A local nonlinear feature structure reduction modal base selection method based on bayes theory

Abstract

This invention discloses a method for selecting reduced modal bases for locally nonlinear characteristic structures based on Bayesian theory. First, a full-degree-of-freedom (FWHM) mechanical analysis model of the locally nonlinear characteristic structure is established, distinguishing between internal and boundary degrees of freedom. The internal degrees of freedom are then condensed using modal synthesis to obtain retained modes. Second, dynamic time-domain response analysis is performed on the FWHM locally nonlinear structural mechanical analysis model to obtain the time-domain dynamic response of the FWHM model. Then, the established retained modes are traversed column-wise, and dynamic time-domain response analysis is performed on the constructed dynamically reduced model. The conditional probability of the time-domain dynamic response of the reduced model is calculated using Bayesian theory within the time-domain dynamic response of the FWHM model. Finally, the conditional probabilities calculated for the reduced models obtained from different mode selections are sorted in descending order. A secondary selection of retained modes is performed by setting an expected probability threshold and plotting the cumulative conditional probability curve, thus determining the final reduced modal base for the locally nonlinear structure. This invention can perform secondary selection on some retained modes that have a significant impact on the result response among the retained master modes of nonlinear structures, thereby constructing a nonlinear structure reduced mode basis with greater similarity, so as to reduce the order of the retained modes. This provides an effective selection method for constructing suitable dynamic reduced basis for nonlinear structures in practical engineering.

Description

Technical Field

[0001] This invention belongs to the field of efficient modeling and analysis of structural dynamics, and particularly relates to a method for selecting reduced modal basis of local nonlinear characteristic structures based on Bayesian theory. Background Technology

[0002] Locally nonlinear structures refer to a class of systems in which certain regions of a complex structure exhibit nonlinear characteristics, primarily manifested as connection nonlinearity and contact nonlinearity. Although the system contains only a few nonlinear degrees of freedom, the overall dynamic characteristics of the structure are significantly affected by them, requiring the solution of differential equations from a large-scale nonlinear mechanical model to obtain the structure's response behavior. Dynamic reduction modeling methods, as an effective approach to reduce the degrees of freedom in nonlinear structural analysis, can reduce a large number of linear degrees of freedom and then recombine them with a small number of locally nonlinear degrees of freedom to construct a reduced model of the locally nonlinear characteristic structure, enabling efficient and high-fidelity solutions to the dynamic response of this structure.

[0003] In the dynamic reduction modeling of linear structures, modal synthesis and data-driven order reduction methods have been well-established for reducing the degrees of freedom in structural analysis. However, considering the existence of local nonlinear characteristics, the dynamic behavior of the structure is closely related to the selection of modes to be retained in the construction of the reduction basis. Current methods for selecting reduction modal basis often use the modal quality and modal contribution of the linear structure as reference indicators, failing to consider the correlation between the analysis response of the nonlinear structure and local nonlinear characteristics. This often results in redundant retained principal modes in the constructed reduction basis, and there is still room for further reduction in analysis time. How to utilize the response data of the locally nonlinear structure and the correlation characteristics of the nonlinear features to propose an effective method for selecting reduction modal basis for nonlinear structures has become a practical engineering simulation analysis problem that needs to be further solved. Summary of the Invention

[0004] Objective of the Invention: To address the above problems, this invention proposes a method for selecting reduced modal basis for structures with local nonlinear characteristics based on Bayesian theory. This method considers the influence of local nonlinear characteristics of the structure on the calculated response of the analysis model. By utilizing the reduced-order model analysis response corresponding to the independent reduced modes and the conditional probability magnitude, cumulative probability, and threshold of the analysis response of the full-degree-of-freedom model, it achieves secondary modal basis selection for structures with local nonlinear characteristics. This provides an effective method for selecting reduced modal basis for structural optimization in engineering structures for rapid analysis of nonlinear dynamic responses.

[0005] Technical solution: This invention provides a method for selecting reduced modal basis of local nonlinear characteristic structures based on Bayesian theory, comprising the following steps:

[0006] (1) Establish a full-degree-of-freedom mechanical analysis model of the local nonlinear characteristic structure, distinguish between internal degrees of freedom and boundary degrees of freedom, and use the modal synthesis method to condense the internal degrees of freedom to obtain the preserved modes.

[0007] (2) Based on the fully free local nonlinear structural mechanics analysis model established in step (1), perform dynamic time-domain response analysis to obtain the time-domain dynamic response of the fully free model.

[0008] (3) Based on the retained modes obtained in step (1), the dynamic time-domain response analysis of the constructed dynamic reduction model is performed, and the conditional probability of the time-domain dynamic response of the reduction model in the time-domain dynamic response of the full-degree-of-freedom model is calculated according to Bayesian theory.

[0009] (4) Based on the conditional probabilities calculated in step (3), the conditional probabilities of the reduced models obtained by selecting different modes are arranged in descending order from high to low, and the cumulative conditional probabilities and curves are plotted. The retained modes are selected a second time by setting the expected probability threshold, and the final nonlinear structure reduced mode basis is determined.

[0010] 2. The method for selecting reduced modal basis of local nonlinear characteristic structure based on Bayesian theory according to claim 1, characterized in that: step (1) includes the following steps:

[0011] (11) Establish a fully free-degree-of-freedom mechanical analysis model of a locally nonlinear structure using mechanical analysis software. f The system matrix (including the mass matrix M, stiffness matrix K, and damping matrix C) and the nonlinear restoring force vector F of the structure are obtained. nl The form, amplitude, and degrees of freedom of the excitation force are determined, and the excitation force F is obtained. ext ;

[0012] (12) The linear degrees of freedom are divided into internal degrees of freedom, and the nonlinear degrees of freedom and the degrees of freedom under the influence of the excitation force are divided into boundary degrees of freedom. The reduced basis T1 and the retained principal mode Φ are constructed for the first time using the modal synthesis method. d .

[0013] 3. The method for selecting reduced modal basis of local nonlinear characteristic structure based on Bayesian theory according to claim 2, characterized in that: step (2) includes the following steps:

[0014] (21) Based on the fully free mechanical analysis model of the local nonlinear structure established in step (11) ℳ f The nonlinear dynamic differential equations are solved using numerical analysis methods, and the time-domain dynamic response data of the l-th degree of freedom at the boundary nodes are extracted, denoted as R. f,i (t).

[0015] 4. The method for selecting reduced modal basis of local nonlinear characteristic structure based on Bayesian theory according to claim 3, characterized in that: step (3) includes the following steps:

[0016] (31) Determine the retained mode Φ established in step (12). d The number of columns J, and the mode Φ extracted each time. d The j-th column of candidate modes φ j Constructing new shrinkage bases Generate a new dynamic reduction model ℳ j , where j=1,…,J;

[0017] (32) For each new dynamic reduction model obtained in step (31) j The time-domain dynamic response was analyzed using numerical analysis methods, and the time-domain dynamic response data of the same node and the same degree of freedom in step (21) were extracted and denoted as R. j,i (t);

[0018] (33) According to Bayesian theory, from the formula Calculate the candidate mode φ j Under conditions of independent selection, its analytical response R j,i (t) and the fully free mechanical analysis model M f Analysis of response R f,i (t) The conditional probability of a match:

[0019]

[0020] Where P j (ℳ f |φ j ) represents the candidate mode φ calculated based on Bayesian theory. j The response matching conditional probability under conditions of independent selection, where σ represents the standard deviation of the analytical response, and n t This indicates the number of time-domain dynamic response data points.

[0021] 5. The method for selecting reduced modal basis of local nonlinear characteristic structure based on Bayesian theory according to claim 4, characterized in that: step (4) includes the following steps:

[0022] (41) Based on the response matching conditional probability P obtained in step (33) j Sort the data in descending order from high to low, and calculate the cumulative conditional probability and ∑P. j Plot the cumulative conditional probability and curve, and set the expected probability threshold;

[0023] (42) Based on the results of step (41), select all candidate modes that meet the probability threshold, realize the secondary selection of the retained modes, and determine the final nonlinear structure reduced mode basis T2.

[0024] Beneficial effects: Compared with the prior art, the present invention provides a method for selecting dynamic reduced modal bases for nonlinear structures that can consider local partial linearity characteristics. The method of the present invention utilizes the conditional probability of the analysis time-domain dynamic response of the reduced-order model corresponding to the independent modal base in the full-degree-of-freedom time-domain dynamic response to measure the influence of the selection of different independent modal bases on the analysis accuracy. It further quantitatively screens out independent modal bases with a high probability of affecting the analysis accuracy of the structure's time-domain dynamic response, and thereby constructs a dynamic reduced modal base matrix containing local nonlinear structural characteristics. This provides an effective method for selecting reduced modal bases for the efficient analysis of structures with local nonlinear characteristics in engineering. Attached Figure Description

[0025] Figure 1 This is a sector-by-sector lumped parameter model of a blade-disk structure with locally cubic nonlinear connections;

[0026] Figure 2 The j-th dynamic reduction model in descending order. j Response matching conditional probability P j ;

[0027] Figure 3 To match cumulative conditional probabilities and curves to responses sorted in descending order. Detailed Implementation

[0028] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments, but the scope of protection of the present invention is not limited to the embodiments described.

[0029] Table 1. Parameter values ​​of the lumped parameter model of the blade-disk structure with local nonlinear connection characteristics.

[0030] <![CDATA[m b1 ]]> 0.125 kg <![CDATA[m d1 ~m d6 ]]> 0.25 kg <![CDATA[m b2 ]]> 0.1 kg <![CDATA[k d ]]> <![CDATA[2×10 4 N / m]]> <![CDATA[k b1 ]]> <![CDATA[6×10 4 N / m]]> <![CDATA[k g ]]> <![CDATA[6×10 4 N / m]]> <![CDATA[k b2 ]]> <![CDATA[2×10 4 N / m]]> <![CDATA[k nl ]]> <![CDATA[2×10 5 N / m 3 ]]>

[0031] This embodiment uses a device with N s The method of the present invention is illustrated using a blade-disc structure with 29 sectors and local nonlinear connection features. Figure 1 The figure shows a lumped parameter model of the blade-disc structure. Each sector of the model has 8 degrees of freedom, for a total of 232 degrees of freedom. Local nonlinear connectivity features act on the mass block m. b1 With m d2 Between these two points, the expression exhibits a cubic nonlinear characteristic, and its nonlinear restoring force is expressed as f. nl =k nl x 3, where x represents the structural displacement response. The parameter values ​​corresponding to the same sign are identical within each sector. The mass and stiffness parameters of the linear structure and the stiffness parameters of the nonlinear structure are listed in Table 1.

[0032] This invention includes the following steps:

[0033] (1) Establish a full-degree-of-freedom mechanical analysis model for the locally nonlinear characteristic structure, distinguish between internal degrees of freedom and boundary degrees of freedom, and use the modal synthesis method to condense the internal degrees of freedom to obtain the preserved modes:

[0034] (11) Using structural dynamics analysis theory and software, such as MATLAB, establish a structural full-degree-of-freedom mechanical analysis model containing local nonlinear characteristics. f The system matrix (including the mass matrix M, stiffness matrix K, and damping matrix C) and the nonlinear restoring force vector F of the structure are obtained. nl The form, amplitude, and degrees of freedom of the excitation force are determined, and the excitation force F is obtained. ext Based on the parameter sign of the i-th sector in the embodiment, the mass matrix M of the i-th sector structure is obtained. i As shown in the formula As shown.

[0035]

[0036] The stiffness matrix K of the i-th sector structure i As shown in the formula As shown.

[0037]

[0038] Using proportional damping, with a damping ratio of 0.5% in the frequency band from 31.6Hz to 155.1Hz, the proportional damping coefficients can be calculated as α = 0.2625 and β = 5.357 × 10⁻⁶. -5 Then the damping matrix C of the i-th sector structure i =αM i +βK i .

[0039] Local nonlinear connectivity characteristics act on mass block m b1 With m d2 Between these two points, the expression exhibits a cubic nonlinear characteristic, and its nonlinear restoring force is expressed as f. nl =k nl x 3 Then the nonlinear restoring force column vector of the i-th sector structure for:

[0040]

[0041] Where, x b1 and x d2 They represent the mass blocks m respectively. b1 and m d2 The displacement response.

[0042] An external excitation force is applied to each sector mass block m b2 The corresponding degrees of freedom are in the form of traveling wave excitation with an amplitude of f. m =30N, operating frequency is f s =34Hz, analysis order is EO=10, then the external excitation force column vector of the i-th sector structure for:

[0043]

[0044] Where π is the constant of pi, and t is the time-domain dynamic response analysis time. Therefore, based on computational mechanics theory, the structural matrix of each sector can be obtained, and then assembled to obtain the fully free-degree-of-freedom mechanical analysis model ℳ. f The corresponding system matrix.

[0045] (12) such as Figure 1 As shown, the linear degrees of freedom within each sector are divided into internal degrees of freedom, and the nonlinear degrees of freedom and the degrees of freedom due to the excitation force are divided into boundary degrees of freedom. The degree of freedom division result of the full-degree-of-freedom mechanical analysis model is shown in the equation. As shown.

[0046]

[0047] In this model, the superscript represents the i-th sector, the subscript represents the degree of freedom corresponding to different mass block symbols within the i-th sector, the superscript T indicates the transpose operation, and x represents the displacement response. The system matrix of the fully free-degree-of-freedom mechanical analysis model is transformed using the degree-of-freedom partitioning results, so that each column of the system matrix corresponds to the equation... The analyzed degrees of freedom correspond to each other, resulting in the block mass matrix and stiffness matrix after the degree of freedom division:

[0048]

[0049] In this context, subscripts II and BB represent the internal and boundary degrees of freedom, respectively, while IB and BI both represent the internal and boundary coupled degrees of freedom. Then, the modal synthesis method is used to construct the first reduced basis T1 and the retained principal mode Φ. d :

[0050]

[0051] Where, Φ d The mass matrix M corresponding to the internal degrees of freedom II and stiffness matrix K II Solving the constructed generalized eigenvalue problem yields a principal mode retention matrix of order N=50, which is constructed by selecting the first 50 eigenvectors from the solution to the generalized eigenvalue problem. Ψ=(K II ) -1 K IB This is the constraint mode matrix. 0 and I represent the zero matrix and identity matrix, respectively.

[0052] Furthermore, according to the Bayesian theory-based method for selecting reduced modal bases of locally nonlinear characteristic structures as described in claim 1, dynamic time-domain response analysis is performed based on the fully free-degree-of-freedom locally nonlinear structural mechanical analysis model established in step (1) to obtain the time-domain dynamic response of the fully free-degree-of-freedom model. The specific steps are as follows:

[0053] (21) Based on the fully free mechanical analysis model of the local nonlinear structure established in step (11) ℳ f The nonlinear dynamic differential equations are solved using the fourth-order Runge-Kutta numerical analysis method. The equations to be solved are:

[0054]

[0055] Where ẍ and ẋ represent the acceleration and velocity response column vectors of the local nonlinear structure, respectively. The analysis time interval in this embodiment is [0, 0.2 s], and the time interval is 1 × 10⁻⁶. -4 s. And extract the displacement and velocity time-domain dynamic response data of the third degree of freedom on the boundary node for conditional probability analysis in step (33). The target time-domain dynamic response of the selected local nonlinear structure full degree of freedom mechanical analysis model is denoted as R. f,3 (t).

[0056] Furthermore, the method for selecting the reduced mode basis based on Bayesian theory of local nonlinear characteristic structures as described in claim 1, traverses the retained modes obtained in step (1) column by column, and performs dynamic time-domain response analysis on the constructed dynamic reduced model. The conditional probability of the time-domain dynamic response of the reduced model in the time-domain dynamic response of the fully free-degree-of-freedom model is calculated according to Bayesian theory. The specific steps are as follows:

[0057] (31) Determine the retained mode Φ established in step (12). d The number of columns J=50, consistent with the order of the retained modes. Then, independent mode extraction is performed iteratively, that is, each time the retained mode Φ is extracted. d The j-th column of candidate modes φj Constructing new shrinkage bases ,

[0058]

[0059] A new dynamic reduction model is constructed based on dynamic reduction theory. j , where j=1,…,J, that is, step (31) determines the candidate modes constructed from 50 independent retained modes.

[0060] (32) For each new dynamic reduction model obtained in step (31) j The time-domain dynamic response of the j-th dynamic reduction model is analyzed using the same fourth-order Runge-Kutta numerical analysis method as in step (21). The equation to be solved for the j-th dynamic reduction model is:

[0061]

[0062] in, , , , , Let the mass matrix, damping matrix, stiffness matrix, nonlinear restoring force vector, and external excitation force vector of the j-th dynamic reduction model be represented respectively, and their expressions are as follows: listed.

[0063]

[0064] Let R represent the acceleration, velocity, and displacement vectors to be solved in the j-th dynamic reduced model, respectively. Extract the time-domain dynamic response data of displacement and velocity of the same mass block with the same degree of freedom from step (21), and construct the response column vector of the third boundary degree of freedom of the j-th reduced model, denoted as R. j,3 (t);

[0065] (33) According to Bayesian theory, from the formula Calculate the candidate mode φ j Under conditions of independent selection, its analytical response R j,3 (t) and the fully free mechanical analysis model ℳ f Analysis of response R f,3 (t) The conditional probability of a match:

[0066]

[0067] Where P j(ℳ f |φ j ) represents the candidate mode φ calculated based on Bayesian theory. j The conditional probability of response matching under conditions of independent selection, where σ represents the response difference R. j,3 (t)-R f,3 The covariance of (t), n t σ and n represent the number of time-domain dynamic response data points, respectively. In this embodiment, σ and n t Given 0.0645 and 80 respectively, the response matching conditional probability of the j-th dynamic reduction model under the condition that 50 candidate modes are independently selected is as follows: Figure 2 As shown.

[0068] Furthermore, according to the Bayesian theory-based method for selecting reduced mode bases of local nonlinear feature structures as described in claim 1, based on the conditional probabilities calculated in step (3), the conditional probabilities of the reduced models obtained by selecting different modes are arranged in descending order from high to low, and the cumulative conditional probability and curve are plotted. A secondary selection of the retained modes is performed by setting an expected probability threshold to determine the final nonlinear structure reduced mode base. The specific steps are as follows:

[0069] (41) Based on the response matching conditional probability P obtained in step (33) j Sort the data in descending order from high to low, and calculate the cumulative conditional probability and ∑P. j And setting the expected probability threshold to 0.1, the cumulative conditional probability curve is compared with the threshold as follows: Figure 3 As shown;

[0070] (42) Based on the result of step (41), select all candidate modes that satisfy the probability threshold, that is, select the 20th, 21st, 38th and 39th order modes to realize the secondary construction of the mode-preserving matrix, Φ d2 =[φ 20 φ 21 φ 38 φ 39 At this point, the order of the retained principal mode is reduced from 50 to 4, thus yielding the local nonlinear structure reduced mode basis matrix T2 after selecting the reduced mode basis based on Bayesian theory:

[0071]

[0072] Furthermore, a local nonlinear structural dynamic reduction model based on the selection of reduced modal basis using Bayesian theory can be obtained. The displacement-time domain dynamic responses at boundary degrees of freedom 1 / 2 / 3 were solved using the fourth-order Runge-Kutta numerical analysis method, specifically for the reduced model retaining 50 principal modes (Model 1), the reduced model retaining 4 principal modes after secondary selection (Model 2), and the full-degree-of-freedom model. The comparison results of the relative error e of the time-domain dynamic response are shown in Table 2, and its calculation formula is as follows: As shown.

[0073]

[0074] Wherein, the subscript l represents the selected l-th boundary degree of freedom, and ||*||2 represents the L2 norm of the response. As can be seen from the table, the dynamic reduction model constructed by the method proposed in this patent can achieve the same analytical accuracy as traditional reduction methods while reducing the order of retained principal modes. This indicates that the method proposed in this patent can eliminate unnecessary redundant modes in the dynamic reduction modeling of local nonlinear structures, and select representative dominant retained principal modes, providing an effective method for selecting secondary modal reduction bases for rapid analysis of nonlinear structural responses in practical engineering.

[0075] Table 2. Time-domain dynamic response error (%) of different reduced models and full-degree-of-freedom models at boundary degrees of freedom 1 / 2 / 3.

[0076] Model 1 0.35 0.39 0.42 Model 2 0.35 0.39 0.42

Claims

1. A method for selecting a reduced modal basis of a local nonlinear feature structure based on Bayesian theory, characterized in that, The method includes the following steps: (1) Establish a full-degree-of-freedom mechanical analysis model of the local nonlinear characteristic structure, distinguish between internal degrees of freedom and boundary degrees of freedom, and use the modal synthesis method to condense the internal degrees of freedom to obtain the preserved modes. (2) Based on the fully free local nonlinear structural mechanics analysis model established in step (1), perform dynamic time-domain response analysis to obtain the time-domain dynamic response of the fully free model. (3) Based on the retained modes obtained in step (1), the dynamic time-domain response analysis of the constructed dynamic reduction model is performed, and the conditional probability of the time-domain dynamic response of the reduction model in the time-domain dynamic response of the full-degree-of-freedom model is calculated according to Bayesian theory. (4) Based on the conditional probabilities calculated in step (3), the conditional probabilities of the reduced models obtained by selecting different modes are arranged in descending order from high to low, and the cumulative conditional probabilities and curves are plotted. The retained modes are selected a second time by setting the expected probability threshold, and the final nonlinear structure reduced mode basis is determined.

2. The method of claim 1, wherein the method is based on Bayesian theory. Step (1) includes the following steps: (11) Establish a fully free-degree-of-freedom mechanical analysis model of a locally nonlinear structure using mechanical analysis software. f The system matrix (including the mass matrix M, stiffness matrix K, and damping matrix C) and the nonlinear restoring force vector F of the structure are obtained. nl The form, amplitude, and degrees of freedom of the excitation force are determined, and the excitation force F is obtained. ext ; (12) The linear degrees of freedom are divided into internal degrees of freedom, and the nonlinear degrees of freedom and the degrees of freedom under the influence of the excitation force are divided into boundary degrees of freedom. The reduced basis T1 and the retained principal mode Φ are constructed for the first time using the modal synthesis method. d .

3. The method of claim 2, wherein the method is based on Bayesian theory. Step (2) includes the following steps: (21) Based on the local nonlinear structural full-degree-of-freedom mechanical analysis model M established in step (11) f The nonlinear dynamic differential equations are solved using numerical analysis methods, and the time-domain dynamic response data of the l-th degree of freedom at the boundary nodes are extracted, denoted as R. f,l (t).

4. The method of claim 3, wherein the method is based on Bayesian theory. Step (3) includes the following steps: (31) Determine the retained mode Φ established in step (12). d The number of columns J, and the mode Φ extracted each time. d The j-th column of candidate modes φ j Constructing new shrinkage bases Generate a new dynamic reduction model ℳ j , where j=1,…,J; (32) For each new dynamic reduction model obtained in step (31) j The time-domain dynamic response was analyzed using numerical analysis methods, and the time-domain dynamic response data of the same node and the same degree of freedom in step (21) were extracted and denoted as R. j,l (t); (33) According to Bayesian theory, from the formula Calculate the candidate mode φ j Under conditions of independent selection, its analytical response R j,i (t) and the fully free mechanical analysis model ℳ f Analysis of response R f,l (t) The conditional probability of a match: Where P j (ℳ f |φ j ) represents the candidate mode φ calculated based on Bayesian theory. j The probability of matching the response under conditions of independent selection, where σ represents the standard deviation of the analytical response, and n t This indicates the number of time-domain dynamic response data points.

5. The method of claim 4, wherein: Step (4) includes the following steps: (41) The response matching condition probability P is obtained according to step (33) j The descending order from high to low is arranged, and the cumulative condition probability sum ∑P is calculated j The cumulative condition probability sum curve is drawn, and the expected probability threshold is set (42) Based on the results of step (41), select all candidate modes that meet the probability threshold, realize the secondary selection of the retained modes, and determine the final nonlinear structure reduced mode basis T2.

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