CN109902357B - Method for analyzing dynamic response sensitivity of complex-variant differential smooth nonlinear structure - Google Patents
Method for analyzing dynamic response sensitivity of complex-variant differential smooth nonlinear structure Download PDFInfo
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Abstract
The invention provides a method for analyzing the dynamic response sensitivity of a nonlinear structure of complex difference, which aims at the sensitivity calculation problem of a smooth nonlinear structure, converts the calculation of a partial derivative into the calculation of a complex domain function value, performs dynamic analysis on the nonlinear structure by perturbing the imaginary part of a design parameter, extracts the imaginary part response of an analysis result, obtains the dynamic response sensitivity of the design parameter and realizes the dynamic response sensitivity analysis of the nonlinear structure. The invention can provide a dynamic response sensitivity analysis method with second-order precision for a nonlinear structure with smooth characteristics, solves the problem of analysis error caused by perturbation step length, and improves the analysis precision of nonlinear inverse problem.
Description
Technical Field
The invention belongs to the field of nonlinear structure optimization, and particularly relates to a method for analyzing dynamic response sensitivity of a smooth nonlinear structure based on complex difference.
Background
In recent years, sensitivity analysis and calculation methods for linear structures have been developed and improved, while non-linear structures are not directly determined by design variables of the structures because the rigidity is related to state variables of the structures, so that corresponding sensitivity analysis is more difficult. The dynamic response sensitivity analysis of the structural parameters is used as an important link in the field of model correction, the sensitivity value can reflect the influence degree and the influence rule of each design parameter of the structure on the structural performance, the accuracy of the model correction result is directly determined by the calculation precision of the sensitivity, and the method for analyzing the dynamic response sensitivity of the nonlinear structure of the complex difference has good engineering practical significance.
The conventional method for sensitivity analysis is a finite difference method, and the finite difference method calculates the partial derivative of the structural performance parameters on the structural design parameters to obtain the influence of each design variable of the structure on the structural performance. However, at the sharp oscillation position of the function partial derivative or when the partial derivative tends to 0 in a larger interval, the method has the problem that the approximate numbers are subtracted and then divided by small numbers, and has poor calculation accuracy and certain limitation.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the problems, the invention provides a method for analyzing the dynamic response sensitivity of a nonlinear structure with complex difference, which analyzes the sensitivity of the nonlinear structure with a plurality of design parameters by constructing the perturbation of the imaginary part of the design parameters, and converts the calculation of the partial derivative into the solution of a complex domain function value, thereby realizing the calculation of the dynamic response sensitivity of the nonlinear structure.
The technical scheme is as follows: in order to realize the purpose of the invention, the technical scheme adopted by the invention is as follows: a method for analyzing the dynamic response sensitivity of a complex differential smooth nonlinear structure comprises the following steps:
(1) performing dynamic modeling on the smooth nonlinear structure to obtain a structural mass matrix, a damping matrix, a rigidity matrix, a nonlinear force vector and an external force vector;
(2) constructing a complex variation differential pickup amount of the design parameter to change the design parameter from a real number to a complex number;
(3) introducing the perturbed design parameters into a smooth nonlinear structure dynamic model, and solving the dynamic model by using a numerical analysis method to obtain a smooth nonlinear dynamic response of the structure;
(4) and extracting the smooth nonlinear dynamic response imaginary part result obtained by the perturbation dynamic model calculation, and calculating the smooth nonlinear dynamic response sensitivity of the corresponding parameters.
Further, in the step (1), dynamic modeling is carried out on the smooth nonlinear structure to obtain a structural mass matrix M, a damping matrix C, a rigidity matrix K and a nonlinear force vector
And an external force vector F (t), the specific steps comprising the steps of:
(1.1) obtaining a mass matrix M, a damping matrix C and a rigidity matrix K of the structure according to the characteristics of the mass, the damping and the rigidity of the structureSmooth non-linear characteristics of a structure to obtain a non-linear force vector
Obtaining an external force vector F (t) according to the external excitation position and the applied load magnitude of the structure, wherein p is a design parameter vector,
representing the structure velocity response and x representing the structure displacement response.
Further, in the step (2), a complex differential pickup amount of the design parameter is constructed to change the design parameter from a real number to a complex number, and the specific steps include the following steps:
(2.1) constructing a complex variation differential pickup amount for the l-th design parameter with respect to the design parameter vector p:
wherein the content of the first and second substances,
denotes the l-th design parameter plI is an imaginary unit, i2＝-1，
Denotes the l-th design parameter plThe complex variation difference of (1) is divided into a shooting amount, and the upper mark Imag represents the meaning of an imaginary part;
(2.2) adding the complex variation difference shooting amount with the original design parameters to obtain the design parameters of a complex number field, and realizing that the design parameters are changed from real numbers to complex numbers:
wherein the content of the first and second substances,
and representing the design parameters of a complex domain obtained after the complex-variant differential perturbation is carried out.
Further, in the step (3), the perturbed design parameters are introduced into a smooth nonlinear structure dynamic model, and the dynamic model is solved by using a numerical analysis method to obtain a smooth nonlinear dynamic response of the structure, wherein the specific steps comprise the following steps:
(3.1) perturbing the complex field design parameters
Substituting original design parameters p into non-linear forceslObtaining perturbed non-linear force vector
And a smooth nonlinear structure dynamics model based on post-perturbation design parameters:
wherein M represents a structural mass matrix, C represents a structural damping matrix, K represents a structural stiffness matrix,
perturbed non-linear force vector, F (t) represents an external force vector,
representing the perturbed parameter vector(s),
and x respectively represents the acceleration response, the speed response and the displacement response of the whole structure and is obtained by utilizing a structural dynamics time domain dynamic response analysis method.
Further, in the step (4), a smooth nonlinear dynamic response imaginary part result obtained by the perturbation dynamic model calculation is extracted, and smooth nonlinear dynamic response sensitivity of the corresponding parameter is calculated, and the specific steps include the following steps:
(4.1) calculating the ith parameter p according to the step (3.1)lMaking a complex transformationAnd (3) carrying out differential perturbation structural dynamic response, including acceleration response, speed response and displacement response, and carrying out complex field Taylor expansion on the perturbed response:
wherein n represents the nth derivative, the high-order terms of the second order or above of the formula (3) are omitted, the imaginary part result in the structural dynamic response data is extracted, and the dynamic response sensitivity of the design parameter is calculated:
wherein the content of the first and second substances,
the representation takes its imaginary part of the dynamic response (#),
indicating the first parameter p of the structural acceleration responselThe sensitivity of (a) to (b) is,
first parameter p representing structural velocity responselThe sensitivity of (a) to (b) is,
indicating the ith parameter p of the structure displacement responselThe superscripts a, v and d respectively represent acceleration, speed and displacement, and when l traverses the number of all design parameters, the dynamic response sensitivity analysis of all design parameters can be realized by adopting the same method based on complex difference.
Has the advantages that: compared with the prior art, the technical scheme of the invention has the following beneficial technical effects:
the invention provides a method for analyzing the dynamic response sensitivity of a complex differential nonlinear structure, which expands design parameters from a real number field to a complex number field by carrying out finite element modeling on the nonlinear structure, constructs the imaginary part shooting amount of the parameters, realizes the calculation of the dynamic response sensitivity of the nonlinear structure with higher precision, and can give the analysis result of the dynamic response sensitivity of the smooth nonlinear structure only by p times of calculation if p design parameters exist.
Drawings
FIG. 1 is a flow chart of an embodiment of the present invention;
FIG. 2 is an analysis structure of an embodiment of the present invention: a smooth non-linear spring-mass structure;
FIG. 3 shows a mass m according to an embodiment of the present invention1With respect to design parameters
Acceleration response sensitivity curve of (a);
FIG. 4 shows a mass m according to an embodiment of the present invention2With respect to design parameters
Acceleration response sensitivity curve of (a);
FIG. 5 shows a mass m according to an embodiment of the present invention3With respect to design parameters
Acceleration response sensitivity curve of (a);
FIG. 6 shows a mass m according to an embodiment of the present invention4With respect to design parameters
Acceleration response sensitivity curve of (a);
FIG. 7 shows a mass m according to an embodiment of the present invention5With respect to design parameters
Acceleration response sensitivity curve of (1).
Detailed Description
The technical solution of the present invention is further described below with reference to the accompanying drawings and examples.
Fig. 1 shows a method for analyzing the dynamic response sensitivity of a complex differential smooth nonlinear structure, which comprises the following steps:
(1) performing dynamic modeling on the smooth nonlinear structure shown in the figure 1 to obtain a structural mass matrix, a damping matrix, a stiffness matrix, a nonlinear force vector and an external force vector;
(2) constructing a complex variation differential pickup amount of the design parameter to change the design parameter from a real number to a complex number;
(3) introducing the perturbed design parameters into a smooth nonlinear structure dynamic model, and solving the dynamic model by using a numerical analysis method to obtain a smooth nonlinear dynamic response of the structure;
(4) and extracting the smooth nonlinear dynamic response imaginary part result obtained by the perturbation dynamic model calculation, and calculating the smooth nonlinear dynamic response sensitivity of the corresponding parameters.
A complex differential structure dynamic response sensitivity analysis method is realized by a flow chart shown in figure 1. Fig. 2 shows a five-degree-of-freedom smooth nonlinear spring-mass model used in the present example, which can simulate a civil house, a mechanical assembly, etc. with smooth nonlinear characteristics. The basic parameters of the smooth nonlinear spring-mass structure are:
TABLE 1 basic parameters of smooth nonlinear spring-mass structure
The design parameters are the nonlinear stiffness coefficient and nonlinear damping coefficient of two groups of spring-damping units in the structure, and a cubic nonlinear spring-damping unit is adopted in the present example. Composed design parameter vector
As shown in fig. 2, the specific values of the design parameters are listed in table 2:
TABLE 2 design parameters for smooth nonlinear spring-mass structure
The specific operation is as follows:
in the step (1), a smooth nonlinear structure is subjected to dynamic modeling to obtain a structural mass matrix, a damping matrix, a stiffness matrix, a nonlinear force vector and an external force vector, and the method specifically comprises the following steps:
(1.1) the nonlinear spring-mass structure shown in fig. 2, considering the undamped state, its damping matrix C is 0, and the expressions of the mass matrix M and stiffness matrix K of the structure are:
wherein m is1～m 55 mass masses for the smooth nonlinear structure shown in fig. 2, as listed in table 1; k is a radical of1～k5The 5 linear stiffness coefficients for the structure shown in fig. 2 are listed in table 1. Obtaining a non-linear force vector from smooth non-linear features of a structure
The arrangement being applied to the mass m3Is x3The initial displacement here can be set according to actual needs, since no external excitation force is applied, i.e. the external force vector F (t) is 0, the expression of the nonlinear force vector is:
where p is a design parameter vector, x1，x2，x4And
are respectively a mass block m1，m2And m4Displacement and velocity.
In the step (2), the complex variation differential pickup amount of the design parameter is constructed, so that the design parameter is changed from a real number to a complex number, and the specific steps comprise the following steps:
(2.1) for the design parameter vector p, for the ith design parameter plThe complex variation differential pickup amount is constructed. In this example, l is 1,2,3,4, which respectively corresponds to four design parameters in the design parameter vector p, and 1 is taken as an example,
the complex variation differential pickup amount is as follows:
wherein the content of the first and second substances,
representing parameters
Is taken as
The value can be set according to actual needs, i is an imaginary unit, i2＝-1。
Representing parameters
The complex variation of (2) is divided into the amount of shooting, and the superscript Imag indicates the meaning of the imaginary part.
And (2.2) adding the complex variation difference shooting amount and the original design parameters to obtain the design parameters of a complex number field, so that the design parameters are changed from real numbers to complex numbers. In this example, the perturbed design parameters obtained when l is 1 are:
wherein the content of the first and second substances,
representing a complex field design parameter obtained after complex differential perturbation is carried out when l is 1;
in the step (3), the perturbed design parameters are brought into a smooth nonlinear structure dynamic model, and the dynamic model is solved by using a numerical analysis method to obtain a smooth nonlinear dynamic response of the structure, wherein the specific steps comprise the following steps:
(3.1) perturbing the complex field design parameters
Substituting original design parameters p into non-linear forceskObtaining perturbed non-linear force vector
In this example, the perturbed parameters are taken as 1
The perturbed nonlinear force vector introduced into the nonlinear force (13) is:
the smooth nonlinear structure dynamic model based on perturbed design parameters is as follows:
wherein M represents a structural mass matrix, C represents a structural damping matrix, K represents a structural rigidity matrix, and F (t) represents an external force vector, which is obtained in the step (1).
Is the perturbed non-linear force vector, determined by equation (11),
representing the perturbed parameter vector(s),
x represents the acceleration response, velocity response and displacement response of the smooth nonlinear structure, respectively. The formula (11), the formula (12) and the formula (16) are taken into the formula (17), the structural dynamic response after the complex differential perturbation is solved by using a structural dynamics time domain dynamic response analysis method, and the common dynamics time domain dynamic response analysis method comprises the following steps: a center difference method, a stepwise integration method, etc.
In the step (4), a smooth nonlinear dynamic response imaginary part result obtained by calculation of the perturbed dynamic model is extracted, and smooth nonlinear dynamic response sensitivity of the corresponding parameter is calculated, and the method specifically comprises the following steps:
(4.1) calculating the ith parameter p according to the step (3.1)lAnd performing complex-domain Taylor expansion on the perturbed response, wherein the structure dynamic response after the complex-domain differential perturbation can be acceleration response, velocity response and displacement response. In this example, the acceleration response is related to the design parameter
Complex field taylor expansion of (a):
where n represents the nth derivative. And (3) extracting the first-order imaginary part response in the formula (18) and calculating the dynamic response sensitivity of the design parameter. In this example, the acceleration response is related to the design parameter knl1Using the first-order imaginary response in equation (18), the dynamic response sensitivity is calculated, for example, as follows:
wherein the content of the first and second substances,
the representation takes its imaginary part of the dynamic response (#),
representing structural acceleration response versus design parameter
The superscript a represents the acceleration, and the acceleration response of 5 mass blocks can be obtained according to the step (3), so that the acceleration response sensitivity of 5 mass blocks can be obtained. FIGS. 3-7 show the acceleration response of 5 masses with respect to
The acceleration response of the five masses can be simultaneously calculated in the calculation of the sensitivity analysis result graph. And when l traverses the number of all design parameters, the dynamic response sensitivity analysis of all design parameters can be realized by adopting the method based on the complex difference, which is the same as the step (2), the step (3) and the step (4).
The structural dynamic response sensitivity of the design parameters is calculated by constructing the imaginary perturbation quantity in the above, i.e. in one analysis. In this example, compared with the finite difference sensitivity analysis method, the structural dynamic response sensitivity analysis method based on the complex difference provided by the patent does not need initial analysis, and the calculation times for realizing the dynamic response sensitivity analysis of 4 design parameters are 4. Compared with the finite difference method, the method has second-order calculation precision.
Claims (3)
1. A method for analyzing the dynamic response sensitivity of a complex differential smooth nonlinear structure is characterized by comprising the following steps:
(1) performing dynamic modeling on the smooth nonlinear structure to obtain a structural mass matrix, a damping matrix, a rigidity matrix, a nonlinear force vector and an external force vector;
(2) constructing the complex variation differential pickup amount of the design parameter to change the design parameter from a real number to a complex number, wherein the specific method comprises the following steps;
(2.1) constructing a complex variation differential pickup amount for the ith design parameter for the design parameter vector p:
wherein the content of the first and second substances,
denotes the l-th design parameter plI is an imaginary unit, i2＝-1，
Denotes the l-th design parameter plThe complex variation difference of (1) is divided into a shooting amount, and the upper mark Imag represents the meaning of an imaginary part;
(2.2) adding the complex variation difference shooting amount with the original design parameters to obtain the design parameters of a complex number field, and realizing that the design parameters are changed from real numbers to complex numbers:
wherein the content of the first and second substances,
representing design parameters of a complex field obtained after complex variation differential perturbation;
(3) introducing the perturbed design parameters into a smooth nonlinear structure dynamic model, and solving the dynamic model by using a numerical analysis method to obtain a smooth nonlinear dynamic response of the structure;
(4) extracting smooth nonlinear dynamic response imaginary part results obtained by calculation of the dynamic model after perturbation, and calculating smooth nonlinear dynamic response sensitivity of corresponding parameters, wherein the specific method comprises the following steps:
(4.1) calculation according to step (3)Obtained for the ith parameter plPerforming structural dynamic response after complex variation differential perturbation, wherein the structural dynamic response comprises acceleration response, speed response and displacement response, and performing complex field Taylor expansion on the perturbed response:
wherein n represents the nth derivative, the high-order terms of the second order or above of the formula (3) are omitted, the imaginary part result in the structural dynamic response data is extracted, and the dynamic response sensitivity of the design parameter is calculated:
wherein the content of the first and second substances,
the representation takes its imaginary part of the dynamic response (#),
indicating the first parameter p of the structural acceleration responselThe sensitivity of (a) to (b) is,
first parameter p representing structural velocity responselThe sensitivity of (a) to (b) is,
indicating the ith parameter p of the structure displacement responselThe superscripts a, v and d respectively represent acceleration, speed and displacement, and when l traverses the number of all design parameters, the dynamic response sensitivity analysis of all design parameters can be realized by adopting the same method based on complex difference.
2. The method for analyzing the dynamic response sensitivity of the smooth nonlinear structure with the complex difference as claimed in claim 1, wherein in the step (1), the method is applied toPerforming dynamic modeling on the smooth nonlinear structure to obtain a structural mass matrix M, a damping matrix C, a stiffness matrix K and a nonlinear force vector
And an external force vector F (t), the specific steps comprising the steps of:
(1.1) obtaining a mass matrix M, a damping matrix C and a stiffness matrix K of the structure according to the mass, damping and stiffness characteristics of the structure, and obtaining a nonlinear force vector according to the smooth nonlinear characteristics of the structure
Obtaining an external force vector F (t) according to the external excitation position and the applied load magnitude of the structure, wherein p is a design parameter vector,
representing the structure velocity response and x representing the structure displacement response.
3. The method for analyzing the sensitivity of the smooth nonlinear structure dynamic response of the complex differential as claimed in claim 2, wherein in the step (3), the perturbed design parameters are introduced into a smooth nonlinear structure dynamic model, and the dynamic model is solved by using a numerical analysis method to obtain the smooth nonlinear dynamic response of the structure, and the specific steps comprise the following steps:
(3.1) perturbing the complex field design parameters
Substituting original design parameters p into non-linear forceslObtaining perturbed non-linear force vector
And a smooth nonlinear structure dynamics model based on post-perturbation design parameters:
wherein M represents a structural mass matrix, C represents a structural damping matrix, K represents a structural stiffness matrix,
perturbed non-linear force vector, F (t) represents an external force vector,
representing the perturbed parameter vector(s),
and x respectively represents the acceleration response, the speed response and the displacement response of the whole structure and is obtained by utilizing a structural dynamics time domain dynamic response analysis method.
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