CN108763611A - A kind of wing structure random eigenvalue analysis method based on probabilistic density evolution - Google Patents
A kind of wing structure random eigenvalue analysis method based on probabilistic density evolution Download PDFInfo
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Abstract
The wing structure random eigenvalue analysis method based on probabilistic density evolution that the invention discloses a kind of, belongs to field of structural design.It fully considers uncertain factor present in large and complex structure, under conditions of sample information abundance, quantification characterization is carried out to uncertain parameter using random device.The finite element equation for establishing free vibration of structures is converted into generalized eigenvalue problem by modal coordinate, and is derived to Eigenvalue Sensitivity.On this basis, characteristic value probability density evolution equation is established, Eigenvalue Sensitivity is introduced into equation, obtains the characteristic value probability density evolution equation based on sensitivity analysis.It is canonical form by introducing new variables by characteristic value probability density evolution equation abbreviation, reducing format using finite difference method and total variance solves characteristic value probability density function.It is preferable that numerical result shows that the characteristic value probability density function that the method for the present invention obtains coincide with monte carlo method, and can significantly reduce and calculate the time.
Description
Technical field
The present invention relates to field of structural design, more particularly to a kind of wing structure random character based on probabilistic density evolution
It is worth analysis method.
Background technology
Structural natural frequencies and characteristic value can provide the information of characterization structural dynamic characteristics, the especially intrinsic frequency of low order
Rate and characteristic value are usually used to the kinetic stability of evaluation structure.Meanwhile considering the structure optimization of intrinsic frequency and characteristic value
It is also very promising in field of structural design.Therefore, structural natural frequencies and characteristic value are generally viewed as in structure design
Important indicator parameter.Structural Eigenvalue Problem has just had received widespread attention since the eighties in last century, up to the present proposes
A variety of methods for solving Structural Eigenvalues, and be successfully applied to Structural Reanalysis, sensitivity analysis and Modifying model etc.
In Practical Project problem.Under the scope of certainty, structural natural frequencies and characteristic value can include mass matrix by that will solve
It is obtained with the generalized eigenvalue equation of stiffness matrix.But there are multi-source uncertain factors in practical structures, such as：(1) material
The uncertainty for expecting parameter makes engineering material since the factors such as manufacturing environment, technical conditions, the multiphase feature of material influence
Elasticity modulus, Poisson's ratio, mass density have uncertainty；(2) uncertainty of geometric dimension, due to manufacture and installation error
Make geometrical scale such as thickness, cross-sectional area etc. that there is uncertainty, the presence of these uncertain factors to make structure feature
Value problem solving has difficulties.
For uncertain caused by material and geometric parameter, generally using random device to the uncertain factor amount of progress
Change characterization, utilizes the distribution characteristics of stochastic variable characterising parameter.To the Structural Eigenvalue Solve problems containing stochastic variable, existing solution
Certainly approach can be divided into three categories：(1) Monte-carlo Simulation Method；(2) response surface method of deploying；(3) Moment equation method.However, covering
Special Carlow analogy method needs to carry out a large amount of sample point analysis, and it is larger to expend computing resource；Response surface method of deploying and square side
Journey method can only handle small uncertain problem.The Matrix perturbation and direct method of analysis of variance proposed in recent years can also solve
Structure random parameters problem, but the above method can only obtain the first-order reliability method information of characteristic value.It is special when needing to obtain structure
When value indicative probability density function, there is presently no effective analysis methods to solve, and limits complicated knot to a certain extent
The development of structure stability analysis technology.In conclusion there is an urgent need for developing one kind, can quickly, accurately to solve structure random parameters general
The new method of rate density function, to overcome conventional method to calculate drawback of long duration, that precision is low.
Invention content
The technical problem to be solved in the present invention is：For conventional process structure random parameters problem method computational efficiency
The problems such as low, characteristic value probability density function is difficult to obtain proposes that a kind of wing structure based on probabilistic density evolution is special at random
Value indicative analysis method.This method fully considers uncertain factor present in large and complex structure, using random device to not true
Determine parameter and carries out quantification characterization.Under the conditions of certainty, free vibration of structures equation is established, is converted by modal coordinate
For generalized eigenvalue problem, and numerical solution is carried out to Eigenvalue Sensitivity.On this basis, characteristic value probability density is established
EVOLUTION EQUATION, and Eigenvalue Sensitivity is introduced into constructed equation, it is close to obtain the characteristic value probability based on sensitivity analysis
Spend EVOLUTION EQUATION.It is canonical form by introducing new variables by characteristic value probability density evolution equation abbreviation, using finite difference
Method and total variance, which reduce format, can obtain characteristic value probability density function.
The present invention solve the technical solution that uses of above-mentioned technical problem for：A kind of wing structure based on probabilistic density evolution
Random eigenvalue analysis method, includes the following steps：
Step (1), the finite element equation for establishing free vibration of structures：
In formula, M is mass matrix, and K is stiffness matrix, and q is generalized coordinates,For generalized acceleration；
Step (2) introduces modal vector u, utilizesFree vibration finite element equation can be turned
Turn to generalized eigenvalue equation：
K { u }-λ M { u }=0 (2)
Step (3), under the conditions of certainty, Structural Eigenvalue λ can be obtained by following determinant equation：
| K- λ M |=0 (3)
Step (4), Structural Eigenvalue λ can be expressed as about the sensitivity of structural parameters：
In formula, λiFor the i-th rank characteristic value, uiFor the i-th rank feature vector, b is structural parameters；
Step (5) establishes Structural Eigenvalue probability density evolution equation, is expressed as form：
In formula, Θ=(Θ1,...,Θs) it is that s ties up random uncertain parameter, p λ Θ (λ, θ, b) are that the joint of (λ, Θ) is general
Rate density function, N are characterized value quantity；When only one characteristic value (when taking N=1) of consideration, formula (5) can be rewritten as：
Step (6), in the domain of variation Ω of uncertain parameter Θ, equably take NtotalA sample point, is denoted as：
Θq(q=1 ..., Ntotal), and it is N that domain of variation Ω, which is divided,totalA subdomain, is denoted as Ωq(q=1 ...,
Ntotal)；
Step (7), by equation (6) in subdomain ΩqInterior integral can obtain：
Step (8), by exchange integral and sequence of solving the derivation, can be by formula (7) abbreviation：
In formula,For the probability density function corresponding to q-th of sample point；
Eigenvalue Sensitivity is introduced into equation (8) by step (9), can obtain the probability of feature based value sensitivity
Density evolution equation：
Step (10) introduces new parameter z=λ (Θq, b) and b, substituting into equation (9) can obtain：
Step (11) determines that primary condition is：
In formula, δ is Dirac function,
Equation (10) is rewritten as following form by step (12)：
In formula, a=λ (Θq,b)；
Step (13) can obtain following difference scheme using finite difference method and total variance reduction format：
In formula,zm=m Δs z (m=0, ± 1 ...), bk=k Δs b (k=0,1 ...),
For current limiter,WithIt is represented by：
Step (14), will be in NtotalIt is calculated at a sample pointSummation, can obtain：
Step (15) takes bk=1, then the expression formula of Structural Eigenvalue probability density function can be obtained：
Wherein, in the step (4), the derivation of Eigenvalue Sensitivity is：
The equation of step (2) is expressed as component form, i.e.,：
K{ui}-λiM{ui}=0 (17)
In formula, λiFor the i-th rank characteristic value, uiFor the i-th rank feature vector.Local derviation is asked to structural parameters b at equation (17) both ends
Number, can obtain following relationship：
In equation (18) both sides while premultiplicationAnd it carries out transposition and converts and can obtain：
Wherein, in the step (7), the concrete mode of exchange integral and sequence of solving the derivation is：
Wherein, in the step (11), can be by primary condition is discrete：
In formula, zm=m Δs z (m=0, ± 1 ...), Δ z indicate the size of mesh opening in the directions z；
Wherein, in the step (13), the difference scheme condition of convergence to be met is：
The beneficial effects of the invention are as follows：
The present invention proposes the Probability Density Evolution Method based on sensitivity, can divide structure random parameters
Analysis obtains Structural Eigenvalue probability density function.When the mean value and standard deviation of given input parameter, what the method for the present invention obtained
The characteristic value probability density function that characteristic value probability density function is obtained with monte carlo method coincide preferably, has higher number
It is worth precision.It, can be in the premise item for fully ensuring that computational accuracy when being analyzed complicated wing structure using the method for the present invention
Under part, the required repeated sample point analysis of traditional Monte Carlo Analogue Method is avoided, is greatly reduced and calculates the time, improves and calculates
Efficiency.The method of the present invention is of great significance for shortening wing structure design cycle, lift structure stability.
Description of the drawings
Fig. 1 is the finite element model of wing structure；
Fig. 2 is the first rank characteristic value probability density function；
Fig. 3 is second-order characteristic value probability density function；
Fig. 4 is third rank characteristic value probability density function；
Fig. 5 is fourth order characteristic value probability density function；
Fig. 6 is the 5th rank characteristic value probability density function；
Fig. 7 is the method implementation process of the present invention.
Specific implementation mode
Hereinafter reference will be made to the drawings, and the design example of the present invention is described in detail.It should be appreciated that selected example only for
Illustrate the present invention, rather than limits the scope of the invention.
(1) using HIRENASD wing structures as object, structural finite element model is established, as shown in Figure 1；
(2) wing structure unit material property parameters are given, as shown in table 1；
1 wing structure material properties parameter of table
In table 1, E is elasticity modulus, and G is shear model, and μ is Poisson's ratio, and ρ is density；
(3) it is uncertain parameter, Normal Distribution, mean value and standard deviation such as table 2 to choose elastic modulus E and density p
It is shown；
The mean value and standard deviation of 2 material properties parameter of table
(4) constant interval of E and ρ [+6 σ of μ 6 σ, μ] is divided into 20 subintervals, then the E and ρ of subinterval boundary are：
In this way, sample point (the E formedi,ρj) share 441；
(5) q-th of sample point (E is takeni,ρj)q, according to (Ei,ρj)qMaterial properties are set, are obtained using business software Nastran
It takes to being applied to sample point (Ei,ρj)qFirst five rank eigenvalue λ1~λ5；
(6) it establishes respectively and is directed to Structural Eigenvalue λ1~λ5Probability density evolution equation：
(7) primary condition can discrete be：
In formula,
(8) finite difference scheme is set as：
In formula,zm=m Δs z (m=0, ± 1 ...), bk=k Δs b (k=0,1 ...),
(9) current limiterIt is set as：
(10) using by finite difference method, meeting the condition of convergenceUnder the premise of, it can be with
It obtains corresponding to sample point (Ei,ρj)qProbability density function
(11) step (5)~(10) are repeated, the corresponding probability density function of all 441 sample points is calculated
Being summed can obtain：
(12) b is takenk=1, then it can calculate structure first five rank eigenvalue λ1~λ5Probability density function expression formula be：
(13) Monte-carlo Simulation Method is utilized, the sample point of N=1000 Normal Distribution is taken, by each sample
This character pair value λ1~λ5Calculating, obtain λ1~λ5Probability density function；
(14) both the above method obtains λ1~λ5Probability density function as shown in figures 2-6, result illustrates the present invention in figure
The result that method obtains is coincide preferable with monte carlo method result；
(15) according to λ1~λ5Probability density function calculates separately mean value (μ) and standard deviation (σ), and the results are shown in Table 3：
3 result of calculation of table compares
From table 3 it can be seen that the max calculation error of two methods is no more than 4%, illustrate that the method for the present invention precision is preferable；
(16) two methods calculating total time-consuming is respectively：TThe method of the present invention=327338.70s, TMonte carlo method=614400.23s.
Time comparing result show the method for the present invention can be reduced by about 50% calculating take, to significantly improve wing structure feature
It is worth the computational efficiency of probability density function.
In conclusion the present invention proposes a kind of structure random eigenvalue analysis method based on probabilistic density evolution.First
Free vibration of structures equation is established, generalized eigenvalue problem is converted by modal coordinate.On this basis, feature is established
It is worth probability density evolution equation, and Eigenvalue Sensitivity is introduced into constructed equation, obtains the spy based on sensitivity analysis
Value indicative probability density evolution equation, characteristic value probability density letter can be obtained by reducing format using finite difference method and total variance
Number.Numerical result shows the characteristic value that the characteristic value probability density function that the method for the present invention obtains is obtained with monte carlo method
Probability density function coincide preferable, and can significantly reduce and calculate the time, to for wing structure Analysis of Vibration Characteristic and
Stability Design provides new approaches.
The specific steps that the above is only the present invention, are not limited in any way protection scope of the present invention, expansible to answer
For Aircraft structural design field, any technical scheme formed by adopting equivalent transformation or equivalent replacement, all falls within this hair
Within bright rights protection scope.
Claims (5)
1. a kind of wing structure random eigenvalue analysis method based on probabilistic density evolution, it is characterised in that：Realize step such as
Under：
Step (1), the finite element equation for establishing wing structure free vibration：
In formula, M is mass matrix, and K is stiffness matrix, and q is generalized coordinates,For generalized acceleration；
Step (2) introduces modal vector u, utilizesFree vibration finite element equation can be converted to extensively
Adopted eigenvalue equation：
K { u }-λ M { u }=0 (2)
Step (3), under the conditions of certainty, Structural Eigenvalue λ can be obtained by following determinant equation：
| K- λ M |=0 (3)
Step (4), Structural Eigenvalue λ can be expressed as about the sensitivity of structural parameters：
In formula, λiFor the i-th rank characteristic value, uiFor the i-th rank feature vector, b is structural parameters；
Step (5) establishes Structural Eigenvalue probability density evolution equation, is expressed as form：
In formula, Θ=(Θ1,...,Θs) it is that s ties up random uncertain parameter, pλΘ(λ, θ, b) is the joint probability density of (λ, Θ)
Function, N are characterized value quantity；When only one characteristic value (when taking N=1) of consideration, formula (5) can be rewritten as：
Step (6), in the domain of variation Ω of uncertain parameter Θ, equably take NtotalA sample point, is denoted asAnd Ω points are N by domain of variationtotalA subdomain, is denoted as Ωq(q=1 ..., Ntotal)；
Step (7), by equation (6) in subdomain ΩqInterior integral can obtain：
Step (8), by exchange integral and sequence of solving the derivation, can be by formula (7) abbreviation：
In formula,For the probability density function corresponding to q-th of sample point；
Eigenvalue Sensitivity is introduced into equation (8) by step (9), can obtain the probability density of feature based value sensitivity
EVOLUTION EQUATION：
Step (10) introduces new parameter z=λ (Θq, b) and b, substituting into equation (9) can obtain：
Step (11) determines that primary condition is：
In formula, δ is Dirac function,
Equation (10) is rewritten as following form by step (12)：
In formula, a=λ (Θq,b)；
Step (13) can obtain following difference scheme using finite difference method and total variance reduction format：
In formula,zm=m Δs z (m=0, ± 1 ...), bk=k Δs b (k=0,1 ...),For current limiter,WithIt is represented by：
Step (14), will be in NtotalIt is calculated at a sample pointSummation, can obtain：
Step (15) takes bk=1, then the expression formula of Structural Eigenvalue probability density function can be obtained：
2. a kind of wing structure random eigenvalue analysis method based on probabilistic density evolution according to claim 1,
It is characterized in that：In the step (4), the derivation of Eigenvalue Sensitivity is：
The equation of step (2) is expressed as component form, i.e.,：
K{ui}-λiM{ui}=0 (17)
In formula, λiFor the i-th rank characteristic value, uiFor the i-th rank feature vector, partial derivative is asked to structural parameters b at equation (17) both ends,
It can obtain following relationship：
In equation (18) both sides while premultiplicationAnd transplant and can obtain：
3. a kind of wing structure random eigenvalue analysis method based on probabilistic density evolution according to claim 1,
It is characterized in that：In the step (7), the concrete mode of exchange integral and sequence of solving the derivation is：
4. a kind of wing structure random eigenvalue analysis method based on probabilistic density evolution according to claim 1,
It is characterized in that：In the step (11), can be by primary condition is discrete：
In formula, zm=m Δs z (m=0, ± 1 ...), Δ z indicate the size of mesh opening in the directions z.
5. a kind of wing structure random eigenvalue analysis method based on probabilistic density evolution according to claim 1,
It is characterized in that：In the step (13), the difference scheme condition of convergence to be met is：
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| CN114429060A (en) * | 2021-12-02 | 2022-05-03 | 中国兵器科学研究院宁波分院 | Method for assessing structure dislocation failure and service life prediction in fatigue vibration |
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| CN114186395B (en) * | 2021-11-23 | 2024-08-02 | 大连理工大学 | Engineering structure system reliability quick calculation method considering multiple randomness |
| CN114429060A (en) * | 2021-12-02 | 2022-05-03 | 中国兵器科学研究院宁波分院 | Method for assessing structure dislocation failure and service life prediction in fatigue vibration |
| CN114429060B (en) * | 2021-12-02 | 2022-12-27 | 中国兵器科学研究院宁波分院 | Method for examining structure dislocation failure and service life prediction in fatigue vibration |
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