CN109933898B - Wallboard aeroelastic stability analysis method considering mixing uncertainty - Google Patents

Wallboard aeroelastic stability analysis method considering mixing uncertainty Download PDF

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CN109933898B
CN109933898B CN201910187129.0A CN201910187129A CN109933898B CN 109933898 B CN109933898 B CN 109933898B CN 201910187129 A CN201910187129 A CN 201910187129A CN 109933898 B CN109933898 B CN 109933898B
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邱志平
郑宇宁
王晓军
王磊
李云龙
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Beihang University
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Abstract

The invention discloses a wallboard aeroelasticity stability analysis method considering mixing uncertainty, belongs to the field of wallboard aeroelasticity design, and aims at a mixing uncertain environment where random variables and interval variables coexist, a random model and an interval model are used for carrying out quantitative characterization on mixing uncertain parameters, and a wallboard aeroelasticity stability analysis model containing the mixing uncertain parameters is established. On the basis, the probability density evolution method is combined with the interval uncertainty propagation analysis method, the random-interval mixed probability density evolution method is provided, the probability statistical characteristics of the boundary of the panel aeroelastic response interval can be estimated when the parameter fluctuation is large, the limitations of the traditional method on the calculation efficiency and the applicability are overcome, and the blank of the research on the analysis of the panel aeroelastic stability under the mixed uncertainty environment is filled.

Description

Wallboard aeroelastic stability analysis method considering mixing uncertainty
Technical Field
The invention belongs to the field of panel aeroelasticity design, and particularly relates to a panel aeroelasticity stability analysis method considering mixing uncertainty.
Background
Aeroelasticity has three remarkable characteristics, one is that it is essentially a fluid-solid coupling problem, the structure elastically deforms under the action of pneumatic force, and the structure deformation in turn changes the boundary of the flow field; secondly, the involved nonlinear factors are many, including the nonlinear factors in the aspects of structure and aerodynamics, such as gap nonlinearity at the control surface hinge and aerodynamic nonlinearity caused by large attack angle flight; in addition, since the aeroelastic system is a complex multidisciplinary coupling system, involving multiple disciplines of pneumatics, structure, heat, etc., there is inevitably an uncertainty factor in the actual aeroelastic system. The sources of uncertainty for the actual panel structure are diverse and are reflected in the following four areas: (1) model errors exist between the established aeroelastic analysis model and an actual object due to the fact that relevant factors are simplified or secondary factors are ignored in the modeling process; (2) uncertainty of material parameters, due to the influence of factors such as manufacturing environment, technical conditions, multiphase characteristics of the material and the like, the elastic modulus, the Poisson ratio and the mass density of the material have uncertainty; (3) uncertainty in geometry, uncertainty in structure geometry such as thickness, cross-sectional area, etc. due to manufacturing and installation errors; (4) uncertainty in load, both aerodynamic and thermal loads acting on the wall structure, are subject to uncertainty due to factors such as measurement conditions, external environment, etc.
Due to the limitations of objective conditions or subjective cognition, designers often face the following two typical engineering problems: firstly, parameters can be clearly divided into two types according to the volume of a test data sample, one type is that test data is sufficient and a probability density function of the test data can be fitted with high precision, and the other type is that the probability density function of the corresponding parameters cannot be obtained due to the fact that the volume of the test data sample is very limited and quantification is carried out by using an interval model. Therefore, a mixed uncertain environment with the coexistence of random variables and interval variables occurs, wherein uncertain parameters with sufficient sample information are defined as random variables, uncertain parameters with limited sample data are defined as interval variables, and the interval boundary of the aeroelastic response of the wall plate is provided with statistical characteristics due to the input of the mixed uncertain parameters.
The existing method for processing the mixing uncertainty is mainly a Monte Carlo simulation method, however, the Monte Carlo simulation method needs a large amount of sample point analysis and consumes a large amount of computing resources. When the probability density function of the boundary of the maximum real part interval of the characteristic value needs to be obtained, no effective analysis method can be provided at present, and the development of the analysis technology of the pneumatic elastic stability of the wall plate is limited to a certain extent. In summary, it is highly desirable to develop a new method capable of rapidly and accurately solving the boundary probability density function of the corresponding generalized eigenvalue interval of the panel aeroelastic equation, so as to overcome the disadvantages of long calculation time and low precision of the conventional method, thereby providing technical support for stability analysis.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: aiming at the problems that the traditional analysis method for the aerodynamic elastic stability of the wallboard containing mixing uncertain parameters is low in calculation efficiency, a characteristic value interval boundary probability density function is difficult to obtain and the like, the analysis method for the aerodynamic elastic stability of the wallboard considering mixing uncertainty is provided. The method aims at a mixed uncertain environment with coexisting random variables and interval variables, quantitative characterization is carried out on mixed uncertain parameters by using a random model and an interval model, and a wallboard aeroelastic stability analysis model containing the mixed uncertain parameters is established. On the basis, the probability density evolution method is combined with the interval uncertainty propagation analysis method, the random-interval mixed probability density evolution method is provided, the probability statistical characteristics of the boundary of the panel aeroelastic response interval can be estimated when the parameter fluctuation is large, and the limit of the traditional method on the calculation efficiency and the applicability is overcome.
The technical scheme adopted by the invention for solving the technical problems is as follows: a method for analyzing aeroelastic stability of a wall panel in consideration of mixing uncertainty, comprising the steps of:
step (1), establishing a panel pneumatic elastic finite element equation containing mixing uncertain parameters:
Figure GDA0003104168800000021
in the formula, alphasto＝(αsto,1,αsto,2,…,αsto,m) Is a random vector, αin＝(αin,1,αin,2,…,αin,l) Is an interval vector, M and l are vector dimensions, M is a wallboard mass matrix, C is a wallboard damping matrix,
Figure GDA0003104168800000022
is a pneumatic damping matrix, K is a wallboard stiffness matrix,
Figure GDA0003104168800000023
is an aerodynamic stiffness matrix, x (t) is a generalized coordinate,
Figure GDA0003104168800000024
in the case of a speed in a broad sense,
Figure GDA0003104168800000025
generalized acceleration, t is time;
step (2) of letting x (t) be x0eλtThe wall plate aeroelasticity finite element equation containing the mixed uncertain parameters can be converted into a generalized characteristic value equation:
A(αsto,αin)u＝λB(αsto,αin)u (2)
in the formula (I), the compound is shown in the specification,
Figure GDA0003104168800000026
Figure GDA0003104168800000027
λ is a generalized eigenvalue;
in the step (3), the maximum real part μ of the eigenvalue can be obtained by the following formula:
μ＝μ(αsto,αin)＝max{Re[λi(A(αsto,αin),B(αsto,αin))]},(i＝1,2,...,2n) (3)
in the formula, Re represents a real part of a characteristic value;
step (4), establishing a probability density evolution equation containing random-interval mixed parameters, wherein the probability density evolution equation is expressed as the following form:
Figure GDA0003104168800000031
Figure GDA0003104168800000032
in the formula (I), the compound is shown in the specification,μand
Figure GDA0003104168800000033
lower and upper intervals of μ, p μ αAnd
Figure GDA0003104168800000034
is prepared from (a)μAlpha) and
Figure GDA0003104168800000035
the joint probability density function of (a) is,
Figure GDA0003104168800000036
step (5) in the uncertain parameter alphastoChange domain omega ofstoTaking N evenlytotalA sample point, denoted as αsto,q(q＝1,...,Ntotal) And will change the domain omegastoIs divided into NtotalIndividual field, denoted as Ωsto,q(q＝1,…,Ntotal)；
Step (6), equations (4) - (5) are placed in the sub-domain omegasto,qInternal integration, one can get:
Figure GDA0003104168800000037
Figure GDA0003104168800000038
by exchanging the integration and derivation order, equations (6) - (7) can be reduced to:
Figure GDA0003104168800000039
Figure GDA00031041688000000310
in the formula (I), the compound is shown in the specification,
Figure GDA00031041688000000311
and
Figure GDA00031041688000000312
is the probability density function corresponding to the qth sample point;
step (8), introduce the fictitious parameter tau, order
Figure GDA00031041688000000313
Substituting equations (8) - (9) can result in:
Figure GDA00031041688000000314
Figure GDA00031041688000000315
step (9), determining the initial conditions as follows:
Figure GDA0003104168800000041
Figure GDA0003104168800000042
where, δ is the dirac function,
Figure GDA0003104168800000043
the following difference format can be obtained by adopting the finite difference method and the total variation reduction format in the step (10):
Figure GDA0003104168800000044
Figure GDA0003104168800000045
in the formula (I), the compound is shown in the specification,
Figure GDA0003104168800000046
Figure GDA0003104168800000047
τk＝kΔτ(k＝0,1,…)，rLAXin order to be a differential grid ratio,
Figure GDA0003104168800000048
in the form of a flow restrictor or flow restrictor,
Figure GDA0003104168800000049
and
Figure GDA00031041688000000410
can be expressed as:
Figure GDA00031041688000000411
step (11) will be at NtotalAt one sample point to calculate
Figure GDA00031041688000000412
Summing, one can get:
Figure GDA00031041688000000413
Figure GDA00031041688000000414
step (12) of taking τ k1, the probability density function expression of the maximum real part μ of the eigenvalue can be obtained:
Figure GDA00031041688000000415
Figure GDA00031041688000000416
step (13) according to
Figure GDA00031041688000000417
If present, is
Figure GDA00031041688000000418
The panel is at risk of flutter failure.
In step (9), the initial condition may be discretized into:
Figure GDA0003104168800000051
Figure GDA0003104168800000052
in the formula (I), the compound is shown in the specification,
Figure GDA0003104168800000053
and
Figure GDA0003104168800000054
is composed of
Figure GDA0003104168800000055
And
Figure GDA0003104168800000056
grid size of direction;
in the step (10), the convergence condition to be satisfied by the difference format is:
Figure GDA0003104168800000057
the invention has the beneficial effects that:
the invention provides a wallboard aeroelasticity stability analysis method considering mixing uncertainty, which can analyze a generalized characteristic value corresponding to a wallboard aeroelasticity equation containing mixing uncertainty parameters, and obtain a probability density function of a boundary of a maximum real part interval of the characteristic value, so as to judge the aeroelasticity stability of a wallboard. The characteristic value real part interval probability density function obtained by the method is well matched with the characteristic value real part interval probability density function obtained by the Monte Carlo method, the calculation time can be greatly reduced, and a new thought is provided for the aeroelastic stability analysis of the wall plate containing mixed uncertain parameters.
Drawings
FIG. 1 is a schematic view of a two-dimensional curved wall panel model;
FIG. 2 is V∞When the speed is 2500m/sμA probability density function of;
FIG. 3 is V∞When the speed is 2500m/s
Figure GDA0003104168800000058
A probability density function of;
FIG. 4 is V∞When the speed is 3200m/sμA probability density function of;
FIG. 5 is V∞When the speed is 3200m/s
Figure GDA0003104168800000059
A probability density function of;
fig. 6 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. The present invention is in the field of aeroelastic design of wall panels, and it should be understood that the examples have been chosen only to illustrate the invention, and not to limit the scope of the invention.
(1) The geometric model of the two-dimensional curved wall plate structure is shown in figure 1;
(2) giving material property parameters and inflow parameters of the curved wall plate structure unit, as shown in table 1;
table 1 curved wall panel structure material properties and incoming flow parameters
Figure GDA0003104168800000061
In Table 1, E is the modulus of elasticity, ρ∞As density of incoming flow, ρsIs a density of curved wall plate, alphasIs the coefficient of thermal expansion;
(3) for the random variable α of Table 1stoFollowing normal distribution, the coefficient of variation cov was taken to be 0.01; for the interval variable alphainThe interval boundary is alphain c[1-β,1-β]Beta is an uncertain factor, and in the present example, beta is 0.05;
(4) e and rho∞The variation interval [ theta-6 sigma, theta +6 sigma ]]Equally divided into 20 subintervals, E and ρ at the subinterval boundaries∞Comprises the following steps:
Figure GDA0003104168800000062
thus, a sample point (E) is formedi,ρ∞j) There were 441 total;
(5) taking the q sample point (E)i,ρ∞j)qAccording to (E)i,ρ∞j)qSetting material and inflow parameters, establishing a aeroelastic finite element equation of the wall plate, and obtaining a corresponding sample point (E) through interval analysisi,ρ∞j)qInterval boundary of maximum real part of eigenvalue ofμAnd
Figure GDA0003104168800000063
(6) set up toμAnd
Figure GDA0003104168800000064
the probability density evolution equation of (a):
Figure GDA0003104168800000065
Figure GDA0003104168800000066
(7) the initial conditions may be discretized as:
Figure GDA0003104168800000067
Figure GDA0003104168800000071
(8) the finite difference format is set as:
Figure GDA0003104168800000072
Figure GDA0003104168800000073
(9) current limiter
Figure GDA0003104168800000074
The following settings are set:
Figure GDA0003104168800000075
(10) repeating the steps (5) to (9), and calculating probability density functions corresponding to all 441 sample points
Figure GDA0003104168800000076
Figure GDA0003104168800000077
Summing them can yield:
Figure GDA0003104168800000078
Figure GDA0003104168800000079
(11) take tau k1, the probability density function expression of the interval boundary with the maximum real part of the eigenvalue can be calculated as:
Figure GDA00031041688000000710
Figure GDA00031041688000000711
(12) taking 10000 sample points which are obeyed normal distribution by using a Monte Carlo simulation method, and corresponding each sample point to the interval boundary of the maximum real part of the characteristic valueμAnd
Figure GDA00031041688000000712
is calculated to obtainμAnd
Figure GDA00031041688000000713
a probability density function of;
(13) obtained by the above two methods under the conditions of inflow velocity of 2500m/s and 3200m/sμAnd
Figure GDA00031041688000000714
the probability density function is shown in fig. 2-5, 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;
(14) the two methods respectively calculate the total time consumption as follows: t isThe method of the invention＝5200s,TMonte Carlo simulation method14210 s. The time comparison result shows that the method can reduce the calculation time consumption, thereby obviously improving the calculation efficiency of the probability density function;
(15) stability analysis can be performed according to FIGS. 2-5 when V is∞When it is 2500m/s, the reason is that
Figure GDA0003104168800000081
The system has no flutter failure risk; when V is∞When 3200m/s, exist
Figure GDA0003104168800000082
I.e. the system has a risk of flutter failure.
In summary, the present invention provides a method for analyzing the aero-elastic stability of a wall panel in consideration of mixing uncertainty. And aiming at a mixed uncertain environment in which random variables and interval variables coexist, carrying out quantitative characterization on mixed uncertain parameters by using a random model and an interval model, and establishing a wallboard aeroelastic stability analysis model containing the mixed uncertain parameters. On the basis, the probability density evolution method is combined with the interval uncertainty propagation analysis method, the random-interval mixed probability density evolution method is provided, the probability statistical characteristics of the boundary of the panel aeroelastic response interval can be estimated when the parameter fluctuation is large, the limitations of the traditional method on the calculation efficiency and the applicability are overcome, and the blank of the research on the analysis of the panel aeroelastic stability under the mixed uncertainty environment is filled.
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 aeroelastic design of aircrafts, and any technical solution formed by equivalent transformation or equivalent replacement falls within the protection scope of the present invention.

Claims (2)

1. A method for analyzing the aeroelastic stability of a wall plate by considering mixing uncertainty is characterized by comprising the following steps: the method comprises the following implementation steps:
step (1), establishing a panel pneumatic elastic finite element equation containing mixing uncertain parameters:
Figure FDA0003081976850000011
in the formula, alphasto＝(αsto,1,αsto,2,…,αsto,h) Is a random vector, αin＝(αin,1,αin,2,…,αin,l) Is an interval vector, h and l are vector dimensions, M is a wallboard mass matrix, C is a wallboard damping matrix,
Figure FDA0003081976850000012
is a pneumatic damping matrix, K is a wall plateThe matrix of the stiffness is then,
Figure FDA0003081976850000013
is an aerodynamic stiffness matrix, x (t) is a generalized coordinate,
Figure FDA0003081976850000014
in the case of a speed in a broad sense,
Figure FDA0003081976850000015
generalized acceleration, t is time;
step (2) of letting x (t) be x0eλtThe wall plate aeroelasticity finite element equation containing the mixed uncertain parameters can be converted into a generalized characteristic value equation:
A(αsto,αin)u＝λB(αsto,αin)u (2)
in the formula (I), the compound is shown in the specification,
Figure FDA0003081976850000016
Figure FDA0003081976850000017
λ is a generalized eigenvalue;
in the step (3), the maximum real part μ of the eigenvalue can be obtained by the following formula:
μ＝μ(αsto,αin)＝max{Re[λi(A(αsto,αin),B(αsto,αin))]},i＝1,2,...,2n (3)
in the formula, Re represents a real part of a characteristic value;
step (4), establishing a probability density evolution equation containing random-interval mixed parameters, wherein the probability density evolution equation is expressed as the following form:
Figure FDA0003081976850000018
Figure FDA0003081976850000019
in the formula (I), the compound is shown in the specification,μand
Figure FDA00030819768500000110
lower and upper intervals of μ, p μ αAnd
Figure FDA00030819768500000111
is prepared from (a)μAlpha) and
Figure FDA00030819768500000112
the joint probability density function of (a) is,
Figure FDA00030819768500000113
step (5) in the uncertain parameter alphastoChange domain omega ofstoTaking N evenlytotalA sample point, denoted as αsto,q，q＝1,...,NtotalAnd will change the domain omegastoIs divided into NtotalIndividual field, denoted as Ωsto,q，q＝1,…,Ntotal；
Step (6), equations (4) - (5) are placed in the sub-domain omegasto,qInternal integration, one can get:
Figure FDA0003081976850000021
Figure FDA0003081976850000022
by exchanging the integration and derivation order, equations (6) - (7) can be reduced to:
Figure FDA0003081976850000023
Figure FDA0003081976850000024
in the formula (I), the compound is shown in the specification,
Figure FDA0003081976850000025
and
Figure FDA0003081976850000026
is the probability density function corresponding to the qth sample point;
step (8), introduce the fictitious parameter tau, order
Figure FDA0003081976850000027
Substituting equations (8) - (9) can result in:
Figure FDA0003081976850000028
Figure FDA0003081976850000029
step (9), determining the initial conditions as follows:
Figure FDA00030819768500000210
Figure FDA00030819768500000211
where, δ is the dirac function,
Figure FDA00030819768500000212
the following difference format can be obtained by adopting the finite difference method and the total variation reduction format in the step (10):
Figure FDA00030819768500000213
Figure FDA0003081976850000031
in the formula (I), the compound is shown in the specification,
Figure FDA0003081976850000032
Figure FDA0003081976850000033
is composed of
Figure FDA0003081976850000034
The size of the grid of directions is such that,
Figure FDA0003081976850000035
Figure FDA0003081976850000036
is composed of
Figure FDA0003081976850000037
Grid size of direction, τk＝kΔτ，k＝0,1,…，rLAXIn order to be a differential grid ratio,
Figure FDA0003081976850000038
in the form of a flow restrictor or flow restrictor,
Figure FDA0003081976850000039
and
Figure FDA00030819768500000310
can be expressed as:
Figure FDA00030819768500000311
step (11) will be at NtotalAt one sample point to calculate
Figure FDA00030819768500000312
Summing, one can get:
Figure FDA00030819768500000313
Figure FDA00030819768500000314
step (12) of taking τk1, the probability density function expression of the maximum real part μ of the eigenvalue can be obtained:
Figure FDA00030819768500000315
Figure FDA00030819768500000316
step (13) according to
Figure FDA00030819768500000317
If present, is
Figure FDA00030819768500000318
The panel is at risk of flutter failure.
2. The method of claim 1 for analyzing aeroelastic stability of a wall panel taking into account mixing uncertainty, wherein: in the step (9), the initial conditions may be discretized into:
Figure FDA00030819768500000319
Figure FDA00030819768500000320
in the formula (I), the compound is shown in the specification,
Figure FDA0003081976850000041
and
Figure FDA0003081976850000042
is composed of
Figure FDA0003081976850000043
And
Figure FDA0003081976850000044
the size of the grid of directions is such that,
Figure FDA0003081976850000045
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