CN109190328B - Multi-parameter structure dynamic response sensitivity analysis method based on mixed finite-complex difference - Google Patents
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Abstract
The invention provides a method for analyzing the dynamic response sensitivity of a multi-parameter structure with mixed finite-complex difference, which aims at the problem of the optimized design of the structure dynamics with multiple parameters, expands design parameters from a real number domain to a complex number domain, performs real part perturbation and imaginary part perturbation on two different design parameters respectively, performs dynamic analysis on the structure, extracts the real part response and the imaginary part response of an analysis result, obtains the dynamic response sensitivity of the two design parameters simultaneously, realizes the analysis of the dynamic response sensitivity of the multi-parameter structure, and provides a quick and effective method for analyzing the dynamic response sensitivity of the multi-parameter structure for engineering application.
Description
Technical Field
The invention belongs to the field of structure dynamics optimization design, and particularly relates to a dynamic response sensitivity analysis method of a mixed finite-complex difference multi-parameter structure.
Background
The dynamic response sensitivity analysis of the structural parameters is used as an important link in the structural dynamics optimization design based on the gradient analysis, and the influence degree and the influence rule of the structural design parameters on the output response are represented. An optimization target can be constructed by utilizing the output response of the structure, and the structure dynamics optimization design is realized according to the sensitivity analysis result of the structure dynamic response.
The existing methods can be divided into a theoretical analytical method, a finite difference method, a complex variation difference method and a semi-analytical method. The finite difference method and the complex variation difference method are simple in theory and easy to apply. However, when analyzing the structural dynamic response sensitivity of multiple parameters, perturbation needs to be performed on each parameter to increase the calculation time of re-analysis, and the analysis efficiency is low. How to realize the sensitivity analysis of a plurality of parameters in one-time analysis, reduce the calculation times, improve the sensitivity analysis efficiency and the structural dynamics optimization efficiency becomes an actual engineering problem to be solved urgently.
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 multi-parameter structure with mixed finite-complex difference, which can be used for simultaneously analyzing the sensitivity of the structure with a plurality of optimized parameters by simultaneously constructing the real part perturbation and the imaginary part perturbation of the optimized parameters, and can effectively improve the analysis efficiency of the dynamic response sensitivity of the multi-parameter 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 of mixed finite-complex differential structure dynamic response sensitivity analysis, the method comprising the steps of:
(1) finite element modeling is carried out on the engineering structure to obtain a structure dynamics control equation, and the equation is solved by utilizing a numerical analysis method of time domain integration to obtain dynamic response before parameter perturbation;
(2) constructing finite difference and complex variation difference shooting quantities for different optimization parameters at the same time, and substituting the parameters after shooting into a structural dynamics control equation to carry out dynamic response solution;
(3) and extracting a real part and an imaginary part in the structure dynamic response data after the parameters perturb, simultaneously calculating the dynamic response sensitivity of different parameters, and constructing a time-course curve of the structure dynamic response sensitivity.
Further, in the step (1), finite element modeling is carried out on the engineering structure to obtain a dynamic control equation of the structure, and the equation is solved by using a numerical analysis method of time domain integration to obtain the structural dynamic response before parameter perturbation, and the specific steps are as follows:
(1.1) carrying out finite element modeling on the engineering structure to obtain a dynamic control equation of the structure:
wherein M represents the mass matrix of the finite element model, C represents the damping matrix of the finite element model, K represents the stiffness matrix of the finite element model, F (t) represents the external excitation vector to which the finite element model is subjected, x (p, t) represents the displacement response vector of the structure,
a velocity response vector representing the structure is generated,
representing an acceleration response vector of the structure, and p represents a design parameter vector;
and (1.2) solving the dynamic control equation by using a numerical analysis method of time domain integration to obtain a structural dynamic response before parameter perturbation, namely a displacement response, a speed response or an acceleration response.
Further, in the step (2), a real part shooting amount and an imaginary part shooting amount are respectively constructed for different optimization parameters, and then a structure dynamics control equation after shooting is solved, and the specific steps are as follows:
(2.1) respectively constructing finite difference and complex variation difference shooting quantities for different optimization parameters:
wherein h and v respectively represent real part perturbation coefficient and imaginary part perturbation coefficient, p1And p2To design the components in the parameter vector p, representing two different parameters thereof,
representing a parameter p1The real part amount of shooting of (a), i.e. the finite difference amount of shooting,
representing a parameter p2The Real part and the imaginary part of the Real-time-based perturbation amount are respectively represented by the superscript Real and Imag, and the perturbed parameters are respectively as follows:
wherein,
and
respectively represents the optimized parameters after finite difference perturbation and complex variation differential perturbation, i represents an imaginary number unit, i represents2Get perturbed parameter vector as-1
(2.2) the perturbed parameters can cause the mass matrix, the damping matrix and the rigidity matrix in the finite element structure to change, and a perturbed structure dynamics control equation is obtained:
wherein,
a perturbed structural mass matrix is represented,
a damping matrix of the structure after perturbation is represented,
expressing the perturbed structural rigidity matrix, and solving a dynamic control equation (4) by using a time domain integral numerical analysis method to obtain a structural dynamic response after parameter perturbation:
further, in the step (3), real part and imaginary part results in the perturbed structural dynamic response data are extracted, and dynamic response sensitivities of corresponding parameters are respectively calculated, wherein the method comprises the following steps:
(3.1) based on the structural dynamic response obtained in (2.2), expanding the structural dynamic response according to a Taylor series:
wherein,
and
first and second partial derivatives of the response (.) to the parameter p, i denotes an imaginary unit, i2＝-1，onRepresenting an n-th order term;
(3.2) neglecting a second order term and an n order term in the formulas (15) - (17), extracting real part and imaginary part results in the structural dynamic response data, and obtaining the dynamic response sensitivity of the corresponding parameters:
wherein Re (.) and
representing the real and imaginary parts of the response (#) respectively,
and
respectively representing parameters p of structure displacement response1And parameter p2The sensitivity of (a) to (b) is,
and
respectively representing parameters p of structure speed response1And parameter p2The sensitivity of (a) to (b) is,
and
respectively representing structural acceleration response parameters p1And parameter p2The superscripts d, v, a represent displacement, velocity and acceleration, respectively.
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 multi-parameter sensitivity analysis method of mixed finite-complex difference, which expands design parameters from a real number field to a complex number field and constructs real part shooting quantity and imaginary part shooting quantity of different parameters at the same time, thereby realizing the analysis of the structural dynamic response sensitivity of a plurality of parameters in one analysis.
Drawings
FIG. 1 is a model diagram of finite element analysis of a structure according to the present invention;
FIG. 2 is a graph of the calculated A-point acceleration response with respect to the parameter c1A graph comparing the results of the sensitivity analysis;
FIG. 3 is a graph of the calculated A-point acceleration response with respect to the parameter k1The sensitivity analysis of (2) is compared with the result graph.
Detailed Description
As shown in fig. 1, a method for analyzing the dynamic response sensitivity of a multi-parameter structure by mixing finite-complex differences comprises the following steps:
(1) carrying out finite element modeling on the structure shown in FIG. 1 to obtain a dynamic control equation of the structure, and solving the equation to obtain the dynamic response of the structure before parameter perturbation;
(2) respectively constructing real part shot amount and imaginary part shot amount for different optimization parameters according to the optimization requirements of the structural dynamics, and then substituting the perturbed parameters into a structural dynamics control equation to carry out dynamic response solution;
(3) and extracting real part and imaginary part results in the perturbed structure dynamic response data, and respectively calculating the structure dynamic response sensitivity time-course curves of the corresponding parameters.
A dynamic response sensitivity analysis method of a mixed finite-complex difference multi-parameter structure is verified by adopting a cantilever beam structure with an elastic support, as shown in figure 1, the parameters of the structure are respectively as follows: the cantilever beam structure has length of 0.2m, circular cross section and radius of 0.005m, and is made of aluminum material with elastic modulus of 70GPa and density of 2.7 × 103kg/m3The poisson ratio μ is 0.3. And the parameters can be set to different values according to different design requirements.
The design parameters are the rigidity coefficient and the damping coefficient of five elastic supports, and the total number of the design parameters is 10, and a formed design parameter vector p ═ c1,k1,c2,k2,c3,k3,c4,k4,c5,k5Specific values of the design parameters are listed in table 1:
TABLE 1 values of design parameters for elastic bearings
The specific operation is as follows:
(1.1) carrying out finite element modeling on the structure, dividing the finite element modeling into 10 beam units, as shown in figure 1, obtaining a mass matrix, a damping matrix and a rigidity matrix according to the materials and geometrical parameters of the structure and the support, and finally obtaining a structure dynamics control equation as follows:
wherein M represents the mass matrix of the finite element model, K represents the stiffness matrix of the finite element model, C represents the damping matrix of the finite element model, and the proportional damping is selected in the present example, wherein C is α M + β K, wherein α is β is 0.0001. F (t) represents the external excitation vector to which the finite element model is subjected, in the present example, 10sin (20t) external excitation is applied in the vertical direction of A point, as shown in FIG. 1, x (p, t) represents the displacement response vector of the structure,
a velocity response vector representing the structure is generated,
an acceleration response vector representing the structure, and p a design parameter vector.
(1.2) solving the equation to obtain the response of the structure before the parameter perturbation, wherein the calculation cut-off time of the structure dynamic response is 0.01 s.
(2) Respectively constructing real part shot amount and imaginary part shot amount for different optimization parameters according to the optimization requirements of the structural dynamics, and then substituting the perturbed parameters into a structural dynamics control equation to carry out dynamic response solution;
(2.1) respectively constructing real part shooting quantity and imaginary part shooting quantity for different optimization parameters:
wherein h and v respectively represent a real part perturbation coefficient and an imaginary part perturbation coefficient, and h-v-10 is selected in the calculation example-3，k1And c1The components in the parameter vector are designed for the structure, representing two different parameters thereof,
and
respectively represent the parameter c1Real part of the amount of shooting, parameter k1The Real and imaginary parts are labeled Real and Imag, respectively. The parameters after perturbation are respectively as follows:
wherein,
and
respectively representing the optimized parameters after real part perturbation and imaginary part perturbation, i represents an imaginary unit, i2Is-1. Perturbed design parameter vector
Wherein
In the case of a real number,
is a plurality of numbers. To this end, the design parameters are expanded from the real-number domain to the complex-number domain.
(2.2) in the present embodiment, the perturbed parameters may cause the damping matrix and the stiffness matrix in the finite element structure to change, so as to obtain a perturbed structural dynamics control equation:
wherein,
represents the perturbed structural mass matrix, which in this example is equal to M in equation (21), i.e.
A damping matrix of the structure after perturbation is represented,
and F (t) representing a structural rigidity matrix after perturbation, wherein the external excitation vector received by the finite element model is the same as the value before parameter perturbation. Solving a dynamic control equation (24) by using a numerical analysis method of time domain integration to obtain a perturbed structural dynamic response:
(3) and extracting real part and imaginary part results in the perturbed structure dynamic response data, and respectively calculating the dynamic response sensitivity time-course curves of the corresponding parameters.
(3.1) based on the structural dynamic response obtained in (2.2), expanding the structural dynamic response according to a Taylor series:
wherein,
and
respectively representing responses (. alpha.). to parameter c1The first and second partial derivatives of (a),
and
respectively representing responses (. alpha.). to the parameter k1I denotes the imaginary unit, i2＝-1，onRepresenting an n-th order term.
(3.2) neglecting second-order terms and n-order terms of the equations (25) - (27), extracting real part and imaginary part results in the structural dynamic response data after A point perturbation, and respectively calculating the dynamic response sensitivity time-course curve of the corresponding parameter:
wherein Re (.) and
representing the real and imaginary parts of the response (#) respectively,
and
respectively representing parameters p of structure displacement response1And parameter p2The sensitivity of (a) to (b) is,
and
respectively representing parameters p of structure speed response1And parameter p2The sensitivity of (a) to (b) is,
and
respectively representing structural acceleration response parameters p1And parameter p2The superscripts d, v, a represent displacement, velocity and acceleration, respectively.
Respectively representing the displacement response, the velocity response and the acceleration response of the point A on the structure. FIG. 2 and FIG. 3 show the acceleration response at point A with respect to the parameter c1And k1The results of the method, the finite difference sensitivity analysis method and the complex variation difference sensitivity analysis method are respectively compared, and the results are consistent.
Claims (2)
1. A method for analyzing the sensitivity of a structure's dynamic response to a mixed finite-complex difference, the method comprising the steps of:
(1) finite element modeling is carried out on the engineering structure to obtain a structure dynamics control equation, and the equation is solved by utilizing a numerical analysis method of time domain integration to obtain dynamic response before parameter perturbation;
(2) constructing finite difference and complex variation difference shooting quantities for different optimization parameters at the same time, and substituting the parameters after shooting into a structural dynamics control equation to carry out dynamic response solution;
the method comprises the following specific steps:
(2.1) respectively constructing finite difference and complex variation difference shooting quantities for different optimization parameters:
wherein h and v respectively represent real part perturbation coefficient and imaginary part perturbation coefficient, p1And p2To design the components in the parameter vector p, representing two different parameters thereof,
representing a parameter p1The real part amount of shooting of (a), i.e. the finite difference amount of shooting,
representing a parameter p2The Real part and the imaginary part of the Real-time-based perturbation amount are respectively represented by the superscript Real and Imag, and the perturbed parameters are respectively as follows:
wherein,
and
respectively represents the optimized parameters after finite difference perturbation and complex variation differential perturbation, i represents an imaginary number unit, i represents2Get perturbed parameter vector as-1
(2.2) the perturbed parameters can cause the mass matrix, the damping matrix and the rigidity matrix in the finite element structure to change, and a perturbed structure dynamics control equation is obtained:
wherein,
a perturbed structural mass matrix is represented,
a damping matrix of the structure after perturbation is represented,
expressing the perturbed structural rigidity matrix, and solving a dynamic control equation (3) by using a time domain integral numerical analysis method to obtain a structural dynamic response after parameter perturbation:
(3) extracting a real part and an imaginary part in the structure dynamic response data after the parameter perturbation, simultaneously calculating the dynamic response sensitivity of different parameters, and constructing a time-course curve of the structure dynamic response sensitivity, wherein the method comprises the following steps:
(3.1) based on the structural dynamic response obtained in (2.2), expanding the structural dynamic response according to a Taylor series:
wherein,
and
first and second partial derivatives of the response (.) to the parameter p, i denotes an imaginary unit, i2＝-1，onRepresenting an n-th order term;
(3.2) neglecting second-order terms and n-order terms in the formulas (4) - (6), extracting real part and imaginary part results in the structural dynamic response data, and obtaining the dynamic response sensitivity of the corresponding parameters:
wherein Re (.) and
representing the real and imaginary parts of the response (#) respectively,
and
respectively representing parameters p of structure displacement response1And parameter p2The sensitivity of (a) to (b) is,
and
respectively representing parameters p of structure speed response1And parameter p2The sensitivity of (a) to (b) is,
and
respectively representing structural acceleration response parameters p1And parameter p2The superscripts d, v, a represent displacement, velocity and acceleration, respectively.
2. The method for analyzing the sensitivity of the structural dynamic response of the hybrid finite-complex difference as claimed in claim 1, wherein in the step (1), finite element modeling is performed on the engineering structure to obtain a structural dynamic control equation, and the equation is solved by using a numerical analysis method of time domain integration to obtain the structural dynamic response before parameter perturbation, and the specific steps are as follows:
(1.1) carrying out finite element modeling on the engineering structure to obtain a dynamic control equation of the structure:
wherein M represents the mass matrix of the finite element model, C represents the damping matrix of the finite element model, K represents the stiffness matrix of the finite element model, F (t) represents the external excitation vector to which the finite element model is subjected, x (p, t) represents the displacement response vector of the structure,
a velocity response vector representing the structure is generated,
representing an acceleration response vector of the structure, and p represents a design parameter vector;
and (1.2) solving the dynamic control equation by using a numerical analysis method of time domain integration to obtain a structural dynamic response before parameter perturbation, namely a displacement response, a speed response or an acceleration response.
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