CN106295159B - A kind of efficient frequency domain estimation method of wind induced structural vibration response based on auto-correlation function - Google Patents
A kind of efficient frequency domain estimation method of wind induced structural vibration response based on auto-correlation function Download PDFInfo
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Abstract
A kind of efficient frequency domain estimation method of wind induced structural vibration response based on auto-correlation function, the present invention relates to the wind induced structural vibrations based on auto-correlation function to respond efficient frequency domain estimation method.The purpose of the present invention is to solve the disadvantages that existing wind-induced response method computational efficiency is low and calculating error is big.Detailed process is：One：Based on pi(t) it calculatesσpi、Rpi(τ) and Cohpij(ω)；Two：Obtain ωmi；Three：Obtain the relevant index k of the structurec, calculate the wind load coherence factor of 2 points of i, j；Four：Extract [M], [K], [C], [R], [I]；Five：[Σ is obtained according to one, two, three and four calculating Analytical Integrationskr]；Six：Form modal response covariance matrix；Seven：Dynamic respond covariance matrix is calculated, and then its covariance matrix arbitrarily responded can be acquired.The present invention responds field for structure wind battle array.
Description
Technical field
The present invention relates to the efficient frequency domain estimation methods 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 link.The vibration characteristics of large and complex structure is difficult to be indicated with the simple vibration shape, to consider the contribution of high order mode
And the coupling between the vibration shape, therefore, the structural dynamic response under wind loads effect is extremely complex.Traditional time domain wind shake
Response analysis needs to input a large amount of wind load time-history information, and using Newmark- β integration methods, calculation amount is larger, and when calculating is required
Memory space is larger, causes computational efficiency low.And traditional frequency domain wind-induced response, input bulk information is also needed, packet
The covariance matrix for including wind load, using numerical integrating, calculating is time-consuming longer, causes computational efficiency low.In addition, traditional frequency domain
Computational methods are difficult to consider modes coupling effect, and it is big to calculate error.
Invention content
The purpose of the present invention is to solve existing wind-induced response method computational efficiency is low and to calculate error big
Disadvantage, 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 and is：
Step 1：The body structure surface pulsating wind pressure time course data p measured based on wind tunnel testi(t), each wind load is calculated to add
The average value of the wind pressure time course data of loading pointThe standard deviation sigma of wind pressure time course datapi, wind pressure time course data auto-correlation function
RpiThe coherent function Coh of (τ) and wind pressure time course datapij(ω)；
Wherein, i, j are any two point in wind load load(ing) point sum Q, and 1≤i≤Q, 1≤j≤Q, Q load for wind load
Point sum is positive integer；
Step 2：Auto-correlation function is fitted using exponential function, obtains the dimensionless wind pressure spectrum of the wind load load(ing) point of i points
The crest frequency ω of curvemi；
From the definition of auto-power spectrum
In formula, Spi(ω) is the wind pressure Power spectral density of each load(ing) point；ω is frequency；τ is the time difference；
Then dimensionless wind pressure stave is shown asIt can obtain,
In formula, Si(ω) composes for dimensionless wind pressure；
Step 3：Coherent function is fitted using exponential function, obtains the relevant index k of the structurec, calculate 2 points of i, j's
Wind load coherence factor；
Step 4：Extract mass matrix [M], stiffness matrix [K], damping matrix [C], wind load load(ing) point and the knot of structure
The transition matrix [R] of structure degree of freedom, the influence face matrix [I] of response；
Model analysis is carried out to structure, solves characteristic equationObtain each rank natural frequency of vibration of structure
ωnk, k=1,2 ..., N, k is the rank number of mode corresponding to the natural frequency of vibration, and N is the Degree of Structure Freedom number, is positive integer；
Take structural eigenvector matrixWhereinFor kth, r first order modes, k,
R=1,2 ... W；W is to calculate selected rank number of mode, and W is positive integer, 10≤W≤N；K, r is natural frequency of vibration ωnk、ωnrInstitute
Corresponding rank number of mode；So thatCalculate the broad sense damping ratio of structure k first order modes
In formula, T is transposed matrix；
Step 5：According to Step 1: Step 2: step 3 and step 4 calculate Analytical Integration；Obtain [Σkr]；
Step 6：Calculate modal response covarianceForm modal response covariance square
Battle array；
Step 7：Calculate dynamic respond covariance matrix [Σx]=[Ψ]T[Σy] [Ψ], and then its arbitrary sound can be acquired
The covariance matrix answered.
Beneficial effects of the present invention are：
This method designs the efficient frequency domain estimation method of large and complex structure wind vibration response based on auto-correlation function, solves to pass
Wind-induced response method of the uniting disadvantage that computational efficiency is low and computational accuracy is poor in large and complex structure analysis.
The wind loads processing that this method first obtains wind tunnel test is the statistic and frequency spectrum parameter for calculating
Information enormously simplifies the input of wind load.Then, in conjunction with modal analysis method, to each vibration shape, to wind in each degree of freedom
Load frequency spectrum carries out simplified Analytical Integration, obtains modal response covariance.Finally, by obtaining wind to the combination of coupled modes
The covariance of vibration response.This method substantially increases computational efficiency while ensureing modal coupling computational accuracy.
The present invention greatly improves the efficiency of large and complex structure wind shake calculating, drop under the premise of ensureing computational accuracy
The low occupied memory space of calculating.The Computing Principle for taking full advantage of Analytical Integration converts numerical integration to parsing product
The algebraic operation divided, avoids the error that numerical integration is brought, while also substantially increasing computational efficiency.In implementation process,
Step is clear, and operability is stronger, is found by instance analysis, and the present invention is significantly increased compared with traditional algorithm efficiency, more practical.
Wind-induced response is carried out to certain fan-shaped plan Long-span Cantilever grid structure using the invention, compared with traditional analysis,
Worst error is no more than 1%, is much better than traditional simplification frequency domain algorithm error (16.7%), and efficiency improves 200 compared with traditional algorithm
Times.
Description of the drawings
Fig. 1 is the flow chart of the present invention；
Fig. 2 a areTake -6.42, σpiTake the knot of 2.48 body structure surface point wind load time-history by step 1 statistical modeling
Fruit schematic diagram, ordinate areFor normalized wind load time-history, abscissa t is time (second), pi(t)
For wind load time-history, pi(t) it is Wind Loads Acting, σpiFor wind load standard deviation；
Fig. 2 b are Rpi(τ)=exp (- ωmi| τ |), ωmiThe body structure surface point wind load time-history of=2.40rad/s is by step
The result schematic diagram of a rapid statistical modeling, ordinate Rpi(τ), Rpi(τ) is the auto-correlation function of wind load, ωmiFor wind load
Frequency, abscissa τ, τ are the time difference (second)；
Fig. 2 c areBody structure surface point wind load time-history press step 1 statistical modeling knot
Fruit schematic diagram, ordinate areFor dimensionless wind load power spectrum, abscissa ω, ω are frequency (arc
Degrees second), Spi(ω) is wind load power spectral density function；
Fig. 3 a are the FEM model schematic diagram that fan-shaped plan overhangs grid structure；
Fig. 3 b are the first order mode (ω that fan-shaped plan overhangs that grid structure is provided by step 2 progress model analysisn1=
7.4rad/s) result schematic diagram；
Fig. 3 c are the second_mode (ω that fan-shaped plan overhangs that grid structure is provided by step 2 progress model analysisn2=
7.5rad/s) result schematic diagram；
Fig. 3 d are the second_mode (ω that fan-shaped plan overhangs that grid structure is provided by step 2 progress model analysisn3=
9.9rad/s) result schematic diagram；
Fig. 4 a are the modal response root mean square result schematic diagram that step 3 calculates, and RMS modal disp are that modal response is equal
Root value, modal order are rank number of mode, and frequency is natural frequency of structures, and bar chart is modal response standard deviation,
Circle is the frequency of each rank mode of structure；
Fig. 4 b are the modal response covariance matrix result schematic diagram that step 4 calculates；
Fig. 5 is algorithm and comparison of the traditional algorithm in precision and efficiency of the present invention.
Specific implementation mode
Specific implementation mode one：Embodiment is described with reference to Fig. 1, present embodiment it is a kind of based on auto-correlation function
Wind induced structural vibration responds efficient frequency domain estimation method detailed process：
Step 1：Structure (such as Long Span Roof Structures (such as two supports steel structural roof measured based on wind tunnel test
Span is more than 36m, and cantilever span is more than the structure of 10m, such as most common stadium, departure hall, conference and exhibition center's exhibition
The roof structure in shop etc.), high-rise building (as height be more than 100m high-level structure)) surface pulsating wind pressure time course data pi
(t), the average value of the wind pressure time course data of each wind load load(ing) point is calculatedThe standard deviation sigma of wind pressure time course datapi, wind pressure when
The auto-correlation function R of number of passes evidencepiThe coherent function Coh of (τ) and wind pressure time course datapij(ω)；
Wherein, i, j are any two point in wind load load(ing) point sum Q, and 1≤i≤Q, 1≤j≤Q, Q load for wind load
Point sum is positive integer；
Step 2：Auto-correlation function is fitted using exponential function, obtains the dimensionless wind pressure spectrum of the wind load load(ing) point of i points
The crest frequency ω of curvemi；
From the definition of auto-power spectrum
In formula, Spi(ω) is the wind pressure Power spectral density of each load(ing) point；ω is frequency；τ is the time difference；
Then dimensionless wind pressure stave is shown asIt can obtain,
In formula, Si(ω) composes for dimensionless wind pressure；
Step 3：Coherent function is fitted using exponential function, obtains the relevant index k of the structurec, calculate 2 points of i, j's
Wind load coherence factor；
Step 4：Extract mass matrix [M], stiffness matrix [K], damping matrix [C], wind load load(ing) point and the knot of structure
The transition matrix [R] of structure degree of freedom, the influence face matrix [I] of response；
Model analysis is carried out to structure, solves characteristic equationObtain each rank natural frequency of vibration of structure
ωnk, k=1,2 ..., N, k is the rank number of mode corresponding to the natural frequency of vibration, and N is the Degree of Structure Freedom number, is positive integer；
Take structural eigenvector matrixWhereinFor kth, r first order modes,
K, r=1,2 ... W；W is to calculate selected rank number of mode, and W is positive integer, is determined according to engineering experience, in general, 10≤W≤
N；K, r is natural frequency of vibration ωnk、ωnrCorresponding rank number of mode；So thatCalculate structure k ranks
The broad sense damping ratio of the vibration shape
In formula, T is transposed matrix；
Step 5：According to Step 1: Step 2: step 3 and step 4 calculate Analytical Integration；Obtain [Σkr]；
Step 6：Calculate modal response covarianceForm modal response covariance square
Battle array；
Step 7：Calculate dynamic respond covariance matrix [Σx]=[Ψ]T[Σy] [Ψ], and then its arbitrary sound can be acquired
The covariance matrix answered.
Specific implementation mode two：The present embodiment is different from the first embodiment in that：Wind is based in the step 1
Test the body structure surface pulsating wind pressure time course data p measured in holei(t), the wind pressure time course data of each wind load load(ing) point is calculated
Average valueThe standard deviation sigma of wind pressure time course datapi, wind pressure time course data auto-correlation function Rpi(τ) and wind pressure time course data
Coherent function Cohpij(ω)；Detailed process is：
It is solved by the mean functions in MATLAB；σpiIt is solved by the std functions in MATLAB；Rpi(τ) is by MATLAB
Xcorr functions solve；Cohpij(ω) is solved by the mscohere functions in MATLAB.
Other steps and parameter are same as the specific embodiment one.
Specific implementation mode three：The present embodiment is different from the first and the second embodiment in that：It is adopted in the step 2
It is fitted auto-correlation function with exponential function, obtains the crest frequency of the dimensionless wind pressure spectral curve of the wind load load(ing) point of i points
ωmi；Detailed process is：
Using exponential function fitting auto-correlation function Rpi(τ)=exp (- ωmi| τ |), obtain the wind load load(ing) point of i points
Dimensionless wind pressure spectral curve crest frequency ωmi。
Other steps and parameter are the same as one or two specific embodiments.
Specific implementation mode four：Unlike one of present embodiment and specific implementation mode one to three：The step 3
It is middle that coherent function is fitted using exponential function, obtain the relevant index k of the structurec, calculate the wind load phase responsibility of 2 points of i, j
Number；Detailed process is：
Coherent function is fitted using exponential functionObtain the relevant index k of the structurec,
Calculate the wind load coherence factor of 2 points of i, j
In formula, DijIndicate that distance between two wind load load(ing) point i, j, U indicate to refer to wind speed, cijFor 2 points of wind load of i, j
Coherence factor is nonnegative real number.
Other steps and parameter are identical as one of specific implementation mode one to three.
Specific implementation mode five：Unlike one of present embodiment and specific implementation mode one to four：The step 5
Middle basis is Step 1: Step 2: step 3 and step 4 calculate Analytical Integration；Obtain [Σkr]；Detailed process is：
According to Step 1: Step 2: step 3 and step 4 calculate Analytical Integration；
In formula,
σijkrFor k, r rank coupled mode i, j point-to-point transmission coupling wind vibration response root mean square；
σpiFor the wind load standard deviation of i points, σpjFor the wind load standard deviation of j points；
ωmiFor the crest frequency of the dimensionless wind pressure spectral curve of the wind load load(ing) point of i points；ωmjFor the wind load of j points
The crest frequency of the dimensionless wind pressure spectral curve of load(ing) point；
ωnkFor the structure k rank natural frequencies of vibration；ωnrFor the structure r rank natural frequencies of vibration；
ξnkFor structure k rank damping ratios；ξnrFor for structure r rank damping ratios；
cijFor 2 points of wind load coherence factor of i, j；
Θ (ω) is the molecule multinomial of integrand；θqFor the polynomial coefficient of molecule；ω2qFor the 2q power of frequency, q
=0,1,2,3；
For the denominator complex polynomails of integrand；For complex frequency；λsFor the coefficient of denominator polynomials；For the s power of complex frequency, s=1~7；；
I1For the molecule determinant of integral；I0For the denominator determinant of integral；
Form W2A Q ranks 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, in general, 10≤W≤N；σ1QkrFor σijkrMiddle i=1, j=Q；σ2QkrFor σijkrMiddle i=2, j=Q；With such
It pushes away, σQQkrFor σijkrMiddle a=Q, b=Q；[Σkr] be k, r rank coupled mode intermediate variable matrix；
The case where first calculating k=r, obtains [Σkr], it calculatesWherein, k=1,
2,…,W；It willIt is ordered as from big to smallWherein, km=1,2 ..., W, m=1,2 ..., W are the serial number of sequence；
In formula,For the variance of kth rank modal response；[Σkk] be kth rank mode intermediate variable matrix, i.e. [Σkr]
The case where middle k=r；
For Z first so thatCalculate k<[the Σ of rkr], according to symmetry principle, [Σrk]=
[Σkr], obtain k>[the Σ of rkr]；
It is described, 1≤Z≤W；1≤n≤W；
For kthmThe variance of rank modal response, i.e.,Take k=kmResult.
Other steps and parameter are identical as one of specific implementation mode one to four.
Specific implementation mode six：Unlike one of present embodiment and specific implementation mode one to five：The molecule is more
The coefficient θ of item formulaqWith the coefficient lambda of denominator polynomialssSpecially：
Multinomial is merged for frequencies omega to the molecule multinomial Θ (ω) about frequencies omega, obtains ω2qCoefficient θq, q
=0,1,2,3；
To about complex frequencyDenominator complex polynomailsFor complex frequencyMerge multinomial, obtainsCoefficient
λs, s=1~7.
Other steps and parameter are identical as one of specific implementation mode one to five.
Specific implementation mode seven：Unlike one of present embodiment and specific implementation mode one to six：The I1And I0Tool
Body formula is：
Work as cijWhen ≠ 0,
Work as cijWhen=0,
Other steps and parameter are identical as one of specific implementation mode one to six.
Specific implementation mode eight：Unlike one of present embodiment and specific implementation mode one to seven：The step 6
Middle calculating modal response covarianceForm modal response covariance matrix；Detailed process
For：
Calculate modal response covarianceForm modal response covariance matrix
Other steps and parameter are identical as one of specific implementation mode one to seven.
Specific implementation mode nine：Unlike one of present embodiment and specific implementation mode one to eight：The step 7
Middle calculating dynamic respond covariance matrix [Σx]=[Ψ]T[Σy] [Ψ], and then its covariance square arbitrarily responded can be acquired
Battle array；Detailed process is：
Calculate dynamic respond covariance matrix [Σx]=[Ψ]T[Σy] [Ψ], and then its association side arbitrarily responded can be acquired
Poor matrix [Σs]=[I]T[Σx][I]。
Other steps and parameter are identical as one of specific implementation mode one to eight.
Beneficial effects of the present invention are verified using following embodiment：
Embodiment one：
A kind of efficient frequency domain estimation method of wind vibration response based on auto-correlation function of the present embodiment is specifically according to following step
Suddenly it prepares：
Practical Project fan-shaped plan overhanging rack example is taken to illustrate and verify.
Step 1~tri-：The wind pressure time-histories that wind tunnel test is measured carries out statistical modeling analysis, as Fig. 2 a, Fig. 2 b, Fig. 2 c,
It is shown, the average of each point, pulsating wind pressure, wind load frequency and relevant index are obtained, and utilize the standard of wind pressure spectral test model
True property.
Step 4：Quality is carried out to finite element model, rigidity, damps the extraction of equal matrix, and carries out model analysis, as a result
As shown in Fig. 3 a, Fig. 3 b, Fig. 3 c, Fig. 3 d.
Step 5：Integral is calculated, forms modal response standard deviation, as shown in fig. 4 a, it can be seen that preceding 2 × 2 coupling feelings
Condition.
Step 6：Modal response covariance matrix is calculated, as a result as shown in Figure 4 b.
Step 7：It is poor to calculate displacement structure response criteria, as shown in Figure 5.
To verify the validity of the invention, also result of calculation and conventional method are compared herein, find traditional wind
Vibration analysis method computational efficiency is relatively low, and uses the traditional frequency domain computational methods for not considering modal coupling, although efficiency is higher,
It is larger (up to 16.7%) to calculate error.While ensureing computational accuracy, computational efficiency greatly promotes method proposed by the present invention.
Illustrate the efficient calculating the method achieve large and complex structure wind vibration response, there is practical value.
The present invention can also have other various embodiments, without deviating from the spirit and substance of the present invention, this field
Technical staff makes various corresponding change and deformations in accordance with the present invention, but these corresponding change and deformations should all belong to
The protection domain of appended claims of the invention.
Claims (9)
1. a kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method, it is characterised in that：One kind is based on certainly
The wind induced structural vibration of correlation function responds efficient frequency domain estimation method detailed process：
Step 1：The body structure surface pulsating wind pressure time course data p measured based on wind tunnel testi(t), each wind load load(ing) point is calculated
Wind pressure time course data average valueThe standard deviation sigma of wind pressure time course datapi, wind pressure time course data auto-correlation function Rpi
The coherent function Coh of (τ) and wind pressure time course datapij(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 is positive integer；
Step 2：Auto-correlation function is fitted using exponential function, obtains the dimensionless wind pressure spectral curve of the wind load load(ing) point of i points
Crest frequency ωmi；
Step 3：Coherent function is fitted using exponential function, obtains the relevant index k of the structurec, calculate the wind load of 2 points of i, j
Coherence factor；
Step 4：Extract mass matrix [M], stiffness matrix [K], damping matrix [C], wind load load(ing) point and the structure of structure certainly
By the transition matrix [R] spent, the influence face matrix [I] of response；
Model analysis is carried out to structure, solves characteristic equationObtain each rank natural frequency of vibration ω of structurenk, 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, is positive integer；
Take structural eigenvector matrixWhereinFor kth, r first order modes, k, r=
1,2,…W；W is to calculate selected rank number of mode, and W is positive integer, 10≤W≤N；K, r is natural frequency of vibration ωnk、ωnrInstitute is right
The rank number of mode answered；So that
Calculate the broad sense damping ratio of structure k first order modes
In formula, T is transposed matrix；
Step 5：According to Step 1: Step 2: step 3 and step 4 calculating Analytical Integration, obtain [Σkr]；
Step 6：Calculate modal response covarianceForm modal response covariance matrix
[Σy]；
Step 7：Calculate dynamic respond covariance matrix [Σx]=[Ψ]T[Σy] [Ψ], and then can acquire what it was arbitrarily responded
Covariance matrix [Σs]。
2. a kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method according to claim 1, special
Sign is：The body structure surface pulsating wind pressure time course data p measured based on wind tunnel test in the step 1i(t), each wind lotus is calculated
Carry the average value of the wind pressure time course data of load(ing) pointThe standard deviation sigma of wind pressure time course datapi, wind pressure time course data auto-correlation
Function RpiThe coherent function Coh of (τ) and wind pressure time course datapij(ω)；Detailed process is：
It is solved by the mean functions in MATLAB；σpiIt is solved by the std functions in MATLAB；Rpi(τ) is by MATLAB
Xcorr functions solve；Cohpij(ω) is solved by the mscohere functions in MATLAB.
3. a kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method according to claim 2, special
Sign is：Auto-correlation function is fitted using exponential function in the step 2, obtains the dimensionless wind of the wind load load(ing) point of i points
Press the crest frequency ω of spectral curvemi；Detailed process is：
Using exponential function fitting auto-correlation function Rpi(τ)=exp (- ωmi| τ |), obtain the immeasurable of the wind load load(ing) point of i points
The crest frequency ω of guiding principle wind pressure spectral curvemi。
4. a kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method according to claim 3, special
Sign is：Coherent function is fitted using exponential function in the step 3, obtains the relevant index k of the structurec, calculate i, j two
The wind load coherence factor of point；Detailed process is：
Coherent function is fitted using exponential functionObtain the relevant index k of the structurec, calculate
I, the wind load coherence factor of 2 points of j
In formula, DijIndicate that distance between two wind load load(ing) point i, j, U indicate to refer to wind speed, cijIt is relevant for 2 points of wind load of i, j
Coefficient is nonnegative real number.
5. a kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method according to claim 4, special
Sign is：According to Step 1: Step 2: step 3 and step 4 calculating Analytical Integration, obtain [Σ in the step 5kr]；Tool
Body process is：
According to Step 1: Step 2: step 3 and step 4 calculate Analytical Integration；
In formula,
σijkrFor k, r rank coupled mode i, j point-to-point transmission coupling wind vibration response root mean square；
σpiFor the wind load standard deviation of i points, σpjFor the wind load standard deviation of j points；
ωmiFor the crest frequency of the dimensionless wind pressure spectral curve of the wind load load(ing) point of i points；ωmjFor the wind load load(ing) point of j points
Dimensionless wind pressure spectral curve crest frequency；
ωnkFor the structure k rank natural frequencies of vibration；ωnrFor the structure r rank natural frequencies of vibration；
ξnkFor structure k rank damping ratios；ξnrFor structure r rank damping ratios；
cijFor 2 points of wind load coherence factor of i, j；
Θ (ω) is the molecule multinomial of integrand；θqFor the polynomial coefficient of molecule；ω2qFor the 2q power of frequency, q=0,
1,2,3；
For the denominator complex polynomails of integrand；For complex frequency；λsFor the coefficient of denominator polynomials；For
The s power of complex frequency, s=0~6；
I1For the molecule determinant of integral；I0For the denominator determinant of integral；
Form W2A Q ranks 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；σ1QkrFor σijkrMiddle i=1, j=Q；σ2QkrFor σijkrMiddle i=2, j=Q；And so on, σQQkrFor σijkrMiddle a=Q, b=Q；
[Σkr] be k, r rank coupled mode intermediate variable matrix；
The case where calculating k=r, obtains [Σkr], it calculatesWherein, k=1,2 ..., W；It willIt is ordered as from big to smallWherein, km=1,2 ..., W, m=1,2 ..., W are the serial number of sequence；
In formula,For the variance of kth rank modal response；[Σkk] be kth rank mode intermediate variable matrix, i.e. [Σkr] in k=
The case where r；
For Z first so thatCalculate k<[the Σ of rkr]；According to symmetry principle, [Σrk]=
[Σkr], obtain k>[the Σ of rkr]；
1≤Z≤W；
For kthmThe variance of rank modal response, i.e.,Take k=kmResult.
6. a kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method according to claim 5, special
Sign is：The polynomial coefficient θ of moleculeqWith the coefficient lambda of denominator polynomialssSpecially：
Multinomial is merged for frequencies omega to the molecule multinomial Θ (ω) about frequencies omega, obtains ω2qCoefficient θq, q=0,
1,2,3；
To about complex frequencyDenominator complex polynomailsFor complex frequencyMerge multinomial, obtainsCoefficient lambdas, s=
0~6.
7. a kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method according to claim 6, special
Sign is：The I1And I0Specifically formula is：
Work as cijWhen ≠ 0,
Work as cijWhen=0,
8. a kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method according to claim 7, special
Sign is：Modal response covariance is calculated in the step 6Form modal response association side
Poor matrix [Σy]；Detailed process is：
Calculate modal response covarianceForm modal response covariance matrix
9. a kind of wind induced structural vibration based on auto-correlation function responds efficient frequency domain estimation method according to claim 8, special
Sign is：Dynamic respond covariance matrix [Σ is calculated in the step 7x]=[Ψ]T[Σy] [Ψ], and then it can be acquired
Anticipate the covariance matrix [Σ respondeds]；Detailed process is：
Calculate dynamic respond covariance matrix [Σx]=[Ψ]T[Σy] [Ψ], and then its covariance square arbitrarily responded can be acquired
Battle array [Σs]=[I]T[Σx][I]。
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