear basis functions in the dictionary set D are selected sequentially to construct the system input vector.

[0021] Determine the integrity index R of the q-th nonlinear basis function. i,q ;

[0022] The nonlinear basis functions corresponding to the maximum values ​​of the integrity indices for multiple degrees of freedom are the nonlinear connections.

[0023] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0024] 1. This invention employs multiple stages for system local nonlinearity location identification, achieving accurate location identification for systems with multiple local nonlinearities;

[0025] 2. This invention does not require multiple measurements to obtain system response data under different excitation levels. It only requires system input and output data under a single broadband excitation force to calculate the local nonlinear position information in the system.

[0026] 3. The present invention has good applicability and is not limited by nonlinear type, number of nonlinearities, or excitation position, and still has good position recognition effect in noisy environments. Attached Figure Description

[0027] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the invention and, together with the description, serve to explain the principles of the invention. It is obvious that the drawings described below are merely some embodiments of the invention, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.

[0028] Figure 1 This is a flowchart illustrating the steps of a method for identifying multiple local nonlinear locations according to the present invention.

[0029] Figure 2 This is a flowchart of a method for identifying multiple local nonlinear locations according to the present invention;

[0030] Figure 3 This is a schematic diagram of a four-degree-of-freedom chain structure with two local nonlinearities in an exemplary embodiment of the multi-local nonlinearity location identification method of the present invention.

[0031] Figure 4 The time history response curves of the input excitation F and displacement of each degree of freedom i for a four-degree-of-freedom chain structure under 25dB noise pollution are shown.

[0032] Figure 5 The top ten singular values ​​and linear exponent NLI of each subsystem under 25dB noise pollution. i ;

[0033] Figure 6 The integrity index for degrees of freedom 1 and 2 in the first iteration under 25dB noise pollution;

[0034] Figure 7 The integrity index for degrees of freedom 1 and 2 in the second iteration under 25dB noise pollution. Detailed Implementation

[0035] Exemplary embodiments will now be described more fully with reference to the accompanying drawings. However, these exemplary embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, they are provided so that the invention will be more comprehensive and complete, and will fully convey the concept of the exemplary embodiments to those skilled in the art. The same reference numerals in the drawings denote the same or similar structures, and therefore their detailed description will be omitted.

[0036] Although relative terms such as "up" and "down" are used in this specification to describe the relative relationship of one component of an icon to another, these terms are used only for convenience, such as according to the orientation of the examples shown in the accompanying drawings. It is understood that if the icon's arrangement is flipped so that it is upside down, the component described as "up" will become the component described as "down." Other relative terms such as "high," "low," "top," "bottom," "left," and "right" also have similar meanings. When a structure is "up" of another structure, it may mean that the structure is integrally formed on the other structure, or that the structure is "directly" mounted on the other structure, or that the structure is "indirectly" mounted on the other structure through another structure.

[0037] The terms “a,” “—,” and “the” are used to indicate the existence of one or more elements / components / etc.; the terms “including” and “having” are used to indicate an open-ended meaning of inclusion and that there may be other elements / components / etc. in addition to the listed elements / components / etc.

[0038] The embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0039] Figure 1 This is a flowchart illustrating the steps of a method for identifying multiple local nonlinear locations according to the present invention, which is implemented in accordance with the following steps:

[0040] Step 1: Under the normal excitation level that induces nonlinearity, obtain a single broadband excitation F and the displacement response x of each degree of freedom i under this single broadband excitation. i (t) and velocity response

[0041] Step 2: Divide the system into subsystems s with degree of freedom i as the center. i And determine the nonlinearity of each subsystem, and combine the actual connection relationship of the system to achieve preliminary nonlinear position identification;

[0042] Step 3: Based on the nonlinear basis functions corresponding to the maximum values ​​of the integrity indices of multiple degrees of freedom, further identify whether there is local nonlinearity in the connections that could not be determined in Step 2.

[0043] Assuming the system has p degrees of freedom, obtain the displacement response x of each degree of freedom i with a single broadband input excitation F.i (t) and velocity response

[0044] Based on the nonlinear system structure, subsystems s are divided with degree of freedom i as the center. i

[0045] s i ={i1,...,i r} (1)

[0046] In the formula, i r Let represent all r degrees of freedom directly connected by springs and dampers with degree i as the center, and 1≤r≤p-1.

[0047] Representing the subsystem as a discrete-time state-space model in general form

[0048]

[0049] Among them, subscript · k Indicates the variable at time t k The values ​​at time points; A, B, C, and D are the system matrix, system input matrix, system output matrix, and direct transfer matrix, respectively; h, y, and u are the system state vector, system output vector, and system input vector, respectively; w and v are the system process noise vector and output measurement noise vector, respectively, and it is assumed that w and v are both zero-mean white noise.

[0050] Iterative formula (2) and written in matrix form

[0051] Y=ΓH+Θ D U+Θ S W+V (3)

[0052] In the formula, Γ is the extended observable matrix; Θ D and Θ S These are the lower triangular Toeplitz matrices representing the deterministic and stochastic aspects of the unknown system, respectively.

[0053] Furthermore, based on the oblique projection, auxiliary variables, and singular value decomposition techniques of the stochastic subspace method, the first ten singular value results for each subsystem were obtained.

[0054]

[0055] In the formula, t is the user-defined maximum order for solving singular values; Let P be the oblique projection matrix, and P = [U T Y T ] T The auxiliary variable is used to eliminate the system input term Θ in equation (4). D U, process noise term Θ SW and the measurement noise term V; W1 and W2 are full-rank weighted matrices.

[0056] Calculate the logarithmic difference between adjacent singular values ​​σ

[0057]

[0058] The calculated differences are normalized to zero mean and unit variance to ensure they conform to a standard normal distribution. The linear exponent is defined as s for the subsystem. i The logarithmic difference between the second and third singular values ​​is normalized to zero mean and unit variance, i.e.

[0059]

[0060] Select threshold C Δ =2. A subsystem is classified as nonlinear if its linear exponent is less than this value; otherwise, it is classified as linear. Combining this with the actual connections within the system allows for the identification of nonlinear locations in structures with single points of local nonlinearity. However, for systems with multiple points of local nonlinearity, questionable connections may arise that cannot be determined.

[0061] Furthermore, a dictionary set D of nonlinear basis functions is established at the points of doubtful connections.

[0062] D = {g1 g2 … g} q … g l} (7)

[0063] In the formula, g q The q-th nonlinear basis function to be considered; l is the number of nonlinear basis functions to be considered as defined by the user.

[0064] The system input vector is constructed by sequentially selecting nonlinear basis functions from the dictionary set.

[0065]

[0066] In the formula, X i (ω) and G q (ω) represent the frequency domain signals obtained by Fourier transform of the displacement response vector of degree i and the nonlinear basis function of term q, respectively; For X i (ω) and G q The system input vector is composed of (ω). Substituting the displacement responses of different degrees of freedom into equation (8) yields different system input vectors.

[0067] Based on the theory of re-coherence functions, the integrity index is defined as the mean of the re-coherence function over a certain frequency range.

[0068]

[0069] in, P FF These are the autopower spectrum matrices for the input and output, respectively; The cross-power spectrum matrix of the input and output; H This represents the Hermitian transpose.

[0070] If the nonlinear basis function is not selected correctly, residuals will appear, resulting in a smaller integrity index value; conversely, the integrity index value will be larger. Therefore, the nonlinear basis function corresponding to the maximum value of the integrity index is selected. The connection corresponding to the selected nonlinear basis function is a nonlinear connection, while the connection not selected is a linear connection, thus realizing a further judgment on the connection that could not be judged in the previous stage. For ease of description, the calculation process of equations (7) to (9) is recorded as one iteration, and the integrity index obtained by substituting the displacement response vector of degree i into equations (8) and (9) is called the integrity index of degree i. It is worth noting that in order to avoid misjudgment caused by overfitting of the noise term, the nonlinear basis functions corresponding to the maximum values ​​of the integrity index of multiple degrees of freedom should be compared. If the same nonlinear basis function is selected for different degrees of freedom in the current iteration, it indicates that the doubtful connection may contain multiple local nonlinearities, and further iteration is needed to select nonlinear basis functions; otherwise, the iteration calculation is stopped.

[0071] To further understand the present invention, Figure 2 This is a flowchart of a method for identifying multiple local nonlinear locations according to the present invention. Figure 2 As shown, under a normal excitation level that induces nonlinearity, a single broadband excitation F and the displacement response x of each degree of freedom i under this single broadband excitation are obtained. i (t) and velocity response Furthermore, based on the system structure, subsystems s are divided. i A discrete-time state-space model is constructed, and then the linear exponential NLI is calculated from the first t singular values ​​of each subsystem. i The linear exponent NLI i The nonlinearity of each subsystem is assessed, and preliminary nonlinearity location identification is achieved by combining the actual existing connections. Furthermore, a dictionary set D of nonlinear basis functions is established for suspected connections. The system input vector is constructed by sequentially selecting nonlinear basis functions from the dictionary set. Subsequently, nonlinear basis functions are selected based on the maximum value of the integrity index, and the corresponding connections are nonlinear connections, thereby achieving accurate location identification of multiple local nonlinearities in the system.

[0072] Exemplary Example:

[0073] Select as Figure 3The four-degree-of-freedom chain structure with two local nonlinearities shown is the research object. The parameters are set as follows: m1 = m2 = m3 = m4 = 1 kg, k1 = k2 = k3 = k4 = 3 × 10⁻⁶. 4 N·m -1 , c1=c2=c3=c4=2N·s·m -1 The system's local nonlinearities are located between degrees of freedom 1 and the ground, and between degrees of freedom 2 and 3. Where, k n1 =5×10 14 N·m -2 c is the coefficient of the second nonlinear term of the nonlinear spring. n2 =5×10 7 N·s 3 ·m -3 is the coefficient of the cubic nonlinear term of the nonlinear damping.

[0074] The excitation signal was selected as Gaussian white noise with a mean of 0 and a variance of 1, with a sampling frequency of 8000 Hz and a sampling duration of 10 s. The obtained input excitation F and displacement responses x for each degree of freedom i were... i (t) and velocity response like Figure 4 As shown. To demonstrate the accuracy of the proposed method in noisy environments, noise with a signal-to-noise ratio of 25 dB was added to the system output.

[0075] Based on the system structure, subsystems s are divided. i A discrete-time state-space model was constructed, divided into four subsystems. For example... Figure 5 The figure shows the top ten singular values ​​and linear exponent NLI of each subsystem under 25dB noise pollution. i .in, Figure 5 (a) Figure 5 (b) Figure 5 (c) and Figure 5 (d) represent subsystem 1, subsystem 2, subsystem 3, and subsystem 4, respectively. Based on the defined threshold, it can be expressed by the linear exponent NLI. i Subsystem 4 is determined to be a linear subsystem, while the remaining subsystems are nonlinear subsystems. Since subsystem 4 is linear and subsystem 3 is nonlinear, it can be inferred that the connection between degrees of freedom 3 and 4 is a linear connection, and the connection between degrees of freedom 2 and 3 is a nonlinear connection. This process narrows down the range of nonlinear location identification. The remaining connections cannot be determined using the above process and require further confirmation.

[0076] Establish a dictionary set D of nonlinear basis functions for the aforementioned questionable connections. Considering second- and third-order nonlinear stiffness and second- and third-order nonlinear damping, the dictionary set D of nonlinear basis functions can be expressed as follows:

[0077]

[0078] The system input vector is constructed by sequentially selecting nonlinear basis functions from the dictionary set.

[0079]

[0080] like Figure 6 As shown, Figure 6 (a) and Figure 6 (b) Integrity indices for degrees of freedom 1 and 2 in the first iteration under 25 dB noise pollution. Observation Figure 6 It can be observed that in the first iteration, both degree of freedom 1 and degree of freedom 2 selected the second nonlinear basis function, that is, the connection between the ground and degree of freedom 1 is a nonlinear connection.

[0081] Since the two degrees of freedom selected the same nonlinear basis function in the first iteration, this indicates that the questionable connection may contain multiple local nonlinearities, requiring further iterations to select nonlinear basis functions. Substituting the displacement responses of degrees of freedom 1 and 2 into equations (8) and (9) respectively, the system input vector is then... like Figure 7 As shown, Figure 7 (a) and Figure 7 (b) Integrity indices for degrees of freedom 1 and 2 in the second iteration under 25 dB noise pollution. Observation Figure 7 It can be observed that there is a discrepancy in the selection of nonlinear basis functions between degrees of freedom 1 and 2 in the second iteration, thus halting the iterative calculation. No nonlinear basis functions were selected during this iteration. Therefore, it can be further confirmed that the connection between degrees of freedom 1 and 2 is a linear connection.

[0082] Based on the above analysis, the connections between the ground and degree of freedom 1, and between degree of freedom 2 and degree of freedom 3, are nonlinear connections, while the remaining connections are linear connections. This determination is consistent with the actual structure of the system in the exemplary embodiment. Therefore, this invention achieves accurate location identification for systems with multiple local nonlinearities.

[0083] This invention employs a multi-stage approach to identify the local nonlinear positions of a system, achieving accurate location identification for systems with multiple local nonlinearities. This invention eliminates the need for multiple measurements to obtain system response data under different excitation levels; it only requires system input-output data under a single broadband excitation force to calculate the local nonlinear position information within the system. This invention has good applicability, is not limited by the type or number of nonlinearities, or the excitation location, and maintains good position identification performance even in noisy environments.

[0084] The above description is merely an overview of the technical solution of this application. In order to better understand the technical means of this application and to implement it according to the contents of the specification, and to make the above and other objects, features and advantages of this application more apparent, specific embodiments of this application are given below. It should be understood that the above general description and the following detailed description are merely exemplary and explanatory, and do not limit the present invention.

[0085] Other embodiments of the invention will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of the invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not disclosed herein. The specification and examples are to be considered exemplary only, and the true scope and spirit of the invention are indicated by the claims.

[0086] It should be understood that the present invention is not limited to the precise structure described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope. The scope of the invention is limited only by the appended claims.

Claims

1. A multiple local nonlinear position identification method, based on a nonlinear system with p degrees of freedom, characterized in that, Includes the following steps: Step 1: Obtain a single broadband excitation at a normal excitation level that induces nonlinearity. and the displacement response of each degree of freedom i under this single broadband excitation and speed response ; Step 2: Divide the system into subsystems centered on degree of freedom i. And determine the nonlinearity of each subsystem, and combine the actual connection relationship of the system to achieve preliminary nonlinear position identification; Step 3: Based on the nonlinear basis functions corresponding to the maximum values ​​of the integrity indices of multiple degrees of freedom, further identify whether there is local nonlinearity in the connections that could not be determined in Step 2; Step 3 further identifies whether the questionable connections in step 2 have local nonlinearity based on the nonlinear basis functions corresponding to the maximum values ​​of the integrity indices of multiple degrees of freedom. Specifically, it includes the following steps: Step 3.1 Establish a dictionary set of nonlinear basis functions at questionable connections. : In the formula, For consideration the first Nonlinear basis functions, The number of user-defined nonlinear basis functions to consider; Step 3.2 Select the dictionary set in sequence. In the case of nonlinear basis functions, construct the system input vector; In the formula, and They represent degrees of freedom. Displacement response vector and the first The frequency domain signal obtained by Fourier transforming a term of nonlinear basis functions; For the reason and The system input vector, composed of different degrees of freedom displacement responses, can be obtained from the above equation as different system input vectors. ; Step 3.3 Determine the first Integrity index of nonlinear basis functions : In the formula, , The autopower spectrum matrices are the input and output power spectra, respectively. The cross-power spectrum matrix of the input and output. This represents the Hermitian transpose. Step 3.4 Compare the nonlinear basis functions corresponding to the maximum values ​​of the integrity indices of multiple degrees of freedom. The corresponding connections are nonlinear connections.

2. The method for identifying multiple local nonlinear locations according to claim 1, characterized in that, Step 2 involves dividing the system into subsystems centered on degree of freedom i. The nonlinearity of each subsystem is determined, and preliminary nonlinearity location identification is achieved by combining the actual connection relationships of the system. The specific steps include the following: Step 2.1 Based on the nonlinear system structure, divide it into subsystems centered on degree of freedom i. : In the formula, i r This represents all nodes directly connected to degree of freedom i via springs and dampers. One degree of freedom, and ; Step 2.2 Construct the discrete-time state-space model of the subsystem and calculate adjacent singular values. logarithmic difference : In the formula, User-defined solution singularities Maximum order of value; The calculated logarithmic differences are normalized to zero mean and unit variance to make them conform to a standard normal distribution. Step 2.3 Calculate the linear exponent of each subsystem The formula is as follows: In the formula, For subsystem The value obtained by normalizing the logarithmic difference between the second and third singular values ​​with zero mean and unit variance; Step 2.4 Based on the linear exponent Determine the nonlinearity of each subsystem: When subsystem Linear exponential Less than the threshold If it is nonlinear, it is determined to be a nonlinear subsystem; otherwise, it is determined to be a linear subsystem. Step 2.5 Based on the nonlinear detection results of each subsystem and the actual connection relationships of the system, perform preliminary nonlinear location identification on the nonlinear system, and identify any questionable connections that cannot be determined.

3. The method for identifying multiple local nonlinear positions according to claim 2, characterized in that, The threshold in step 2.4 .

Images