CN117763856A - Binary chebyshev differential Lagrangian multiplier interval analysis method - Google Patents
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Abstract
The invention provides a binary chebyshev differential Lagrangian multiplier interval analysis method, which comprises the steps of initializing upper and lower boundary information of interval uncertain parameters of a multi-body system and correlation information between the interval parameters, grouping the uncertain parameters according to the correlation of the interval parameters, determining a corresponding correlation coefficient matrix, and constructing a multi-ellipsoid model; calculating derivative information of response about uncertain parameters based on a binary chebyshev difference method, wherein the derivative information comprises a first-order partial derivative and a second-order partial derivative; calculating the upper and lower boundaries of the system output interval response by using a Lagrangian multiplier method and a second-order Taylor polynomial based on derivative information of the response of the multi-body system about the uncertain parameters; calculating correlation coefficients between response components based on derivative information of the output response about the uncertain parameter, and obtaining a covariance matrix of the output response; and constructing an uncertain domain of the output response according to the covariance matrix and the system output response boundary. The method can accurately quantify the influence of parameter uncertainty on the system response.
Description
Technical Field
The invention relates to the technical field of mechanical multi-body system dynamics analysis, in particular to a binary chebyshev differential Lagrange multiplier interval analysis method.
Background
With the development of the aerospace technology, modern aerospace institutions progress towards the targets of high precision, high stability and high reliability, the requirements on the mechanical environment of the aerospace institutions are more and more severe, and the accurate dynamic modeling and analysis of the aerospace institutions are of great significance in improving the overall design level of the spacecraft. Therefore, accurate modeling and dynamic behavior prediction of a complex multi-body system become hot spot directions of multi-body system dynamic research, and are important points of attention of engineering technicians.
However, in actual engineering, due to the influence of factors such as manufacturing process, processing error, external environment, incomplete cognition and the like, the parameters of the physical model of the multi-body system inevitably show uncertainty. In order to increase the robustness of the multi-body system model to obtain a more accurate and reliable system dynamics response that is consistent with engineering reality, it is necessary to consider the effect of uncertainty of parameters on the system response in the multi-body system dynamics analysis. In the case where a plurality of section parameters are coupled due to a physical phenomenon, independence between section parameters may also cause overestimation of the response boundary of the system when correlation between the parameters is not considered.
Disclosure of Invention
In order to solve the problems in the prior art, the application provides a binary chebyshev differential Lagrange multiplier interval analysis method, which takes a plurality of related interval parameters and independent interval parameters into consideration in the multi-body system dynamics response analysis to carry out system response boundary analysis and response correlation analysis, so that the influence of parameter uncertainty on system response can be more accurately quantified.
In order to achieve the above objective, the present application provides a binary chebyshev differential lagrangian multiplier interval analysis method, which includes the following steps:
step 1, initializing upper and lower boundary information of interval uncertain parameters of a multi-body system and correlation information between the interval parameters, grouping the uncertain parameters according to the correlation of the interval parameters, determining a corresponding correlation coefficient matrix, and constructing a multi-ellipsoid model;
step 2, calculating the derivative information of response about uncertain parameters based on a binary chebyshev difference method, wherein the derivative information comprises a first-order partial derivative and a second-order partial derivative;
step 3, calculating the upper and lower boundaries of the system output interval response by adopting a Lagrangian multiplier method and a second-order Taylor polynomial based on derivative information of the response of the multi-body system about the uncertain parameters;
step 4, calculating correlation coefficients among response components based on derivative information of the output response about the uncertain parameters, and obtaining a covariance matrix of the output response;
and 5, constructing an uncertain domain of the output response according to the covariance matrix and the system output response boundary.
In some embodiments, in the step 2, derivative information of the response about the uncertain parameter is calculated based on a binary chebyshev differential method, including a first partial derivative and a second partial derivative, and the specific procedures are as follows:
for the multidimensional response function y (α), the binary function decomposition is expressed as:
wherein n is the number of interval parameters;for corresponding to interval parameter->And->Is a binary response function of y (alpha) k ) For interval parameter alpha k A unitary response function; y is 0 Response of the system at the midpoint of the uncertainty parameter;
based on the binary function decomposition, the response function in equation (9) is related to the kth uncertainty parameter α k The first partial derivative of (2) is:
in the method, in the process of the invention,a first partial derivative which is a binary response function; />A partial derivative that is a unitary response function; it can be seen that calculating the first derivative of the response function with respect to the uncertainty parameter requires the first derivatives of n-1 binary functions and the first derivative of one unitary function;
similarly, response function vs. α k The second derivative of (2) is represented by equation (11):
further, the second order mixed partial derivative of the response function is expressed by equation (10) versus α k The bias derivative is expressed as follows:
as can be seen from the above equation, the second order hybrid derivative of the response function with respect to the uncertain parameter only needs to be derived from the corresponding binary function with respect to the uncertain parameter;
to calculate the derivatives of the unitary and binary response functions with respect to the uncertainty parameters, the unitary and binary response functions are approximated using chebyshev polynomials, in particular, the unitary response function may be approximated using chebyshev polynomials:
in the method, in the process of the invention,coefficients of a unitary chebyshev polynomial, p being the polynomial order; for a pair ofEquation (13) for uncertainty parameter α k Obtaining first order partial derivative, and obtaining:
in the method, in the process of the invention,thus, it is possible to obtain:
wherein ζ is a standard interval variable; similarly, equation (13) is described with respect to uncertainty parameter α k The second partial derivative is calculated to obtain the second partial derivative of the unitary response function, which is expressed as:
wherein:
since the chebyshev polynomial is defined on [ -1,1], ζ is zero in calculating the derivative at the midpoint of the parameters for the interval; similarly, approximation of the binary response function using chebyshev polynomials can be expressed as:
in the method, in the process of the invention,coefficients that are binary cut-ratio polynomials; h is subscript->Number of zeros; for formula (18) about->The first derivative and the second derivative are calculated to obtain:
the second derivative of the uncertainty parameter is:
further, the second order mixed derivative of the binary function is expressed as:
the binary chebyshev differential Lagrange multiplier interval analysis method has the beneficial effects that in the multi-body system dynamics response analysis, a plurality of related interval parameters and independent interval parameters are simultaneously considered to carry out system response boundary analysis and response correlation analysis, so that the influence of parameter uncertainty on system response can be more accurately quantified.
Drawings
FIG. 1 illustrates a flow chart of a binary Chebyshev differential Lagrangian multiplier interval analysis method in an embodiment.
Fig. 2 shows a schematic view of a slider-crank mechanism in an embodiment.
Fig. 3 shows an uncertainty map obtained by MCCA and BCDLM at t=0.47 s of the speed response of the crank block mechanism, where fig. 3 (a) is an uncertainty response field constituted by crank angular velocity and connecting rod angular velocity, and fig. 3 (b) is an uncertainty response field constituted by crank angular velocity and slider velocity.
Detailed Description
The following describes the embodiments of the present application further with reference to the accompanying drawings.
As shown in fig. 1, the binary chebyshev differential lagrangian multiplier interval analysis method according to the present application includes the following steps:
step 1, initializing upper and lower boundary information of interval uncertain parameters of a multi-body system and correlation information between the interval parameters, grouping the uncertain parameters according to the correlation of the interval parameters, determining a corresponding correlation coefficient matrix, and constructing a multi-ellipsoid model.
And 2, calculating the derivative information of response about the uncertain parameters based on a binary Chebyshev difference method, wherein the derivative information comprises a first partial derivative and a second partial derivative.
And step 3, calculating the upper and lower boundaries of the system output interval response by using a Lagrangian multiplier method and a second-order Taylor polynomial based on derivative information of the response of the multi-body system about the uncertain parameters.
And 4, calculating a correlation coefficient between response components based on derivative information of the output response about the uncertain parameters, and obtaining a covariance matrix of the output response.
And 5, constructing an uncertain domain of the output response according to the covariance matrix and the system output response boundary.
Specifically, in the step 1, the process of constructing the multi-ellipsoid model is as follows:
the uncertain parameters are regarded as interval variables which are independent from each other in the interval model, however, physical parameters in an actual system are mutually influenced, and correlation exists between the physical parameters. There is a positive correlation between the strength and stiffness of the component materials produced by casting, while the width and thickness tend to exhibit a negative correlation due to the effect of gravity. To describe the correlation between interval variables, a non-probabilistic ellipsoid model was developed. The non-probability ellipsoid model adopts a multidimensional ellipsoid to describe uncertain domains of a plurality of interval variables, any implementation of uncertain parameters is contained in a set of ellipsoid model descriptions, but the occurrence probability of the uncertain parameters is unknown, and the size and the correlation degree of the uncertain parameters are characterized by the shape and the geometric parameters of the ellipsoid.
Section vector α= [ α ] for a plurality of section variables 1 ,α 2 ,...,α n ] T Wherein alpha is i Is the ithInterval variable and satisfyα i And->Respectively interval variable alpha i Lower and upper boundaries of (2); />A section formed by upper and lower boundaries; n is the number of interval variables. When the interval variables are independent of each other, the uncertain domain is an "n-dimensional box", which can be expressed as:
if the interval variables are related to each other, the uncertainty domain formed by the interval uncertainty parameters can be described by an n-dimensional ellipsoid, which is expressed as follows:
(α-α c ) T W α (α-α c )≤1 (2)
in the method, in the process of the invention,is a mid-point vector of the interval parameter and is also a center point of the n-dimensional ellipsoid; w (W) α The characteristic matrix is a characteristic matrix of an ellipsoid model and is a symmetrical positive definite matrix, and the characteristic matrix determines the size and the direction of the ellipsoid.
In general, geometric features of an ellipsoid model can be determined by a minimum volume method based on sample information of uncertain parameters, and a feature matrix of the ellipsoid model can be further obtained. The method has the problems of high calculation cost, reduced precision and the like when being applied to the condition of more interval variables. The relation between the feature matrix and the interval variable covariance matrix is as follows:
W α ＝C α -1 (3)
wherein C is α Covariance matrix, which is interval variable, is expressed as:
in the method, in the process of the invention,is the interval variable alpha i And interval variable alpha j Covariance between.
The ellipsoid model adopts the correlation coefficient to represent the variable alpha of any interval i And alpha j The correlation between them is recorded asThe relationship with covariance is as follows:
in the method, in the process of the invention,for the interval radius, can be defined by->And (5) calculating to obtain the product.
For a multi-ellipsoid model, all interval uncertainty parameter vectors alpha can be divided into w groups according to the correlation information among interval parameters and recorded asThe uncertainty domain of each group is described by an ellipsoid model. For any one of the groups, an interval midpoint vector, an interval radius vector similar to the interval variable may be defined. In the multi-ellipsoid model, the ith group α i Inter-interval parameters within (i=1, 2,., w) are related to each other, but independent of the interval parameters in the other groups. Referring to an ellipsoid model, for the ith group of multi-ellipsoid models, the uncertainty domain composed of its interval uncertainty parameters can be described by the ellipsoid model as:
wherein w is the number of ellipsoidal models in the multi-ellipsoidal model;the mid-point vector of the interval parameter of the ith ellipsoid model; w (W) α,i Is the characteristic matrix of the ith ellipsoid model, and can be obtained by the covariance matrix C of the ellipsoid model α,i Is obtained by inverse matrix of (i.e.)
Recording deviceThe multi-dimensional ellipsoidal convex model can be expressed as:
based on the multi-ellipsoid model, the uncertain domain of the interval parameters can be expressed as:
in the step 2, derivative information of response about uncertain parameters is calculated based on a binary chebyshev differential method, wherein the derivative information comprises a first partial derivative and a second partial derivative, and the specific process is as follows:
for a multidimensional response function y (α), the use of a binary function decomposition can be expressed as:
wherein n is the number of interval parameters;for corresponding to interval parameter->And->Is a binary response function of y (alpha) k ) For interval parameter alpha k A unitary response function; y is 0 Is the response of the system at the midpoint of the uncertainty parameter.
Based on the binary function decomposition, the response function in equation (9) is related to the kth uncertainty parameter α k The first partial derivative of (2) is:
in the method, in the process of the invention,a first partial derivative which is a binary response function; />Is the partial derivative of the unitary response function. It can be seen that calculating the first derivative of the response function with respect to the uncertainty parameter requires the first derivatives of n-1 binary functions and the first derivative of one unitary function.
Similarly, response function vs. α k The second derivative of (2) can be represented by equation (11):
further, the second order mixed partial derivative of the response function may be expressed as the equation (10) versus α k The bias derivative is expressed as follows:
as can be seen from the above equation, the second order hybrid derivative of the response function with respect to the uncertainty parameter only needs to be derived from the corresponding binary function with respect to the uncertainty parameter.
To calculate the derivatives of the unitary and binary response functions with respect to the uncertainty parameter, chebyshev polynomials may be used to approximate the unitary and binary response functions. Specifically, using chebyshev polynomials, the unitary response function can be approximated as:
in the method, in the process of the invention,the coefficients of the unitary chebyshev polynomial are represented by p, the polynomial order. For equation (13) about uncertainty parameter α k Obtaining first order partial derivative, and obtaining:
in the method, in the process of the invention,thus, it is possible to obtain:
wherein ζ is a standard interval variable. Similarly, equation (13) is described with respect to uncertainty parameter α k The second partial derivative is calculated to obtain the second partial derivative of the unitary response function, which is expressed as:
wherein:
notably, since the chebyshev polynomial is defined on [ -1,1], ζ is zero in the formula when calculating the derivative at the midpoint of the parameters with respect to the interval. Similarly, approximation of the binary response function using chebyshev polynomials can be expressed as:
in the method, in the process of the invention,coefficients that are binary cut-ratio polynomials; h is subscript->Zero number of zeros. For formula (18) about->The first derivative and the second derivative are calculated to obtain:
the second derivative of the uncertainty parameter is:
further, the second order hybrid derivative of the binary function can be expressed as:
in the step 3, based on derivative information of the response of the multi-body system about the uncertain parameter, calculating the upper and lower boundaries of the response of the output interval of the system by using a Lagrangian multiplier method and a second-order Taylor polynomial, wherein the specific process is as follows:
based on the second-order taylor expansion polynomial and ignoring higher-order terms, the system convex response is expressed as:
in the formula g c The first derivative of the response function at the mid-point vector in the interval can be expressed as:
H c the matrix of the response component versus uncertainty parameter can be expressed as:
when parameter uncertainty propagation with an uncertainty domain being a convex set is performed, since the convex model of the uncertainty domain is controlled to be an inequality, for calculating an extremum of a response function under a multi-ellipsoid model, a Lagrangian multiplier is introduced to the response component function and a Lagrangian function L (delta) is defined as follows:
in the method, in the process of the invention,the matrix of the Hairs matrix for which the response component corresponding to the ith ellipsoid model is uncertain parameters can be obtained by grouping the pairs H by ellipsoid model c The block is obtained as follows:
wherein w is the number of ellipsoidal models in the multi-ellipsoidal model;the matrix is the matrix of the response component corresponding to the ith ellipsoid model. W (W) α,i The feature matrix is the i-th ellipsoid model; delta i The deviation vector for the interval parameter in the ith ellipsoid model can be defined by +.>Grouping according to grouping information of uncertain parameters; lambda (lambda) i Is the Lagrangian multiplier corresponding to the ith ellipsoid model in the multi-ellipsoid models.
For inequality-containing optimization, according to the K-T condition, for the ith ellipsoid model, the following two cases are satisfied when the response function takes an extremum:
(1) If the extreme point is inside an uncertainty domain, the lagrangian multiplier satisfies λ=0 and has the following equation:
solving the above equation, we can obtain:
at this time, the extremum of the response function is:
(2) If the extreme point is located on the boundary of an uncertain domain, the lagrangian multiplier λ+.0 at this time, the following requirements apply:
solving the above equation to obtain solutions of at least two Lagrangian multipliers, and thusExtreme points corresponding to itAt this time, the upper and lower boundaries of the response function under the second extremum condition are:
in the method, in the process of the invention,the solution of (s-1) is the solution of equation (29). The upper and lower boundaries of the response component are:
wherein:
in the step 4, a correlation coefficient between response components is calculated based on derivative information of the output response about the uncertain parameter, and a covariance matrix of the output response is obtained, which is specifically as follows:
consider a response comprising N response components y k (k=1, 2,., N) a multi-body system, outputting a response y k And y l The covariance of (c) can be defined as:
wherein b is 1 Is a constant; omega shape y In response to an indeterminate domain. δy k And delta y l Can be obtained by the following formula:
in the method, in the process of the invention,and->Is the response value of the system at the vector of points in the uncertainty parameter.
Based on the first-order taylor polynomial expansion, then:
wherein b is 2 Is a constant. "Det" is a determinant value taking operation, and "diag" represents a diagonal matrix.
Considering that the interval parameters between different ellipsoid models in the multi-ellipsoid model are independent of each other, the right side of equation (33) can be expressed as:
in the method, in the process of the invention,an uncertain domain of an ith ellipsoid model in the multi-ellipsoid models; omega shape α Is an uncertain domain of a multi-ellipsoidal convex model.
For a single ellipsoidal model, there are:
in b 3,i Is constant.
Combined formula (37) and formula (33), can be obtained:
where "Trace" represents the Trace operation of the matrix.
Further, the response component y k And y l The correlation coefficient between them can be expressed as:
where k, l=1, 2,..n is the number of system responses,and->Respectively the response components y k And y l Column vectors of the first partial derivative of the interval parameter vector in the ith ellipsoid male die at the middle point of the interval parameter vector; />And->Respectively the response components y k And y l And (3) a sea-sequoyins matrix formed by second-order partial derivatives of the i-th ellipsoidal male die middle section parameter vector at the middle point of the i-th ellipsoidal male die middle section parameter vector. Cov (y) k ,y l ) For covariance of the response components, covariance matrix construction of the output response is available.
The binary chebyshev differential lagrangian subinterval analysis method can be used for multi-body system response correlation analysis of general correlation interval parameters. In this embodiment, a slider-crank mechanism is described as an application. The crank block mechanism is used as a typical multi-body system and is commonly applied to the research of uncertain analysis examples. Next, taking the crank block mechanism shown in fig. 2 as an example, validity verification of the response analysis method of the multi-body system with related interval parameters is carried out. The parameters of the crank block mechanism are shown in table 1.
Table 1 crank slider mechanism parameter table
Considering the length, width and height of the crank and the connecting rod as interval uncertainty parameters, wherein the uncertainty of each interval parameter is 1%; the correlation coefficient between the different related interval parameters of the crank and the connecting rod is 0.01, and the characterization of the correlation between the crank and the connecting rod interval parameters is realized by adopting a double-ellipsoid model. And verifying a result obtained by BCDLM (binary Chebyshev-Lagrange multiplier) by adopting a Monte Carlo correlation analysis Method (MCCA), and simultaneously providing a distribution condition of sample point response, wherein the number of samples of the MCCA convex scanning method is 5.
The uncertainty domain obtained by MCCA and BCDLM at t=0.47 s of the speed response of the crank block mechanism is shown in fig. 3, where fig. 3 (a) is an uncertainty domain composed of crank angular velocity and connecting rod angular velocity, and fig. 3 (b) is an uncertainty domain composed of crank angular velocity and block velocity.
As can be seen from fig. 3, the response uncertainty domains constructed by the two methods are substantially identical, thus illustrating the effectiveness of the BCDLM method. A strong negative correlation is present between the velocity response components compared to a weak negative correlation of the displacement response. As can be seen from fig. 3, most samples fall in the center of the elliptical domain, while when response correlation is not considered, the underlying assumption is that the response samples fall uniformly in the multidimensional boxes constructed at the upper and lower boundaries of the response, and obviously, the uncertainty domains of the multidimensional boxes are more conservative. Thus, for a multi-body system containing multiple interval parameters, considering response correlation can construct a more accurate response uncertainty domain.
According to the binary chebyshev differential Lagrange multiplier interval analysis method, in the multi-body system dynamics response analysis, a plurality of related interval parameters and independent interval parameters are simultaneously considered to conduct system response boundary analysis and response correlation analysis, and the influence of parameter uncertainty on system response can be accurately quantified.
The foregoing is only a preferred embodiment of the present application, but the scope of the present application is not limited thereto, and any person skilled in the art, within the scope of the present application, should make equivalent substitutions or modifications according to the technical solution and the concept of the present application, and should be covered by the scope of the present application.
Claims (2)
1. A binary chebyshev differential Lagrangian multiplier interval analysis method is characterized in that: the method comprises the following steps:
step 1, initializing upper and lower boundary information of interval uncertain parameters of a multi-body system and correlation information between the interval parameters, grouping the uncertain parameters according to the correlation of the interval parameters, determining a corresponding correlation coefficient matrix, and constructing a multi-ellipsoid model;
step 2, calculating the derivative information of response about uncertain parameters based on a binary chebyshev difference method, wherein the derivative information comprises a first-order partial derivative and a second-order partial derivative;
step 3, calculating the upper and lower boundaries of the system output interval response by adopting a Lagrangian multiplier method and a second-order Taylor polynomial based on derivative information of the response of the multi-body system about the uncertain parameters;
step 4, calculating correlation coefficients among response components based on derivative information of the output response about the uncertain parameters, and obtaining a covariance matrix of the output response;
and 5, constructing an uncertain domain of the output response according to the covariance matrix and the system output response boundary.
2. The binary chebyshev differential lagrangian subinterval analysis method according to claim 1, wherein: in the step 2, derivative information of response about uncertain parameters is calculated based on a binary chebyshev differential method, wherein the derivative information comprises a first partial derivative and a second partial derivative, and the specific process is as follows:
for the multidimensional response function y (α), the binary function decomposition is expressed as:
wherein n is the number of interval parameters;for corresponding to interval parameter->And->Is a binary response function of y (alpha) k ) For interval parameter alpha k A unitary response function; y is 0 Response of the system at the midpoint of the uncertainty parameter;
based on the binary function decomposition, the response function in equation (9) is related to the kth uncertainty parameter α k The first partial derivative of (2) is:
in the method, in the process of the invention,a first partial derivative which is a binary response function; />A partial derivative that is a unitary response function; it can be seen that calculating the first derivative of the response function with respect to the uncertainty parameter requires the first derivatives of n-1 binary functions and the first derivative of one unitary function;
similarly, response function vs. α k The second derivative of (2) is represented by equation (11):
further, the second order mixed partial derivative of the response function is expressed by equation (10) versus α k The bias derivative is expressed as follows:
as can be seen from the above equation, the second order hybrid derivative of the response function with respect to the uncertain parameter only needs to be derived from the corresponding binary function with respect to the uncertain parameter;
to calculate the derivatives of the unitary and binary response functions with respect to the uncertainty parameters, the unitary and binary response functions are approximated using chebyshev polynomials, in particular, the unitary response function may be approximated using chebyshev polynomials:
in the method, in the process of the invention,coefficients of a unitary chebyshev polynomial, p being the polynomial order; for equation (13) about uncertainty parameter α k Obtaining first order partial derivative, and obtaining:
in the method, in the process of the invention,thus, it is possible to obtain:
wherein ζ is a standard interval variable; similarly, equation (13) is described with respect to uncertainty parameter α k The second partial derivative is calculated to obtain the second partial derivative of the unitary response function, which is expressed as:
wherein:
since the chebyshev polynomial is defined on [ -1,1], ζ is zero in calculating the derivative at the midpoint of the parameters for the interval; similarly, approximation of the binary response function using chebyshev polynomials can be expressed as:
in the method, in the process of the invention,coefficients that are binary cut-ratio polynomials; h is subscript->Number of zeros; for formula (18) about->The first derivative and the second derivative are calculated to obtain:
the second derivative of the uncertainty parameter is:
further, the second order mixed derivative of the binary function is expressed as:
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