CN101587007A - Output-only wavelet analytical method for recognizing flexible bridge structure kinetic parameter - Google Patents

Abstract

The invention provides an output-only wavelet analytical method for recognizing flexible bridge structure kinetic parameter, which includes the steps of: measuring random environment vibration response signal of structure for detection from the flexible bridge structure; selecting a certain signal measurement point with relatively obvious vibration movement respond on the structure as a reference point, estimating the measurement signal and the reference point signal for executing relevant respond; executing point-to-point wavelet analyze and modal decoupling for relevant response signals; recognizing modal parameter using wavelet coefficient on camber line point of the wavelet amount figure. Further, when the known stochastic excited power spectrum intensity is S0, the physical parameter of the structure is calculated. The method provided by the invention meets the need of low-frequency vibration test for bridge structure, and removes dependence for excitation measurement in the recognizing process and improves accuracy of structural damping recognition.

Description

The output-only wavelet analytical method of identification flexible bridge structure kinetic parameter

Technical field

The invention belongs to the vibration monitoring of engineering structure field, relate to the identification of structural dynamic parameter,, propose the method that adopts the output-only wavelet analysis to come the recognition structure kinetic parameter at the key issue in the flexible bridge structure dynamic test.

Background technology

Flexible bridge structure yo-yo effect under the dynamic load functions such as vehicle of fluctuating wind, earth pulsation, operation is obvious, it is carried out the identification of kinetic parameter all have great significance for dynamic response analysis, vibration control, operation monitoring and the Performance Evaluation of structure.Traditionally, adopt Fourier transform to come the recognition structure modal parameter usually.Carry out convolution by simple harmonic wave, the time-domain signal that collects can be transformed in the frequency domain original signal and different frequency, thus the frequency content in the announcement signal.Yet Fourier transform is only applicable to the analysis of steady-state signal.For unstable state or momentary signal, the time-frequency characteristic that Fourier transform can't signal acquisition.This is because Fourier transform projects to the serial data of measuring on the sinusoidal wave function of stable state, and the unlimited vibration of sinusoidal wave function and unattenuated, thereby the time domain local characteristics of signal has just on average been fallen during conversion of signals.In order to remedy this defective, recent researchers have been primarily focused on the signal analysis technology with time frequency analysis function.Signal processing technology based on wavelet transformation can provide multiple dimensioned time domain and frequency domain resolution, is applicable to the analysis to random signal.Staszewski ^[1]The method of utilization small wave converting method estimating system damping ratio from the free damping signal of many-degrees of freedom system has been proposed.People such as Ruzzene ^[2]The utilization small wave converting method is estimated the natural frequency of vibration of many-degrees of freedom system and by actual measurement acceleration responsive Signal Processing having been verified the accuracy of method.People such as Roberston ^[3]Proposed a cover wavelet basis deconvolution technology and used it for from the configuration forces vibration signal to estimate impulse response signal.People such as Piomo ^[4]Develop the utilization small wave converting method and estimated the method for Mode Shape and used the modal mass index evaluation reliability of estimating.People such as Chang ^[5]Then by verification experimental verification the accuracy and the validity of wavelet basis Modal Parameter Identification technology.Though above-mentioned research has all obtained success in various degree, structure institute excited target can be surveyed or controlled in the above-mentioned research.Yet for the large-scale flexible bridge structure, structure is in uncontrollable random environment exciting field usually, the suffered environmental excitation of structure not still at random and also immeasurability, this has proposed stern challenge for the system identification of structure.In addition, during the identification of flexible bridge structure dynamic test and kinetic parameter, also can run into the main mode of structure, also need to adopt suitable method to solve in problems such as low-frequency range dense distribution and the noisy amount of structural environment vibration-testing signal are big.

1.Staszewski，W.J.“Identification?of?damping?in?MDOF?systems?using?time-scale?decomposition.”Journal?ofSound?and?Vibration，1997；203(2)，283-305.

2.Ruzzene，M.，Fasana，A.，Garibaldi，L.and?Piombo，B.“Natural?frequencies?and?dampings?identificationusing?wavelet?transform：application?to?real?data.”Mechanical?Systems?and?Signal?Processing，1997；11(2)，207-218.

3.Piombo，B.A.D.，Fasana，A.，Marchesiello，S.and?Ruzzene，M.“Modeling?and?identification?of?the?dynamicresponse?ofa?supported?bridge.”Mechanical?Systems?andSignal?Processing，2000；14(1)，75-89.

4.Kijewski，T.and?Kareem，A.“Wavelet?transform?for?system?identification?in?civil?engineering.”Computer-Aided?Civil?and?Infrastructure?Engineering，2003；18，339-355.

5.Chang，C.C.，Sun，Z.and?Li，N.“Identification?of?structural?dynamic?properties?using?wavelet?transform.”The?First?International?Conference?on?Structural?Health?Monitoring?and?Intelligent?Infrastructure，November13-15，2003，Tokyo，Japan.

Summary of the invention

The objective of the invention is at the deficiencies in the prior art, a kind of output-only wavelet analytical method of discerning flexible bridge structure kinetic parameter is proposed, at immesurable arbitrary excitation after the match, identify the modal parameters such as frequency, damping ratio and the vibration shape of structure by only test and analytical structure response signal, for the kinetic parameter of flexible bridge structure is provided by the feasible approach that provides.

For reaching above purpose, solution of the present invention is:

A kind of output-only wavelet analytical method of discerning flexible bridge structure kinetic parameter vibrates the noisy response based on only output measurement and auto-correlation response are estimated and adopted asymptotic wavelet analysis method to identify the modal parameter and the physical parameter of flexible bridge structure from the structure random environment.

Further, it may further comprise the steps:

1) utilize the existing structure vibration test technology to measure the random environment vibration response signal of structure from flexible bridge structure;

2) choose on the structure relatively more significant certain the signal measurement point of vibratory response and be reference point, measuring-signal and reference point signal are carried out the relevant response estimation;

3) wavelet analysis is carried out in the pointwise of relevant response signal, the mode decoupling zero;

4) utilize the wavelet coefficient at small echo spirogram crestal line place to carry out the identification of modal parameter.

Its power spectrum intensity at known arbitrary excitation is S ₀Situation under, the physical parameter of structure is calculated.

Described relevant response estimates to comprise structure auto-correlation and simple crosscorrelation response estimation, adopts the average method of multisample; Perhaps satisfy under the ergodic hypothesis, adopt single sample along the average method of time shaft in structure random vibration response signal.

After described measuring-signal relevant response is estimated, carry out continuation and handle, its width according to the time domain resolution of wavelet transformation and time frequency window determines that relevant response bears continuation length.

It also comprises the frequency domain resolution of analysis wavelet conversion, and when providing flexible bridge structure low-frequency range mode dense distribution in view of the above in order to realize the full decoupled of mode, the formula that the wavelet mother function centre frequency is provided with.

Relevant response is estimated

Suppose that a n system with one degree of freedom is subjected to the effect of zero-mean white Gaussian noise excitation f, the equation of motion of system can be expressed as

$M\stackrel{\·\·}{Z}+C\stackrel{\·}{Z}+\mathrm{KZ}=\mathrm{Df}---\left(1\right)$

M in the formula, C, K are respectively architecture quality, damping and stiffness matrix; D is an exciting point position vector; Z=[z ₁z ₂Z _n] ^TBe the displacement response vector,

Then represent structure acceleration and speed responsive vector respectively.Supposing the system is the linear scaling damping system, and its displacement response vector can be expressed as according to the mode superposition principle

$Z=\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\Φ}}_j{q}_j---\left(2\right)$

Φ in the formula _jAnd q _jBe respectively the vibration shape vector sum broad sense modal coordinate of j rank mode.The equation of motion of j rank mode can be expressed as after the decoupling zero

${\stackrel{\·\·}{q}}_j+2{\mathrm{\ζ}}_j{\mathrm{\ω}}_j{\stackrel{\·}{q}}_j+{{\mathrm{\ω}}_j}^2{q}_j=\frac{d_j}{m_j}f---\left(3\right)$

ω in the formula _j, ζ _jAnd m _jBe respectively and be the j rank natural frequency of vibration, damping ratio and modal mass,

q _jAcceleration, speed responsive and the displacement response of representing j rank broad sense modal coordinate respectively; J rank mode participates in coefficient $d_j={\mathrm{\Φ}}_j^{T}D.$ If the power spectrum intensity of exciting force f is S ₀, q _jStable state autocorrelation function response R _jCan be expressed as

$R_j\left(t\right)=B_j{e}^{-{\mathrm{\ζ}}_j{\mathrm{\ω}}_{j}t}\cos ({\stackrel{\‾}{\mathrm{\ω}}}_{j}t-{\mathrm{\θ}}_j)---\left(4\right)$

$B_j=\frac{\mathrm{\π}{d_j}^2{S}_0}{2{m_j}^2{\mathrm{\ζ}}_j{{\mathrm{\ω}}_j}^3\sqrt{1-{{\mathrm{\ζ}}_j}^2}},$ ${\stackrel{\‾}{\mathrm{\ω}}}_j={\mathrm{\ω}}_j\sqrt{1-{{\mathrm{\ζ}}_j}^2},$ ${\mathrm{\θ}}_j={\tan }^{-1}\frac{{\mathrm{\ζ}}_j}{\sqrt{1-{{\mathrm{\ζ}}_j}^2}}---(5a,b,c)$

ω in the formula _jBe j rank damped frequency; θ _jBe phasing degree, j rank.Obtained the stable state autocorrelation function response at structure broad sense modal coordinate place, but the g of system with h degree of freedom between the stable state simple crosscorrelation to respond approximate representation as follows

$R_{\mathrm{gh}}\left(t\right)\≅\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\φ}}_j^g{\mathrm{\φ}}_j^h{R}_j\left(t\right)---\left(6\right)$

φ in the formula _j ^gAnd φ _j ^hG and h value for structure j first order mode vector.This formula has been ignored the cross term between different modalities.This being similar in structure two adjacent mode satisfied as lower inequality | ω _i-ω _jThe max{ ζ of |＞＞ _iω _i, ζ _jω _j(set up during i ≠ j).

For structurally associated response R _GhEstimation, can adopt the average method of multisample, satisfy in structure random vibration response signal and also can adopt single sample under the ergodic hypothesis along the average method of time shaft.

Asymptotic wavelet analysis

Wavelet transformation exactly a time domain can be amassed signal x (t) be transformed into time-yardstick plane (a, b) a kind of linear transformation method on.This process can be expressed as follows

$W_x(a,b)=\frac{1}{\sqrt{a}}\underset{-\∞}{\overset{\∞}{\∫}}x\left(t\right)w*\left(\frac{b-t}{a}\right)\mathrm{dt}---\left(7\right)$

In the formula, * represents complex conjugate; A is the scale parameter that the control analysis window stretches; B is the translation parameters of location wavelet function; And mother wavelet function w (t) is a decay fast, and average is zero wave function.

Generally speaking, the signal x (t) in the real number field can be represented as following canonical form uniquely arbitrarily,

In the formula, amplitude and phase function to (A (t),

) be called as x (t) standard right.What asymptotic signal referred to is exactly the relative pace of change of amplitude that class signal much smaller than phase change speed.They satisfy

Concerning asymptotic signal, the wavelet conversion coefficient W in the formula (7) _x(a b) can try to achieve by scalar product is approximate according to the stable state phase method.Here it is asymptotic wavelet analysis.Under little damping situation, because ω _j＞＞ζ _jω _jSo, R _jSatisfy the condition of asymptotic signal.

Though mother wavelet function has a variety of, be not to be used for asymptotic wavelet analysis.In the research work of this method, use be Morlet mother wavelet function as follows

$w\left(t\right)=e^{i{\mathrm{\ω}}_{0}t}{e}^{-t^2/2}---\left(10\right)$

In the formula, ω ₀It is the centre frequency of mother wavelet function.The residing circular frequency of scale parameter and small echo has following one-to-one relationship a=ω _o/ ω.For the Morlet small echo, by asymptotic Wavelet Analysis Theory, the wavelet conversion coefficient of signal x (t) can be approximately

Wherein, the modulus of wavelet coefficient, phase value are

The wavelet basis system identification

Wavelet transformation is carried out in formula (6) both sides can be got

$W_{R_{\mathrm{gh}}}(a,t)=\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\φ}}_j^g{\mathrm{\φ}}_j^h{W}_{R_j}(a,t)---\left(13\right)$

The wavelet conversion coefficient of j rank auto-correlation response in the formula

Can obtain by comparison expression (4) and formula (8) and substitution formula (11)

$W_{R_j}(a,t)=\sqrt{a}{B}_j{e}^{-{\mathrm{\ζ}}_j{\mathrm{\ω}}_{j}t}{e}^{-{(a{\stackrel{\‾}{\mathrm{\ω}}}_j-{\mathrm{\ω}}_o)}^2}{e}^{i({\stackrel{\‾}{\mathrm{\ω}}}_{j}t-{\mathrm{\θ}}_j)}---\left(14\right)$

Amplitude is got on formula (14) substitution formula (13) and both sides can be got

$\left|W_{R_{\mathrm{gh}}}(a,t)\right|=\underset{j=1}{\overset{n}{\mathrm{\Σ}}}\left|{\mathrm{\φ}}_j^g\right|\left|{\mathrm{\φ}}_j^h\right|\sqrt{a}{B}_j{e}^{-{\mathrm{\ζ}}_j{\mathrm{\ω}}_{j}t}{e}^{-{(a{\stackrel{\‾}{\mathrm{\ω}}}_j-{\mathrm{\ω}}_o)}^2}---\left(15\right)$

By following formula as can be known, for a structural system with n rank mode, the wavelet coefficient amplitude of its relevant response all can have n local peaking along the yardstick axle at any time.These local peaking's yardsticks or be called the crestal line yardstick for time-invariant system for constant.By formula (15) as can be known, there are following relation in system's j rank damped frequency and j rank crestal line yardstick

${\stackrel{\‾}{\mathrm{\ω}}}_j=\frac{{\mathrm{\ω}}_o}{a_j},$ j＝1，2，...，n (16)

Like this, when system damping frequency sparse distribution, formula (14) but wavelet coefficient approximate representation at j rank crestal line yardstick place be

$W_{R_{\mathrm{gh}}}(a_j,t)\≅{\mathrm{\φ}}_j^g{\mathrm{\φ}}_j^h\sqrt{a_j}{B}_j{e}^{-{\mathrm{\ζ}}_j{\mathrm{\ω}}_{j}t}{e}^{i({\stackrel{\‾}{\mathrm{\ω}}}_{j}t-{\mathrm{\θ}}_j)}---\left(17\right)$

Based on this formula, can obtain following two formulas

$-{\mathrm{\ζ}}_j{\mathrm{\ω}}_j\≅\frac{d}{\mathrm{dt}}\ln \left|W_{R_{\mathrm{gh}}}(a_j,t)\right|---\left(18\right)$

${\stackrel{\‾}{\mathrm{\ω}}}_j={\mathrm{\ω}}_j\sqrt{1-{{\mathrm{\ζ}}_j}^2}\≅\frac{d}{\mathrm{dt}}(\∠W_{R_{\mathrm{gh}}}(a_j,t))---\left(19\right)$

Utilize these two equatioies can obtain structure j rank natural frequency of vibration ω _jAnd dampingratio _jIn principle, the structure auto-correlation response that only need estimate a measuring point place just can be estimated the n rank natural frequency of vibration and the damping ratio of structure simultaneously.For the estimation of vibration shape vector, then need to estimate simultaneously earlier to respond with reference to the auto-correlation response at measuring point place and other measuring points and with reference to the simple crosscorrelation between measuring point.Be not difficult to find out that by formula (17) the structure j rank Mode Shape ratio at measuring point h and measuring point g place can be obtained by following formula:

$\frac{{\mathrm{\φ}}_j^h}{{\mathrm{\φ}}_j^g}\≅\frac{W_{R_{\mathrm{gh}}}(a_j,t)}{W_{R_{\mathrm{gg}}}(a_j,t)}---\left(20\right)$

If the power spectrum strength S of arbitrary excitation ₀Also be known, can also estimate quality, rigidity and the damping matrix of structure according to the methods below.At first with formula (5a) substitution formula (17) and make t=0, the j rank modal mass of structure can be determined by following formula:

$m_j=\sqrt{\frac{\sqrt{a_j}\left|{\mathrm{\φ}}_j^g\right|\left|{\mathrm{\φ}}_j^h\right|\mathrm{\π}{d_j}^2{S}_0}{2{\mathrm{\ζ}}_j{{\mathrm{\ω}}_j}^3\left|W_{R_{\mathrm{gh}}}(a_j,0)\right|\sqrt{1-{{\mathrm{\ζ}}_j}^2}}}---\left(21\right)$

J rank broad sense modal stiffness coefficient k then _jWith modal damping c _jCan determine by following formula:

$k_j=m_j{\mathrm{\ω}}_j^2;$ c _j＝2ζ _jm _jω _j (22a，b)

After having obtained structure broad sense mode physical parameter and Mode Shape vector, can easily obtain rigidity, quality and the damping matrix of structure.Because structure steady state speed and acceleration relevant response are respectively the second order and the quadravalence derivative of structure stable state displacement relevant response, for speed and amount of acceleration measured value, carry out similar analysis and also can obtain modal parameters and physical parameter.

Wavelet frequency domain resolution: modal separation

If mother wavelet function ψ (t) and Fourier transform thereof remember that to the requirement that ψ (ω) satisfies window function its time-frequency window center is (t ^*, ω ^*), width is respectively Δ t _ψWith Δ ω _ψTo arbitrary parameter to (a, b), the small echo kernel function ${\mathrm{\ψ}}_{a,b}\left(t\right)=\frac{1}{\sqrt{\left|a\right|}}\mathrm{\ψ}\left(\frac{t-b}{a}\right)$ Corresponding time frequency window is respectively

[b+at ^*-|a|Δt _ψ，b+at ^*+|a|Δt _ψ]，[ω ^*/a-Δω _ψ/|a|，ω ^*/a+Δω _ψ/|a|] (23，24)

That is to say that the time frequency window of continuous wavelet transform is variable rectangular on the time frequency plane, its time-frequency window width is respectively 2|a| Δ t _ψ, 2 Δ ω _ψ/ | a|, the shape of time-frequency window changes along with scale parameter a.The time-frequency resolution of this time frequency window arbitrarily adjustable characteristic is the unique property that wavelet transformation is different from other time frequency analyzing tool such as short time discrete Fourier transform.Concerning mode identification, comprised the modal information of different frequency corresponding to the wavelet conversion coefficient of different scale value, by wavelet transformation, each the rank modal information in the structure vibration signals is separated.This separation is converted into a plurality of single-order mode identification problems with a multistage mode identification problem originally, greatly reduces the difficulty of identification problem.Though wavelet transformation has very strong modal separation function, its separating power depends on its frequency domain resolution to a great extent.The frequency resolution degree is high more, and separating power is just strong more.Concerning system with intensive mode, how the reasonable frequency resolution is set separates these intensive mode, especially when one or several energy in these mode is very low, how to separate very crucial.

With following Morlet mother wavelet function is example,

$g\left(t\right)=e^{i{\mathrm{\ω}}_{0}t}{e}^{-t^2/2}---\left(25\right)$

In essence, it is a Gaussian window trigonometric function, and the centre frequency of its trigonometric function is ω ₀, the time frequency window width is respectively $\mathrm{\Δ}{t}_{\mathrm{\ψ}}=\frac{1}{\sqrt{2}},$ $\mathrm{\Δ}{\mathrm{\ω}}_{\mathrm{\ψ}}=\frac{1}{\sqrt{2}},$ Corresponding Fourier transform is:

$G\left(\mathrm{a\ω}\right)=\frac{1}{\sqrt{2\mathrm{\π}}}{e}^{-\frac{1}{2}{(\mathrm{a\ω}-{\mathrm{\ω}}_0)}^2}---\left(26\right)$

By above-mentioned frequency-domain expression as can be known, to selected arbitrarily scale parameter a, when following formula is got maximal value, there is following formula to set up: a=ω ₀/ ω, the time-frequency resolution of corresponding small echo convolution kernel function is Δ ω _a=Δ ω _ψ/ a; Δ t _a=a Δ t _ψTherefore, centre frequency is the distinguishable value Δ of the little wave frequency ω at ω place _aFor

$\mathrm{\Δ}{\mathrm{\ω}}_a=\frac{\mathrm{\ω}}{\sqrt{2}{\mathrm{\ω}}_0}---\left(27\right)$

As can be seen from the above equation, the centre frequency ω of mother wavelet function ₀Big more, the distinguishable value of its frequency is more little, and the frequency resolution degree is high more.Guarantee that two frequencies are ω ₁And ω ₂The separation of mode, centre frequency ω ₀Minimum value to satisfy following formula:

${\mathrm{\ω}}_0=\left(2\mathrm{\α}\right)\frac{{\mathrm{\ω}}_{\mathrm{1,2}}}{\sqrt{2}\mathrm{\Δ}{\mathrm{\ω}}_{\mathrm{1,2}}}---\left(28\right)$

Δ ω in the formula _1,2=| ω ₁-ω ₂|, ω _1,2=(ω ₁+ ω ₂The overlapping degree of adjacent two Gaussian analysis windows of being allowed by the Morlet small echo is represented in)/2, parameter alpha.For most of linear systems, desirable α=2.

Small echo time domain resolution: the processing of boundary effect

The time domain resolution of wavelet analysis is directly related with the Boundary Effect problem of wavelet transformation.According to above derivation as can be known, centre frequency is the wavelet transformation time frequency window at ω place, and its time domain window half width is

$\mathrm{\Δ}{t}_a=\frac{\mathrm{\ω}}{\sqrt{2}\mathrm{\ω}}---\left(29\right)$

Be placed on when this time-frequency window center on the time zero of TIME HISTORY SIGNAL and along the time shaft forward [0, Δ t _a] in the scope when mobile, half-window overlay area, a time frequency window left side can not supply the data of convolution with part, thereby causes the distortion of boundary wavelet transformation.The boundary effect of wavelet transformation is very unfavorable to signal Processing, in order to address this problem, has produced a kind of technology of continuation by name.This technology is duplicated prolongation with the beginning and the end portion of research signal with other data, the signal that research institute is concerned about is positioned in the middle of the signal after the prolongation, so in the wavelet transformation process, that part of signal that only is extended is subjected to the influence of boundary effect, thereby reaches the purpose of preserving all information of original signal.When carrying out the continuation processing, the signal that is prolonged must satisfy continuity and symmetric requirement.

When the present invention estimates in relevant response, adopted the Boundary Effect problem that exists in the following formula manipulation wavelet transformation,

$R_{\mathrm{gh}}\left(\mathrm{i\Δ\τ}\right)=\underset{N\→\∞}{\lim }\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{z}_g\left(\mathrm{k\Δ\τ}\right)z_h[(k+i+1)\mathrm{\Δ\τ}-\mathrm{\β\Δ}{t}_a]$ (i＝0，1，2，...，(T+2βΔt _a)/Δτ)(30)

β Δ t in the formula _aBe boundary extension length; Δ t _aFor when institute's analytic signal reaches low-limit frequency, with the maximum time domain half width that produces, the maximum magnitude of its expression boundary effect influence; The levels of precision that β will reach according to research institute is selected, and β gets 4 and gets final product generally speaking.When continuation is born in relevant response, the provable signal that prolongs satisfies continuity and symmetric requirement, having confirmed that structurally associated sound after the negative continuation would not introduce singular point at boundary and also can not introduce new frequency content, is the feasible method of handling the wavelet transformation Boundary Effect problem.

Owing to adopted such scheme, the present invention to have following characteristics: based on wavelet analysis solved systemicly that structure random environment exciting field of living in flexible bridge structure dynamic test and the mode identification can not be surveyed, the main mode of structure is in key issues such as low-frequency range dense distribution, the noisy amount of structural environment vibration-testing signal are big.It provides the intensive mode of adaptive time-frequency mode decoupling zero ability recognition structure, the restriction of having avoided traditional mode recognition technology that excitation is measured simultaneously.

Description of drawings

Fig. 1 discerns the process flow diagram of flexible bridge structure kinetic parameter for the present invention.

Embodiment

The present invention is further illustrated below in conjunction with the accompanying drawing illustrated embodiment.

Suppose that a flexible bridge structure vibrates under zero-mean white Gaussian noise arbitrary excitation, the response that records any 2 g of structure, h place is z _g, z _h, then the simple crosscorrelation at structure two measuring point places responds R _GhCan adopt the average method of multisample to estimate, satisfy also can adopt under the ergodic hypothesis in structure random vibration response signal and estimate along the average method of time shaft as the sample that places an order

$R_{\mathrm{gh}}\left(\mathrm{i\Δ\τ}\right)=\underset{N\→\∞}{\lim }\frac{1}{N}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{z}_g\left(\mathrm{k\Δ\τ}\right)z_h[(k+i+1)\mathrm{\Δ\τ}-\mathrm{\β\Δ}{t}_a]$ (i＝0，1，2，...，(T+2βΔt _a)/Δτ)(31)

Δ τ is the sampling time increment in the formula, β Δ t _aBe boundary extension length; Δ t _aFor when institute's analytic signal reaches low-limit frequency, with the maximum time domain half width that produces, the maximum magnitude of its expression boundary effect influence; The levels of precision that β will reach according to research institute is selected, and β gets 4 and gets final product generally speaking.In the estimation procedure, need to adopt of the convergence of sufficiently long test signal to guarantee that the structurally associated response is estimated.The structurally associated response signal of estimating is carried out following wavelet transformation, obtain corresponding small echo spirogram

$W_{R_{\mathrm{gh}}}(a,b)=\frac{1}{\sqrt{a}}\underset{-\∞}{\overset{\∞}{\∫}}{R}_{\mathrm{gh}}\left(t\right){{\mathrm{\ψ}}^{*}}_{a,b}\left(t\right)\mathrm{dt}---\left(32\right)$

In the formula, * represents complex conjugate; A is the scale parameter that the control analysis window stretches; B is the translation parameters of location wavelet function; And mother wavelet function ψ (t) is a decay fast, and average is zero wave function.For a structural system with n rank mode, the small echo spirogram of its relevant response all can have n local peaking along the yardstick axle at any time, these local peaking's yardsticks or be called the crestal line yardstick have following relation for constant and system's j rank damped frequency and j rank crestal line yardstick for time-invariant system

${\stackrel{\‾}{\mathrm{\ω}}}_j=\frac{{\mathrm{\ω}}_o}{a_j},$ (j＝1，2，...，n) (33)

The amplitude of the wavelet coefficient by extracting crestal line yardstick correspondence

And phase place Information can estimate the j rank natural frequency of vibration ω of structure according to following formula _jAnd dampingratio _j:

${\mathrm{\ω}}_j=\sqrt{{\left\{\frac{d}{\mathrm{dt}}\right[\∠W_{R_{\mathrm{gh}}}(a_j,b)\left]\right\}}^2+{\left\{\frac{d}{\mathrm{dt}}\right[\ln \left|W_{R_{\mathrm{gh}}}(a_j,b)\right|\left]\right\}}^2},$ ${\mathrm{\ξ}}_j=\frac{-\frac{d}{\mathrm{dt}}\left[\ln \right|W_{R_{\mathrm{gh}}}(a_j,b)\left|\right]}{{\mathrm{\ω}}_j}---\left(\mathrm{34,35}\right)$

In theory, the structure auto-correlation response that only need estimate a measuring point place just can be estimated the n rank natural frequency of vibration and the damping ratio of structure simultaneously.For the estimation of vibration shape vector, then need to estimate simultaneously earlier to respond with reference to the auto-correlation response at measuring point g place and other arbitrary measuring point h and with reference to the simple crosscorrelation between measuring point, structure j rank Mode Shape compares φ _j ^h/ φ _j ^gCan obtain by following formula:

$\frac{{\mathrm{\φ}}_j^h}{{\mathrm{\φ}}_j^g}\≅\frac{W_{R_{\mathrm{gh}}}(a_j,t)}{W_{R_{\mathrm{gg}}}(a_j,t)}---\left(36\right)$

If the power spectrum strength S of arbitrary excitation ₀Also be known, can also estimate modal mass, rigidity and the damping matrix of structure according to formula (21), (22).After having obtained structure broad sense mode physical parameter and Mode Shape vector, can easily obtain rigidity, quality and the damping matrix of structure.

The above-mentioned description to embodiment is can understand and apply the invention for ease of those skilled in the art.The person skilled in the art obviously can easily make various modifications to these embodiment, and needn't pass through performing creative labour being applied in the General Principle of this explanation among other embodiment.Therefore, the invention is not restricted to the embodiment here, those skilled in the art should be within protection scope of the present invention for improvement and modification that the present invention makes according to announcement of the present invention.

Claims (6)

1, a kind of output-only wavelet analytical method of discerning flexible bridge structure kinetic parameter is characterized in that: vibrate the noisy response based on only output measurement and auto-correlation response are estimated and adopted asymptotic wavelet analysis method to identify the modal parameter and the physical parameter of flexible bridge structure from the structure random environment.

2, the output-only wavelet analytical method of identification flexible bridge structure kinetic parameter as claimed in claim 1, it is characterized in that: it may further comprise the steps:

1) measures the random environment vibration response signal of structure from flexible bridge structure;

2) choose on the structure relatively more significant certain the signal measurement point of vibratory response and be reference point, measuring-signal and reference point signal are carried out the relevant response estimation;

3) wavelet analysis is carried out in the pointwise of relevant response signal, the mode decoupling zero;

4) utilize the wavelet coefficient at small echo spirogram crestal line place to carry out the identification of modal parameter.

3, the output-only wavelet analytical method of identification flexible bridge structure kinetic parameter as claimed in claim 2 is characterized in that: its power spectrum intensity at known arbitrary excitation is S ₀Situation under, the physical parameter of structure is calculated.

4, the output-only wavelet analytical method of identification flexible bridge structure kinetic parameter as claimed in claim 2 is characterized in that: described relevant response estimates to comprise structure auto-correlation and simple crosscorrelation response estimation, adopts the average method of multisample; Perhaps satisfy under the ergodic hypothesis, adopt single sample along the average method of time shaft in structure random vibration response signal.

5, the output-only wavelet analytical method of identification flexible bridge structure kinetic parameter as claimed in claim 2, it is characterized in that: after described measuring-signal relevant response is estimated, carry out continuation and handle, its width according to the time domain resolution of wavelet transformation and time frequency window determines that relevant response bears continuation length.

6, the output-only wavelet analytical method of identification flexible bridge structure kinetic parameter as claimed in claim 2, it is characterized in that: it also comprises the frequency domain resolution of analysis wavelet conversion, and when providing flexible bridge structure low-frequency range mode dense distribution in view of the above in order to realize the full decoupled of mode, the formula that the wavelet mother function centre frequency is provided with.

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