CN106096239A - Random dynamic loads decomposition technique based on trigonometric function orthogonal basis - Google Patents

Abstract

The invention discloses a kind of random dynamic loads decomposition technique based on trigonometric function orthogonal basis, comprise the following steps: 1, determine average and the auto-covariance matrix of random dynamic loads；2, select trigonometric function to solve Equations of The Second Kind Fredholm integral equation as orthogonal basis, calculate eigenvalue and the characteristic vector of auto-covariance matrix, and obtain the truncation number of eigenvalue；3, use trigonometric function orthogonal basis to decompose the characteristic vector of auto-covariance matrix, and calculate the participation factors of orthogonal basis；4, launch to decompose random dynamic loads based on KL.Method disclosed by the invention can be carried out steadily and the decomposition of non-stationary random dynamic loads, and the method can improve decomposition efficiency on the basis of ensureing Decomposition Accuracy simultaneously.

Description

Random dynamic loads decomposition technique based on trigonometric function orthogonal basis

Technical field

The present invention relates to Random dynamic loads decomposition technique field, be specifically related to a kind of based on trigonometric function orthogonal basis random Dynamic loading decomposition technique.

Background technology

Engineering structure is subjected to definitiveness dead load and dynamic loading, and bears Uncertain Stochastic dynamic loading, such as: Atmospheric turbulance, noise, road roughness, earthquake and wind load etc..Random dynamic loads is commonly divided into steady random load and non- Steadily random load.In engineering, overwhelming majority arbitrary excitations are non-stationary ecitation, but for the ease of computational analysis, as well as The limitation of calculation and analysis methods, is often reduced to stationary random excitation non-stationary random excitation, but such simplified way Follow-up random dynamic response analysis can be brought appreciable error.

Analyze for the ease of random dynamic response, particularly the analysis of non-stationary random dynamic response, often Random dynamic loads is decomposed For a series of definitiveness stochastic variable.At present, the decomposition of random dynamic loads frequently with Karhunen-Loeve (KL) and The spectrum STOCHASTIC FINITE ELEMENT technology such as Polynomial Chaos (PC) expansion, wherein KL launches is again one of conventional method.When adopting When decomposing the auto-covariance function of random dynamic loads with KL expansion technique, orthogonal basis function is commonly used to solve Equations of The Second Kind Fredholm integration.But when using different basic functions, solving precision and the efficiency of Equations of The Second Kind Fredholm integration can be completely Different.Therefore, the selection of basic function has large effect for Decomposition Accuracy and the efficiency of Random dynamic loads, selects suitable base Function is most important for the decomposition of random dynamic loads.

Summary of the invention

Goal of the invention: for problems of the prior art, the invention discloses a kind of on guarantee Decomposition Accuracy basis On can improve again the random dynamic loads decomposition technique of decomposition efficiency, this technology can be used for steadily and non-stationary random dynamic loads Decomposition.

Technical scheme: the invention discloses a kind of random dynamic loads decomposition technique, comprise the steps:

(1) average and the auto-covariance matrix of random dynamic loads are determined；

(2) select trigonometric function as orthogonal basis, solve Equations of The Second Kind Fredholm integral equation, it is thus achieved that auto-covariance matrix Eigenvalue and characteristic vector and the truncation number of eigenvalue；

(3) use orthogonal basis to decompose the characteristic vector of auto-covariance matrix, and calculate the participation factors of orthogonal basis；

(4) launch to decompose random dynamic loads based on KL.

Further, the mean μ (t) of random dynamic loads X (t) and auto-covariance matrix C (t in described step (1)₁,t₂) Computing formula is:

μ (t)=E [X (t)] (1)

C(t₁,t₂)=E [(X (t₁)-μ(t₁))(X(t₂)-μ(t₂))] (2)

Wherein t, t₁,t₂Being time variable, expectation is asked in E [] expression.

Further, described step (2) comprises the following steps:

201, trigonometric function h is selected_kT () is as orthogonal basis；

202, Equations of The Second Kind Fredholm integral equation is solved, it is thus achieved that the eigenvalue of auto-covariance matrix and characteristic vector；Its Middle Equations of The Second Kind Fredholm integral equation is:

M Φ=Λ N Φ (3)

In formula, the element of matrix M isElement in matrix N is The element of matrix Λ is Λ_ij=δ_ijλ_i, matrix Φ=[φ₁(t),φ₂(t),...,φ_i(t),...,φ_m(t)]^T, φ_i(t) be Auto-covariance matrix C (t₁,t₂) the i-th rank characteristic vector, λ_iIt is φ_i(t) characteristic of correspondence value, t_minAnd t_maxIt is respectively and analyzes The bound of time, δ_ijFor Kronecker function, definition is such as formula (4)；I, j=1,2 ..., m, m be random dynamic loads time Between step number；

${\mathrm{\δ}}_{ij}=\left\{\begin{array}{cc}0,& ifi\≠j\\ 1,& ifi=j\end{array}\right.---\left(4\right)$

203, obtaining truncation number n of eigenvalue, front n eigenvalue sum the most from large to small is more than all eigenvalue sums 95% time, block at n-th order.

Further, described trigonometric function h_kT () is semisinusoidal and half cosine function, its expression formula is:

Wherein L is the half of analysis time；M is the time step number of random dynamic loads.

Further, described trigonometric function h_kT () is the most sinusoidal and full cosine function, its expression formula is:

Wherein L is the half of analysis time；M is the time step number of random dynamic loads.

Further, characteristic vector φ in step (3)_iT () uses orthogonal basis h_kT () decomposes, calculate the ginseng of orthogonal basis With factor d_kiEmploying formula (7):

${\mathrm{\φ}}_i\left(t\right)=\underset{k=1}{\overset{n}{\Σ}}{d}_{ki}{h}_k\left(t\right)---\left(7\right)$

Further, step (4) launches to be decomposed into random dynamic loads X (t) formula (8) based on KL:

$X\left(t\right)=\mathrm{\μ}\left(t\right)+\underset{i=1}{\overset{n}{\Σ}}\sqrt{{\mathrm{\λ}}_i}{\mathrm{\ξ}}_i\left(\underset{k=1}{\overset{n}{\Σ}}{d}_{ki}{h}_k\left(t\right)\right)---\left(8\right)$

Wherein ξ_iRepresent the stochastic variable of one group of standard normal, have average be 0, variance be the character of 1.

Beneficial effect: the invention discloses a kind of random dynamic loads decomposition technique based on trigonometric function orthogonal basis, be A kind of not only can guarantee that Decomposition Accuracy but also the random dynamic loads decomposition technique of decomposition efficiency can be improved, be that one can be decomposed simultaneously Steadily random dynamic loads can decompose again the decomposition technique of non-stationary random dynamic loads.

Accompanying drawing explanation

Fig. 1 is the logical procedure diagram of the inventive method.

Detailed description of the invention

Below in conjunction with the accompanying drawings and detailed description of the invention, it is further elucidated with the present invention.

With average for zero, auto-covariance is exponential form, time a length of 1s, time step number is that the random dynamic loads of 1000 is Example, uses the random dynamic loads decomposition technique based on trigonometric function orthogonal basis of the present invention to decompose, comprises the following steps:

Step 1: determine mean μ (t) and the auto-covariance matrix C (t of random dynamic loads X (t)₁,t₂), respectively such as formula (9) With formula (10):

μ (t)=0 (9)

$C(t_1,t_2)=10e^{-10|t_1-t_2|}---\left(10\right)$

Step 2: select trigonometric function h_kT () solves Equations of The Second Kind Fredholm integral equation as orthogonal basis, it is thus achieved that self tuning The eigenvalue λ of variance matrix_iWith characteristic vector φ_iT truncation number n of () and eigenvalue, specifically comprises the following steps that

201, trigonometric function orthogonal basis can be with the semisinusoidal shown in selecting type (5) and half cosine function, expression formula such as formula (11):

$h_k\left(t\right)=\left\{\begin{array}{cc}1& k=1\\ cos\mathrm{\π}kt& k=2,4,...,1000\\ sin\mathrm{\π}(k-1)t& k=3,5,...,999\end{array}\right.---\left(11\right)$

Trigonometric function orthogonal basis can be with the most sinusoidal and full cosine function shown in selecting type (6), expression formula such as formula (12):

$h_k\left(t\right)=\left\{\begin{array}{cc}1& k=1\\ cos\frac{\mathrm{\π}k}{2}t& k=2,4,...,1000\\ sin\frac{\mathrm{\π}(k-1)}{2}t& k=3,5,...,999\end{array}\right.---\left(12\right)$

In formula (11) and formula (12) analysis time be 1s, L be 0.5.

202, Equations of The Second Kind Fredholm integral equation is solved, it is thus achieved that the eigenvalue of auto-covariance matrix and characteristic vector；The Two class Fredholm integral equations are as follows:

M Φ=Λ N Φ

Wherein the element of matrix M isElement in matrix N is The element of matrix Λ is Λ_ij=δ_ijλ_i, matrix Φ=[φ₁(t),φ₂(t),...,φ_i(t),...,φ₁₀₀₀(t)]^T, φ_i(t) For auto-covariance matrix C (t₁,t₂) the i-th rank characteristic vector, λ_iIt is φ_i(t) characteristic of correspondence value；I, j=1,2 ..., 1000。

203, the most front 40 rank eigenvalue sums are 0.95, so n takes 40.

Step 3: by characteristic vector φ of auto-covariance matrix_iT () uses orthogonal basis h_kT () decomposes, and calculate orthogonal The participation factors d of base_ki, such as formula (13)；

${\mathrm{\φ}}_i\left(t\right)=\underset{k=1}{\overset{40}{\Σ}}{d}_{ki}{h}_k\left(t\right)---\left(13\right)$

Step 4: launch to decompose random dynamic loads X (t) by formula (8) based on Karhunen-Loeve (KL), as Following formula:

$X\left(t\right)=0+\underset{i=1}{\overset{40}{\Σ}}\sqrt{{\mathrm{\λ}}_i}{\mathrm{\ξ}}_i\left(\underset{k=1}{\overset{40}{\Σ}}{d}_{ki}{h}_k\left(t\right)\right)---\left(14\right)$

Wherein, ξ_iRepresent the stochastic variable of one group of standard normal, have average be 0, variance be the character of 1.

Claims (7)

1. a random dynamic loads decomposition technique based on trigonometric function orthogonal basis, it is characterised in that comprise the following steps:

(1) average and the auto-covariance matrix of random dynamic loads are determined；

(2) select trigonometric function as orthogonal basis, solve Equations of The Second Kind Fredholm integral equation, it is thus achieved that the spy of auto-covariance matrix Value indicative and characteristic vector and the truncation number of eigenvalue；

(3) use orthogonal basis to decompose the characteristic vector of auto-covariance matrix, and calculate the participation factors of orthogonal basis；

(4) launch to decompose random dynamic loads based on KL.

Random dynamic loads decomposition technique the most according to claim 1, it is characterised in that stochastic and dynamic in described step (1) The mean μ (t) of load X (t) and auto-covariance matrix C (t₁,t₂) computing formula is:

μ (t)=E [X (t)]

C(t₁,t₂)=E [(X (t₁)-μ(t₁))(X(t₂)-μ(t₂))]

Wherein t, t₁,t₂Being time variable, expectation is asked in E [] expression.

Random dynamic loads decomposition technique the most according to claim 1, it is characterised in that described step (2) includes following step Rapid:

201, trigonometric function h is selected_kT () is as orthogonal basis；

202, Equations of The Second Kind Fredholm integral equation is solved, it is thus achieved that the eigenvalue of auto-covariance matrix and characteristic vector；Wherein Two class Fredholm integral equations are:

M Φ=Λ N Φ

In formula, the element of matrix M isElement in matrix N is The element of matrix Λ is Λ_ij=δ_ijλ_i, matrix Φ=[φ₁(t),φ₂(t),...,φ_i(t),...,φ_m(t)]^T, φ_i(t) be Auto-covariance matrix C (t₁,t₂) the i-th rank characteristic vector, λ_iIt is φ_i(t) characteristic of correspondence value, t_minAnd t_maxIt is respectively and analyzes The bound of time, δ_ijFor Kronecker function；I, j=1,2 ..., m, m are the time step number of random dynamic loads；

203, obtaining truncation number n of eigenvalue, front n eigenvalue sum the most from large to small is more than all eigenvalue sums When 95%, block at n-th order.

Random dynamic loads decomposition technique the most according to claim 3, it is characterised in that described trigonometric function h_kT () is half Sine and half cosine function, its expression formula is:

Wherein L is the half of analysis time；M is the time step number of random dynamic loads.

Random dynamic loads decomposition technique the most according to claim 3, it is characterised in that described trigonometric function h_kT () is complete Sinusoidal and full cosine function, its expression formula is:

Wherein L is the half of analysis time；M is the time step number of random dynamic loads.

Random dynamic loads decomposition technique the most according to claim 1, it is characterised in that characteristic vector in described step (3) φ_iT () uses orthogonal basis h_kT () decomposes, calculate the participation factors d of orthogonal basis_kiEmploying following formula:

${\mathrm{\φ}}_i\left(t\right)=\underset{k=1}{\overset{n}{\Σ}}{d}_{ki}{h}_k\left(t\right)$

Random dynamic loads decomposition technique the most according to claim 1, it is characterised in that based on KL exhibition in described step (4) Open and random dynamic loads X (t) be decomposed into following formula:

$X\left(t\right)=\mathrm{\μ}\left(t\right)+\underset{i=1}{\overset{n}{\Σ}}\sqrt{{\mathrm{\λ}}_i}{\mathrm{\ξ}}_i\left(\underset{k=1}{\overset{n}{\Σ}}{d}_{ki}{h}_k\right(t\left)\right)$

Wherein, ξ_iRepresenting the stochastic variable of one group of standard normal, having average is 0, and variance is the character of 1.