CN106295159A - A kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method - Google Patents

Abstract

A kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method, the present invention relates to wind induced structural vibration based on auto-correlation function and responds efficient frequency domain estimation method.The invention aims to solve the shortcoming that existing wind-induced response method computational efficiency is low and calculating error is big.Detailed process is: one: based on p_iT () calculates σ_pi、R_pi(τ) and Coh_pij(ω)；Two: obtain ω_mi；Three: obtain the relevant index k of this structure_c, calculate the wind load coherence factor of i, j 2；Four: extract [M], [K], [C], [R], [I]；Five: calculate Analytical Integration according to one, two, three and four and obtain [Σ_kr]；Six: form modal response covariance matrix；Seven: calculate dynamic respond covariance matrix, and then its covariance matrix arbitrarily responded can be tried to achieve.The present invention responds field for structure wind battle array.

Description

A kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method

Technical field

The present invention relates to the efficient frequency domain estimation method of wind vibration response based on auto-correlation function.

Background technology

Wind-induced response is large and complex structure (such as Long Span Roof Structures, high-rise building etc.) wind force proofing design One of important step.The vibration characteristics of large and complex structure is difficult to represent by the simple vibration shape, the contribution of high order mode to be considered And the coupling between the vibration shape, therefore, the structural dynamic response under wind loads effect is extremely complex.Traditional time domain wind shakes Response analysis, needs to input a large amount of wind load time-history information, uses Newmark-β integration method, and amount of calculation is relatively big, required during calculating Memory space is relatively big, causes computational efficiency low.And traditional frequency domain wind-induced response, need also exist for inputting bulk information, bag Include the covariance matrix of wind load, use numerical integrating, calculate the longest, cause computational efficiency low.Additionally, traditional frequency domain Computational methods are difficult to consider modes coupling effect, calculate error big.

Summary of the invention

The invention aims to solve that existing wind-induced response method computational efficiency is low and to calculate error big Shortcoming, and propose a kind of wind induced structural vibration based on auto-correlation function and respond efficient frequency domain estimation method.

A kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method detailed process:

Step one: the body structure surface pulsating wind pressure time course data p recorded based on wind tunnel test_iT (), calculates each wind load and adds The meansigma methods of the blast time course data of loading pointThe standard deviation sigma of blast time course data_pi, the auto-correlation function of blast time course data R_pi(τ) and the coherent function Coh of blast time course data_pij(ω)；

Wherein, i, j are any two point in wind load load(ing) point sum Q, and 1≤i≤Q, 1≤j≤Q, Q are that wind load loads Point sum, for positive integer；

Step 2: use exponential function matching auto-correlation function, obtains the dimensionless blast spectrum of the wind load load(ing) point of i point The crest frequency ω of curve_mi；

From the definition of auto-power spectrum

$\frac{S_{pi}\left(\mathrm{\ω}\right)}{{\mathrm{\σ}}_{pi}^2}=\frac{1}{\mathrm{\π}}{\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}{R}_{pi}\left(\mathrm{\τ}\right)\exp (-i\mathrm{\ω}\mathrm{\τ})d\mathrm{\τ}=\frac{2}{\mathrm{\π}}\·\frac{{\mathrm{\ω}}_{mi}}{{\mathrm{\ω}}_{mi}^2+{\mathrm{\ω}}^2}$

In formula, S_pi(ω) it is the blast Power spectral density of each load(ing) point；ω is frequency；τ is the time difference；

Then dimensionless blast stave is shown asCan draw,

In formula, S_i(ω) it is dimensionless blast spectrum；

Step 3: use exponential function matching coherent function, obtain the relevant index k of this structure_c, calculate i, j's 2 Wind load coherence factor；

Step 4: extract the mass matrix [M] of structure, stiffness matrix [K], damping matrix [C], wind load load(ing) point and knot The transition matrix [R] of structure degree of freedom, the face that affects matrix [I] of response；

Structure is carried out model analysis, solves characteristic equationObtain each rank natural frequency of vibration of structure ω_nk, k=1,2 ..., N, k are the rank number of mode corresponding to the natural frequency of vibration, and N is the Degree of Structure Freedom number, for positive integer；

Take structural eigenvector matrixWhereinFor kth, r first order mode, k, R=1,2 ... W；W is to calculate selected rank number of mode, and W is positive integer, 10≤W≤N；K, r are natural frequency of vibration ω_nk、ω_nrInstitute Corresponding rank number of mode；MakeThe broad sense damping ratio of computation structure k first order mode

In formula, T is transposed matrix；

Step 5: calculate Analytical Integration according to step one, step 2, step 3 and step 4；Obtain [Σ_kr]；

Step 6: calculate modal response covarianceForm modal response covariance square Battle array；

Step 7: calculate dynamic respond covariance matrix [Σ_x]=[Ψ]^T[Σ_y] [Ψ], and then it is the loudest to try to achieve it The covariance matrix answered.

The invention have the benefit that

This method designs the large and complex structure efficient frequency domain estimation method of wind vibration response based on auto-correlation function, solves to pass The system wind-induced response method shortcoming that computational efficiency is low and computational accuracy is poor in large and complex structure is analyzed.

First the wind loads that wind tunnel test obtains is processed as the statistic for calculating and frequency spectrum parameter by the method Information, enormously simplify the input of wind load.Then, in conjunction with modal analysis method, to each vibration shape, to wind on each degree of freedom Load frequency spectrum carries out the Analytical Integration simplified, and obtains modal response covariance.Finally, by the combination of coupled modes is obtained wind The covariance of vibration response.The method substantially increases computational efficiency while ensureing modal coupling computational accuracy.

The present invention is ensureing on the premise of computational accuracy, is greatly improved large and complex structure wind and shakes the efficiency calculated, fall Low calculate shared by memory space.Take full advantage of the Computing Principle of Analytical Integration, numerical integration is converted into parsing long-pending Point algebraic operation, it is to avoid the error that numerical integration is brought, also substantially increase computational efficiency simultaneously.In implementation process, Step is clear and definite, and operability is relatively strong, is found by instance analysis, and the present invention relatively traditional algorithm efficiency significantly increases, more practical. Use this invention that certain fan-shaped plan Long-span Cantilever grid structure is carried out wind-induced response, compared with traditional analysis, Maximum error is less than 1%, is much better than traditional simplification frequency domain algorithm error (16.7%), and efficiency relatively traditional algorithm improves 200 Times.

Accompanying drawing explanation

Fig. 1 is the flow chart of the present invention；

Fig. 2 a isTake-6.42, σ_piTake the body structure surface point wind load time-history of 2.48 by the knot of step one statistical modeling Really schematic diagram, vertical coordinate isFor normalized wind load time-history, abscissa is t, for time (second), p_i(t) For wind load time-history, p_iT () is Wind Loads Acting, σ_piFor wind load standard deviation；

Fig. 2 b is R_pi(τ)=exp (-ω_mi| τ |), ω_miThe body structure surface point wind load time-history of=2.40rad/s is by step The result schematic diagram of a rapid statistical modeling, vertical coordinate is R_pi(τ), R_pi(τ) it is the auto-correlation function of wind load, ω_miFor wind load Frequency, abscissa be τ, τ be the time difference (second)；

Fig. 2 c isBody structure surface point wind load time-history by the knot of step one statistical modeling Really schematic diagram, vertical coordinate isFor dimensionless wind load power spectrum, abscissa be ω, ω be frequency (arc Degrees second), S_pi(ω) it is wind load power spectral density function；

Fig. 3 a is that fan-shaped plan is encorbelmented the FEM (finite element) model schematic diagram of grid structure；

Fig. 3 b is that fan-shaped plan grid structure of encorbelmenting carries out, by step 2, the first order mode (ω that model analysis provides_n1= 7.4rad/s) result schematic diagram；

Fig. 3 c is that fan-shaped plan grid structure of encorbelmenting carries out, by step 2, the second_mode (ω that model analysis provides_n2= 7.5rad/s) result schematic diagram；

Fig. 3 d is that fan-shaped plan grid structure of encorbelmenting carries out, by step 2, the second_mode (ω that model analysis provides_n3= 9.9rad/s) result schematic diagram；

Fig. 4 a is the modal response root-mean-square result schematic diagram that step 3 calculates, and RMS modal disp is that modal response is equal Root value, modal order is rank number of mode, and frequency is natural frequency of structures, and bar diagram is modal response standard deviation, Circle is the frequency of structure each order mode state；

Fig. 4 b is the modal response covariance matrix result schematic diagram that step 4 calculates；

Fig. 5 is algorithm and the traditional algorithm comparison in precision and efficiency of the present invention.

Detailed description of the invention

Detailed description of the invention one: combine Fig. 1 and present embodiment is described, present embodiment a kind of based on auto-correlation function Wind induced structural vibration responds efficient frequency domain estimation method detailed process:

Step one: (such as Long Span Roof Structures is (such as two supports steel structural roof for the structure recorded based on wind tunnel test Span is more than 36m, the cantilever span structure more than 10m, such as most common stadium, departure hall, conference and exhibition center's exhibition The roof structure in shop etc.), high-rise building (such as highly high-level structure more than 100m)) surface pulsating wind pressure time course data p_i T (), calculates the meansigma methods of the blast time course data of each wind load load(ing) pointThe standard deviation sigma of blast time course data_pi, blast time The auto-correlation function R of number of passes evidence_pi(τ) and the coherent function Coh of blast time course data_pij(ω)；

Wherein, i, j are any two point in wind load load(ing) point sum Q, and 1≤i≤Q, 1≤j≤Q, Q are that wind load loads Point sum, for positive integer；

Step 2: use exponential function matching auto-correlation function, obtains the dimensionless blast spectrum of the wind load load(ing) point of i point The crest frequency ω of curve_mi；

From the definition of auto-power spectrum

$\frac{S_{pi}\left(\mathrm{\ω}\right)}{{\mathrm{\σ}}_{pi}^2}=\frac{1}{\mathrm{\π}}{\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}{R}_{pi}\left(\mathrm{\τ}\right)\exp (-i\mathrm{\ω}\mathrm{\τ})d\mathrm{\τ}=\frac{2}{\mathrm{\π}}\·\frac{{\mathrm{\ω}}_{mi}}{{\mathrm{\ω}}_{mi}^2+{\mathrm{\ω}}^2}$

In formula, S_pi(ω) it is the blast Power spectral density of each load(ing) point；ω is frequency；τ is the time difference；

Then dimensionless blast stave is shown asCan draw,

In formula, S_i(ω) it is dimensionless blast spectrum；

Step 3: use exponential function matching coherent function, obtain the relevant index k of this structure_c, calculate i, j's 2 Wind load coherence factor；

Step 4: extract the mass matrix [M] of structure, stiffness matrix [K], damping matrix [C], wind load load(ing) point and knot The transition matrix [R] of structure degree of freedom, the face that affects matrix [I] of response；

Structure is carried out model analysis, solves characteristic equationObtain each rank natural frequency of vibration of structure ω_nk, k=1,2 ..., N, k are the rank number of mode corresponding to the natural frequency of vibration, and N is the Degree of Structure Freedom number, for positive integer；

Take structural eigenvector matrixWhereinFor kth, r first order mode, k, R=1,2 ... W；W is to calculate selected rank number of mode, and W is positive integer, determines according to engineering experience, generally, and 10≤W≤N； K, r are natural frequency of vibration ω_nk、ω_nrCorresponding rank number of mode；MakeComputation structure k rank shake The broad sense damping ratio of type

In formula, T is transposed matrix；

Step 5: calculate Analytical Integration according to step one, step 2, step 3 and step 4；Obtain [Σ_kr]；

Step 6: calculate modal response covarianceForm modal response covariance square Battle array；

Step 7: calculate dynamic respond covariance matrix [Σ_x]=[Ψ]^T[Σ_y] [Ψ], and then it is the loudest to try to achieve it The covariance matrix answered.

Detailed description of the invention two: present embodiment is unlike detailed description of the invention one: based on wind in described step one The body structure surface pulsating wind pressure time course data p recorded is tested in hole_iT (), calculates the blast time course data of each wind load load(ing) point Meansigma methodsThe standard deviation sigma of blast time course data_pi, the auto-correlation function R of blast time course data_pi(τ) with blast time course data Coherent function Coh_pij(ω)；Detailed process is:

Solved by the mean function in MATLAB；σ_piSolved by the std function in MATLAB；R_pi(τ) by MATLAB Xcorr function solve；Coh_pij(ω) solved by the mscohere function in MATLAB.

Other step and parameter are identical with detailed description of the invention one.

Detailed description of the invention three: present embodiment is unlike detailed description of the invention one or two: adopt in described step 2 With exponential function matching auto-correlation function, obtain the crest frequency of the dimensionless blast spectral curve of the wind load load(ing) point of i point ω_mi；Detailed process is:

Use exponential function matching auto-correlation function R_pi(τ)=exp (-ω_mi| τ |), obtain the wind load load(ing) point of i point The crest frequency ω of dimensionless blast spectral curve_mi。

Other step and parameter are identical with detailed description of the invention one or two.

Detailed description of the invention four: present embodiment is unlike one of detailed description of the invention one to three: described step 3 Middle employing exponential function matching coherent function, obtains the relevant index k of this structure_c, calculate the wind load phase responsibility of i, j 2 Number；Detailed process is:

Use exponential function matching coherent functionObtain the relevant index k of this structure_c, Calculate the wind load coherence factor of i, j 2

In formula, D_ijRepresenting two wind load load(ing) point i, j spacings, U represents with reference to wind speed, c_ijWind load for i, j 2 Coherence factor, for nonnegative real number.

Other step and parameter are identical with one of detailed description of the invention one to three.

Detailed description of the invention five: present embodiment is unlike one of detailed description of the invention one to four: described step 5 Middle according to step one, step 2, step 3 and step 4 calculating Analytical Integration；Obtain [Σ_kr]；Detailed process is:

Analytical Integration is calculated according to step one, step 2, step 3 and step 4；

$\begin{array}{c}{\mathrm{\σ}}_{ijkr}=\frac{2}{\mathrm{\π}}{\mathrm{\σ}}_{pi}{\mathrm{\σ}}_{pj}\sqrt{{\mathrm{\ω}}_{mi}{\mathrm{\ω}}_{mj}}{\∫}_0^{\mathrm{\∞}}\frac{\[({\mathrm{\ω}}^2-{\mathrm{\ω}}_{nk}^2)({\mathrm{\ω}}^2-{\mathrm{\ω}}_{nr}^2)-4{\mathrm{\ξ}}_{nk}{\mathrm{\ξ}}_{nr}{\mathrm{\ω}}_{nk}{\mathrm{\ω}}_{nr}{\mathrm{\ω}}^2\]\·({\mathrm{\ω}}^2+{\mathrm{\ω}}_{mi}{\mathrm{\ω}}_{mj})}{\[{({\mathrm{\ω}}^2-{\mathrm{\ω}}_{nk}^2)}^2+4{\mathrm{\ξ}}_{nk}^2{\mathrm{\ω}}_{nk}^2{\mathrm{\ω}}^2\]\·\[{({\mathrm{\ω}}^2-{\mathrm{\ω}}_{nr}^2)}^2+4{\mathrm{\ξ}}_{nr}^2{\mathrm{\ω}}_{nr}^2{\mathrm{\ω}}^2\]\·({\mathrm{\ω}}_{mi}^2+{\mathrm{\ω}}^2)\·({\mathrm{\ω}}_{mj}^2+{\mathrm{\ω}}^2)\·\[1+{\left(c_{ij}\mathrm{\ω}\right)}^2\]}d\mathrm{\ω}\\ =\frac{2}{\mathrm{\π}}{\mathrm{\σ}}_{pi}{\mathrm{\σ}}_{pj}\sqrt{{\mathrm{\ω}}_{mi}{\mathrm{\ω}}_{mj}}{\∫}_0^{\mathrm{\∞}}\frac{\mathrm{\Θ}\left(\mathrm{\ω}\right)}{\mathrm{\Λ}(-\sqrt{-1}\mathrm{\ω})\·\mathrm{\Λ}\left(\sqrt{-1}\mathrm{\ω}\right)}d\mathrm{\ω}={\mathrm{\σ}}_{pi}{\mathrm{\σ}}_{pj}\sqrt{{\mathrm{\ω}}_{mi}{\mathrm{\ω}}_{mj}}\·\frac{I_1}{I_0}\end{array}$

In formula,

σ_ijkrFor k, r rank coupled mode at the coupling wind vibration response root-mean-square of i, j point-to-point transmission；

σ_piFor the wind load standard deviation of i point, σ_pjWind load standard deviation for j point；

ω_miThe crest frequency of dimensionless blast spectral curve for the wind load load(ing) point of i point；ω_mjWind load for j point The crest frequency of the dimensionless blast spectral curve of load(ing) point；

ω_nkFor the structure k rank natural frequency of vibration；ω_nrFor the structure r rank natural frequency of vibration；

ξ_nkFor structure k rank damping ratio；ξ_nrFor for structure r rank damping ratio；

c_ijWind load coherence factor for i, j 2；

Θ (ω) is the molecule multinomial of integrand；θ_qFor the polynomial coefficient of molecule；ω^2qFor the 2q power of frequency, q =0,1,2,3；

Denominator complex polynomails for integrand；For complex frequency；λ_sCoefficient for denominator polynomials；For the s power of complex frequency, s=1～7；；

I₁Molecule determinant for integration；I₀Denominator determinant for integration；

Form W²Individual Q rank square formation, is expressed as

In formula, i, j=1,2 ... Q；K, r=1,2 ... W；W is to calculate selected rank number of mode, and W is positive integer, according to Engineering experience determines, generally, and 10≤W≤N；σ_1QkrFor σ_ijkrMiddle i=1, j=Q；σ_2QkrFor σ_ijkrMiddle i=2, j=Q；With this type of Push away, σ_QQkrFor σ_ijkrMiddle a=Q, b=Q；[Σ_kr] it is the intermediate variable matrix of k, r rank coupled mode；

First calculate the situation of k=r, obtain [Σ_kr], calculateWherein, k=1, 2,…,W；WillIt is ordered as from big to smallWherein, k_m=1,2 ..., W, m=1,2 ..., W is the sequence number of sequence；

In formula,Variance for kth rank modal response；[Σ_kk] be the intermediate variable matrix of kth order mode state, i.e. [Σ_kr] The situation of middle k=r；

For front Z item so thatCalculate [the Σ of k < r_kr], according to symmetry principle, [Σ_rk]= [Σ_kr], obtain k > [the Σ of r_kr]；

Described, 1≤Z≤W；1≤n≤W；

For kth_mThe variance of rank modal response, i.e.Take k=k_mResult.

Other step and parameter are identical with one of detailed description of the invention one to four.

Detailed description of the invention six: present embodiment is unlike one of detailed description of the invention one to five: described molecule is many The coefficient θ of item formula_qCoefficient lambda with denominator polynomials_sParticularly as follows:

Molecule multinomial Θ (ω) about frequencies omega is merged multinomial for frequencies omega, obtains ω^2qCoefficient θ_q, q =0,1,2,3；

To about complex frequencyDenominator complex polynomailsFor complex frequencyMerge multinomial, obtainCoefficient λ_s, s=1～7.

Other step and parameter are identical with one of detailed description of the invention one to five.

Detailed description of the invention seven: present embodiment is unlike one of detailed description of the invention one to six: described I₁And I₀Tool Body formula is:

Work as c_ijWhen ≠ 0,

$I_1=\left|\begin{array}{ccccccc}0& 0& 0& {\mathrm{\&theta;}}_3& {\mathrm{\&theta;}}_2& {\mathrm{\&theta;}}_1& {\mathrm{\&theta;}}_0\\ -c_{ij}& {\mathrm{\&lambda;}}_5& -{\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0& 0& 0\\ 0& -{\mathrm{\&lambda;}}_6& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& {\mathrm{\&lambda;}}_0& 0& 0\\ 0& c_{ij}& -{\mathrm{\&lambda;}}_5& {\mathrm{\&lambda;}}_3& -{\mathrm{\&lambda;}}_1& 0& 0\\ 0& 0& {\mathrm{\&lambda;}}_6& -{\mathrm{\&lambda;}}_4& {\mathrm{\&lambda;}}_2& -{\mathrm{\&lambda;}}_0& 0\\ 0& 0& -c_{ij}& {\mathrm{\&lambda;}}_5& -{\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0\\ 0& 0& 0& -{\mathrm{\&lambda;}}_6& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& {\mathrm{\&lambda;}}_0\end{array}\right|,$

$I_0=c_{ij}\&CenterDot;\left|\begin{array}{ccccccc}{\mathrm{\&lambda;}}_6& -{\mathrm{\&lambda;}}_4& {\mathrm{\&lambda;}}_2& -{\mathrm{\&lambda;}}_0& 0& 0& 0\\ -c_{ij}& {\mathrm{\&lambda;}}_5& -{\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0& 0& 0\\ 0& -{\mathrm{\&lambda;}}_6& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& {\mathrm{\&lambda;}}_0& 0& 0\\ 0& c_{ij}& -{\mathrm{\&lambda;}}_5& {\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0& 0\\ 0& 0& {\mathrm{\&lambda;}}_6& -{\mathrm{\&lambda;}}_4& {\mathrm{\&lambda;}}_2& -{\mathrm{\&lambda;}}_0& 0\\ 0& 0& -c_{ij}& {\mathrm{\&lambda;}}_5& -{\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0\\ 0& 0& 0& -{\mathrm{\&lambda;}}_6& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& {\mathrm{\&lambda;}}_0\end{array}\right|;$

Work as c_ijWhen=0,

$I_1=\left|\begin{array}{cccccc}0& 0& {\mathrm{\&theta;}}_3& {\mathrm{\&theta;}}_2& {\mathrm{\&theta;}}_1& {\mathrm{\&theta;}}_0\\ -1& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& 0& 0& 0\\ 0& -{\mathrm{\&lambda;}}_5& {\mathrm{\&lambda;}}_3& -{\mathrm{\&lambda;}}_1& 0& 0\\ 0& 1& -{\mathrm{\&lambda;}}_4& {\mathrm{\&lambda;}}_2& -{\mathrm{\&lambda;}}_0& 0\\ 0& 0& {\mathrm{\&lambda;}}_5& -{\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0\\ 0& 0& -1& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& {\mathrm{\&lambda;}}_0\end{array}\right|$

$I_0=\left|\begin{array}{cccccc}{\mathrm{\&lambda;}}_5& -{\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0& 0& 0\\ -1& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& 0& 0& 0\\ 0& -{\mathrm{\&lambda;}}_5& {\mathrm{\&lambda;}}_3& -{\mathrm{\&lambda;}}_1& 0& 0\\ 0& 1& -{\mathrm{\&lambda;}}_4& {\mathrm{\&lambda;}}_2& -{\mathrm{\&lambda;}}_0& 0\\ 0& 0& {\mathrm{\&lambda;}}_5& -{\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0\\ 0& 0& -1& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& {\mathrm{\&lambda;}}_0\end{array}\right|.$

Other step and parameter are identical with one of detailed description of the invention one to six.

Detailed description of the invention eight: present embodiment is unlike one of detailed description of the invention one to seven: described step 6 Middle calculating modal response covarianceForm modal response covariance matrix；Detailed process For:

Calculate modal response covarianceForm modal response covariance matrix

Other step and parameter are identical with one of detailed description of the invention one to seven.

Detailed description of the invention nine: present embodiment is unlike one of detailed description of the invention one to eight: described step 7 Middle calculating dynamic respond covariance matrix [Σ_x]=[Ψ]^T[Σ_y] [Ψ], and then its covariance square arbitrarily responded can be tried to achieve Battle array；Detailed process is:

Calculate dynamic respond covariance matrix [Σ_x]=[Ψ]^T[Σ_y] [Ψ], and then its association side arbitrarily responded can be tried to achieve Difference matrix [Σ_s]=[I]^T[Σ_x][I]。

Other step and parameter are identical with one of detailed description of the invention one to eight.

Employing following example checking beneficial effects of the present invention:

Embodiment one:

The present embodiment efficient frequency domain estimation method of a kind of wind vibration response based on auto-correlation function is specifically according to following step Rapid preparation:

Take Practical Project fan-shaped plan rack example of encorbelmenting to illustrate and verify.

Step one～three: blast time-histories wind tunnel test recorded carries out statistical modeling analysis, as Fig. 2 a, Fig. 2 b, Fig. 2 c, Shown in, obtain average, pulsating wind pressure, the wind load frequency of each point and relevant index, and utilize the standard of blast spectral test model Really property.

Step 4: FEM (finite element) model is carried out quality, rigidity, the extraction of damping equal matrix, and carries out model analysis, result As shown in Fig. 3 a, Fig. 3 b, Fig. 3 c, Fig. 3 d.

Step 5: calculate integration, forms modal response standard deviation, as shown in fig. 4 a, it can be seen that the coupling feelings of front 2 × 2 Condition.

Step 6: calculating modal response covariance matrix, result is as shown in Figure 4 b.

Step 7: computation structure dynamic respond standard deviation, as shown in Figure 5.

For verifying the effectiveness of this invention, the most also result of calculation is contrasted with traditional method, found tradition wind Vibration analysis method computational efficiency is relatively low, and uses the traditional frequency domain computational methods not considering modal coupling, although efficiency is higher, but Calculate error relatively big (reaching 16.7%).The method that the present invention proposes is while ensureing computational accuracy, and computational efficiency is greatly promoted. Illustrate to the method achieve the efficient calculating of large and complex structure wind vibration response, there is practical value.

The present invention also can have other various embodiments, in the case of without departing substantially from present invention spirit and essence thereof, and this area Technical staff is when making various corresponding change and deformation according to the present invention, but these change accordingly and deformation all should belong to The protection domain of appended claims of the invention.

Claims (9)

1. a wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method, it is characterised in that: a kind of based on certainly The wind induced structural vibration of correlation function responds efficient frequency domain estimation method detailed process:

Step one: the body structure surface pulsating wind pressure time course data p recorded based on wind tunnel test_iT (), calculates each wind load load(ing) point The meansigma methods of blast time course dataThe standard deviation sigma of blast time course data_pi, the auto-correlation function R of blast time course data_pi (τ) and the coherent function Coh of blast time course data_pij(ω)；

Wherein, i, j are any two point in wind load load(ing) point sum Q, and 1≤i≤Q, 1≤j≤Q, Q are that wind load load(ing) point is total Number, for positive integer；

Step 2: use exponential function matching auto-correlation function, obtain the dimensionless blast spectral curve of the wind load load(ing) point of i point Crest frequency ω_mi；

Step 3: use exponential function matching coherent function, obtain the relevant index k of this structure_c, calculate the wind load of i, j 2 Coherence factor；

Step 4: extract the mass matrix [M] of structure, stiffness matrix [K], damping matrix [C], wind load load(ing) point with structure certainly By the transition matrix [R] spent, the face that affects matrix [I] of response；

Structure is carried out model analysis, solves characteristic equationObtain each rank natural frequency of vibration ω of structure_nk, k =1,2 ..., N, k are the rank number of mode corresponding to the natural frequency of vibration, and N is the Degree of Structure Freedom number, for positive integer；

Take structural eigenvector matrixWhereinFor kth, r first order mode, k, r= 1,2,…W；W is to calculate selected rank number of mode, and W is positive integer, 10≤W≤N；K, r are natural frequency of vibration ω_nk、ω_nrInstitute is right The rank number of mode answered；Make

The broad sense damping ratio of computation structure k first order mode

In formula, T is transposed matrix；

Step 5: calculate Analytical Integration according to step one, step 2, step 3 and step 4, obtain [Σ_kr]；

Step 6: calculate modal response covarianceForm modal response covariance matrix [Σ_y]；

Step 7: calculate dynamic respond covariance matrix [Σ_x]=[Ψ]^T[Σ_y] [Ψ], and then can try to achieve what it arbitrarily responded Covariance matrix [Σ_s]。

A kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method, and it is special Levy and be: the body structure surface pulsating wind pressure time course data p recorded based on wind tunnel test in described step one_iT (), calculates each wind lotus Carry the meansigma methods of the blast time course data of load(ing) pointThe standard deviation sigma of blast time course data_pi, the auto-correlation of blast time course data Function R_pi(τ) and the coherent function Coh of blast time course data_pij(ω)；Detailed process is:

Solved by the mean function in MATLAB；σ_piSolved by the std function in MATLAB；R_pi(τ) by MATLAB Xcorr function solves；Coh_pij(ω) solved by the mscohere function in MATLAB.

A kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method, and it is special Levy and be: described step 2 uses exponential function matching auto-correlation function, obtains the dimensionless wind of the wind load load(ing) point of i point The crest frequency ω of pressure spectral curve_mi；Detailed process is:

Use exponential function matching auto-correlation function R_pi(τ)=exp (-ω_mi| τ |), obtain wind load load(ing) point immeasurable of i point The crest frequency ω of guiding principle blast spectral curve_mi。

A kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method, and it is special Levy and be: described step 3 uses exponential function matching coherent function, obtains the relevant index k of this structure_c, calculate i, j two The wind load coherence factor of point；Detailed process is:

Use exponential function matching coherent functionObtain the relevant index k of this structure_c, calculate The wind load coherence factor of i, j 2

In formula, D_ijRepresenting two wind load load(ing) point i, j spacings, U represents with reference to wind speed, c_ijWind load for i, j 2 is concerned with Coefficient, for nonnegative real number.

A kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method, and it is special Levy and be: described step 5 calculates Analytical Integration according to step one, step 2, step 3 and step 4, obtains [Σ_kr]；Tool Body process is:

Analytical Integration is calculated according to step one, step 2, step 3 and step 4；

$\begin{array}{c}{\mathrm{\&sigma;}}_{ijkr}=\frac{2}{\mathrm{\&pi;}}{\mathrm{\&sigma;}}_{pi}{\mathrm{\&sigma;}}_{pj}\sqrt{{\mathrm{\&omega;}}_{mi}{\mathrm{\&omega;}}_{mj}}{\&Integral;}_0^{\mathrm{\&infin;}}\frac{\&lsqb;({\mathrm{\&omega;}}^2-{\mathrm{\&omega;}}_{nk}^2)({\mathrm{\&omega;}}^2-{\mathrm{\&omega;}}_{nr}^2)-4{\mathrm{\&xi;}}_{nk}{\mathrm{\&xi;}}_{nr}{\mathrm{\&omega;}}_{nk}{\mathrm{\&omega;}}_{nr}{\mathrm{\&omega;}}^2\&rsqb;\&CenterDot;({\mathrm{\&omega;}}^2+{\mathrm{\&omega;}}_{mi}{\mathrm{\&omega;}}_{mj})}{\&lsqb;{({\mathrm{\&omega;}}^2-{\mathrm{\&omega;}}_{nk}^2)}^2+4{\mathrm{\&xi;}}_{nk}^2{\mathrm{\&xi;}}_{nk}^2{\mathrm{\&omega;}}^2\&rsqb;\&CenterDot;\&lsqb;{({\mathrm{\&omega;}}^2-{\mathrm{\&omega;}}_{nr}^2)}^2+4{\mathrm{\&xi;}}_{nr}^2{\mathrm{\&omega;}}_{nr}^2{\mathrm{\&omega;}}^2\&rsqb;\&CenterDot;({\mathrm{\&omega;}}_{mi}^2+{\mathrm{\&omega;}}^2)\&CenterDot;({\mathrm{\&omega;}}_{mj}^2+{\mathrm{\&omega;}}^2)\&CenterDot;\&lsqb;1+{\left(c_{ij}\mathrm{\&omega;}\right)}^2\&rsqb;}d\mathrm{\&omega;}\\ =\frac{2}{\mathrm{\&pi;}}{\mathrm{\&sigma;}}_{pi}{\mathrm{\&sigma;}}_{pj}\sqrt{{\mathrm{\&omega;}}_{mi}{\mathrm{\&omega;}}_{mj}}{\&Integral;}_0^{\mathrm{\&infin;}}\frac{\mathrm{\&Theta;}\left(\mathrm{\&omega;}\right)}{\mathrm{\&Lambda;}(-\sqrt{-1}\mathrm{\&omega;})\&CenterDot;\mathrm{\&Lambda;}\left(\sqrt{-1}\mathrm{\&omega;}\right)}d\mathrm{\&omega;}={\mathrm{\&sigma;}}_{pi}{\mathrm{\&sigma;}}_{pj}\sqrt{{\mathrm{\&omega;}}_{mi}{\mathrm{\&omega;}}_{mj}}\&CenterDot;\frac{I_1}{I_0}\end{array}$

In formula,

σ_ijkrFor k, r rank coupled mode at the coupling wind vibration response root-mean-square of i, j point-to-point transmission；

σ_piFor the wind load standard deviation of i point, σ_pjWind load standard deviation for j point；

ω_miThe crest frequency of dimensionless blast spectral curve for the wind load load(ing) point of i point；ω_mjWind load load(ing) point for j point The crest frequency of dimensionless blast spectral curve；

ω_nkFor the structure k rank natural frequency of vibration；ω_nrFor the structure r rank natural frequency of vibration；

ξ_nkFor structure k rank damping ratio；ξ_nrFor structure r rank damping ratio；

c_ijWind load coherence factor for i, j 2；

Θ (ω) is the molecule multinomial of integrand；θ_qFor the polynomial coefficient of molecule；ω^2qFor the 2q power of frequency, q=0, 1,2,3；

Denominator complex polynomails for integrand；For complex frequency；λ_sCoefficient for denominator polynomials；For The s power of complex frequency, s=0-6；；

I₁Molecule determinant for integration；I₀Denominator determinant for integration；

Form W²Individual Q rank square formation, is expressed as

In formula, i, j=1,2 ... Q；K, r=1,2 ... W；W is to calculate selected rank number of mode, and W is positive integer, 10≤W≤ N；σ_1QkrFor σ_ijkrMiddle i=1, j=Q；σ_2QkrFor σ_ijkrMiddle i=2, j=Q；By that analogy, σ_QQkrFor σ_ijkrMiddle a=Q, b=Q； [Σ_kr] it is the intermediate variable matrix of k, r rank coupled mode；

Calculate the situation of k=r, obtain [Σ_kr], calculateWherein, k=1,2 ..., W；WillIt is ordered as from big to smallWherein, k_m=1,2 ..., W, m=1,2 ..., W is the sequence number of sequence；

In formula,Variance for kth rank modal response；[Σ_kk] be the intermediate variable matrix of kth order mode state, i.e. [Σ_krK=in] The situation of r；

For front Z item so thatCalculate [the Σ of k < r_kr]；According to symmetry principle, [Σ_rk]= [Σ_kr], obtain k > [the Σ of r_kr]；

Described, 1≤Z≤W；

For kth_mThe variance of rank modal response, i.e.Take k=k_mResult.

A kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method, and it is special Levy and be: described molecule polynomial coefficient θ_qCoefficient lambda with denominator polynomials_sParticularly as follows:

Molecule multinomial Θ (ω) about frequencies omega is merged multinomial for frequencies omega, obtains ω^2qCoefficient θ_q, q=0, 1,2,3；

To about complex frequencyDenominator complex polynomailsFor complex frequencyMerge multinomial, obtainCoefficient lambda_s, s= 0-6。

A kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method, and it is special Levy and be: described I₁And I₀Concrete formula is:

Work as c_ijWhen ≠ 0,

$I_1=\left|\begin{array}{ccccccc}0& 0& 0& {\mathrm{\&theta;}}_3& {\mathrm{\&theta;}}_2& {\mathrm{\&theta;}}_1& {\mathrm{\&theta;}}_0\\ -c_{ij}& {\mathrm{\&lambda;}}_5& -{\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0& 0& 0\\ 0& -{\mathrm{\&lambda;}}_6& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& {\mathrm{\&lambda;}}_0& 0& 0\\ 0& c_{ij}& -{\mathrm{\&lambda;}}_5& {\mathrm{\&lambda;}}_3& -{\mathrm{\&lambda;}}_1& 0& 0\\ 0& 0& {\mathrm{\&lambda;}}_6& -{\mathrm{\&lambda;}}_4& {\mathrm{\&lambda;}}_2& -{\mathrm{\&lambda;}}_0& 0\\ 0& 0& -c_{ij}& {\mathrm{\&lambda;}}_5& -{\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0\\ 0& 0& 0& -{\mathrm{\&lambda;}}_6& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& {\mathrm{\&lambda;}}_0\end{array}\right|,$

$I_0=c_{ij}\&CenterDot;\left|\begin{array}{ccccccc}{\mathrm{\&lambda;}}_6& -{\mathrm{\&lambda;}}_4& {\mathrm{\&lambda;}}_2& -{\mathrm{\&lambda;}}_0& 0& 0& 0\\ -c_{ij}& {\mathrm{\&lambda;}}_5& -{\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0& 0& 0\\ 0& -{\mathrm{\&lambda;}}_6& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& {\mathrm{\&lambda;}}_0& 0& 0\\ 0& c_{ij}& -{\mathrm{\&lambda;}}_5& {\mathrm{\&lambda;}}_3& -{\mathrm{\&lambda;}}_1& 0& 0\\ 0& 0& {\mathrm{\&lambda;}}_6& -{\mathrm{\&lambda;}}_4& {\mathrm{\&lambda;}}_2& -{\mathrm{\&lambda;}}_0& 0\\ 0& 0& -c_{ij}& {\mathrm{\&lambda;}}_5& -{\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0\\ 0& 0& 0& -{\mathrm{\&lambda;}}_6& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& {\mathrm{\&lambda;}}_0\end{array}\right|;$

Work as c_ijWhen=0,

$I_1=\left|\begin{array}{cccccc}0& 0& {\mathrm{\&theta;}}_3& {\mathrm{\&theta;}}_2& {\mathrm{\&theta;}}_1& {\mathrm{\&theta;}}_0\\ -1& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& 0& 0& 0\\ 0& -{\mathrm{\&lambda;}}_5& {\mathrm{\&lambda;}}_3& -{\mathrm{\&lambda;}}_1& 0& 0\\ 0& 1& -{\mathrm{\&lambda;}}_4& {\mathrm{\&lambda;}}_2& -{\mathrm{\&lambda;}}_0& 0\\ 0& 0& {\mathrm{\&lambda;}}_5& -{\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0\\ 0& 0& -1& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& {\mathrm{\&lambda;}}_0\end{array}\right|$

$I_0=\left|\begin{array}{cccccc}{\mathrm{\&lambda;}}_5& -{\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0& 0& 0\\ -1& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& 0& 0& 0\\ 0& -{\mathrm{\&lambda;}}_5& {\mathrm{\&lambda;}}_3& -{\mathrm{\&lambda;}}_1& 0& 0\\ 0& 1& -{\mathrm{\&lambda;}}_4& {\mathrm{\&lambda;}}_2& -{\mathrm{\&lambda;}}_0& 0\\ 0& 0& {\mathrm{\&lambda;}}_5& -{\mathrm{\&lambda;}}_3& {\mathrm{\&lambda;}}_1& 0\\ 0& 0& -1& {\mathrm{\&lambda;}}_4& -{\mathrm{\&lambda;}}_2& {\mathrm{\&lambda;}}_0\end{array}\right|.$

A kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method, and it is special Levy and be: described step 6 calculates modal response covarianceForm modal response association side Difference matrix [Σ_y]；Detailed process is:

Calculate modal response covarianceForm modal response covariance matrix

A kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method, and it is special Levy and be: described step 7 calculates dynamic respond covariance matrix [Σ_x]=[Ψ]^T[Σ_y] [Ψ], and then can be tried to achieve it Covariance matrix [the Σ of meaning response_s]；Detailed process is:

Calculate dynamic respond covariance matrix [Σ_x]=[Ψ]^T[Σ_y] [Ψ], and then its covariance square arbitrarily responded can be tried to achieve Battle array [Σ_s]=[I]^T[Σ_x][I]。