CN108491578A - A kind of Random dynamic loads recognition methods based on perturbative matrix - Google Patents
A kind of Random dynamic loads recognition methods based on perturbative matrix Download PDFInfo
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Abstract
The Random dynamic loads recognition methods based on perturbative matrix that the present invention provides a kind of, structure containing uncertain parameters is carried out with multiple modal test under operating mode, calculate uncertainty rigidity, quality and damping matrix, to uncertain rigidity, quality and damping matrix are unfolded, calculate Green's function matrix, establish the uncertain kinetic model based on perturbative matrix, measure the random file response sample of the structure containing uncertain parameters under Random dynamic loads effect, utilize the mean value of suffered Random dynamic loads in random file response sample mean value identification structure, calculate the approximation for only considering random file response covariance caused by dynamic loading randomness, identification obtains the statistical nature of Random dynamic loads.Utilize the method for the present invention, the uncertainty of dynamic response, structural system and dynamic loading can be considered simultaneously, the statistical nature of structure dynamic loading is obtained using actual measurement dynamic response specimen discerning, abundant dynamic loading information can be provided for engineering structure, be more advantageous to the security evaluation and optimization design of engineering structure.
Description
Technical field
The present invention relates to a kind of dynamic loading indirect identification methods, and in particular to a kind of Random dynamic loads recognition methods.
Background technology
Dynamic loading suffered by engineering structure is most important to the design and security evaluation of structure.In many cases, some engineerings
External load suffered by structure is difficult to obtain by method measured directly, such as suffered aerodynamic loading, sea in aircraft flight course
The contact load etc. of wheel and ground on storm load, driving vehicle suffered by foreign platform, therefore, using surveying dynamic response in structure
Carrying out indirect gain dynamic loading information becomes a kind of technology being widely adopted, also referred to as dynamic loading indirect identification technology.
Current dynamic loading indirect identification method does not consider the uncertainty of structural system parameter mostly, uses certainty
The structure that can be obtained by single measurement of Dynamic Load Identification method on dynamic response identification when time acting on the dynamic load in structure
Lotus.However, dynamic loading such as wind load, seismic (seismal, aerodynamic loading, acoustic loads etc. suffered on engineering structure, Chang Cheng
Now random feature, what certainty Dynamic Load Identification method obtained only works as time approximation of effect dynamic loading, can not provide reality
The statistical nature of border dynamic loading is easy since dynamic loading information understands comprehensively not cause " cross and design " in structured design process
Or " owing design "；On the other hand, engineering structure itself is made due to the discreteness of error and material itself in manufacture and measurement
The systematic parameter for obtaining structure is sometimes not appropriate for being described with a determining value, does not consider the uncertainty of systematic parameter
It can lead to the distortion even mistake of Dynamic Load Identification result.
Invention content
Goal of the invention：In view of the above-mentioned deficiencies in the prior art, it is an object of the present invention to provide a kind of limited based on perturbation stochastic
The Random dynamic loads recognition methods of member, this method can consider the uncertainty of dynamic loading and systematic parameter, utilize reality simultaneously
Geodesic structure dynamic response sample identifies the statistical nature that Random dynamic loads are acted on structural system.
Technical solution：The Random dynamic loads recognition methods based on perturbative matrix that the present invention provides a kind of, including
Following steps：
(1) structure containing uncertain parameters is carried out with multiple modal test under operating mode, obtains uncertain system parameter
Random distribution field；
(2) K-L expansion is carried out to system random parameter field, calculates uncertainty rigidity, quality and damping matrix；
(3) uncertain rigidity, quality and damping matrix are unfolded using perturbation method, calculate Green's function matrix,
Establish the uncertain kinetic model based on perturbative matrix；
(4) the random file response sample of the structure containing uncertain parameters under Random dynamic loads effect is measured；
(5) mean value of suffered Random dynamic loads in random file response sample mean value identification structure is utilized；
(6) structure random file when only considering systematic parameter uncertainty is solved using the Random dynamic loads mean value of identification
Response；
(7) approximation for only considering random file response covariance caused by dynamic loading randomness is calculated；
(8) identification obtains the statistical nature of Random dynamic loads.
Further, in step (2) uncertain stiffness matrix K, mass matrix M and damping matrix C respectively by element stiffness
Matrix Ke, element mass matrix MeWith unit damping matrix CeAssemble：
Element mass matrix MeWith element stiffness matrix KeIt is calculate by the following formula respectively：
In formula,WithThe respectively desired value of density and elastic parameter, λρWithRespectively contain uncertain density parameter
Covariance function eigenvalue and eigenfunction, λDWithThe spy of covariance function respectively containing uncertain elastic parameter
Value indicative and characteristic function, KρAnd KDRespectively density and elastic parameter K-L block after item number, HeAnd BeThe respectively shape of unit
Jacobian matrix and strain matrix, D are elastic matrix, veFor unit volume, ξiFor orthogonal standard gaussian stochastic variable, under
Mark i-th of ingredient that i indicates randomness part；
Unit damping matrix CeAccording to Rayleigh damping model, by MeAnd KeIt is calculated.
Further, step (3) includes the following steps：
(31) the Green's function matrix G (η) of uncertain kinetic model is calculated：
Green's function matrix G (η) is assembled using the forms of Green's function g (t, η) according to the following formula：
Time t in Green's function g (t, η) is wherein subjected to discrete, tQIndicate the Q discrete time in Green's function
Step；Green's function matrix is unfolded using perturbation method：
Wherein, ηdThe mean value for the stochastic variable η for including in systematic parameter and dynamic loading is represented, n is the quantity of stochastic variable,
Gi' be Green's function matrix the corresponding coefficient matrix in randomness part；
(32) matrix G (η when uncertain parameters take mean value are calculatedd)：
Uncertain parameters take Green's function D when mean valued(t) it is obtained by solving following equation：
In formula, subscript d represents certainty part, i.e., acquired results when stochastic variable takes desired, and δ (t) is impulse response letter
The transposition of number, T representing matrixes or vector.
According to formula (3), g (t, η) is replaced with into Dd(t), it can solve and obtain matrix G (ηd)；
(33) Green's function matrix randomness part G is calculatedi′；
In formula, Mi、CiAnd KiThe respectively randomness part of mass matrix M, damping matrix C and stiffness matrix K, by unit
Mass matrix Me, unit damping matrix CeWith element stiffness matrix KeAssembling obtains；
Solution formula (6) respectively obtains stochastic variable ξiThe variation D of caused response when variationi(t), according to formula (3), by g
(t, η) replaces with Di(t), it can solve and obtain matrix Gi′；
(34) Random dynamic loads and random file response are used into chaos polynomial expansion, established limited based on perturbation stochastic
The uncertainty structure kinetic model of member.
Further, step (5) utilizes random file response sample mean valueIdentify suffered Random dynamic loads in structure
Mean valueSpecific method be：
Further, step (6) is based on the kinetic model based on perturbative matrix that step (3) is established, solution side
Journey (8) calculates Random dynamic loads mean valueAct on the corresponding vector u of random file response on uncertain system(P)
(t)：
In formula, P is the item number of the chaos polynomial expansion of system random file response, and,
Wherein, Ψj, Ψk(j, k=1,2 ... P) is respectively jth and k rank chaos multinomials,<·>Expectation is asked in expression.
Further, step (7) includes the following steps：
(71) uncertain parameters are asked and it is expected that corresponding system is defined as deterministic system, by the equal of Random dynamic loads
Value is defined as corresponding certainty dynamic loading；Remember covariance matrix of the lower uncertain system of Random dynamic loads effect with kinematical displacement
Dynamic displacement covariance matrix for [R], the lower corresponding deterministic system of Random dynamic loads effect is [R]d, corresponding certainty dynamic loading
The dynamic displacement covariance matrix of the lower uncertain system of effect is [R]s, above three covariance matrix is with following relational expression：
In formula,WithIt indicates respectively only by dynamic loading uncertainty and only caused by systematic parameter uncertainty
Uncertain dynamic respond is turned into the projection vector in random space in jth rank chaos multinomial；
(72) since the uncertainty of dynamic loading and uncertain the two correlation of systematic parameter are weak, formula (10) equation is left
Side Section 3 contributes very little, calculates [R]dApproximation：
[R]d≈[R]-[R]s (11)。
Further, step (8) includes the following steps：
(81) covariance matrix [R] is soughtdCharacteristic valueAnd feature vectorCalculate its K-L vectorsIdentify the vector f corresponding to Random dynamic loads in structure(j)(t)：
(82) vector corresponding to Random dynamic loads is utilized to solve the mean μ F (t) and variance of Random dynamic loads
Advantageous effect：1, existing Dynamic Load Identification method does not consider the uncertainty of structural system parameter mostly, knows
The precision of other result will be completely dependent on when time dynamic response measurement accuracy and structural system modeling accuracy, be obtained using duplicate measurements
Dynamic response identification dynamic loading result it is inconsistent, it is difficult to give engineering staff to provide accurate dynamic loading information, be also unfavorable for determining
Plan；Using the method for the present invention, the uncertainty of dynamic response, structural system and dynamic loading can be considered simultaneously, utilize the actual measurement sound of something astir
It answers specimen discerning to obtain the statistical nature of structure dynamic loading, abundant dynamic loading information can be provided for engineering structure, more favorably
In the security evaluation and optimization design of engineering structure；
2, existing Dynamic Load Identification method is only used for simple structure mostly, and the present invention is according to the Green's function of system
The Random dynamic loads information acted in identifying system, it is effective to improve computational efficiency and anti-noise ability and can be used in complicated work
Random dynamic loads identification in journey structure.
Description of the drawings
Fig. 1 is the logical procedure diagram of the method for the present invention；
Fig. 2 is FEM model schematic diagram in embodiment；
Fig. 3 is the comparison diagram of load mean value and reference value that identification obtains；
Fig. 4 is the comparison diagram of load variance and reference value that identification obtains.
Specific implementation mode
Technical solution of the present invention is described in detail below, but protection scope of the present invention is not limited to the implementation
Example.
The composite material cantilever beam structure for including uncertain parameters for one, as shown in Figure 1, using the method for the present invention base
In surveying the statistical nature for moving displacement specimen discerning and going out suffered Random dynamic loads in the structure on beam, following steps are specifically included：
S1, carry out with multiple modal test, modal test under operating mode for a batch of more composite material cantilever beams
System is using conventional mode experiment system；Structural finite element model as shown in Figure 2 is established, cantilever beam is divided into 14 sections
Point, number 1~14 is respectively cantilever beam node serial number wherein in figure, and it is close that the model modification method based on frequency response function obtains structure
The sample for spending parameter and elastic parameter, calculates the statistical property of structural parameters, including mean value and covariance；
S2, systematic parameter random field is unfolded using K-L expansion, calculates uncertainty rigidity, quality and damping
Matrix；Wherein element mass matrix and element stiffness matrix can be calculated by following formula：
In formula：I is cantilever beam cross sectional moment of inertia.
Unit damping matrix C is calculated according to Rayleigh damping modele。
Assembling obtains the certainty part K of global stiffness, quality and damping matrixd、MdAnd Cd, and uncertain part
Ki、MiAnd Ci。
S3, uncertain rigidity, quality and the damping matrix of system are unfolded using perturbation method, and are calculated not
The Green's function matrix of certainty structure.
Green's function matrix G (η) can be assembled using the forms of Green's function g (t, η) according to the following formula：
Time t in Green's function g (t, η) is wherein subjected to discrete, tQIndicate the Q discrete time in Green's function
Step.Green's function matrix is unfolded using perturbation method：
Wherein, ξiFor orthogonal standard gaussian stochastic variable, n is the quantity of structural uncertainty parameter
Wherein, ηdRepresent the mean value for the stochastic variable η for including in systematic parameter and dynamic loading, Gi' it is Green's function matrix
The corresponding coefficient matrix in randomness part.
Green's function matrix can be calculated by formula (7) and (8)：
Solution formula (7) and (8) obtain Dd(t) and Di(t), the g (t, η) in alternate form (4) can solve to obtain Green respectively
The mean value G of Jacobian matrix0With randomness part coefficient of correspondence Gi′。
S4, apply the single-point arbitrary excitation for giving statistical property to every cantilever beam using vibrator, use laser displacement
Meter measure set point on beam with kinematical displacement, calculated with kinematical displacement mean value using with kinematical displacement sample
S5, suffered Random dynamic loads mean value in random file response sample mean value identification structure is utilized
S6, the random dynamic response of structure is unfolded using chaos multinomial, is established based on perturbative matrix not
Certainty kinetic model solves equation (10), calculates Random dynamic loads mean valueIt acts on uncertain system
The corresponding vector u of random file response(j)(t), wherein equation (10) is：
In formula, P is the item number after the chaos polynomial expansion of system random file response.
Wherein, Ψj, Ψk(j, k=1,2 ... P) is respectively jth and k rank chaos multinomials,<·>Expectation is asked in expression.
S7, the lower uncertain system (cantilever beam) of note Random dynamic loads effect are [R] with the covariance matrix of kinematical displacement,
The dynamic displacement covariance matrix of the lower corresponding deterministic system of Random dynamic loads effect is [R]d, correspond under certainty dynamic load effect
The dynamic displacement covariance matrix of uncertain system is [R]s, above three covariance matrix is with following relational expression：
In formula,WithIt indicates respectively only by dynamic loading uncertainty and only caused by systematic parameter uncertainty
Uncertain dynamic respond is turned into the projection vector in random space in jth rank chaos multinomial；
Uncertainty due to dynamic loading and weaker, formula (12) equation as uncertain the two Relativity of systematic parameter
Left side Section 3 relative contribution very little calculates [R]dApproximation：
[R]d≈[R]-[R]s (13)
S8, covariance matrix [R] is soughtdCharacteristic valueAnd feature vectorCalculate its K-L vectorsIdentify the vector corresponding to Random dynamic loads in structure：
The mean μ of Random dynamic loads is solved using the vector corresponding to Random dynamic loadsF(t) and variance
It is set forth in Fig. 3 and Fig. 4 when the coefficient of variation of random systematical and Random dynamic loads amplitude is equal to
10%, the Random dynamic loads mean value and variance identified using technology in the present invention is compared with reference value, is illustrated in the present invention
Technology can utilize the random file in structure to respond the statistical nature that Random dynamic loads are recognized accurately.
The present invention can utilize the dynamic response data of actual measurement to identify the dynamic loading information acted in structure, not only can be because same
When consider systematic parameter uncertainty and load randomness, and for a fairly large number of labyrinth of degree of freedom also have compared with
Good applicability has important theory and application value.
Claims (7)
1. a kind of Random dynamic loads recognition methods based on perturbative matrix, it is characterised in that：Include the following steps：
(1) structure containing uncertain parameters is carried out with multiple modal test under operating mode, obtain uncertain system parameter with
Machine distribution field；
(2) K-L expansion is carried out to system random parameter field, calculates uncertainty rigidity, quality and damping matrix；
(3) uncertain rigidity, quality and damping matrix are unfolded using perturbation method, calculate Green's function matrix, established
Uncertain kinetic model based on perturbative matrix；
(4) the random file response sample of the structure containing uncertain parameters under Random dynamic loads effect is measured；
(5) mean value of suffered Random dynamic loads in random file response sample mean value identification structure is utilized；
(6) the structure random file solved when only considering systematic parameter uncertainty using the Random dynamic loads mean value of identification is rung
It answers；
(7) approximation for only considering random file response covariance caused by dynamic loading randomness is calculated；
(8) identification obtains the statistical nature of Random dynamic loads.
2. the Random dynamic loads recognition methods according to claim 1 based on perturbative matrix, it is characterised in that：Step
Suddenly in (2) uncertain stiffness matrix K, mass matrix M and damping matrix C respectively by element stiffness matrix Ke, element quality square
Battle array MeWith unit damping matrix CeAssemble：
Element mass matrix MeWith element stiffness matrix KeIt is calculate by the following formula respectively：
In formula,WithThe respectively desired value of density and elastic parameter, λρWithAssociation respectively containing uncertain density parameter
The eigenvalue and eigenfunction of variance function, λDWithThe characteristic value of covariance function respectively containing uncertain elastic parameter
And characteristic function, KρAnd KDRespectively density and elastic parameter K-L block after item number, HeAnd BeThe respectively shape function of unit
Matrix and strain matrix, D are elastic matrix, veFor unit volume, ξiFor orthogonal standard gaussian stochastic variable, subscript i tables
Show i-th of ingredient of randomness part；
Unit damping matrix CeAccording to Rayleigh damping model, by MeAnd KeIt is calculated.
3. the Random dynamic loads recognition methods according to claim 2 based on perturbative matrix, it is characterised in that：Step
Suddenly (3) include the following steps：
(31) the Green's function matrix G (η) of uncertain kinetic model is calculated：
Green's function matrix G (η) is assembled using the forms of Green's function g (t, η) according to the following formula：
Time t in Green's function g (t, η) is wherein subjected to discrete, tQIndicate the Q discrete time step in Green's function；Profit
Green's function matrix is unfolded with perturbation method：
Wherein, ηdThe mean value for the stochastic variable η for including in systematic parameter and dynamic loading is represented, n is the quantity of stochastic variable, Gi' be
The corresponding coefficient matrix in randomness part of Green's function matrix；
(32) matrix G (η when uncertain parameters take mean value are calculatedd)：
Uncertain parameters take Green's function D when mean valued(t) it is obtained by solving following equation：
In formula, subscript d represents certainty part, i.e., acquired results when stochastic variable takes desired, and δ (t) is impulse response function, T
The transposition of representing matrix or vector.
According to formula (3), g (t, η) is replaced with into Dd(t), it can solve and obtain matrix G (ηd)；
(33) Green's function matrix randomness part G is calculatedi′；
In formula, Mi、CiAnd KiThe respectively randomness part of mass matrix M, damping matrix C and stiffness matrix K, by element quality
Matrix Me, unit damping matrix CeWith element stiffness matrix KeAssembling obtains；
Solution formula (6) respectively obtains stochastic variable ξiThe variation D of caused response when variationi(t), according to formula (3), by g (t,
η) replace with Di(t), it can solve and obtain matrix Gi′；
(34) Random dynamic loads and random file response are used into chaos polynomial expansion, established based on perturbative matrix
Uncertainty structure kinetic model.
4. the Random dynamic loads recognition methods according to claim 3 based on perturbative matrix, it is characterised in that：Step
Suddenly (5) utilize random file response sample mean valueIdentify the mean value of suffered Random dynamic loads in structureSpecific side
Method is：
5. the Random dynamic loads recognition methods according to claim 3 or 4 based on perturbative matrix, feature exist
In：Step (6) be based on step (3) establish the kinetic model based on perturbative matrix, solve equation (8), calculate with
Maneuver load mean valueAct on the corresponding vector u of random file response on uncertain system(P)(t)：
In formula, P is the item number of the chaos polynomial expansion of system random file response, and,
Wherein, Ψj, Ψk(j, k=1,2 ... P) is respectively jth and k rank chaos multinomials,<·>Expectation is asked in expression.
6. the Random dynamic loads recognition methods according to claim 5 based on perturbative matrix, it is characterised in that：Step
Suddenly (7) include the following steps：
(71) uncertain parameters are asked and it is expected that corresponding system is defined as deterministic system, the mean value of Random dynamic loads is determined
Justice is corresponding certainty dynamic loading；Uncertain system is with the covariance matrix of kinematical displacement under note Random dynamic loads effect
The dynamic displacement covariance matrix of [R], the lower corresponding deterministic system of Random dynamic loads effect are [R]d, corresponding certainty dynamic loading work
It is [R] with the dynamic displacement covariance matrix of lower uncertain systems, above three covariance matrix is with following relational expression：
In formula,WithIndicate only by dynamic loading uncertainty and only not true caused by systematic parameter uncertainty respectively
Qualitative dynamic respond is turned into the projection vector in random space in jth rank chaos multinomial；
(72) since the uncertainty of dynamic loading and uncertain the two correlation of systematic parameter are weak, formula (10) equation left side the
Three contribution very littles, calculate [R]dApproximation：
[R]d≈[R]-[R]s (11)。
7. the Random dynamic loads recognition methods according to claim 6 based on perturbative matrix, it is characterised in that：Step
Suddenly (8) include the following steps：
(81) covariance matrix [R] is soughtdCharacteristic valueAnd feature vectorCalculate its K-L vectorsIdentify the vector f corresponding to Random dynamic loads in structure(j)(t)：
(82) vector corresponding to Random dynamic loads is utilized to solve the mean μ of Random dynamic loadsF(t) and variance
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| CN113392545B (en) * | 2021-05-31 | 2024-01-30 | 东南大学 | Time domain distribution random dynamic load identification method considering spatial correlation |
| CN115859796A (en) * | 2022-11-24 | 2023-03-28 | 东南大学 | Multi-target structure safety monitoring sensor arrangement method, equipment and storage medium |
| CN115859796B (en) * | 2022-11-24 | 2024-01-26 | 东南大学 | Multi-target structure safety monitoring sensor arrangement method, equipment and storage medium |
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