CN106096101A - A kind of consideration construction geometry random dynamic response of nonlinear non-stationary analyzes method - Google Patents

Abstract

The invention discloses a kind of consideration construction geometry random dynamic response of nonlinear non-stationary and analyze method, including: (1) is calculated its average and auto-covariance matrix according to non-stationary random dynamic loads；(2) calculate the eigenvalue of auto-covariance matrix and characteristic vector, and obtain eigenvalue and characteristic vector block exponent number；(3) launch and Latin Hypercube Sampling method based on KL, non-stationary random dynamic loads is decomposed, it is thus achieved that the random sample of one group of non-stationary random dynamic loads；(4) setting up the FEM (finite element) model of structure, and use transient state analyzing method, the random sample of calculating non-stationary random dynamic loads is as the receptance function under load, and is calculated variance and auto-covariance function according to this receptance function.Instant invention overcomes tradition non-stationary random dynamic response analysis can only the limitation of linear structure.

Description

A kind of consideration construction geometry random dynamic response of nonlinear non-stationary analyzes method

Technical field

The present invention relates to the random dynamic response of non-stationary and analyze method, particularly relate to a kind of consideration construction geometry nonlinear non- The most random dynamic response analyzes method.

Background technology

Engineering structure may be born when reality is on active service steadily or the random dynamic loads of non-stationary, such as: gust load, Turbulent boundary layer load, wind load and seismic (seismal etc..But in actual applications, analyze method due to the random dynamic response of non-stationary Limitation, often non-stationary random dynamic loads is reduced to steady random dynamic loads, but such simplified way can be right Follow-up random dynamic response analysis brings very important error.Therefore, in random dynamic response is analyzed, necessary consideration carries The non-stationary property of lotus.

At present for the dynamic response analysis under non-stationary random excitation, often Random dynamic loads is used Karhunen-Loeve And the spectrum STOCHASTIC FINITE ELEMENT technology such as Polynomial Chaos (PC) expansion is decomposed into a series of definitiveness stochastic variable (KL) After, use Monte Carlo method to carry out dynamic response analysis, but the method is only capable of solving the linear random of structure under small deformation at present Dynamic response is analyzed, it is impossible to consider that the construction geometry caused due to large deformation is non-linear.Engineering exists a lot of flexible structure, example Such as wing, satellite, bridge, power transmission tower etc., owing to the rigidity of structure is less, under random dynamic loads construction geometry non-linear because of Element can significantly change the response characteristic of structure.In the last few years in fields such as civil engineering, machinery and Aero-Space, geometry is non- The influence research of structural dynamic characteristics is received much concern by linear factor.Consider that construction geometry is nonlinear it is therefore proposed that a kind of The random dynamic response of non-stationary is analyzed method and is had very important engineer applied value.

Summary of the invention

Goal of the invention: the present invention provides a kind of and considers that the construction geometry random dynamic response of nonlinear non-stationary analyzes method, Solve current method be only capable of linear structure carry out the random dynamic response of non-stationary analyze confinement problems, solution is by no means simultaneously The suitability problem that the most random dynamic response is analyzed.

Technical scheme: the consideration construction geometry of the present invention random dynamic response of nonlinear non-stationary analyzes method bag Include:

(1) it is calculated its average and auto-covariance matrix according to non-stationary random dynamic loads；

(2) calculate the eigenvalue of auto-covariance matrix and characteristic vector, and obtain eigenvalue and characteristic vector block rank Number；

(3) launch and Latin Hypercube Sampling method based on KL, non-stationary random dynamic loads is decomposed, it is thus achieved that one group non- The steadily random sample of random dynamic loads；

(4) set up the FEM (finite element) model of structure, and use transient state analyzing method, calculate non-stationary random dynamic loads with Press proof this as the receptance function under load, and be calculated variance and auto-covariance function according to this receptance function.

Further, in described step (1), the computing formula of average and auto-covariance matrix is:

The average of non-stationary random dynamic loads F (t) is: μ (t)=E [F (t)]；

Auto-covariance matrix is: C (t₁,t₂)=E [(F (t₁)-μ(t₁))(F(t₂)-μ(t₂))]；

Wherein, t₁、t₂For time variable, expected value is sought in E [] expression.

Further, described step (2) specifically includes:

(21) time t is divided into m time period { [t_k-1,t_k] | k=1,2 ..., m}；Wherein the value of m more than or etc. In random dynamic loads time step number；

(22) sectioned basis functions is generated according to the time period divided, and as orthogonal basis；Wherein, sectioned basis functions is:

(23) Equations of The Second Kind Fredholm integral equation is solved according to described orthogonal basis, it is thus achieved that the eigenvalue of auto-covariance matrix And characteristic vector；

Wherein, Equations of The Second Kind Fredholm integral equation is: M φ=Λ N φ；In formula, the element in matrix φ be characterized to Amount φ_iT (), the element in matrix M isElement in matrix N is Element in matrix Λ is Λ_ij=δ_ijλ_i, t_minAnd t_maxIt is respectively the bound of analysis time, C (t₁,t₂) it is that non-stationary is random The auto-covariance matrix of load, δ_ijFor Kronecker function, λ_iFor auto-covariance matrix C (t₁,t₂) the i-th rank eigenvalue, i, j =1,2 ..., m；

(24) obtain eigenvalue and characteristic vector blocks exponent number n, i.e. before n rank eigenvalue λ_iSum is more than all eigenvalues Sum 95% time, block at n-th order.

Further, described step (3) specifically includes:

(31) Latin Hypercube Sampling method is used, it is thus achieved that one group of standard normal random sample ξ_s；Wherein, standard normal with This ξ of press proof_sAverage be 0 variance be 1；

(32) combine KL and launch to be decomposed into random dynamic loads F (t) the random sample f of one group of random dynamic loads_p (t):

$f_p\left(t\right)=\mathrm{\μ}\left(t\right)+\underset{s=1}{\overset{n}{\Σ}}\sqrt{{\mathrm{\λ}}_s}{\mathrm{\φ}}_s\left(t\right){\mathrm{\ξ}}_s$

In formula, μ (t) represent random dynamic loads F (t) average, n for blocking exponent number, λ_sS for auto-covariance matrix Rank eigenvalue, φ_sT () represents λ_sCharacteristic of correspondence vector, p is the random sample number of random dynamic loads, and more than 1000.

Further, described step (4) specifically includes:

(41) set up the FEM (finite element) model of structure, and use the nonlinear transient analysis method in business finite element software, Calculate random sample f_pT () is as the receptance function X under load_p(t)；

(42) receptance function X is calculated_pThe variance of (t) and auto-covariance function；Wherein,

Variance

Auto-covariance function

Beneficial effect: compared with prior art, its remarkable advantage is the present invention: provide a kind of consideration construction geometry non-thread Property the random dynamic response of non-stationary analyze method, expanded the random dynamic response of current non-stationary analyze method research range, can To solve to consider the random dynamic response analysis of the nonlinear non-stationary of construction geometry；Meanwhile, the random sound of something astir of current non-stationary has been expanded The object of study that should analyze, can solve the random dynamic response analysis of the non-stationary for complex nonlinear structure.

Accompanying drawing explanation

Fig. 1 is the schematic flow sheet of one embodiment of the present of invention；

Fig. 2 is cantilever beam schematic diagram；

Fig. 3 is the dynamic respond standard deviation of cantilever beam beam-ends；

Fig. 4 is the dynamic respond auto-covariance function of cantilever beam beam-ends.

Detailed description of the invention

As it is shown in figure 1, the present embodiment specifically includes following steps:

(1) it is calculated its average and auto-covariance matrix according to non-stationary random dynamic loads.

Wherein, the average of non-stationary random dynamic loads F (t) is: μ (t)=E [F (t)]；Auto-covariance matrix is: C (t₁,t₂)=E [(F (t₁)-μ(t₁))(F(t₂)-μ(t₂))]；In formula, t₁、t₂For time variable, expected value is sought in E [] expression.

With certain cantilever beam structure as object of study (as shown in Figure 2), its geometric parameter and material parameter are as shown in table 1.Execute Adding average is zero, and auto-covariance is the random uniform Random dynamic loads of modulation index form, and load step number is 1000, load duration For 1s.Then the average of random dynamic loads is μ (t)=0, and auto-covariance matrix is:

The material parameter of table 1 cantilever beam

Parameter	Numerical value
		Quality	10686kg/m
Elastic modelling quantity	5.0175×1010Pa
		Diameter	3m
Damping ratio	0.005
		Length	18m

(2) calculate the eigenvalue of auto-covariance matrix and characteristic vector, and obtain eigenvalue and characteristic vector block rank Number.

Step (2) specifically includes following steps:

(21) time t is divided into m time period { [t_k-1,t_k] | k=1,2 ..., m}；Wherein the value of m more than or etc. In random dynamic loads time step number.

Such as, still as a example by cantilever beam structure, time 1s can be divided into 1000 time period { [t_k-1,t_k] | k=1, 2,…,1000}

(22) sectioned basis functions is generated according to the time period divided, and as orthogonal basis；Wherein, sectioned basis functions is:

(23) Equations of The Second Kind Fredholm integral equation is solved according to described orthogonal basis, it is thus achieved that the eigenvalue of auto-covariance matrix And characteristic vector.

Wherein, Equations of The Second Kind Fredholm integral equation is: M φ=Λ N φ；In formula, the element in matrix φ be characterized to Amount φ_iT (), the element in matrix M isElement in matrix N is Element in matrix Λ is Λ_ij=δ_ijλ_i, t_minAnd t_maxIt is respectively the bound of analysis time, C (t₁,t₂) it is that non-stationary is random The auto-covariance matrix of load, δ_ijFor Kronecker function, λ_iFor auto-covariance matrix C (t₁,t₂) the i-th rank eigenvalue, φ_i T () is λ_iCharacteristic of correspondence vector, i, j=1,2 ..., m.

(24) obtain eigenvalue and characteristic vector blocks exponent number n, i.e. before n rank eigenvalue λ_iSum is more than all eigenvalues Sum 95% time, block at n-th order.

Such as, as a example by cantilever beam structure, front 40 rank eigenvalue sums are 0.97, so blocking at the 40th rank, n=4.

(3) launch and Latin Hypercube Sampling method based on KL, non-stationary random dynamic loads is decomposed, it is thus achieved that one group non- The steadily random sample of random dynamic loads.

Concrete, step (3) comprises the following steps:

(31) Latin Hypercube Sampling method is used, it is thus achieved that one group of standard normal random sample ξ_s；Wherein, standard normal with This ξ of press proof_sAverage be 0 variance be 1.

(32) combine KL and launch to be decomposed into random dynamic loads F (t) the random sample f of one group of random dynamic loads_p (t):

$f_p\left(t\right)=\mathrm{\μ}\left(t\right)+\underset{s=1}{\overset{n}{\Σ}}\sqrt{{\mathrm{\λ}}_s}{\mathrm{\φ}}_s\left(t\right){\mathrm{\ξ}}_s$

In formula, μ (t) represent random dynamic loads F (t) average, n for blocking exponent number, λ_sS for auto-covariance matrix Individual eigenvalue, φ_sT () represents λ_sCharacteristic of correspondence vector, p is the random sample number of random dynamic loads, and more than 1000.

Such as, as a example by cantilever beam structure, the random sample of random dynamic loadsp Value 1000.

(4) set up the FEM (finite element) model of structure, and use transient state analyzing method, calculate non-stationary random dynamic loads The receptance function that random sample obtains as load, and it is calculated variance and auto-covariance function according to this receptance function.

Concrete, step (4) specifically includes:

(41) set up the FEM (finite element) model of structure, and use the nonlinear transient analysis method in business finite element software, Calculate random sample f_pT () is as the receptance function X under load_p(t)；Wherein, the FEM (finite element) model setting up structure is existing skill Art, does not do concrete introduction at this；

(42) receptance function X is calculated_pThe variance of (t) and auto-covariance function；Wherein,

Variance

Auto-covariance function

By varianceAnd auto-covariance functionIt is expressed as the form of figure respectively such as Fig. 3 and Fig. 4 institute Show.

Claims (5)

1. one kind considers that the construction geometry random dynamic response of nonlinear non-stationary analyzes method, it is characterised in that the method includes:

(1) it is calculated its average and auto-covariance matrix according to non-stationary random dynamic loads；

(2) calculate the eigenvalue of auto-covariance matrix and characteristic vector, and obtain eigenvalue and characteristic vector block exponent number；

(3) launch and Latin Hypercube Sampling method based on KL, non-stationary random dynamic loads is decomposed, it is thus achieved that one group of non-stationary The random sample of random dynamic loads；

(4) set up the FEM (finite element) model of structure, and use transient state analyzing method, calculate non-stationary random dynamic loads with press proof This is as the receptance function under load, and is calculated variance and auto-covariance function according to this receptance function.

The consideration construction geometry the most according to claim 1 random dynamic response of nonlinear non-stationary analyzes method, its feature It is: in described step (1), the computing formula of average and auto-covariance matrix is:

The average of non-stationary random dynamic loads F (t) is: μ (t)=E [F (t)]；

Auto-covariance matrix is: C (t₁,t₂)=E [(F (t₁)-μ(t₁))(F(t₂)-μ(t₂))]；

Wherein, t₁、t₂For time variable, expected value is sought in E [] expression.

The consideration construction geometry the most according to claim 1 random dynamic response of nonlinear non-stationary analyzes method, its feature It is: described step (2) specifically includes:

(21) time t is divided into m time period { [t_k-1,t_k] | k=1,2 ..., m}；Wherein the value of m is more than or equal to random Dynamic load time step number；

(22) sectioned basis functions is generated according to the time period divided, and as orthogonal basis；Wherein, sectioned basis functions is:K=1,2 ..., m；

(23) Equations of The Second Kind Fredholm integral equation is solved according to described orthogonal basis, it is thus achieved that the eigenvalue of auto-covariance matrix and spy Levy vector；

Wherein, Equations of The Second Kind Fredholm integral equation is: M φ=Λ N φ；In formula, the element in matrix φ is characterized vector φ_i T (), the element in matrix M isElement in matrix N is Element in matrix Λ is Λ_ij=δ_ijλ_i, t_minAnd t_maxIt is respectively the bound of analysis time, C (t₁,t₂) it is that non-stationary is random The auto-covariance matrix of load, δ_ijFor Kronecker function, λ_iFor auto-covariance matrix C (t₁,t₂) the i-th rank eigenvalue, i, j =1,2 ..., m；

(24) obtain eigenvalue and characteristic vector blocks exponent number n, i.e. before n rank eigenvalue λ_iSum is more than all eigenvalue sums 95% time, block at n-th order.

The consideration construction geometry the most according to claim 1 random dynamic response of nonlinear non-stationary analyzes method, its feature It is: described step (3) specifically includes:

(31) Latin Hypercube Sampling method is used, it is thus achieved that one group of standard normal random sample ξ_s；Wherein, standard normal is with press proof This ξ_sAverage be 0 variance be 1；

(32) combine KL and launch to be decomposed into random dynamic loads F (t) the random sample f of one group of random dynamic loads_p(t):

$f_p\left(t\right)=\mathrm{\μ}\left(t\right)+\underset{s=1}{\overset{n}{\Σ}}\sqrt{{\mathrm{\λ}}_s}{\mathrm{\φ}}_s\left(t\right){\mathrm{\ξ}}_s$

In formula, μ (t) represent random dynamic loads F (t) average, n for blocking exponent number, λ_sS rank for auto-covariance matrix are special Value indicative, φ_sT () represents λ_sCharacteristic of correspondence vector, p is the random sample number of random dynamic loads, and more than 1000.

The consideration construction geometry the most according to claim 1 random dynamic response of nonlinear non-stationary analyzes method, its feature It is: described step (4) specifically includes:

(41) set up the FEM (finite element) model of structure, and use the nonlinear transient analysis method in business finite element software, calculate Random sample f_pT () is as the receptance function X under load_p(t)；

(42) receptance function X is calculated_pThe variance of (t) and auto-covariance function；Wherein,

Variance

Auto-covariance function