CN108763611B - Wing structure random eigenvalue analysis method based on probability density evolution - Google Patents

Wing structure random eigenvalue analysis method based on probability density evolution Download PDF

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CN108763611B
CN108763611B CN201810261867.0A CN201810261867A CN108763611B CN 108763611 B CN108763611 B CN 108763611B CN 201810261867 A CN201810261867 A CN 201810261867A CN 108763611 B CN108763611 B CN 108763611B
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邱志平
郑宇宁
王晓军
王磊
祝博
刘峰
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Abstract

The invention discloses a random eigenvalue analysis method for a wing structure based on probability density evolution, and belongs to the field of structural design. And (3) sufficiently considering uncertain factors existing in a large-scale complex structure, and carrying out quantitative characterization on uncertain parameters by using a random method under the condition of sufficient sample information. And establishing a finite element equation of the free vibration of the structure, converting the finite element equation into a generalized characteristic value problem through a modal coordinate, and deducing the sensitivity of the characteristic value. On the basis, a characteristic value probability density evolution equation is established, and the sensitivity of the characteristic value is introduced into the equation to obtain the characteristic value probability density evolution equation based on sensitivity analysis. And (3) simplifying the characteristic value probability density evolution equation into a standard form by introducing new variables, and solving a characteristic value probability density function by adopting a finite difference method and a total variation reduction format. The numerical result shows that the probability density function of the characteristic value obtained by the method is better matched with the Monte Carlo method, and the calculation time can be greatly reduced.

Description

Wing structure random eigenvalue analysis method based on probability density evolution
Technical Field
The invention relates to the field of structural design, in particular to a method for analyzing a random eigenvalue of a wing structure based on probability density evolution.
Background
The structure natural frequencies and eigenvalues can give information characterizing the structure dynamics, in particular the low order natural frequencies and eigenvalues, which are typically used to evaluate the dynamic stability of the structure. Meanwhile, the structural optimization considering the natural frequency and the characteristic value has a good application prospect in the field of structural design. Therefore, the structural natural frequency and the characteristic value are generally regarded as important index parameters in the structural design. The problem of structural characteristic values has attracted much attention since the last 80 th century, and various methods for solving the structural characteristic values have been proposed so far and have been successfully applied to practical engineering problems such as structural re-analysis, sensitivity analysis and model correction. In the deterministic category, the structural natural frequencies and eigenvalues can be obtained by solving a generalized eigenvalue equation containing a mass matrix and a stiffness matrix. However, there are multiple source uncertainties in practical structures, such as: (1) uncertainty of material parameters, and due to the influence of factors such as manufacturing environment, technical conditions, multiphase characteristics of materials and the like, uncertainty exists in the elastic modulus, Poisson's ratio and mass density of engineering materials; (2) uncertainty of geometric dimensions, uncertainty of geometric dimensions of structures such as thickness, cross-sectional area and the like due to manufacturing and installation errors, and the existence of uncertainty factors makes the solving of the structure characteristic value problem difficult.
For uncertainty caused by materials and geometric parameters, a random method is generally adopted to carry out quantitative characterization on uncertainty factors, and random variables are utilized to describe distribution characteristics of the parameters. For solving the problem of structural characteristic values containing random variables, the existing solving approaches can be divided into three categories: (1) a monte carlo simulation method; (2) a response surface deployment method; (3) a method of moment equations. However, the monte carlo simulation method requires a large number of sample point analyses, which consumes a large amount of computing resources; the response surface expansion method and the moment equation method can only handle small uncertainty problems. The matrix perturbation method and the direct variance analysis method proposed in recent years can also solve the problem of the structure random eigenvalue, but the method can only obtain the first-order second-order moment information of the eigenvalue. When a structure characteristic value probability density function needs to be obtained, no effective analysis method can be provided at present, and the development of a complex structure stability analysis technology is limited to a certain extent. In summary, it is highly desirable to develop a new method capable of rapidly and accurately solving the probability density function of the random eigenvalue of the structure, so as to overcome the disadvantages of long calculation time and low precision of the conventional method.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: aiming at the problems that the traditional method for processing the random characteristic value of the structure is low in calculation efficiency, the probability density function of the characteristic value is difficult to obtain and the like, the method for analyzing the random characteristic value of the wing structure based on probability density evolution is provided. The method fully considers uncertain factors existing in a large-scale complex structure, and carries out quantitative characterization on uncertain parameters by using a random method. Under the condition of certainty, a structure free vibration equation is established, the structure free vibration equation is converted into a generalized characteristic value problem through a modal coordinate, and the sensitivity of the characteristic value is solved numerically. On the basis, a characteristic value probability density evolution equation is established, and the sensitivity of the characteristic value is introduced into the established equation to obtain the characteristic value probability density evolution equation based on sensitivity analysis. The characteristic value probability density evolution equation is simplified into a standard form by introducing new variables, and a characteristic value probability density function can be obtained by adopting a finite difference method and a total variation reduction format.
The technical scheme adopted by the invention for solving the technical problems is as follows: a wing structure random eigenvalue analysis method based on probability density evolution comprises the following steps:
step (1), establishing a finite element equation of the free vibration of the structure:
Figure BDA0001610432360000021
wherein M is a mass matrix, K is a stiffness matrix, q is a generalized coordinate,
Figure BDA0001610432360000022
generalized acceleration;
step (2) introducing a modal vector u, utilizing
Figure BDA0001610432360000023
The free vibration finite element equation can be converted into a generalized eigenvalue equation:
K{u}-λM{u}＝0 (2)
step (3), under the deterministic condition, the structural characteristic value lambda can be obtained through the following determinant equation:
|K-λM|＝0 (3)
in step (4), the sensitivity of the structure characteristic value λ with respect to the structure parameter can be expressed as:
Figure BDA0001610432360000024
in the formula, λiIs the ith order eigenvalue, uiIs the ith order eigenvector, b is the structural parameter;
step (5), establishing a structure characteristic value probability density evolution equation, which is expressed as the following form:
Figure BDA0001610432360000025
wherein Θ ═ is (Θ)1,...,Θs) For s-dimensional random uncertainty parameters, p λ Θ (λ, θ, b) is a joint probability density function of (λ, Θ), and N is the number of eigenvalues; when only one eigenvalue is considered (i.e., when N is taken to be 1), equation (5) may be rewritten as:
Figure BDA0001610432360000026
step (6), uniformly taking N in a variation domain omega of the uncertain parameter thetatotalSample points, noted:
Θq(q＝1,...,Ntotal) And the variation domain omega is divided into NtotalIndividual field, denoted as Ωq(q＝1,...,Ntotal)；
Step (7) putting equation (6) in sub-domain omegaqInternal integration, one can get:
Figure BDA0001610432360000031
step (8), by exchanging the integration and derivation order, equation (7) can be simplified as:
Figure BDA0001610432360000032
in the formula (I), the compound is shown in the specification,
Figure BDA0001610432360000033
is the probability density function corresponding to the qth sample point;
step (9), introducing the sensitivity of the characteristic value into equation (8) to obtain a probability density evolution equation based on the sensitivity of the characteristic value:
Figure BDA0001610432360000034
step (10), introducing a new parameter z ═ λ (Θ)qB) b, substituting into equation (9) can result in:
Figure BDA0001610432360000035
step (11), determining the initial conditions as follows:
Figure BDA0001610432360000036
where, δ is the dirac function,
Figure BDA0001610432360000037
step (12), rewriting equation (10) into the following form:
Figure BDA0001610432360000038
wherein a ═ λ (Θ)q,b)；
And (13) obtaining the following difference format by adopting a finite difference method and a total variation reduction format:
Figure BDA0001610432360000039
in the formula (I), the compound is shown in the specification,
Figure BDA00016104323600000310
zm＝mΔz(m＝0,±1,…)，bk＝kΔb(k＝0，1，…)，
Figure BDA00016104323600000311
in the form of a flow restrictor or flow restrictor,
Figure BDA00016104323600000312
and
Figure BDA00016104323600000313
can be expressed as:
Figure BDA0001610432360000041
step (14) of adding N tototalAt one sample point to calculate
Figure BDA0001610432360000042
Summing, one can get:
Figure BDA0001610432360000043
step (15), bk1, the expression of the probability density function of the structural feature value can be obtained:
Figure BDA0001610432360000044
in the step (4), the derivation process of the sensitivity of the eigenvalue is as follows:
expressing the equation of step (2) in component form, namely:
K{ui}-λiM{ui}＝0 (17)
in the formula, λiIs the ith order eigenvalue, uiIs the ith order feature vector. The partial derivative of the structural parameter b across equation (17) yields the following relationship:
Figure BDA0001610432360000045
simultaneous left multiplication on both sides of equation (18)
Figure BDA0001610432360000046
And the term shift transformation is carried out to obtain:
Figure BDA0001610432360000047
in the step (7), the specific manner of exchanging the integration and derivation orders is as follows:
Figure BDA0001610432360000048
Figure BDA0001610432360000049
wherein, in the step (11), the initial condition may be discretized into:
Figure BDA00016104323600000410
in the formula, zmM Δ z (m ═ 0, ± 1, …), Δ z representing the grid size in the z direction;
whereinIn the step (13), the convergence condition to be satisfied by the difference format is:
Figure BDA00016104323600000411
the invention has the beneficial effects that:
the invention provides a sensitivity-based probability density evolution method, which can analyze the structure random characteristic value and obtain a structure characteristic value probability density function. When the mean value and the standard deviation of the input parameters are given, the characteristic value probability density function obtained by the method is well matched with the characteristic value probability density function obtained by the Monte Carlo method, and the numerical precision is high. When the method is used for analyzing the complex wing structure, the repeated sample point analysis required by the traditional Monte Carlo simulation method can be avoided on the premise of fully ensuring the calculation precision, the calculation time is greatly reduced, and the calculation efficiency is improved. The method has important significance for shortening the design period of the wing structure and improving the structural stability.
Drawings
FIG. 1 is a finite element model of a wing structure;
FIG. 2 is a first order eigenvalue probability density function;
FIG. 3 is a second order eigenvalue probability density function;
FIG. 4 is a third order eigenvalue probability density function;
FIG. 5 is a fourth order eigenvalue probability density function;
FIG. 6 is a fifth order eigenvalue probability density function;
fig. 7 is a flow chart of the method implementation of the present invention.
Detailed Description
Hereinafter, a design example of the present invention will be described in detail with reference to the accompanying drawings. It should be understood that the examples are chosen only for the purpose of illustrating the invention and are not intended to limit the scope of the invention.
(1) Establishing a structural finite element model by taking a HIRENAD wing structure as an object, as shown in figure 1;
(2) giving wing structural element material property parameters as shown in table 1;
TABLE 1 wing structural Material Property parameters
Figure BDA0001610432360000051
In table 1, E is the elastic modulus, G is the shear model, μ is the poisson's ratio, ρ is the density;
(3) selecting the elastic modulus E and the density rho as uncertain parameters, and following normal distribution, wherein the mean value and the standard deviation are shown in table 2;
TABLE 2 mean and standard deviation of Material Property parameters
Figure BDA0001610432360000061
(4) The variation interval [ mu 6 sigma, mu +6 sigma ] of E and rho is equally divided into 20 subintervals, so that E and rho at the subinterval boundary are as follows:
Figure BDA0001610432360000062
thus, a sample point (E) is formedi,ρj) There were 441 total;
(5) taking the q sample point (E)i,ρj)qAccording to (E)i,ρj)qSetting material properties, using commercial software Nastran to obtain corresponding application sample points (E)i,ρj)qFirst five order eigenvalues λ1～λ5；
(6) Respectively establishing a structural characteristic value lambda1～λ5Probability density evolution equation:
Figure BDA0001610432360000063
(7) the initial conditions may be discretized as:
Figure BDA0001610432360000064
in the formula (I), the compound is shown in the specification,
Figure BDA0001610432360000065
(8) the finite difference format is set as:
Figure BDA0001610432360000066
in the formula (I), the compound is shown in the specification,
Figure BDA0001610432360000067
zm＝mΔz(m＝0,±1,…)，bk＝kΔb(k＝0，1，…)，
Figure BDA0001610432360000068
(9) current limiter
Figure BDA0001610432360000071
The following settings are set:
Figure BDA0001610432360000072
(10) by using finite difference method, the convergence condition is satisfied
Figure BDA0001610432360000073
On the premise that (E) corresponding to the sample point can be obtainedi,ρj)qProbability density function of
Figure BDA0001610432360000074
(11) Repeating the steps (5) to (10), and calculating probability density functions corresponding to all 441 sample points
Figure BDA0001610432360000075
Summing them can yield:
Figure BDA0001610432360000076
(12) get bkIf 1, the first five-order eigenvalue λ of the structure can be calculated1～λ5The probability density function expression of (a) is:
Figure BDA0001610432360000077
(13) using a Monte Carlo simulation method, taking 1000 sample points which are subjected to normal distribution, and corresponding a characteristic value lambda to each sample point1～λ5Is calculated to obtain lambda1～λ5A probability density function;
(14) the above two methods obtain lambda1～λ5The probability density function is shown in fig. 2-6, and the results in the figure show that the results obtained by the method of the invention are better matched with the results obtained by the Monte Carlo method;
(15) according to λ1～λ5Probability density function, calculating mean (μ) and standard deviation (σ), respectively, and the results are shown in table 3:
TABLE 3 comparison of the results
Figure BDA0001610432360000078
Figure BDA0001610432360000081
As can be seen from Table 3, the maximum calculation error of the two methods does not exceed 4%, which indicates that the method of the invention has better precision;
(16) the two methods respectively calculate the total time consumption as follows: t isThe method of the invention＝327338.70s,TMonte Carlo method614400.23 s. The time comparison result shows that the method can reduce the calculation time consumption by about 50 percent, thereby obviously reducing the calculation time consumptionThe calculation efficiency of the probability density function of the wing structure characteristic value is improved.
In summary, the invention provides a structure random eigenvalue analysis method based on probability density evolution. Firstly, a structure free vibration equation is established, and the structure free vibration equation is converted into a generalized characteristic value problem through a modal coordinate. On the basis, a characteristic value probability density evolution equation is established, the sensitivity of the characteristic value is introduced into the established equation, the characteristic value probability density evolution equation based on sensitivity analysis is obtained, and a characteristic value probability density function can be obtained by adopting a finite difference method and a total variation reduction format. The numerical result shows that the characteristic value probability density function obtained by the method is well matched with the characteristic value probability density function obtained by the Monte Carlo method, and the calculation time can be greatly reduced, so that a new thought is provided for the vibration characteristic analysis and stability design of the wing structure.
The above are only the specific steps of the present invention, and the protection scope of the present invention is not limited at all, and the present invention can be extended to be applied in the field of aircraft structure design, and any technical solution formed by equivalent transformation or equivalent replacement falls within the protection scope of the present invention.

Claims (5)

1. A wing structure random eigenvalue analysis method based on probability density evolution is characterized by comprising the following steps: the method comprises the following implementation steps:
step (1), establishing a finite element equation of free vibration of the wing structure:
Figure FDA0003397404300000011
wherein M is a mass matrix, K is a stiffness matrix, q is a generalized coordinate,
Figure FDA0003397404300000012
generalized acceleration;
step (2) introducing a modal vector u, utilizing
Figure FDA0003397404300000013
The free vibration finite element equation can be converted into a generalized eigenvalue equation:
K{u}-λM{u}＝0 (2)
step (3), under the deterministic condition, the structural characteristic value lambda can be obtained through the following determinant equation:
|K-λM|＝0 (3)
in step (4), the sensitivity of the structure characteristic value λ with respect to the structure parameter can be expressed as:
Figure FDA0003397404300000014
in the formula, λiIs the ith order eigenvalue, uiIs the ith order eigenvector, b is the structural parameter;
step (5), establishing a structure characteristic value probability density evolution equation, which is expressed as the following form:
Figure FDA0003397404300000015
wherein Θ ═ is (Θ)1,...,Θs) For the s-dimensional random uncertainty parameter, pλΘ(lambda, theta, b) is a joint probability density function of (lambda, theta), and N is the number of eigenvalues; when only one eigenvalue is considered, i.e. taking N ═ 1, equation (5) can be rewritten as:
Figure FDA0003397404300000016
step (6), uniformly taking N in a variation domain omega of the uncertain parameter thetatotalSample points, denoted as Θq，q＝1,...,NtotalAnd the variation domain omega is divided into NtotalIndividual field, denoted as Ωq，q＝1,...,Ntotal；
Step (7) putting equation (6) in sub-domain omegaqInternal integration, one can get:
Figure FDA0003397404300000017
step (8), by exchanging the integration and derivation order, equation (7) can be simplified as:
Figure FDA0003397404300000018
in the formula (I), the compound is shown in the specification,
Figure FDA0003397404300000021
is the probability density function corresponding to the qth sample point;
step (9), introducing the sensitivity of the characteristic value into equation (8) to obtain a probability density evolution equation based on the sensitivity of the characteristic value:
Figure FDA0003397404300000022
step (10), introducing a new parameter z ═ λ (Θ)qB) b, substituting into equation (9) yields:
Figure FDA0003397404300000023
step (11), determining the initial conditions as follows:
Figure FDA0003397404300000024
where, δ is the dirac function,
Figure FDA0003397404300000025
step (12), rewriting equation (10) into the following form:
Figure FDA0003397404300000026
wherein a ═ λ (Θ)q,b)；
And (13) obtaining the following difference format by adopting a finite difference method and a total variation reduction format:
Figure FDA0003397404300000027
in the formula (I), the compound is shown in the specification,
Figure FDA0003397404300000028
zm＝mΔz，m＝0,±1,...，bk＝kΔb，k＝0,1,…，
Figure FDA0003397404300000029
in the form of a flow restrictor or flow restrictor,
Figure FDA00033974043000000210
and
Figure FDA00033974043000000211
can be expressed as:
Figure FDA00033974043000000212
step (14) of adding N tototalAt one sample point to calculate
Figure FDA00033974043000000213
Summing, one can get:
Figure FDA00033974043000000214
step (15), bk1, the expression of the probability density function of the structural feature value can be obtained:
Figure FDA00033974043000000215
2. the method for analyzing the random eigenvalue of the wing structure based on probability density evolution of claim 1, wherein: in the step (4), the derivation process of the sensitivity of the eigenvalue is as follows:
expressing the equation of step (2) in component form, namely:
K{ui}-λiM{ui}＝0 (17)
in the formula, λiIs the ith order eigenvalue, uiFor the ith order eigenvector, the partial derivative of the structural parameter b is calculated at both ends of equation (17), and the following relationship can be obtained:
Figure FDA0003397404300000031
simultaneous left multiplication on both sides of equation (18)
Figure FDA0003397404300000032
And the item shifting is carried out to obtain:
Figure FDA0003397404300000033
3. the method for analyzing the random eigenvalue of the wing structure based on probability density evolution of claim 1, wherein: in the step (7), the specific manner of exchanging the integration and derivation orders is as follows:
Figure FDA0003397404300000034
Figure FDA0003397404300000035
4. the method for analyzing the random eigenvalue of the wing structure based on probability density evolution of claim 1, wherein: in the step (11), the initial condition may be discretized into:
Figure FDA0003397404300000036
in the formula, zmWhere m Δ z (m is 0, ± 1, …), Δ z represents the grid size in the z direction.
5. The method for analyzing the random eigenvalue of the wing structure based on probability density evolution of claim 1, wherein: in the step (13), the convergence condition to be satisfied by the difference format is:
Figure FDA0003397404300000037
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