An efficient method for predicting dynamic response of uncertain flexible multi-body systems

Abstract

The present invention discloses an efficient method for predicting the dynamic response of an uncertain flexible multi-body system, which belongs to the field of dynamics of uncertain flexible multi-body systems. The implementation method of the present invention is: using the absolute node coordinate formula ANCF to establish the differential-algebraic equations of the dynamics of a deterministic flexible multi-body system. Introducing the uncertainty parameters into the equations of the flexible multi-body system model to form uncertain differential-algebraic equations. Using the signal decomposition method, the output response of the system is decomposed into a plurality of single-component signals in the form of vibrations and a trend term signal, and then each single-component signal is characterized in the form of instantaneous amplitude and instantaneous phase. Based on compressed sensing, sparse polynomial proxy models of instantaneous amplitude, instantaneous phase and trend term are established at each moment. According to the constructed proxy model, a high-precision and high-efficiency prediction of the dynamic response of an uncertain flexible multi-body system is achieved. The present invention is particularly suitable for the prediction of the dynamic response of large and complex systems.

Description

Technical Field

[0001] The present invention relates to an efficient prediction method for the dynamic response of an uncertain flexible multi-body system, and in particular to a prediction method for the dynamic response of a flexible multi-body system with uncertain parameters approximated by a polynomial proxy model based on compressed sensing and signal decomposition methods, and belongs to the field of dynamics of uncertain flexible multi-body systems.

[0002] Background Methods

[0003] One of the most important subsystems in the field of mechanical engineering is the multi-body system. In practical engineering methods, the dynamics of a multi-rigid system, in which each component of the system is assumed to be a rigid body, is no longer applicable to current research needs. Therefore, the theory of the dynamics of a flexible multi-body system has been developed and is often used to explain complex dynamic properties, thereby providing theoretical guidance for the development and processing of the system, and has high engineering value.

[0004] Since the material properties, dynamic parameters, loads, boundary conditions and initial conditions of the system are generally uncertain, and the accuracy of the model parameters directly affects the accuracy of the prediction and control of the system's dynamic behavior. If the uncertainty parameters cannot be effectively identified, the system's dynamic prediction results may be biased or even seriously wrong, and in serious cases, the entire system may fail. For example, the analysis results of the crash of the Schiaparelli probe of the European Space Agency (ESA) pointed out that the control system of the probe did not take into account the uncertainty of the angle between the parachute and the lander during landing. Therefore, in order to provide a strong guarantee for the reliability analysis and design of product performance, engineers need to reasonably analyze the uncertainties in the system to ensure the normal operation of the engineering system.

[0005] Uncertainty analysis, also known as uncertainty propagation analysis, is a method for positively evaluating the impact of uncertain parameters on the static and dynamic responses of a system. It has gradually become a basic means and important method for analyzing uncertainty factors. However, with the increasing complexity of engineering system design, the computational cost of system models has become increasingly high, which has brought many problems to uncertainty propagation, such as "low precision" and "dimensionality disaster".

[0006] In response to the "low precision" problem, Wan and Xu et al. proposed a multi-subinterval method, which divides the original uncertainty interval into multiple subintervals and establishes a low-order proxy model for each subinterval. When the uncertain system runs for a long time, the low-order proxy model cannot meet the needs, so these methods do not fundamentally solve the problem. Cui et al. introduced the adaptive signal decomposition method into the field of uncertainty analysis and combined the polynomial proxy model with the adaptive signal decomposition method. This method overcomes the shortcomings of the traditional polynomial proxy model method in processing periodic time domain signals and greatly improves the accuracy of the calculation. However, this method ignores the computational efficiency of the response of complex engineering systems. When there are many uncertain parameters, even if a low-order proxy model is used, the "dimensionality curse" problem still exists for complex flexible multi-body systems.

[0007] Therefore, considering the computational accuracy and efficiency of complex flexible multi-body systems with uncertainties over a long period of time is a research topic that urgently needs to be broken through at this stage. Summary of the invention

[0008] It is difficult to ensure high prediction accuracy for complex flexible multi-body systems in the engineering field, especially the dynamic responses of flexible multi-body systems with uncertain parameters, over a long period of time. At the same time, its computational cost is expensive. The methodological problem to be solved by an efficient method for predicting the dynamic responses of uncertain flexible multi-body systems disclosed in the present invention is: a sparse polynomial proxy model of the dynamic responses of uncertain flexible multi-body systems is established based on compressed sensing and signal decomposition methods. Compared with the existing method for predicting the dynamic responses of flexible multi-body systems, the prediction accuracy of the dynamic responses over a long period of time can be improved, and at the same time, its prediction efficiency can be effectively improved. The present invention is applied to large and complex systems in the engineering field that are large in scale, have high dimensions of input uncertain parameters, and have strong nonlinear characteristics, to solve the prediction of related uncertain responses and related engineering problems in the engineering field.

[0009] The present invention is achieved through the following method scheme:

[0010] The present invention discloses an efficient method for predicting the dynamic response of an uncertain flexible multi-body system, which uses the absolute node coordinate formula ANCF to establish the differential-algebraic equations of the dynamics of a deterministic flexible multi-body system. First, the uncertainty parameters are introduced into the equations of the flexible multi-body system model to form uncertain differential-algebraic equations. Secondly, the output response of the system is decomposed into multiple single-component signals in the form of vibrations and a trend term signal using a signal decomposition method, and then each single-component signal is characterized in the form of instantaneous amplitude and instantaneous phase. Thirdly, based on compressed sensing, sparse polynomial proxy models of instantaneous amplitude, instantaneous phase and trend term are established at each moment. Finally, based on the constructed proxy model, a high-precision and high-efficiency prediction of the dynamic response of an uncertain flexible multi-body system is achieved.

[0011] The present invention discloses an efficient method for predicting the dynamic response of an uncertain flexible multi-body system, comprising the following steps:

[0012] Step 1: When the structural parameters of the flexible multi-body system are deterministic, the absolute node coordinate formula ANCF is used to establish the differential-algebraic equations of the dynamics of the flexible multi-body system.

[0013] The differential-algebraic equation of the flexible multi-body system dynamics established in step 1 is shown in formula (1):

[0014]

[0015] Where M is the system generalized mass matrix, q is the system generalized coordinate, F is the system generalized elastic force, Φ=0 and Φ q are the system constraint equations and the Jacobi matrix for generalized coordinates, λ is the Lagrange multiplier, and Q is the generalized external force of the system.

[0016] Step 2: When the structural parameters of the flexible multi-body system are uncertain, the uncertain parameters are introduced into the equations of the flexible multi-body system model to establish uncertain differential-algebraic equations. The uncertain parameters include random parameters or interval parameters.

[0017] The uncertainty differential-algebraic equation established in step 2 is shown in formula (2):

[0018]

[0019] in, are n random or interval uncertainty parameter variables. When is a random uncertainty parameter variable, without loss of generality, It obeys the standard normal distribution N(0,1); when When is an interval uncertainty parameter variable, without loss of generality, The meanings of the remaining matrices and vectors are the same as those in step 1. The matrices and vectors with a superscript “~” indicate that they are matrices and vectors containing uncertainty parameters. N groups of sample points are selected from the above, and the system output response corresponding to each sample can be expressed as

[0020] Step 3: Use the signal decomposition method to decompose each output response of the system into multiple single-component signals in the form of vibration and a trend term signal, and characterize each single-component signal in the form of instantaneous amplitude and instantaneous phase.

[0021] The HHT method is used to decompose each output response of the system, which mainly includes empirical mode decomposition and Hilbert transform. The specific implementation method is:

[0022] Empirical mode decomposition decomposes the original signal into the sum of multiple eigenmode functions and the residual signal:

[0023]

[0024] in, represents the eigenmode function, Represents the trend term signal. According to the Hilbert transform, the instantaneous amplitude and instantaneous phase of each eigenmode function are solved, and the uncertainty output response is:

[0025]

[0026] in and They represent the instantaneous amplitude and instantaneous phase of the kth eigenmode function corresponding to the jth system output response, and both the instantaneous amplitude and the instantaneous phase are related to the uncertainty parameter function.

[0027] Step 4: Based on compressed sensing, at each moment, polynomial proxy models of instantaneous amplitude, instantaneous phase and trend term signals are established. Based on compressed sensing, the sparse coefficient matrix of each polynomial proxy model is solved. According to the type of uncertainty parameters, the obtained polynomial proxy model is used to obtain the dynamic response of the uncertain flexible multi-body system, that is, to achieve the prediction of the dynamic response of the uncertain flexible multi-body system.

[0028] According to the type of uncertainty parameters, the obtained polynomial proxy model is used to obtain the dynamic response of the uncertain flexible multi-body system. The implementation method is:

[0029] When the uncertainty parameter is an interval parameter, the interval scanning method is used to scan the interval uncertainty parameter with enough sample points to obtain the upper and lower bounds of the response of the uncertain flexible multi-body system dynamics.

[0030] When the uncertainty parameters are random parameters, the mean value of the dynamic response of the uncertain flexible multi-body system can be efficiently predicted according to the derived analytical formula of the mean value of the system response.

[0031] Step 4.1: For the instantaneous amplitude and trend term signals obtained in step 3, a polynomial proxy model is established.

[0032]

[0033]

[0034] in, represents the undetermined coefficient in the polynomial surrogate model for the kth instantaneous amplitude, γ r The undetermined coefficients in the polynomial surrogate model representing the trend term, Represents uncertainty parameter The corresponding polynomial basis functions, A and R are the cutoff numbers of the expanded terms of the polynomial surrogate model of the instantaneous amplitude and trend terms, respectively.

[0035] Step 4.2: Perform phase unwrapping on the instantaneous phase and establish a polynomial proxy model based on compressed sensing.

[0036] Since the instantaneous phase obtained by the Hilbert transform solution jumps within [-π, π], it is necessary to use a phase unwrapping algorithm to make the phase value monotonic, and then establish a polynomial proxy model for the unwrapped instantaneous phase.

[0037]

[0038] in, represents the undetermined coefficient in the polynomial proxy model of the kth instantaneous amplitude, and B is the number of truncated terms in the expanded terms of the polynomial proxy model of the instantaneous phase.

[0039] Step 4.3: Based on compressed sensing, obtain the sparse approximate solution of the unknown coefficients of the polynomial proxy model (5)-(7).

[0040] Step 4.3.1: Based on compressed sensing, the coefficient matrix α = [α 0 α 1 … α A-1 ] T , β=[β 0 β 1 …β B-1 ] T and γ=[γ 0 γ 1 …γ R-1 ] T The approximate solution of as well as By solving the following three optimization problems:

[0041]

[0042]

[0043]

[0044] Among them, 1 ,Ψ 2 ,Ψ 3 is the measurement matrix, a=[a 1 a 2 …a N ] T , and r = [r 1 r 2 …r N ] T Respectively represent N groups of instantaneous amplitude response, instantaneous phase response and trend term response. Where ||α|| 0 =W 1 ,||β|| 0 =W 2 ,||γ|| 0 =W 3 The non-zero elements in the coefficient matrices α, β, and γ are W 1 , W 2 , W 3 And W 1 , W 2 , W 3 It also represents the sparsity of the coefficient matrices α, β, and γ, and is used to measure the sparsity of the coefficient matrices α, β, and γ.

[0045] Step 4.3.2: Considering that equations (8)-(10) are NP-hard problems, it is difficult to solve them with an exact algorithm, and we can only seek an effective approximate algorithm. Therefore, we can obtain an approximate solution by solving the following optimization problem: and

[0046]

[0047]

[0048]

[0049] Among them, ε 1 , ε 2 , ε 3 Indicates the tolerance of prediction accuracy, ε 1 , ε2 , ε 3 If the value of is set too large, the calculation result will be of low precision, and if it is set too small, it will lead to overfitting. Therefore, it is very important to select a suitable precision tolerance value, and ε 1 , ε 2 , ε 3 Depends on the sparsity W 1 , W 2 , W 3 .

[0050] Step 4.3.3: Obtain reasonable sparsity through the cross-validation method in machine learning and solve the approximate solution of equation (9).

[0051] Preferably, a greedy algorithm is used to obtain approximate solutions to the optimization problems (11)-(13). The greedy algorithm includes orthogonal matching pursuit, compressed sampling matching pursuit and subspace pursuit.

[0052] Step 4.4: According to step 4.3, the sparse approximate solution of the unknown coefficients is obtained to predict the dynamic response of the uncertain flexible multi-body system.

[0053] According to the sparse polynomial proxy models of the instantaneous amplitude, instantaneous phase and trend term signals obtained in step 4.3, the polynomial proxy model of the response of the uncertain flexible multi-body system dynamics is further combined:

[0054]

[0055] Step 4.4.1: When the uncertainty parameter is an interval parameter, use the interval scanning method to By taking enough sample points to scan the polynomial proxy model (14), we can obtain the upper and lower bounds of the response of the uncertain flexible multi-body system dynamics.

[0056] Step 4.4.2: When the uncertainty parameters are random parameters, the mean of the dynamic response of the uncertain flexible multi-body system is efficiently predicted based on the derived analytical formula for the mean of the system response.

[0057] When the system contains two random uncertain parameters, and the probability density function of the random uncertain parameters obeys the normal distribution, the implementation method of step 4.4.2 is:

[0058] Using the second-order chaotic polynomial proxy model to approximate the instantaneous amplitude, instantaneous phase and trend term signal of the system, we have

[0059]

[0060]

[0061] in, and denote the instantaneous amplitude and instantaneous phase of the kth (k=1, ..., m) eigenmode function; Indicates trend item; represents the Hermite polynomial basis function. Then the analytical formula for the mean response of the uncertain flexible multi-body system dynamics is

[0062]

[0063] in

[0064] E[r]=γ 0 (19)

[0065] For a given k (k = 1, ..., m), we have

[0066]

[0067] According to the adaptive signal decomposition, the total number of eigenmode functions of N groups of different output responses is different. For the convenience of prediction, the first two groups of data remain unchanged, and the third group and the following groups are all merged into the third group, that is, let m=3.

[0068] According to the analytical formula (18) of the response mean of the dynamics of the uncertain flexible multi-body system, the prediction of the response mean of the dynamics of the random uncertain flexible multi-body system is realized.

[0069] It also includes step 5: according to steps 1 to 4, a high-efficiency and high-precision prediction of the dynamic response of a flexible multi-body system with uncertain parameters is achieved, and the analysis and prediction results are applied to large-scale complex systems in the engineering field with large calculation scale, high dimensionality of input uncertain parameters and strong nonlinear characteristics, so as to solve the prediction of related uncertain responses in the engineering field.

[0070] Beneficial effects:

[0071] 1. The present invention discloses an efficient method for predicting the dynamic response of an uncertain flexible multi-body system. By adaptively decomposing the output response, the system dynamic response maintains a high prediction accuracy over a long time range. The nonlinearity of the proxy model using the signal decomposition method will increase with time, although a high prediction accuracy can be achieved using a low-order proxy model. However, when the dimension of the input uncertainty parameter is large, the "curse of dimensionality" problem still exists for large and complex flexible multi-body systems. Therefore, a sparse polynomial proxy model based on compressed sensing and signal decomposition methods is used. While maintaining a high prediction accuracy, the number of sampling points required is less than the number of degrees of freedom of the undetermined coefficients of the polynomial proxy model, which effectively improves the prediction efficiency and reduces the prediction cost.

[0072] 2. The present invention discloses an efficient method for predicting the dynamic response of an uncertain flexible multi-body system, which is applicable to large-scale complex systems with large computational scale, high dimension of input uncertain parameters and strong nonlinear characteristics in the engineering field. Compared with other current methods, the method proposed in the present invention has higher prediction accuracy, less time consumption, higher efficiency and lower cost, and is more suitable for complex engineering structure systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Figure 1 A flowchart of an efficient method for predicting dynamic response of an uncertain flexible multi-body system according to the present invention;

[0074] Figure 2 Three different calculation methods are used, including direct polynomial surrogate model method, scanning method and sparse polynomial surrogate model method based on compressed sensing and signal decomposition method, to predict the upper bound of the displacement response x of the Duffing oscillator with interval uncertainty parameters.

[0075] Figure 3 Three different calculation methods are used, including direct polynomial surrogate model method, scanning method and sparse polynomial surrogate model method based on compressed sensing and signal decomposition method, to predict the lower bound of the displacement response x of the Duffing oscillator with interval uncertainty parameters.

[0076] Figure 4 The mean value of the displacement response x of the Duffing vibrator with random uncertainty parameters is predicted by using four different calculation methods, including a direct polynomial proxy model method, a Monte Carlo simulation, a Monte Carlo simulation of the polynomial proxy model of the method proposed by the present invention, and a mean analytical formula derived by the present invention;

[0077] Figure 5 It is a schematic diagram of the flexible double pendulum system;

[0078] Figure 6 Three different calculation methods are used, including direct polynomial surrogate model method, scanning method and sparse polynomial surrogate model method based on compressed sensing and signal decomposition method, to predict the upper bound of the X-direction displacement of the end point of the flexible double pendulum with interval uncertainty parameters.

[0079] Figure 7 Three different calculation methods are used, including direct polynomial surrogate model method, scanning method and sparse polynomial surrogate model method based on compressed sensing and signal decomposition method, to predict the lower bound of the X-direction displacement of the end point of a flexible double pendulum with interval uncertainty parameters.

[0080] Figure 8The mean value of the X-direction displacement of the end point of the flexible double pendulum with random uncertainty parameters is predicted by using three different calculation methods, including direct polynomial proxy model method, direct Monte Carlo simulation of the original model, and the mean analytical formula derived by the present invention;

[0081] Fig. 9 for Figure 8 Local comparison of the three calculation methods. DETAILED DESCRIPTION

[0082] In order to better illustrate the purpose and advantages of the present invention, the invention is further described below with reference to the accompanying drawings and examples.

[0083] Embodiment 1:

[0084] Reference Figure 1 This embodiment discloses an efficient method for predicting the dynamic response of an uncertain flexible multi-body system, and the specific implementation steps are as follows:

[0085] Predict the dynamic response of a Duffing oscillator with interval uncertainty:

[0086] Step 1: Take the Duffing oscillator as an example to illustrate the effectiveness of the method proposed in the present invention. The control equation of the Duffing oscillator is:

[0087]

[0088] Where ω=20,ε=0.01,h=8000,Ω=100. Consider when the parameter is an interval uncertainty parameter, that is where η 1 =[-1,1],η 2 =[-1,1],η 3 =[-1,1],η 4 =[-1, 1].

[0089] Step 2: Determine the sample points. The interval uncertainty parameters correspond to the selection of Chebyshev or Legendre polynomial basis. Therefore, when the order of the polynomial expansion is p, the sample points of the interval variables should be selected from the roots of the p+1 order Chebyshev or Legendre polynomial, and the number of sample points N can be less than the number of degrees of freedom of the undetermined coefficients in the polynomial surrogate model.

[0090] Step 3: Use the HHT method to decompose each group of sample responses and obtain the instantaneous amplitude, instantaneous phase and trend term corresponding to each group of sample responses. Then, at each moment, sparse polynomial proxy models of instantaneous amplitude, instantaneous phase and trend term are established respectively.

[0091] Step 4: Use the subspace tracking algorithm and cross-validation method to obtain the sparsity of the sparse polynomial proxy model of instantaneous amplitude, instantaneous phase and trend terms, and then solve the unknown coefficients.

[0092] Step 5: After combining the sub-agent models, the interval scanning method is used to obtain the upper and lower bounds of the output response of the Duffing oscillator system, such as Figure 2 and Figure 3 As shown. At the same time, the upper and lower bounds of the output response of the Duffing oscillator system are calculated by using the direct proxy model method and directly performing interval scanning on the original system. From the numerical simulation results, it can be found that the direct proxy model method is invalid when calculating the dynamic response over a long period of time. The dynamic response results obtained by directly performing interval scanning on the original system are almost consistent with the results obtained by the prediction method proposed in the present invention.

[0093] Predict the dynamic response of a Duffing oscillator with stochastic uncertainty:

[0094] Step 1: The control equation of the Duffing oscillator is shown in Equation (21). Considering that the system parameters are random uncertainty parameters, that is, where ξ 1 ~N(0,1),ξ 2 ~N(0,1),ξ 3 ~N(0,1),ξ 4 ~N(0,1).

[0095] Step 2: Determine the sample points. The Hermite polynomial basis is selected corresponding to the random uncertainty parameters. Therefore, when the order of the polynomial expansion is p, the sample points of the random variables should be selected from the roots of the p+1 order Hermite polynomial, and the number of sample points N can be less than the number of degrees of freedom of the undetermined coefficients in the polynomial proxy model.

[0096] Steps 3 and 4 are the same as above.

[0097] Step 5: After combining the sub-proxy models, the mean value of the output response of the Duffing oscillator system is calculated using the analytical method proposed in the present invention, such as Figure 4 As shown. At the same time, the direct proxy model method and the Monte Carlo simulation of the original system are used to calculate the mean of the output response of the Duffing oscillator system. The results show that the direct proxy model method is also ineffective when calculating the dynamic response over a long period of time. The dynamic response results obtained by directly performing Monte Carlo simulation on the original system are almost consistent with the results obtained by the analytical method proposed in the present invention, and the prediction accuracy is high.

[0098] Embodiment 2:

[0099] Reference Figure 1This embodiment discloses an efficient method for predicting the dynamic response of an uncertain flexible multi-body system, and the specific implementation steps are as follows:

[0100] Predict the dynamic response of a flexible double-pendulum system with interval uncertainty:

[0101] Step 1: Take the flexible double pendulum system as an example to illustrate the effectiveness of the method proposed in the present invention. Figure 5 As shown. Pendulum length L 1 =L 2 =1m, the cross section is of side length L y =L z = 0.02m square, material density is ρ = 7800kg / m 3 , Young's modulus is E = 2.07e11Pa, Poisson's ratio is v = 0.3, and the acceleration due to gravity is g = [0 0 9.81] T m / s, the initial state is horizontal. Now consider the case where the length of the double pendulum is an interval parameter: where η 1 =[1,1],η 2 = [1, 1]. Five ANCF beam elements with the same parameters are used to model each beam of the flexible body system.

[0102] Step 2: Determine the base point. The interval uncertainty parameter corresponds to the selection of the Chebyshev or Legendre polynomial basis. Therefore, when the order of the polynomial expansion is p, the base point of the interval variable should be selected from the roots of the p+1 order Chebyshev or Legendre polynomial, and the number of base points N can be less than the number of degrees of freedom of the undetermined coefficients in the polynomial surrogate model.

[0103] Step 3: Different from Example 1, the displacement signal at the end point of the flexible double pendulum system is less smooth. Therefore, the response signal needs to be smoothed first, and the smoothed signal of each group of sample responses is decomposed by HHT to obtain the corresponding instantaneous amplitude, instantaneous phase and trend term. In order to mitigate the "endpoint effect", before performing HHT decomposition, a mirror extension is performed at the M points of the first and last signals, and the size of M can be selected according to the actual situation.

[0104] Step 4: Then, at each moment, sparse polynomial proxy models of instantaneous amplitude, instantaneous phase and trend term are established respectively. At the same time, a sparse polynomial proxy model is also established for the residual signal obtained by subtracting the original signal from the smoothed signal of each group of sample responses.

[0105] Step 5: Use the subspace tracking algorithm and cross-validation method to obtain the sparsity of the sparse polynomial proxy model of the instantaneous amplitude, instantaneous phase and trend term as well as the residual signal, and then solve the unknown coefficients.

[0106] Step 6: After combining the sub-proxy models, the interval scanning method is used to obtain the upper and lower bounds of the X-direction displacement response of the end points of the flexible double pendulum system, such as Figure 6 and Figure 7 As shown. The results obtained by using the direct proxy model method and directly scanning the original system at the same time are compared and it is found that the direct proxy model method is ineffective when calculating the dynamic response over a long period of time. The results obtained by the method proposed in the present invention and directly scanning the original system at the interval are closer, indicating the effectiveness of the method proposed in the present invention.

[0107] Predict the dynamic response of a flexible double-pendulum system with stochastic uncertainty:

[0108] Step 1: Take the flexible double pendulum system as an example to illustrate the effectiveness of the method proposed in the present invention. The deterministic parameters of the flexible double pendulum system are consistent with the above. Now consider the case where the pendulum length of the double pendulum is a random parameter: where ξ 1 ~N(0,1),ξ 2 ~N(0,1). Five ANCF full-parameter beam elements are used to model each beam of the flexible body system.

[0109] Step 2: Determine the sample points. The Hermite polynomial basis is selected corresponding to the random uncertainty parameters. Therefore, when the order of the polynomial expansion is p, the sample points of the interval variables should be selected from the roots of the p+1 order Hermite polynomial, and the number of sample points N can be less than the number of degrees of freedom of the undetermined coefficients in the polynomial surrogate model.

[0110] Step 3: Since the displacement signal at the end point of the flexible double pendulum system is more uneven, the response signal also needs to be smoothed first. At the same time, in order to mitigate the "end point effect", before the HHT decomposition, a mirror extension is performed at the M points of the first and last signals. The size of M can be selected according to the actual situation. For the specific process, please refer to step 3 of the interval uncertainty flexible double pendulum system.

[0111] Step 4 and step 5 are consistent with steps 4 and 5 of the above interval uncertainty flexible double pendulum system.

[0112] Step 6: After combining the sub-agent models, the analytical method proposed in the present invention is used to calculate the mean value of the X-direction displacement output response of the end point of the random uncertainty flexible double pendulum system, such as Figure 8 As shown. At the same time, the direct proxy model method and the Monte Carlo simulation of the original system are used to calculate the mean value of the X-direction displacement output response of the system end point. Since the computational cost of Monte Carlo simulation of the flexible double pendulum system is too high, the numerical simulation results are only studied for the first 5 seconds of the flexible double pendulum system movement. Fig. 9The results show that when calculating the dynamic response over a long period of time, the results of directly performing Monte Carlo simulation on the original system are closer to the results obtained by the analytical method proposed in the present invention than the direct proxy model method, which illustrates the effectiveness of the method proposed in the present invention.

[0113] The method disclosed in this embodiment overcomes the shortcoming of the traditional calculation method in processing the low accuracy of the dynamic response of the system over a long time range, making the prediction result more effective and effectively improving its prediction efficiency.

[0114] According to steps 1 to 6, efficient prediction of the dynamics of uncertain flexible multi-body systems is achieved, and the analysis and prediction results can be applied to large and complex systems in the engineering field with large calculation scales, high dimensions of input uncertain parameters, and strong nonlinear characteristics, thereby solving related uncertainty response prediction and related engineering problems in the field of flexible multi-body systems.

[0115] The above specific description further illustrates the purpose, method scheme and beneficial effects of the invention in detail. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the invention should be included in the protection scope of the invention.

Claims

1. An efficient method for predicting the dynamic response of an uncertain flexible multi-body system, characterized by: The following steps are included: Step 1: When the structural parameters of the flexible multi-body system are deterministic, the absolute node coordinate formula ANCF is used to establish the differential-algebraic equations of the dynamics of the flexible multi-body system; The differential-algebraic equation of the flexible multi-body system dynamics established in step 1 is shown in formula (1): Where M is the system generalized mass matrix, q is the system generalized coordinate, F is the system generalized elastic force, Φ=0 and Φ q are the system constraint equations and the Jacobi matrix for generalized coordinates, λ is the Lagrange multiplier, and Q is the generalized external force of the system; Step 2: When the structural parameters of the flexible multi-body system are uncertain, the uncertain parameters are introduced into the equations of the flexible multi-body system model to establish uncertain differential-algebraic equations; the uncertain parameters include random parameters or interval parameters; The uncertainty differential-algebraic equation established in step 2 is shown in formula (2): in, are n random or interval uncertainty parameter variables; when When is a random uncertainty parameter variable, It follows the standard normal distribution N(0,1); when When is an interval uncertainty parameter variable, The meanings of the remaining matrices and vectors are the same as in step 1, with superscripts " ～ " indicates that it is a matrix and vector containing uncertainty parameters; from the uncertainty parameters N groups of sample points are selected from the above, and the system output response corresponding to each sample can be expressed as Step 3: Decompose each output response of the system into multiple single-component signals in the form of vibration and a trend term signal by using the signal decomposition method, and characterize each single-component signal in the form of instantaneous amplitude and instantaneous phase; Step 4: Based on compressed sensing, at each moment, polynomial proxy models of instantaneous amplitude, instantaneous phase and trend term signals are established respectively; based on compressed sensing, the sparse coefficient matrix of each polynomial proxy model is solved; according to the type of uncertainty parameters, the obtained polynomial proxy model is used to obtain the dynamic response of the uncertain flexible multi-body system, that is, to achieve the prediction of the dynamic response of the uncertain flexible multi-body system; According to the type of uncertainty parameters, the obtained polynomial surrogate model is used to obtain the dynamic response of the uncertain flexible multi-body system; When the uncertainty parameter is an interval parameter, the interval scanning method is used to scan the interval uncertainty parameter with enough sample points to obtain the upper and lower bounds of the response of the uncertain flexible multi-body system dynamics. When the uncertainty parameters are random parameters, the mean value of the dynamic response of the uncertain flexible multi-body system can be efficiently predicted according to the analytical formula of the mean value of the system response established by deduction. Step 4 is implemented as follows: Step 4.1: Establish a polynomial proxy model for the instantaneous amplitude and trend term signals obtained in step 3; in, represents the undetermined coefficient in the polynomial surrogate model for the kth instantaneous amplitude, γ r The undetermined coefficients in the polynomial surrogate model representing the trend term, Represents uncertainty parameter The corresponding polynomial basis functions, A and R are the cutoff numbers of the expanded terms of the polynomial surrogate model of the instantaneous amplitude and trend terms, respectively; Step 4.2: Perform phase unwrapping on the instantaneous phase and establish a polynomial proxy model based on compressed sensing; Since the instantaneous phase obtained by Hilbert transform jumps within [-π,π], it is necessary to use the phase unwrapping algorithm to make the phase value monotonic, and then establish a polynomial proxy model for the unwrapped instantaneous phase; in, represents the undetermined coefficient in the polynomial proxy model of the kth instantaneous amplitude, B is the number of truncated terms in the expanded terms of the polynomial proxy model of the instantaneous phase; Step 4.3: Based on compressed sensing, obtain the sparse approximate solution of the unknown coefficients of the polynomial proxy model (5)-(7); Step 4.3.1: Based on compressed sensing, the coefficient matrix α = [α0 α1 … α A-1 ] T , β=[β0 β1 … β B-1 ] T and γ=[γ0 γ1 … γ R-1 ] T The approximate solution of as well as By solving the following three optimization problems: Among them, Ψ1, Ψ2, Ψ3 are measurement matrices, a=[a1 a2 … a N ] T , and r = [r1 r2… r N ] T Respectively represent N groups of instantaneous amplitude response, instantaneous phase response and trend term response; where ||α||0=W1, ||β||0=W2, ||γ||0=W3 means that there are W1, W2, W3 non-zero elements in the coefficient matrix α, β, γ, respectively, and W1, W2, W3 also represent the sparsity of the coefficient matrix α, β, γ, which is used to measure the sparsity of the coefficient matrix α, β, γ; Step 4.3.2: Considering that equations (8)-(10) are NP-hard problems, it is difficult to solve them using exact algorithms, and only effective approximate algorithms can be sought; therefore, an approximate solution is obtained by solving the following optimization problem: and Among them, ε1, ε2, ε3 represent the tolerance of prediction accuracy. If the values ​​of ε1, ε2, ε3 are set too large, the accuracy of the calculation results will be low, and if they are set too small, it will lead to overfitting. Therefore, it is very important to select a suitable accuracy tolerance value, and ε1, ε2, ε3 depend on the sparsity W1, W2, W3 respectively. Step 4.3.3: Obtain a reasonable sparsity through the cross-validation method in machine learning and solve the approximate solution of equation (9); Step 4.4: According to step 4.3, the sparse approximate solution of the unknown coefficients is obtained to predict the dynamic response of the uncertain flexible multi-body system; The method further includes step 5: realizing high-efficiency and high-precision prediction of the dynamic response of a flexible multi-body system with uncertain parameters according to steps 1 to 4, and applying the prediction results to large-scale complex systems with large calculation scale, high dimension of input uncertain parameters and strong nonlinear characteristics in the engineering field, so as to solve the prediction of relevant uncertain responses in the engineering field; The HHT method is used to decompose each output response of the system, which mainly includes empirical mode decomposition and Hilbert transform; the specific implementation method is: Empirical mode decomposition decomposes the original signal into the sum of multiple eigenmode functions and the residual signal: in, represents the eigenmode function, Represents the trend term signal; according to the Hilbert transform, the instantaneous amplitude and instantaneous phase of each eigenmode function are solved, and the uncertainty output response is: in and They represent the instantaneous amplitude and instantaneous phase of the kth eigenmode function corresponding to the jth system output response, and both the instantaneous amplitude and the instantaneous phase are related to the uncertainty parameter function.

2. The efficient method for predicting the dynamic response of an uncertain flexible multi-body system according to claim 1, characterized in that: A greedy algorithm is used to obtain approximate solutions to the optimization problems (11)-(13); the greedy algorithm includes orthogonal matching pursuit, compressed sampling matching pursuit and subspace pursuit.

3. The efficient method for predicting the dynamic response of an uncertain flexible multi-body system according to claim 2, characterized in that: Step 4.4 is implemented as follows: According to the sparse polynomial proxy models of the instantaneous amplitude, instantaneous phase and trend term signals obtained in step 4.3, the polynomial proxy model of the response of the uncertain flexible multi-body system dynamics is further combined: Step 4.4.1: When the uncertainty parameter is an interval parameter, use the interval scanning method to Take enough sample points to scan the polynomial proxy model (14) to obtain the upper and lower bounds of the response of the uncertain flexible multi-body system dynamics; Step 4.4.2: When the uncertainty parameters are random parameters, the mean of the dynamic response of the uncertain flexible multi-body system is efficiently predicted based on the derived analytical formula for the mean of the system response.

4. The efficient method for predicting the dynamic response of an uncertain flexible multi-body system according to claim 3, characterized in that: Step 4.4.2 is implemented as follows: When the system contains two random uncertain parameters, and the probability density function of the random uncertain parameters obeys the normal distribution, the implementation method of step 4.4.2 is: Using the second-order chaotic polynomial proxy model to approximate the instantaneous amplitude, instantaneous phase and trend term signal of the system, we have in, and denote the instantaneous amplitude and instantaneous phase of the given kth (k=1,...,m) eigenmode function respectively; Indicates trend item; represents the Hermite polynomial basis function; then the analytical formula for the mean response of the uncertain flexible multi-body system dynamics is in, E[r]=γ0 (19) For a given k (k = 1, ..., m), we have According to the adaptive signal decomposition, the total number of eigenmode functions of N groups of different output responses is different; for the convenience of prediction, the first two groups of data remain unchanged, and the third group and the following groups are all merged into the third group, that is, let m = 3; According to the analytical formula (18) of the response mean of the dynamics of the uncertain flexible multi-body system, the prediction of the response mean of the dynamics of the random uncertain flexible multi-body system is realized.

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