CN109902357A - A kind of multiple smooth nonlinear organization dynamic response Sensitivity Analysis Method divided that is deteriorated - Google Patents
A kind of multiple smooth nonlinear organization dynamic response Sensitivity Analysis Method divided that is deteriorated Download PDFInfo
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Abstract
The present invention provides a kind of multiple nonlinear organization dynamic response Sensitivity Analysis Methods divided that is deteriorated, this method is directed to the Calculation of Sensitivity problem of smooth nonlinear organization, the calculating of partial derivative is converted into the calculating of complex domain functional value, by carrying out imaginary part perturbation to design parameter, dynamic analysis is carried out to nonlinear organization, the imaginary part of extraction and analysis result responds, and obtains the dynamic response sensitivity of design parameter, realizes the dynamic response sensitivity analysis of nonlinear organization.The present invention can provide the dynamic response Sensitivity Analysis Method with second order accuracy for the nonlinear organization with smooth features, solve the problems, such as the analytical error as caused by perturbation step-length, improve the analysis precision of Nonlinear inverse problem.
Description
Technical field
The invention belongs to nonlinear structural optimization field more particularly to a kind of multiple smooth nonlinear organization sounds of something astir divided that is deteriorated
Answer Sensitivity Analysis Method.
Background technique
The analysis and Calculation of Sensitivity of linear structure developed in recent years it is more perfect, and nonlinear organization due to
Rigidity is related with the state variable of structure, is directly determined by the design variable of structure, therefore corresponding sensitivity analysis
Also just more difficult.The dynamic response sensitivity analysis of structural parameters as the important link in Modifying model field, sensitivity
Numerical value can reflect each design parameter of structure to the influence degree and affecting laws of structural behaviour, and the computational accuracy of sensitivity is straight
Connect the accuracy for determining Modifying model result, a kind of multiple nonlinear organization dynamic response Sensitivity Analysis Method being deteriorated point has
Preferable practical meaning in engineering.
The conventional method of sensitivity analysis is finite difference calculus, and finite difference calculus is by calculating structural behaviour parameter to structure
The partial derivative of design parameter obtains influence of each design variable of structure to structural behaviour.But at function partial derivative sharply oscillation
Or partial derivative is when larger section tends to 0, this method there are problems that after approximate number is subtracted each other again divided by fractional value, computational accuracy compared with
Difference has certain limitation.
Summary of the invention
Goal of the invention: in view of the above problems, the present invention proposes a kind of multiple nonlinear organization dynamic response sensitivity divided that is deteriorated
Analysis method, is perturbed by the imaginary part of tectonic sieving parameter, carries out sensitivity to the nonlinear organization with multiple design parameters
Analysis, the calculating of partial derivative are converted into the solution of complex domain functional value, realize the calculating of nonlinear organization dynamic response sensitivity.
Technical solution: to achieve the purpose of the present invention, the technical scheme adopted by the invention is that: a kind of multiple light divided that is deteriorated
Sliding nonlinear organization dynamic response Sensitivity Analysis Method, method includes the following steps:
(1) Dynamic Modeling is carried out to smooth nonlinear organization, obtains architecture quality matrix, damping matrix, stiffness matrix,
Nonlinear force vector and external force vector；
(2) the multiple variation of tectonic sieving parameter divides Perturbation, and design parameter is made to become plural number from real number；
(3) it brings the design parameter after perturbation into smooth nonlinear structural dynamics model, and utilizes numerical analysis method
The kinetic model is solved, the smooth Nonlinear Dynamic response of structure is obtained；
(4) the smooth Nonlinear Dynamic response imaginary results that kinetic model after perturbing is calculated are extracted, corresponding ginseng is calculated
Several smooth Nonlinear Dynamic response sensitivities.
Further, in step (1), Dynamic Modeling is carried out to smooth nonlinear organization, obtains architecture quality matrix M,
Damping matrix C, stiffness matrix K, nonlinear force vectorWith external force vector F (t), specific steps include following step
It is rapid:
(1.1) the mass matrix M, damping matrix C and rigidity of structure are obtained according to the quality of structure, damping, rigidity characteristic
Matrix K obtains nonlinear force vector according to the smooth nonlinear characteristic of structureAccording to the external drive position of structure
External force vector F (t) is obtained with magnitude of load is applied, wherein p is design parameter vector,Indicate that structure speed responsive, x indicate
Displacement structure response.
Further, in step (2), the multiple variation of tectonic sieving parameter divides Perturbation, becomes design parameter from real number
Plural number, specific steps include the following steps:
(2.1) it is directed to design parameter vector p, divides Perturbation to first of multiple be deteriorated of design parameter construction:
Wherein,Indicate first of design parameter plMultiple change perturbation coefficient, i is imaginary unit, i2=-1,It indicates
First of design parameter plMultiple variation divide Perturbation, subscript Imag indicates the meaning of imaginary part；
(2.2) divide Perturbation to be added with former design parameter multiple variation, obtain the design parameter of complex field, realize design ginseng
Number becomes plural number from real number:
Wherein,Indicate that carrying out multiple be deteriorated divides the design parameter of the complex field obtained after perturbation.
Further, in step (3), the design parameter after perturbation is brought into smooth nonlinear structural dynamics model, and
The kinetic model is solved using numerical analysis method, obtains the smooth Nonlinear Dynamic response of structure, specific steps packet
Include following steps:
(3.1) by the complex field design parameter after perturbationIt brings into non-linear force and replaces former design parameter pl, taken the photograph
Nonlinear force vector after dynamicWith the smooth nonlinear structural dynamics model based on design parameter after perturbation:
Wherein, M indicates that architecture quality matrix, C indicate that structural damping matrix, K indicate structural stiffness matrix,
Nonlinear force vector after perturbation, F (t) indicate external force vector,Parameter vector after indicating perturbation,X is respectively indicated
Integrally-built acceleration responsive, speed responsive and dynamic respond are acquired using Structural Dynamics time domain dynamic response analysis method.
Further, in step (4), the smooth Nonlinear Dynamic response imaginary part that kinetic model is calculated after perturbing is extracted
As a result, calculating the smooth Nonlinear Dynamic response sensitivity of corresponding parameter, specific steps include the following steps:
(4.1) according to step (3.1) be calculated to first of parameter plThe structure sound of something astir after carrying out the multiple point perturbation that is deteriorated
It answers, includes acceleration responsive, speed responsive and dynamic respond, the response after perturbation is subjected to complex field Taylor expansion:
Wherein, n indicates n-th order derivative, ignores the higher order term of formula (4) second-order or more, extracts in structure dynamic response data
Imaginary results, calculate the dynamic response sensitivity of design parameter:
Wherein,Expression takes its imaginary part part to dynamic response (*),Indicate structure acceleration response to first of parameter
pkSensitivity,Indicate structure speed responsive to first of parameter plSensitivity,Indicate displacement structure response response to l
A parameter plSensitivity, subscript a, v, d respectively indicate acceleration, speed and displacement, when l traverses the number of all design parameters
When, the dynamic response sensitivity analysis to all design parameters can be thus achieved based on the multiple method divided that is deteriorated using identical.
The utility model has the advantages that compared with prior art, technical solution of the present invention has following advantageous effects:
The present invention proposes a kind of multiple nonlinear organization dynamic response Sensitivity Analysis Method divided that is deteriorated, by non-linear knot
Structure carries out finite element modeling, design parameter is expanded from real number field to complex field, the imaginary part Perturbation of constructing variable, realizes
The calculating of the nonlinear organization dynamic response sensitivity of degree of precision, if there is p design parameter, it is only necessary to which p calculating can give
The analysis result of the dynamic response sensitivity of smooth nonlinear organization out.
Detailed description of the invention
Fig. 1 is implementation flow chart of the invention；
Fig. 2 is that the embodiment of the present invention analyzes structure: smooth nonlinear spring mass structure；
Fig. 3 is mass block m in the embodiment of the present invention1About design parameterAcceleration responsive sensitivity curve；
Fig. 4 is mass block m in the embodiment of the present invention2About design parameterAcceleration responsive sensitivity curve；
Fig. 5 is mass block m in the embodiment of the present invention3About design parameterAcceleration responsive sensitivity curve；
Fig. 6 is mass block m in the embodiment of the present invention4About design parameterAcceleration responsive sensitivity curve；
Fig. 7 is mass block m in the embodiment of the present invention5About design parameterAcceleration responsive sensitivity curve.
Specific embodiment
Further description of the technical solution of the present invention with reference to the accompanying drawings and examples.
Fig. 1 illustrates a kind of multiple smooth nonlinear organization dynamic response sensitivity analysis side divided that is deteriorated proposed by the present invention
Method, this method comprises the following steps:
(1) Dynamic Modeling is carried out to smooth nonlinear organization shown in FIG. 1, obtains architecture quality matrix, damping matrix,
Stiffness matrix, nonlinear force vector and external force vector；
(2) the multiple variation of tectonic sieving parameter divides Perturbation, and design parameter is made to become plural number from real number；
(3) it brings the design parameter after perturbation into smooth nonlinear structural dynamics model, and utilizes numerical analysis method
The kinetic model is solved, the smooth Nonlinear Dynamic response of structure is obtained；
(4) the smooth Nonlinear Dynamic response imaginary results that kinetic model after perturbing is calculated are extracted, corresponding ginseng is calculated
Several smooth Nonlinear Dynamic response sensitivities.
A kind of multiple structure dynamic response Sensitivity Analysis Method divided that is deteriorated, implementation flow chart are as shown in Figure 1.Fig. 2 is shown
Smooth nonlinear spring-quality model of five degree of freedom used by this example, the model can be simulated with smooth non-linear
The structures such as the building house of feature, Automatic manual transmission.The basic parameter of smooth nonlinear spring-quality structure is respectively as follows:
The basic parameter of the smooth nonlinear spring-quality structure of table 1
Design parameter is the non-linear rigidity coefficient and nonlinear damping coefficients of two groups of spring-damper units in structure, this
Nonlinear Cubic spring-damper unit is used in example.The design parameter vector of compositionSuch as Fig. 2
It is shown, the specific value of design parameter as listed in table 2:
The design parameter of the smooth nonlinear spring-quality structure of table 2
Concrete operations are as follows:
In step (1), Dynamic Modeling is carried out to smooth nonlinear organization, obtains architecture quality matrix, damping matrix, just
Matrix is spent, nonlinear force vector and external force vector, specific steps include the following steps:
(1.1) nonlinear spring-quality structure as shown in Figure 2 considers undamped state, damping matrix C=0, knot
The mass matrix M and stiffness matrix K expression formula of structure are as follows:
Wherein, m1~m5 is 5 mass block quality of smooth nonlinear organization shown in Fig. 2, as listed in table 1；K1~k5 is
5 linear rigidity coefficients of structure shown in Fig. 2, as listed in table 1.According to the smooth nonlinear characteristic of structure obtain non-linear force to
AmountSetting is applied to mass block m3On initial displacement be x3=0.05m, initial displacement herein can be according to reality
Border needs to be arranged, due to not applying external drive power, i.e. external force vector F (t)=0, the expression formula of nonlinear force vector are as follows:
Wherein, p is design parameter vector, x1, x2, x4WithRespectively mass block m1, m2And m4Displacement and speed
Degree.
In step (2), the multiple variation of tectonic sieving parameter divides Perturbation, so that design parameter is become plural number from real number, specifically
Step includes the following steps:
(2.1) it is directed to design parameter vector p, to first of design parameter plMultiple be deteriorated of construction divides Perturbation.L in this example
1,2,3,4 are taken, four design parameters in design parameter vector p are corresponded respectively to, for taking l=1,It is deteriorated again
Divide Perturbation are as follows:
Wherein,Expression parameterMultiple change perturbation coefficient, be taken asThe value can be according to reality
Border needs to be arranged, and i is imaginary unit, i2=-1.Expression parameterMultiple variation divide Perturbation, subscript Imag indicates empty
The meaning in portion.
(2.2) divide Perturbation to be added with former design parameter multiple variation, obtain the design parameter of complex field, realize design ginseng
Number becomes plural number from real number.Design parameter in this example, after the perturbation that when l=1 obtains are as follows:
Wherein,Indicate the design parameter that the complex field obtained after the multiple point perturbation that is deteriorated is carried out when l=1；
In step (3), the design parameter after perturbation is brought into smooth nonlinear structural dynamics model, and utilize numerical value point
Analysis method solves the kinetic model, obtains the smooth Nonlinear Dynamic response of structure, and specific steps include the following steps:
(3.1) by the complex field design parameter after perturbationIt brings into non-linear force and replaces former design parameter pk, taken the photograph
Nonlinear force vector after dynamicFor this example takes l=1, by the parameter after perturbationIt brings into non-linear
Nonlinear force vector after being perturbed in power (13) are as follows:
Smooth nonlinear structural dynamics model based on design parameter after perturbation are as follows:
Wherein, M indicates that architecture quality matrix, C indicate that structural damping matrix, K indicate that structural stiffness matrix, F (t) indicate outer
Portion's force vector is obtained by step (1).For the nonlinear force vector after perturbation, determined by formula (11),It indicates
Parameter vector after perturbation,X respectively indicates the acceleration responsive, speed responsive and dynamic respond of smooth nonlinear organization.
By formula (11), formula (12) and formula (16) bring formula (17) into, solve multiple be deteriorated using Structural Dynamics time domain dynamic response analysis method
Structure dynamic response after dividing perturbation, commonly using dynamics time domain dynamic response analysis method includes: central difference method, step by step integration
Deng.
In step (4), the smooth Nonlinear Dynamic response imaginary results that kinetic model is calculated after perturbing are extracted, are calculated
The smooth Nonlinear Dynamic response sensitivity of corresponding parameter, specific steps include the following steps:
(4.1) according to step (3.1) be calculated to first of parameter plThe structure sound of something astir after carrying out the multiple point perturbation that is deteriorated
It answers, can be acceleration responsive, speed responsive and dynamic respond, the response after perturbation is subjected to complex field Taylor expansion.This calculation
With acceleration responsive about design parameter in exampleComplex field Taylor expansion for:
Wherein, n indicates n-th order derivative.Single order imaginary part response in formula (18) is extracted, the dynamic response spirit of design parameter is calculated
Sensitivity.With acceleration responsive about design parameter in this exampleDynamic response Calculation of Sensitivity for, using in formula (18)
The response of single order imaginary part, calculates dynamic response sensitivity, and mode is as follows:
Wherein,Expression takes its imaginary part part to dynamic response (*),Indicate structure acceleration response to design parameterDynamic response sensitivity, subscript a respectively indicates acceleration, can be dynamic in the hope of the acceleration of 5 mass blocks according to step (3)
Response, and then acquire 5 mass acceleration dynamic response sensitivity.Fig. 3-Fig. 7 show the acceleration dynamic response of 5 mass blocks
AboutSensitivity analysis result figure, calculate when, the acceleration responsive of five mass blocks can be found out simultaneously.Work as l
When traversing the number of all design parameters, using step (2), step (3) and step (4) are identical based on the multiple method for being deteriorated and dividing
It can realize the dynamic response sensitivity analysis to all design parameters.
More than, i.e., in primary analysis, the structure dynamic response sensitivity of design parameter is calculated by construction imaginary part Perturbation.
In this example, mentioned using this patent it is a kind of based on the multiple structure dynamic response Sensitivity Analysis Method being deteriorated point, compared to having
Differential sensitivity analysis method is limited, initial analysis is not needed, realizes the calculating time of the dynamic response sensitivity analysis of 4 design parameters
Number is 4 times.And compared with finite difference calculus, this method has second order computational accuracy.
Claims (5)
1. a kind of multiple smooth nonlinear organization dynamic response Sensitivity Analysis Method being deteriorated point, which is characterized in that this method includes
Following steps:
(1) Dynamic Modeling is carried out to smooth nonlinear organization, obtains architecture quality matrix, damping matrix, stiffness matrix is non-thread
Property force vector and external force vector；
(2) the multiple variation of tectonic sieving parameter divides Perturbation, and design parameter is made to become plural number from real number；
(3) design parameter after perturbation is brought into smooth nonlinear structural dynamics model, and using numerical analysis method to this
Kinetic model is solved, and the smooth Nonlinear Dynamic response of structure is obtained；
(4) the smooth Nonlinear Dynamic response imaginary results that kinetic model after perturbing is calculated are extracted, corresponding parameter is calculated
Smooth Nonlinear Dynamic response sensitivity.
2. a kind of multiple smooth nonlinear organization dynamic response Sensitivity Analysis Method divided that is deteriorated as described in claim 1, special
Sign is, in step (1), carries out Dynamic Modeling to smooth nonlinear organization, obtains architecture quality matrix M, damping matrix C,
Stiffness matrix K, nonlinear force vectorWith external force vector F (t), specific steps include the following steps:
(1.1) the mass matrix M, damping matrix C and stiffness matrix of structure are obtained according to the quality of structure, damping, rigidity characteristic
K obtains nonlinear force vector according to the smooth nonlinear characteristic of structureAccording to the external drive position of structure and apply
Magnitude of load is added to obtain external force vector F (t), wherein p is design parameter vector,Indicate that structure speed responsive, x indicate structure
Dynamic respond.
3. a kind of multiple smooth nonlinear organization dynamic response Sensitivity Analysis Method divided that is deteriorated as claimed in claim 2, special
Sign is, in step (2), the multiple variation of tectonic sieving parameter divides Perturbation, and design parameter is made to become plural number from real number, specific to walk
Suddenly include the following steps:
(2.1) it is directed to design parameter vector p, divides Perturbation to first of multiple be deteriorated of design parameter construction:
Wherein, hplIndicate first of design parameter plMultiple change perturbation coefficient, i is imaginary unit, i2=-1,It indicates first
Design parameter plMultiple variation divide Perturbation, subscript Imag indicates the meaning of imaginary part；
(2.2) divide Perturbation to be added with former design parameter multiple variation, obtain the design parameter of complex field, realization design parameter from
Real number becomes plural number:
Wherein,Indicate that carrying out multiple be deteriorated divides the design parameter of the complex field obtained after perturbation.
4. a kind of multiple smooth nonlinear organization dynamic response Sensitivity Analysis Method divided that is deteriorated as claimed in claim 3, special
Sign is, in step (3), brings the design parameter after perturbation into smooth nonlinear structural dynamics model, and utilize numerical value point
Analysis method solves the kinetic model, obtains the smooth Nonlinear Dynamic response of structure, and specific steps include the following steps:
(3.1) by the complex field design parameter after perturbationIt brings into non-linear force and replaces former design parameter pl, after being perturbed
Nonlinear force vectorWith the smooth nonlinear structural dynamics model based on design parameter after perturbation:
Wherein, M indicates that architecture quality matrix, C indicate that structural damping matrix, K indicate structural stiffness matrix,Perturbation
Nonlinear force vector afterwards, F (t) indicate external force vector,Parameter vector after indicating perturbation,X respectively indicates entirety
Acceleration responsive, speed responsive and the dynamic respond of structure are acquired using Structural Dynamics time domain dynamic response analysis method.
5. the multiple smooth nonlinear organization dynamic response Sensitivity Analysis Method divided that is deteriorated as claimed in claim 4, feature exist
In in step (4), the smooth Nonlinear Dynamic response imaginary results that kinetic model is calculated after extraction perturbation are calculated and corresponded to
The smooth Nonlinear Dynamic response sensitivity of parameter, specific steps include the following steps:
(4.1) according to step (3.1) be calculated to first of parameter plStructure dynamic response after carrying out the multiple point perturbation that is deteriorated, packet
Containing acceleration responsive, speed responsive and dynamic respond, the response after perturbation is subjected to complex field Taylor expansion:
Wherein, n indicates n-th order derivative, ignores the higher order term of formula (4) second-order or more, extracts the void in structure dynamic response data
Portion is as a result, calculate the dynamic response sensitivity of design parameter:
Wherein,Expression takes its imaginary part part to dynamic response (*),Indicate structure acceleration response to first of parameter pk's
Sensitivity,Indicate structure speed responsive to first of parameter plSensitivity,Indicate displacement structure response response to first
Parameter plSensitivity, subscript a, v, d respectively indicate acceleration, speed and displacement, when l traverses the number of all design parameters
When, the dynamic response sensitivity analysis to all design parameters can be thus achieved based on the multiple method divided that is deteriorated using identical.
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