CN102222219A - Structural concentration modal parameter recognition method based on Moret wavelet transformation - Google Patents

Abstract

The invention relates to a structural concentration modal parameter recognition method based on Moret wavelet transformation. A wavelet amplitude curve and a wavelet phase curve are analyzed by using an improved modal parameter recognition method on the basis of the Moret wavelet transformation theory so that various orders of modal frequency and modal damping of a structure are calculated. According to the invention, criterions of civil engineering structure concentration modals are quantitatively defined, a method for determining structural frequency according to wavelet accumulating energy spectrum is provided, a wavelet center frequency optimizing algorithm is established, and the process of carrying out the damping ratio recognition on applied wavelet transformation is improved. On the basis, the invention provides a full set of flow of the structural concentration modal parameter recognition method based on the Moret wavelet transformation. The method has the advantages of improving the precision of wavelet transformation recognition structural modal parameters, effectively avoiding the influence caused by end effects, and overcoming the subjectivity, the experience and the blindness of selecting a wavelet transformation centre frequency and a wavelet amplitude fitting region, thereby being capable of being widely applied and popularized.

Description

Based on the intensive Modal Parameters Identification of the structure of Moret wavelet transformation

Technical field

The present invention is a kind of method that is applied to Modal Parameter Identification, especially a kind of method that is applied to the parameter recognition of intensive mode in the civil engineering structure.

Background technology

The small echo variation is proposed by Moret and Grossman jointly with the eighties in 20th century the earliest ^[1], the thought that Mallat proposed multiscale analysis in 1989 has been set up the small wave converting method of signal decomposition and reconstruct ^[2]Daubechies ^[3-4]Proposed the orthogonal wavelet of one group of finite support, set up the relation between the analysis of wavelet analysis and discrete signal, and obtained using widely.Wavelet transformation provides a variable T/F window, narrows down automatically when the analysis of high frequency signal, and broadens automatically during the analysing low frequency signal.Modal parameter is comprising the principal character of system, and accurately each rank modal parameter of recognition system has very important significance to the engineering research tool.Wavelet transformation has good time-domain analysis ability, has all obtained successful application in many scientific domains.Many good characteristics of wavelet transformation have determined it in the Modal Parameter Identification field wide application prospect to be arranged.

Systematic parameter identification based on wavelet analysis more and more receives publicity, and utilizes wavelet transformation to carry out modal parameters identification and often sees.And by numerous lists of references as can be known, wavelet analysis is one of effective way of carrying out the system mode parameter recognition.As M.Ruzzene and W.J.Staszewski ^[5-6]Utilize continuous wavelet transform identification model frequency and damping ratio, P.Argoul etc. ^[7]Introduce the Cauchy small echo and be used for discerning model frequency, the vibration shape and damping ratio, T-P.Le etc. ^[8]Freely responding of system carried out continuous wavelet transform, recognition structure modal parameter.People such as J.Lardies and J.Slavic ^[9-10]Also study Modal Parameter Identification problem based on continuous wavelet transform.

Yet in intensive Modal Parameter Identification, the selection of wavelet center frequency and bandwidth has crucial influence to accuracy of identification.Though Chinese scholars is carried out the intensive modal parameter of structure to application Moret wavelet transformation and was also carried out research ^[11], but, still have some problems still unresolved or do not cause enough attention in this field:

When (1) using the intensive modal parameter of Moret small wave converting method recognition structure, the definition of intensive mode remains to be quantized more accurately, and the relation of mode aliasing degree and damping ratio is still waiting clearly.

(2) whether the value of the value of wavelet transformation centre frequency and wavelet amplitude fit interval is rationally bigger to the influence of modal parameters accuracy of identification.And choosing of the two often has subjectivity.

(3) use the Moret wavelet transformation and carry out modal parameters identification, do not form the normalized flow process of utilizing the Moret wavelet transformation to carry out modal parameters identification of a cover as yet.

List of references:

[1] Grosssman.A, Moret .J.Decomposition of Handy Function into Square Integrable Wavelets of Consistant Shape[C] .SIAM, 1984,15 (4): 736-783

[2]S.G.Mallat.A?Theory?for?Multiresolution?Signal?Decomposition：the?Wavelet?Representation[C].IEEE?Transactions?on?Pattern?Analysis?and?Machine?Intelligence，1989，11(7)：674～693.

[3]I.Daubechies.Orthonarmal?Bases?of?Compactly?Supported?Wavelets[J].Communication?in?Pure?and?Applied?Mathematics，1988，(41)：909～996.

[4]Daubecies?I.Ten?Lecture?on?Wavelet.Society?for?Industrial?and?Applied?Mathematics?Philadelphia，Pennsylvania，1992.

[5]M.Ruzzene，A.Fasana，L.Garibaldi?and?B.Piombo.Natural?Frequencies?and?Dampings?Identification?Using?Wavelet?Transform：Application?to?Real?Data[J].Mechanical?Systems?and?Signal?Processing，1997，11(2)：207～218.

[6]W.J.Staszewski.Identification?of?Damping?in?MDOF?Systems?Using?Time-Scale?Decomposition[J].Journal?of?Sound?and?Vibration，1997，203(2)：283～305.

[7]P.Argoul，T-P.Le.Instantaneous?Indicators?of?Structural?Behavior?Based?on?Continuous?Cauchy?Wavelet?Transform[J].Mechanical?Systems?and?Signal?Processing，2003，17：243～250.

[8]T-P.Le，P.Argoul.Continuous?Wavelet?Transform?for?Modal?Identification?Using?Free?Decay?Response[J].Journal?of?Sound?and?Vibration，2004，277：73～100.

[9]J.Lardies，S.Gouttebroze.Identification?of?Modal?Parameters?Using?the?Wavelet?Transform[J].International?Journal?of?Mechanical?Sciences，2002，44：2263～2283.

[10]J.Slavic，I.Simonovski，M.Boltezar.Damping?Identification?Using?a?Continuous?Wavelet?Transform：Application?to?Real?Data[J].Journal?of?Sound?and?Vibration，2003，262：291～307.

[11]Lardies?J，Ta?M?N，Berthither?M.Modal?parameter?estimation?based?on?the?wavelet?transform?of?output?data[J].Archive?of?Applied?Mechanic，2004，73：718-733.

Summary of the invention

Technical matters: the purpose of this invention is to provide the intensive Modal Parameters Identification of a kind of structure based on the Moret wavelet transformation, emphasis quantizes definition to the intensive mode standard of civil engineering structure, improved the method for wavelet transformation recognition structure modal parameter on this basis, proposed the optimized Algorithm of wavelet center frequency, thereby utilized it to carry out method based on the intensive Modal Parameter Identification of structure of Moret wavelet transformation.

Technical scheme: for realizing above technical purpose, the intensive Modal Parameters Identification of the structure based on the Moret wavelet transformation of the present invention is:

1) uses the Moret small echo, get f ₀For wavelet center frequency conversion is carried out in structure free damping response and obtained small echo index W _gX (a, b), wherein, a is the small echo contraction-expansion factor, b is the small echo shift factor;

2) interior in b ∈ [β Δ t, T-β Δ t] scope to small echo index W _gX (a b) adds up and draws small echo cumlative energy spectrum AES, and the computing formula of small echo cumlative energy spectrum AES is as follows:

$\mathrm{AES}\left(f\right)=\underset{b=\mathrm{\β\Δt}}{\overset{T-\mathrm{\β\Δt}}{\mathrm{\Σ}}}\left|\mathrm{Wab}(a,b)\right|=\underset{t=\mathrm{\β\Δt}}{\overset{T-\mathrm{\β\Δt}}{\mathrm{\Σ}}}\left|\mathrm{Wab}(f,t)\right|$

F=f in the formula ₀/ a is a frequency, and t=b is the time, and T is the free damping response time,

β=3, f _D1Be the structure first stage structure vibration natural frequency.What the frequency of obtaining AES maximum point correspondence was each rank of structure has a damped vibration natural frequency f _Di

3) according to each rank model frequency f of structure _DiJudge whether adjacent mode is intensive mode, the index ω of intensive mode and sparse mode is distinguished in definition _Ri=f _{N (i+1)}/ f _Ni≈ f _{D (i+1)}/ f _Di＞1, f wherein _Ni, f _{N (i+1)}Be respectively structure i rank and i+1 rank undamped oscillation natural frequency, f _Di, f _{D (i+1)}Be respectively structure i rank and i+1 rank the damped vibration natural frequency is arranged; For general civil engineering structure, if ω _r＞1.12, then adjacent mode is sparse mode, illustrates that adjacent mode can discern, and next step carries out step 4); Otherwise then it is intensive mode, need carry out damping ratio and discern in advance, and next step carries out step 5);

4) sparse modal damping is discerned than pre-: wavelet transformation, wavelet center frequency are carried out in the free damping response

F wherein _{I, i+1}Be structure i order frequency or i+1 frequency, Δ f _{I, i+1}Poor for both is to eliminate " limit end effect ", sets wavelet amplitude linear analysis interval at t ∈ [β Δ t, T-β Δ t], in β=3; To small echo index amplitude at [t ₁, t ₂] carry out linear fit, the damping ratio of recognition structure in the interval

T wherein ₁=max[β Δ t, t (ln|W _gX (a _k, t) | _Max)], t ₂=min[T-β Δ t, t (ln|W _gX (a _k, t) | _Min)];

5) intensive modal damping is discerned than pre-: wavelet transformation, wavelet center frequency are carried out in the free damping response

If wavelet transformation index amplitude curve is at t ₁Near in slope to undergo mutation be reversion, then this intensive mode cannot decoupling zero; Otherwise, then can decoupling zero, wavelet transformation, wavelet center frequency are carried out in the free damping response

To small echo index amplitude at [t ₁, t ₂] carry out linear fit, the damping ratio of recognition structure in the interval

6) computing center's frequency minima:

At first calculate ${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000069099530000021.png,HDA0000069099530000022.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\frac{({{\mathrm{\&xi;}}_2}^2-{{\mathrm{\&xi;}}_1}^2){{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000069099530000021.png,HDA0000069099530000022.png"\; state="\{\{state\}\}">r</figure-callout>}}}^2}{2{(1-2{{\mathrm{\&xi;}}_1}^2)}^2}-\frac{{(\sqrt{1-{{\mathrm{\&xi;}}_2}^2}{\mathrm{\&omega;}}_r-\sqrt{1-{{\mathrm{\&xi;}}_1}^2})}^2}{2(1-{{\mathrm{\&xi;}}_1}^2)},$

${\mathrm{<figure-callout\; id="4"\; label="r"\; filenames="HDA0000069099530000033.png"\; state="\{\{state\}\}">r</figure-callout>}}_4=\frac{({{\mathrm{\&xi;}}_1}^2-{{\mathrm{\&xi;}}_2}^2)}{2{(1-2{{\mathrm{\&xi;}}_2}^2)}^2{{\mathrm{\&omega;}}_r}^2}-\frac{{(\sqrt{1-{{\mathrm{\&xi;}}_1}^2}{\mathrm{\&omega;}}_r-\sqrt{1-{{\mathrm{\&xi;}}_2}^2})}^2}{2(1-{{\mathrm{\&xi;}}_1}^2){{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000069099530000021.png,HDA0000069099530000022.png"\; state="\{\{state\}\}">r</figure-callout>}}}^2}$

Then with the damping ratio of trying to achieve The substitution following formula, and in conjunction with inequality

With

Get the minimum value f that wavelet center frequency need satisfy the decoupling zero condition _0min

7) choosing of wavelet center frequency is optimized calculating, gets

Analyze,, calculate the fitting a straight line L (ln|W of wavelet amplitude for each centre frequency _gX (a _k, t) |), and calculate fitting a straight line L (ln|W _gX (a _k, t) |) standard deviation sigma=std (ln|W _gX (a _i, t) |-L (ln|W _gX (a _i, t) |)), a _iIt is the scale factor of i rank mode correspondence;

8) identification modal parameter: the i rank and the i+1 rank wavelet center frequency f that get the minimum value correspondence of σ _0-i, f _0-(i+1), signal is carried out wavelet transformation respectively, try to achieve modal parameter end value f _Ni, f _{N (i+1)}, ξ _i, ξ _I+1

Beneficial effect: at some defectives of the intensive modal parameter of traditional wavelet method recognition structure, the present invention has carried out quantizing definition to the standard of intensive mode, and the relation of clear and definite mode aliasing degree and damping ratio is carried out; Proposed to determine the method for structure frequency, set up the wavelet center frequency optimized Algorithm, improved and improved using the method that wavelet transformation carries out damping ratio identification according to small echo cumlative energy spectrum; A whole set of flow process based on the intensive Modal Parameters Identification of structure of Moret wavelet transformation has been proposed on this basis.This method flow has good operability, higher advantages such as parameter recognition precision, and has antinoise interference performance preferably.

Description of drawings

Fig. 1 is a process flow diagram of the present invention,

Fig. 2 is three types of synoptic diagram of adjacent mode,

Fig. 3 is ω _r=1.03, ω _r=1.02, ω _r=1.01 o'clock decoupling zero interval,

Fig. 4 (a) (b) is respectively wavelet amplitude figure into the bigger mode of the less mode of damping ratio in can not the intensive mode of decoupling zero and damping ratio,

Fig. 5 (a) (b) is respectively linear comparison diagram and the logarithm comparison diagram that auto-power spectrum method and small echo cumlative energy spectrum has damped frequency identification,

Fig. 6 is wavelet amplitude curve and Wavelet Phase curve,

Fig. 7 is f ₀-σ curve.

Embodiment

(1) the judgement index of the intensive mode of definition civil engineering structure.

By small echo cumlative energy spectrum, the structural vibrations natural frequency is can priori known (to be ω _rPriori is known).If ω _r＞1.12, then adjacent mode is sparse mode; Otherwise, then be intensive mode.If mode is intensive mode, then undergo mutation (reversion) by near slope in wavelet transformation index amplitude curve is initial value, promptly whether exist surging the interference to judge whether intensive mode can decoupling zero.Concrete steps are as follows:

For system with several degrees of freedom, the small echo index of its free damping response is:

$W_{g}x(a,b)\&ap;\frac{\sqrt{2\mathrm{\&pi;a}}}{2}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\left(A_i{e}^{{\mathrm{\&alpha;}}_i+{\mathrm{j\&beta;}}_i}\right)---\left(1\right)$

Wherein: α _i=-ξ _iω _NiB-(1/2) [(1-2 ξ _i ²) ω _Ni ²a ²+ ω ₀ ²-2 ω ₀ω _DiA],

β _i＝-ξ _iω _niω _dia ²+ω ₀ω _naξ _i+ω _dib-φ _i

A _i, φ _i, ξ _i, ω _Di, ω _NiBe respectively system's i rank mode amplitude, phasing degree, damping ratio, damping inherent circular frequency and undamped inherent circular frequency are arranged; A is the small echo contraction-expansion factor, and b is the small echo shift factor.

Natural frequency for adjacent mode differs bigger, and damping ratio separately also hour, when a gets certain fixed value

The time, | W _gX (a, b) | obtain maximum value.Because k rank mode is to the contribution maximum of wavelet coefficient, the wavelet coefficient values of other mode correspondences is very little, can ignore.And then can be in the hope of modal parameters.

If but the natural frequency of two mode is very approaching, and damping ratio separately is also bigger, then is presented as two peak mutual superposition at frequency domain, so just can't use as above algorithm realization to multi-modal separation, and then influences the precision of multi-modal parameter recognition ^[1]Document [2] adopts a frequency closeness factor gamma=(f ₂-f ₁)/(f ₂+ f ₁) weigh the frequency of adjacent mode near degree.But, frequency closeness one regularly, mode aliasing degree is also relevant with damping ratio, the relational expression of therefore only considering frequency is not enough to correctly weigh the degree that influences each other between mode.

I). sparse mode and intensive mode

At first discuss, wavelet transformation carried out in the free damping response get with the two-freedom system:

$W_{g}x(a,b)\&ap;\frac{\sqrt{2\mathrm{\&pi;a}}}{2}\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}\left(A_i{e}^{{\mathrm{\&alpha;}}_i+{\mathrm{j\&beta;}}_i}\right)---\left(2\right)$

Wherein: α _i=-ξ _iω _NiB-(1/2) [(1-2 ξ _i ²) ω _Ni ²a ²+ ω ₀ ²-2 ω ₀ω _DiA]

β _i＝-ξ _iω _niω _dia ²+ω ₀ω _naξ _i+ω _dib-φ _i

In order to allow the small echo index decoupling zero of adjacent mode, need make | W _gx ₁(a ₁, b) |＞＞| W _gx ₂(a ₁, b) |, and | W _gx ₁(a ₂, b) |＜＜| W _gx ₂(a ₂, b) |, wherein $a_i=\frac{{\mathrm{\&omega;}}_0\sqrt{1-{{\mathrm{\&xi;}}_i}^2}}{{\mathrm{\&omega;}}_{\mathrm{ni}}(1-2{{\mathrm{\&xi;}}_i}^2)}(i=\mathrm{1,2}){.}^{\left[3\right]}$

Make ω _r=ω _N2/ ω _N1＞1, then get by formula (2):

$\left|W_g{x}_1(a_1,b)\right|=\frac{\sqrt{2\mathrm{\&pi;}{a}_1}{\mathrm{<figure-callout\; id="1"\; label="A"\; filenames="HDA0000069099530000031.png,HDA0000069099530000033.png"\; state="\{\{state\}\}">A</figure-callout>}}_1}{2}{e}^{-{\mathrm{\&xi;}}_1{\mathrm{\&omega;}}_{n1}b+{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000069099530000031.png,HDA0000069099530000033.png"\; state="\{\{state\}\}">r</figure-callout>}}_1{{\mathrm{\&omega;}}_0}^2},$ $\left|W_g{x}_2(a_1,b)\right|=\frac{\sqrt{2\mathrm{\&pi;}{a}_1}{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="HDA0000069099530000021.png,HDA0000069099530000022.png"\; state="\{\{state\}\}">A</figure-callout>}}_2}{2}{e}^{-{\mathrm{\&xi;}}_2{\mathrm{\&omega;}}_{n2}b+({\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000069099530000031.png,HDA0000069099530000033.png"\; state="\{\{state\}\}">r</figure-callout>}}_1+r_2){{\mathrm{\&omega;}}_0}^2}---\left(3\right)$

$\left|W_g{x}_1(a_2,b)\right|=\frac{\sqrt{2\mathrm{\&pi;}{a}_2}{\mathrm{<figure-callout\; id="1"\; label="A"\; filenames="HDA0000069099530000031.png,HDA0000069099530000033.png"\; state="\{\{state\}\}">A</figure-callout>}}_1}{2}{e}^{-{\mathrm{\&xi;}}_1{\mathrm{\&omega;}}_{n1}b+({\mathrm{<figure-callout\; id="3"\; label="r"\; filenames="HDA0000069099530000033.png"\; state="\{\{state\}\}">r</figure-callout>}}_3+r_4){{\mathrm{\&omega;}}_0}^2},$ $\left|W_g{x}_2(a_2,b)\right|=\frac{\sqrt{2\mathrm{\&pi;}{a}_2}{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="HDA0000069099530000021.png,HDA0000069099530000022.png"\; state="\{\{state\}\}">A</figure-callout>}}_2}{2}{e}^{-{\mathrm{\&xi;}}_2{\mathrm{\&omega;}}_{n2}b+{\mathrm{<figure-callout\; id="3"\; label="r"\; filenames="HDA0000069099530000033.png"\; state="\{\{state\}\}">r</figure-callout>}}_3{{\mathrm{\&omega;}}_0}^2}---\left(4\right)$

Wherein ${\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="HDA0000069099530000031.png,HDA0000069099530000033.png"\; state="\{\{state\}\}">r</figure-callout>}}_1=\frac{{{\mathrm{\&xi;}}_1}^2}{2(1-2{{\mathrm{\&xi;}}_1}^2)},$ ${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000069099530000021.png,HDA0000069099530000022.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\frac{({{\mathrm{\&xi;}}_2}^2-{{\mathrm{\&xi;}}_1}^2){{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000069099530000021.png,HDA0000069099530000022.png"\; state="\{\{state\}\}">r</figure-callout>}}}^2}{2{(1-2{{\mathrm{\&xi;}}_1}^2)}^2}-\frac{{(\sqrt{1-{{\mathrm{\&xi;}}_2}^2}{\mathrm{\&omega;}}_r-\sqrt{1-{{\mathrm{\&xi;}}_1}^2})}^2}{2(1-{{\mathrm{\&xi;}}_1}^2)}---\left(5a\right)$

${\mathrm{<figure-callout\; id="3"\; label="r"\; filenames="HDA0000069099530000033.png"\; state="\{\{state\}\}">r</figure-callout>}}_3=\frac{{{\mathrm{\&xi;}}_2}^2}{2(1-2{{\mathrm{\&xi;}}_2}^2)},$ ${\mathrm{<figure-callout\; id="4"\; label="r"\; filenames="HDA0000069099530000033.png"\; state="\{\{state\}\}">r</figure-callout>}}_4=\frac{({{\mathrm{\&xi;}}_1}^2-{{\mathrm{\&xi;}}_2}^2)}{2{(1-2{{\mathrm{\&xi;}}_2}^2)}^2{{\mathrm{\&omega;}}_r}^2}-\frac{{(\sqrt{1-{{\mathrm{\&xi;}}_1}^2}{\mathrm{\&omega;}}_r-\sqrt{1-{{\mathrm{\&xi;}}_2}^2})}^2}{2(1-{{\mathrm{\&xi;}}_1}^2){{\mathrm{\&omega;}}_r}^2}---\left(5b\right)$

For decoupling zero, following condition must be set up:

$\frac{\left|W_g{x}_1(a_1,b)\right|}{\left|W_g{x}_2(a_1,b)\right|}=\frac{A_1}{A_2}{e}^{({\mathrm{\&xi;}}_2{\mathrm{\&omega;}}_{n2}-{\mathrm{\&xi;}}_1{\mathrm{\&omega;}}_{n1})b-{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000069099530000021.png,HDA0000069099530000022.png"\; state="\{\{state\}\}">r</figure-callout>}}_2{{\mathrm{\&omega;}}_0}^2}>>1---\left(6\right)$

$\frac{\left|W_g{x}_2(a_2,b)\right|}{\left|W_g{x}_1(a_2,b)\right|}=\frac{A_2}{A_1}{e}^{({-\mathrm{\&xi;}}_2{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000069099530000021.png,HDA0000069099530000022.png"\; state="\{\{state\}\}">n</figure-callout>}2}+{\mathrm{\&xi;}}_1{\mathrm{\&omega;}}_{n1})b-{\mathrm{<figure-callout\; id="4"\; label="r"\; filenames="HDA0000069099530000033.png"\; state="\{\{state\}\}">r</figure-callout>}}_4{{\mathrm{\&omega;}}_0}^2}>>1---\left(7\right)$

For intensive mode, because | (ξ ₂ω _N2-ξ ₁ω _N1) the suitable I of value of b| ignores, so can simplify processing (supposition A to formula (6) (7) ₁=A ₂):

$\frac{\left|W_g{x}_1(a_1,b)\right|}{\left|W_g{x}_2(a_1,b)\right|}\&ap;e^{-{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000069099530000021.png,HDA0000069099530000022.png"\; state="\{\{state\}\}">r</figure-callout>}}_2{{\mathrm{\&omega;}}_0}^2}\&GreaterEqual;10---\left(8\right)$

$\frac{\left|W_g{x}_2(a_2,b)\right|}{\left|W_g{x}_1(a_2,b)\right|}\&ap;e^{-{\mathrm{<figure-callout\; id="4"\; label="r"\; filenames="HDA0000069099530000033.png"\; state="\{\{state\}\}">r</figure-callout>}}_4{{\mathrm{\&omega;}}_0}^2}\&GreaterEqual;10---\left(9\right)$

Coefficient r ₂And r ₄Only relevant with the inherent characteristic of structure.Only have at r ₂＜0 and r ₄Under＜0 the condition, ω ₀Obtain enough big, feasible-r ₂ω ₀ ²With-r ₄ω ₀ ²Get enough greatly, can make inequality (8) (9) set up.

Suppose that intensive mode can decoupling zero, then by r ₂＜0 and r ₄＜0 can get ω _rPermissive condition:,

1) ξ ₁＞ξ ₂The time, ${\mathrm{\&omega;}}_r>\sqrt{\frac{1-{{\mathrm{\&xi;}}_2}^2}{1-{{\mathrm{\&xi;}}_1}^2}}+\sqrt{\frac{{{\mathrm{\&xi;}}_1}^2-{{\mathrm{\&xi;}}_2}^2}{(1-{{\mathrm{\&xi;}}_1}^2)(1-2{{\mathrm{\&xi;}}_1}^2)}}$ (because of ω _r＞1, so abandon

${\mathrm{\&omega;}}_r<\sqrt{\frac{1-{{\mathrm{\&xi;}}_2}^2}{1-{{\mathrm{\&xi;}}_1}^2}}-\sqrt{\frac{{{\mathrm{\&xi;}}_1}^2-{{\mathrm{\&xi;}}_2}^2}{(1-{{\mathrm{\&xi;}}_1}^2)(1-2{{\mathrm{\&xi;}}_1}^2)}})$

2) ξ ₁＜ξ ₂The time, ${\mathrm{\&omega;}}_r>\frac{\sqrt{1-{{\mathrm{\&xi;}}_1}^2}}{\sqrt{1-{{\mathrm{\&xi;}}_2}^2}-\sqrt{\frac{{{\mathrm{\&xi;}}_2}^2-{{\mathrm{\&xi;}}_1}^2}{1-2{{\mathrm{\&xi;}}_1}^2}}}$ (because of ω _r＞1, so abandon ${\mathrm{\&omega;}}_r<\frac{\sqrt{1-{{\mathrm{\&xi;}}_1}^2}}{\sqrt{1-{{\mathrm{\&xi;}}_2}^2}+\sqrt{\frac{{{\mathrm{\&xi;}}_2}^2-{{\mathrm{\&xi;}}_1}^2}{1-2{{\mathrm{\&xi;}}_1}^2}}}$ )

Comprehensive various countries standard, the damping ratio of steel construction are generally between 0.01-0.02, and the damping ratio of reinforced concrete structure is generally 0.03-0.08, so the patent applicant is only to (ξ ₁, ξ ₂) ∈ [0,0.1] ²Analyze discussion.Work as ω _r＞1.12 o'clock, (ξ ₁, ξ ₂) decoupling zero zone cover whole zone, so define ω _r＞1.12 o'clock is sparse mode, ω _r≤ 1.12 o'clock is intensive mode.

But ii). the intensive mode of decoupling zero and can not the intensive mode of decoupling zero

By top discussion analysis, ω _r≤ 1.12 o'clock is intensive mode.Convolution (5)-(9) can obtain ω _r=1.03, ω _r=1.02, ω _rCan use (the ξ of small wave converting method decoupling zero at=1.01 o'clock ₁, ξ ₂) distributed areas.As shown in Figure 3, along with ω _rReduce, the decoupling zero interval is drawn close gradually to the miter angle bisector, the decoupling zero interval reduces gradually.Adjacent two rank modal dampings are than outside the decoupling zero interval time, and mode aliasing degree is serious more, cause the intensive mode can't decoupling zero (as Fig. 2 (b)).

Find: ω _rOne regularly, and damping ratio is big more, and the width in decoupling zero interval is more little, i.e. the condition of decoupling zero is just harsh more; In addition, the damping ratio difference of adjacent mode is big more, and mode aliasing degree is serious more.

When using the parameter recognition of Moret wavelet method recognition structure, should note: when adjacent two rank mode were intensive mode, when having only the structure self-characteristic to satisfy certain condition, decoupling zero just became possibility.Otherwise, even ω ₀Value is enough big, and method of wavelet also can't be with intensive mode decoupling zero.

At intensive mode, has only the ω of working as _r, ξ ₁, ξ ₂Satisfy r ₂＜0 and r ₄＜0 o'clock, intensive mode can decoupling zero (as Fig. 2 (c)).Otherwise intensive mode can not decoupling zero (as Fig. 2 (b)).But,, use so can suppose earlier that intensive mode can decoupling zero because the damping ratio of adjacent mode is to know in advance $f_{}$ ${}_0=\mathrm{\&alpha;}\sqrt{2}{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000069099530000031.png,HDA0000069099530000033.png"\; state="\{\{state\}\}">\; <figure-callout\; id="2"\; label="f"\; filenames="HDA0000069099530000021.png,HDA0000069099530000022.png"\; state="\{\{state\}\}">f</figure-callout>\; </figure-callout>}}_{\mathrm{1,2}}/\left(2\mathrm{\&pi;\&Delta;}{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA0000069099530000031.png,HDA0000069099530000033.png"\; state="\{\{state\}\}">f</figure-callout>}}_{\mathrm{1,2}}\right){(\mathrm{\&alpha;}=2)}^{\left[4\right]}$ Carrying out small echo changes.But the intensive mode (ω of decoupling zero _r=1.10, f ₁=1.10Hz, f ₂=1.21Hz, ξ ₁=0.014, ξ ₂=0.012) and can not the intensive mode (ω of decoupling zero _r=1.10, f ₁=1.10Hz, f ₂=1.21Hz, ξ ₁=0.014, ξ ₂=0.070) example demonstration (seeing Table 1).

But intensive mode of table 1 decoupling zero and recognition result that can not the intensive mode of decoupling zero

But during intensive mode decoupling zero, small echo index adjacent peaks is distinguished obviously, the wavelet amplitude line smoothing, and slope is obvious, and damping ratio identification is accurately.In the time of can not decoupling zero, small echo adjacent peaks aliasing be serious.In the adjacent mode, the mode wavelet amplitude line smoothing that damping ratio is less, slope is obvious, and damping ratio identification is accurate, as Fig. 4 (a).But there be surging the interference in the bigger mode wavelet amplitude curve of damping ratio, and as Fig. 4 (b): there is sudden change in slope at initial position; But along with the increase of t, curve is gradually level and smooth, and the less mode small echo index of damping ratio this moment is seriously polluted to it.

As shown in table 1, recognition result shows when intensive mode can not decoupling zero, and the less modal damping of damping ratio is more accurate than identification, and the bigger modal damping of damping ratio is than discerning serious distortion, and the result who identifies is the smaller damping ratio of adjacent modal damping.

By above analysis discussion, draw to draw a conclusion:

(1) ω _r=1.12 is the distinguishing limit of sparse mode (as Fig. 2 (a)) and intensive mode (as Fig. 2 (b) and (c));

(2) at intensive mode, has only the ω of working as _r, ξ ₁, ξ ₂Satisfy r ₂＜0 and r ₄＜0 o'clock, intensive mode can decoupling zero (as Fig. 2 (c)).Otherwise intensive mode can not decoupling zero (as Fig. 2 (b)).

Comprehensive above conclusion proposes adjacent Modal Parameter Identification flow process (as Fig. 1 II part):

By auto-power spectrum method or small echo cumlative energy spectrum, the structural vibrations natural frequency is can priori known (to be ω _rPriori is known).If ω _r＞1.12, then adjacent mode is sparse mode; Otherwise, then be intensive mode.If mode is intensive mode, then whether exist surging the interference to judge whether intensive mode can decoupling zero by the wavelet transformation indicatrix.

(2) proposition and the checking of small echo cumlative energy spectrum

The abstracting method of present crestal line commonly used is to extract local maximum in the small echo spirogram, and this kind method is better to the extraction effect of linear ridges, but is subject to the interference of noise and " limit end effect ", and operand is big.Some scholars adopt marginal spectrometry, though this method operand is less, equally also have been subjected to the interference of " limit end effect ".The patent applicant proposes the ridge point that the feasible method of a kind of convenience is extracted the small echo spirogram: small echo cumlative energy spectrum (Accumulated-Energy Spectrum is called for short AES).For the time-histories response of many-degrees of freedom system, its concentration of energy is near the little wave crest of each natural frequency behind the wavelet transformation.Therefore to small echo mark sense frequency axis projection summation, near each rank vibration natural frequency bigger peak value will inevitably appear.And The noise is at random (white noise has covered whole time domain), can eliminate the influence of partial noise by the adding up on time domain of small echo index.Simultaneously, consider the pollution of elimination " limit end effect ", can only take the wavelet transformation index and on [β Δ t, T-β Δ t] (β=3), analyze the wavelet transformation index.

$\mathrm{AES}\left(f\right)=\underset{b=\mathrm{\&beta;\&Delta;t}}{\overset{T-\mathrm{\&beta;\&Delta;t}}{\mathrm{\&Sigma;}}}\left|\mathrm{Wab}(a,b)\right|=\underset{t=\mathrm{\&beta;\&Delta;t}}{\overset{T-\mathrm{\&beta;\&Delta;t}}{\mathrm{\&Sigma;}}}\left|\mathrm{Wab}(f,t)\right|---\left(10\right)$

Use auto-power spectrum method and small echo cumlative energy spectrum to carry out frequency identification respectively to certain structural vibration response signal, the result as shown in Figure 5.The frequency location that both identify does not almost have difference.But dotted line indicates as can be seen from figure, on the 4th rank and the 8th rank mode damped frequency 13.08Hz and 50.00Hz is arranged, and peakedness ratio auto-power spectrum method is more obvious in the small echo cumlative energy spectral curve.And the amplitude difference that the small echo cumlative energy is composed each peak value does not have the auto-power spectrum method so obvious yet, and is easier to the identification of frequency like this.Simultaneously, for intensive mode, the effect that the AES method is separated mode is good more.Shown in mark oval among Fig. 5, small echo cumlative energy spectrum is easy to distinguish structure the 1st rank and there are damped vibration natural frequency 5.90Hz and 6.96Hz in the 2nd rank.And the side frequency of auto-power spectrum method identification modal coupling is comparatively difficult.

(3) the wavelet amplitude fit interval chooses

For eliminating the influence of " limit end effect ", analystal section should be selected (getting β=3) in [β Δ t, T-β Δ t].Be necessary for the descending branch of wavelet amplitude between extracting because of branch again, so fit interval should be at [t (ln|W _gX (a _k, t) | _Max, t (ln|W _gX (a _k, t) | _Min))].To sum up, make the small echo fit interval be [t ₁, t ₂], t wherein ₁=max[β Δ t, t (ln|W _gX (a _k, t) | _Max)], t ₂=min[T-β Δ t, t (ln|W _gX (a _k, t) | _Min)].

(4) example checking:

The numerical example with a two-freedom is an example below, illustrates how to carry out based on the intensive Modal Parameter Identification of the structure of Moret wavelet transformation.

Construct the intensive modal system free vibration of a double freedom signal:

$x\left(t\right)=\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{A}_i{e}^{-2\mathrm{\&pi;}{\mathrm{\&xi;}}_i{f}_{i}t}\cos (2\mathrm{\&pi;}\sqrt{1-{{\mathrm{\&xi;}}_i}^2}{f}_{i}t-{\mathrm{\&phi;}}_i)+\mathrm{noise}\left(t\right)---\left(11\right)$

Parameter is in the formula: A ₁=A ₂=1, f ₁=1.10Hz, f ₂=1.21Hz, ξ ₁=0.014, ξ ₂=0.012, φ ₁=φ ₂=0, noise (t) is the gaussian random white noise.

(1) uses the Moret small echo, structure free damping response x (t) is carried out conversion obtain small echo index W _gX (a, b)

(2) interior in b ∈ [β Δ t, T-β Δ t] scope to small echo index W _gX (a b) adds up and draws small echo cumlative energy spectrum AES, and the computing formula of small echo cumlative energy spectrum AES is as follows:

$\mathrm{AES}\left(f\right)=\underset{b=\mathrm{\&beta;\&Delta;t}}{\overset{T-\mathrm{\&beta;\&Delta;t}}{\mathrm{\&Sigma;}}}\left|\mathrm{Wab}(a,b)\right|=\underset{t=\mathrm{\&beta;\&Delta;t}}{\overset{T-\mathrm{\&beta;\&Delta;t}}{\mathrm{\&Sigma;}}}\left|\mathrm{Wab}(f,t)\right|$

T=102.4s in the formula, β=3, f _D1=1.102Hz (can obtain) by auto-power spectrum method priori.According to each rank of maximum point recognition structure of AES damped vibration natural frequency f arranged _D1=1.102Hz ' f _D2=1.212Hz °

(3) each rank model frequency f of structure that draws according to step (2) _D1, f _D2Judge whether adjacent mode is intensive mode.ω _r=1.0098＜1.12, then structure the 1st, 2 rank are intensive mode, need carry out damping ratio and discern in advance.

(4) intensive modal damping is than pre-identification.X (t) is carried out wavelet transformation, wavelet center frequency $f_{}$ ${}_0=2\sqrt{2}{f}_{i,i+1}/\left(2\mathrm{\&pi;\&Delta;}{f}_{i,i+1}\right)=4.5016\mathrm{Hz},$ Observe the wavelet transformation indicatrix, there be not surging the interference in curve, and this intensive mode can decoupling zero.To small echo index amplitude at [t ₁, t ₂] carry out linear fit, the frequency of recognition structure and damping ratio in the interval.T wherein ₁=max[β Δ t, t (ln|W _gX (a _k, t) | _Max)]=15s, t ₂=min[T-β Δ t, t (ln|W _gX (a _k, t) | _Min)]=80s.(as Fig. 6)

(5) computing center's frequency minima.The damping ratio that top step is tried to achieve

The substitution inequality

With

In find the solution, wavelet center frequency need satisfy the minimum value f of decoupling zero condition _0min=3.768Hz.Wherein

${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000069099530000021.png,HDA0000069099530000022.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\frac{({{\mathrm{\&xi;}}_2}^2-{{\mathrm{\&xi;}}_1}^2){{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000069099530000021.png,HDA0000069099530000022.png"\; state="\{\{state\}\}">r</figure-callout>}}}^2}{2{(1-2{{\mathrm{\&xi;}}_1}^2)}^2}-\frac{{(\sqrt{1-{{\mathrm{\&xi;}}_2}^2}{\mathrm{\&omega;}}_r-\sqrt{1-{{\mathrm{\&xi;}}_1}^2})}^2}{2(1-{{\mathrm{\&xi;}}_1}^2)}=-0.0050,$

${\mathrm{<figure-callout\; id="4"\; label="r"\; filenames="HDA0000069099530000033.png"\; state="\{\{state\}\}">r</figure-callout>}}_4=\frac{({{\mathrm{\&xi;}}_1}^2-{{\mathrm{\&xi;}}_2}^2)}{2{(1-2{{\mathrm{\&xi;}}_2}^2)}^2{{\mathrm{\&omega;}}_r}^2}-\frac{{(\sqrt{1-{{\mathrm{\&xi;}}_1}^2}{\mathrm{\&omega;}}_r-\sqrt{1-{{\mathrm{\&xi;}}_2}^2})}^2}{2(1-{{\mathrm{\&xi;}}_1}^2){{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HDA0000069099530000021.png,HDA0000069099530000022.png"\; state="\{\{state\}\}">r</figure-callout>}}}^2}=-0.0041$

(6) choosing of wavelet center frequency is optimized calculating.Get $f_{}$ ${}_0\&Element;[{\mathrm{<figure-callout\; id="0"\; label="f"\; filenames="HDA0000069099530000022.png,HDA0000069099530000023.png"\; state="\{\{state\}\}">f</figure-callout>}}_{0\min },\mathrm{\&alpha;}\sqrt{2}{f}_{i,i+1}/\left(2\mathrm{\&pi;\&Delta;}{f}_{i,i+1}\right)](\mathrm{\&alpha;}=4)$ Analyze i.e. [3.77Hz, 9.00Hz], the fitting a straight line L (ln|W that each centre frequency in the scope is found the solution wavelet amplitude _gX (a _k, t) |)

(7) basis of calculation difference σ=std (ln|W _gX (a _i, t) |-L (ln|W _gX (a _i, t) |)), a _iIt is the scale factor of i rank mode correspondence.

(8) identification modal parameter.According to f ₀-σ curve (as Fig. 7) is tried to achieve the 1st rank and the 2nd rank wavelet center frequency f of the minimum value correspondence of σ _0-1=7.4325Hz, f _0-2=5.8619Hz carries out wavelet transformation with them for centre frequency, draws its wavelet amplitude curve respectively.And then recognition structure damping ratio (as table 2).As shown in Table 2, the method recognition result is better, and has antinoise interference performance preferably.

The contrast of table 2 modal parameters recognition result

List of references:

[1]Staszewski?WJ.Identification?of?damping?in?MDOF?systems?using?time-scale?decomposition[J].Journal?of?Sound?and?Vibration，1997，203(2)：283-305.

[2]Lardies?J，Ta?M?N，Berthither?M.Modal?parameter?estimation?based?on?the?wavelet?transform?of?output?data[J].Archive?of?Applied?Mechanic，2004，73：718-733.

[3] Huang Tianli, Lou Menglin. the application [J] of wavelet transformation in intensive modal structure parameter recognition. vibration and impact .2006,25 (5): 149-152.

[4] Ren Weixin, Han Jiangang, Sun Zengshou. the application [M] of wavelet analysis in civil engineering structure. Beijing: China Railway Press, 2006.

Claims (1)

1. intensive Modal Parameters Identification of the structure based on the Moret wavelet transformation is characterized in that this method may further comprise the steps:

1) uses the Moret small echo, get f ₀For wavelet center frequency conversion is carried out in structure free damping response and obtained small echo index W _gX (a, b), wherein, a is the small echo contraction-expansion factor, b is the small echo shift factor;

2) interior in b ∈ [β Δ t, T-β Δ t] scope to small echo index W _gX (a b) adds up and draws small echo cumlative energy spectrum AES, and the computing formula of small echo cumlative energy spectrum AES is as follows:

$\mathrm{AES}\left(f\right)=\underset{b=\mathrm{\&beta;\&Delta;t}}{\overset{T-\mathrm{\&beta;\&Delta;t}}{\mathrm{\&Sigma;}}}\left|\mathrm{Wab}(a,b)\right|=\underset{t=\mathrm{\&beta;\&Delta;t}}{\overset{T-\mathrm{\&beta;\&Delta;t}}{\mathrm{\&Sigma;}}}\left|\mathrm{Wab}(f,t)\right|$

F=f in the formula ₀/ a is a frequency, and t=b is the time, and T is the free damping response time,

β=3, f _D1Be the structure first stage structure vibration natural frequency.What the frequency of obtaining AES maximum point correspondence was each rank of structure has a damped vibration natural frequency f _Di

3) according to each rank model frequency f of structure _DiJudge whether adjacent mode is intensive mode, the index ω of intensive mode and sparse mode is distinguished in definition _Ri=f _{N (i+1)}/ f _Ni≈ f _{D (i+1)}/ f _Di＞1, f wherein _Ni, f _{N (i+)}Be respectively structure i rank and i+1 rank undamped oscillation natural frequency, f _Di, f _{D (i+1)}Be respectively structure i rank and i+1 rank the damped vibration natural frequency is arranged; For general civil engineering structure, if ω _r＞1.12, then adjacent mode is sparse mode, illustrates that adjacent mode can discern, and next step carries out step 4); Otherwise then it is intensive mode, need carry out damping ratio and discern in advance, and next step carries out step 5);

4) sparse modal damping is discerned than pre-: wavelet transformation, wavelet center frequency are carried out in the free damping response

F wherein _{I, i+1}Be structure i order frequency or i+1 frequency, Δ f _{I, i+1}Poor for both is to eliminate " limit end effect ", sets wavelet amplitude linear analysis interval at t ∈ [β Δ t, T-β Δ t], in β=3; To small echo index amplitude at [t ₁, t ₂] carry out linear fit, the damping ratio of recognition structure in the interval T wherein ₁=max[β Δ t, t (ln|W _gX (a _k, t) | _Max)], t ₂=min[T-β Δ t, t (ln|W _gX (a _k, t) | _Min)];

5) intensive modal damping is discerned than pre-: wavelet transformation, wavelet center frequency are carried out in the free damping response

If wavelet transformation index amplitude curve is at t ₁Near in slope to undergo mutation be reversion, then this intensive mode cannot decoupling zero; Otherwise, then can decoupling zero, wavelet transformation, wavelet center frequency are carried out in the free damping response

To small echo index amplitude at [t ₁, t ₂] carry out linear fit, the damping ratio of recognition structure in the interval

6) computing center's frequency minima:

At first calculate $r_2=\frac{({{\mathrm{\&xi;}}_2}^2-{{\mathrm{\&xi;}}_1}^2){{\mathrm{\&omega;}}_r}^2}{2{(1-2{{\mathrm{\&xi;}}_1}^2)}^2}-\frac{{(\sqrt{1-{{\mathrm{\&xi;}}_2}^2}{\mathrm{\&omega;}}_r-\sqrt{1-{{\mathrm{\&xi;}}_1}^2})}^2}{2(1-{{\mathrm{\&xi;}}_1}^2)},$

$r_4=\frac{({{\mathrm{\&xi;}}_1}^2-{{\mathrm{\&xi;}}_2}^2)}{2{(1-2{{\mathrm{\&xi;}}_2}^2)}^2{{\mathrm{\&omega;}}_r}^2}-\frac{{(\sqrt{1-{{\mathrm{\&xi;}}_1}^2}{\mathrm{\&omega;}}_r-\sqrt{1-{{\mathrm{\&xi;}}_2}^2})}^2}{2(1-{{\mathrm{\&xi;}}_1}^2){{\mathrm{\&omega;}}_r}^2}$

Then with the damping ratio of trying to achieve

The substitution following formula, and in conjunction with inequality

With

Get the minimum value f that wavelet center frequency need satisfy the decoupling zero condition _0min

7) choosing of wavelet center frequency is optimized calculating, gets

Analyze,, calculate the fitting a straight line L (ln|W of wavelet amplitude for each centre frequency _gX (a _k, t) |), and calculate fitting a straight line L (ln|W _gX (a _k, t) |) standard deviation sigma=std (ln|W _gX (a _i, t) |-L (ln|W _gX (a _i, t) |)), a _iIt is the scale factor of i rank mode correspondence;

8) identification modal parameter: the i rank and the i+1 rank wavelet center frequency f that get the minimum value correspondence of σ _0-i, f _0-(i+1), signal is carried out wavelet transformation respectively, try to achieve modal parameter end value f _Ni, f _{N (i+1)}, ξ _i, ξ _I+1

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