CN107702789A - Using the body structure surface transient state vibration velocity method for reconstructing of particle plane vibration speed measurement - Google Patents
Using the body structure surface transient state vibration velocity method for reconstructing of particle plane vibration speed measurement Download PDFInfo
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Abstract
The invention discloses a kind of body structure surface transient state vibration velocity method for reconstructing using particle plane vibration speed measurement, and two holographic facets conformal with body structure surface and equivalent source face are respectively arranged in structural outer and inside；Gather the time domain normal direction particle vibration velocity signal on holographic facet；Using the time-domain equivalent relation that majorant for integral is established between the time domain normal direction particle vibration velocity on holographic facet and the time domain normal direction vibration velocity of body structure surface as medium in a steady stream；The time domain normal direction particle vibration velocity on holographic facet and known Green functions are recycled, all time-domain equivalents of all moment majorant for integral in a steady stream is solved by iteration inversion process；It is final that using the time-domain equivalent solved, majorant for integral calculates the normal direction vibration velocity of each moment body structure surface in a steady stream.The inventive method provides higher reconstruction precision using time domain particle vibration velocity as input, to rebuild body structure surface transient state vibration velocity；This method possesses the ability of processing random appearance structure, the measurement available for complex appearance body structure surface transient state vibration velocity in Practical Project.
Description
Technical field
The present invention relates to the technical field of noise class in Speciality of Physics, and in particular to a kind of knot using particle plane vibration speed measurement
Structure surface transient vibration velocity method for reconstructing.
Background technology
Body structure surface vibration information not only can aid in structure improvement and fail-safe analysis, and be also that research power source is known
Other and fault diagnosis important input quantity.Generally using laser vibration measurer come measurement structure surface vibration velocity, but laser vibration measurer
Need vibrational structure surface reflection characteristic good, or need vibrational structure easily to paste reflective sheeting to strengthen its surface reflection spy
Property.But this demand tends not to enough satisfactions in many structures, such as surface high-temp equipment, surface electromagnetic equipment and abnormal
Shape surface etc..Therefore need to obtain accurate vibration velocity signal using a kind of non-optical measuring method.Sound source in vibration, because
This can reconstruct the vibration velocity signal of body structure surface by acoustic method.Near field acoustic holography (NAH) is exactly a kind of such acoustics
Method, it reconstructs the vibratory output of body structure surface by measuring the acoustics amount in sound field.Up to the present, domestic and foreign scholars have been
It is proposed a variety of NAH methods：M.Martarelli etc. and C.Pezerat etc. proposed at 2006 and 2009 based on two dimension respectively
The NAH of spatial Fourier transform carrys out the vibration velocity of the resonant frequency of reconstruction plate, mode and plate surface, and with laser-Doppler vibration measuring
The measurement result of instrument matches, and illustrates the validity of NAH vibration measuring methods；H.C.Lu was utilized based on Helmholtz most in 2007
The NAH of small square law realizes the reconstruction of clamped plate surface vibration velocity；J.G.In etc. and E.G.Williams etc. is utilized and is based on border
The NAH methods of member realize the reconstruction of freezer compressor and aircraft fuselage skin vibration velocity respectively；Bi Chuanxing etc. utilizes distributed point side
The surface vibration velocity that the NAH of boundary source method realizes Roots vaccum pump under running status is rebuild.But the reconstruction of above-mentioned NAH vibration measurings method
As a result all it is to be presented with the vibration mode in single-frequency or single narrow frequency range, analyzes, be not used to applied to steady-state vibration
The reconstruction of transient oscillation.Transient oscillation not only needs the reconstruction some time to inscribe mode of oscillation distribution, and needs to obtain body structure surface
The vibration characteristics of point how change over time and change.Therefore, X.Z.Zhang etc. proposed in 2012 based on acoustic pressure with
The real-time NAH vibration measurings method of vibration velocity impulse response function, and rebuild the surface transient vibration velocity of one piece of impact iron plate.In order to reduce
Wrap around error etc., L.Geng etc. caused by two-dimensional space Fourier transformation propose in real-time NAH is superimposed based on Time Domain Planar ripple
The shock plate surface transient acceleration method for reconstructing of method.In order to realize that spherical structure surface vibration velocity is rebuild, S.F.Wu etc. is proposed
Transient state HELS methods.But these methods are suitable only for planar structure or spherical structure.
The content of the invention
In order to realize, complex appearance body structure surface vibration velocity is rebuild under any time, is shaken the invention provides one kind using particle
The transient state vibration velocity method for reconstructing of speed measurement.For this method using particle vibration velocity as input, reconstruction precision is high, and can handle arbitrary shape
Structure, and then suitable for the time-varying vibration characteristics of analysis project labyrinth in practice.
The present invention is to solve the technical scheme that technical problem uses to be：A kind of body structure surface wink using particle plane vibration speed measurement
State vibration velocity method for reconstructing, is carried out as follows；
Step 1, in vibrational structure outwardly and inwardly it is respectively arranged holographic facet H and equivalent source face E, and holographic facet H and equivalent
Source face E shape and body structure surface S is conformal；It is respectively uniformly distributed on described holographic facet H, equivalent source face E and body structure surface S
M measurement point, K time-domain equivalent source and L normal direction vibration velocity reconstruction point；Examination network size, equivalent source face E on holographic facet H
On equivalent source sizing grid it is identical with the reconstruction sizing grid on body structure surface S, and the examination network point position on holographic facet H
Put, the equivalent source position on the E of equivalent source face it is identical with the reconstruction point position on body structure surface S；Gather upper m-th of the survey of holographic facet H
Measure the time domain normal direction particle vibration velocity v of pointHm(t)；
Step 2, the time domain normal direction particle vibration velocity v according to formula (1) structure upper m-th of the measurement point of t holographic facet HHm(t) with
Time-domain equivalent majorant for integral in a steady streamAnd time-domain equivalent majorant for integral derivative in a steady streamBetween relation：
In formula (1), τHmk=t-RHmk/ c, τHmkRepresent between upper m-th of the measurement point of holographic facet H and k-th of time-domain equivalent source
Time delay, n is body structure surface normal unit vector, and ρ is Media density, and c is the spread speed of sound, RHmkRepresent holographic
The distance between upper m-th of the measurement point of face H and k-th time-domain equivalent source,Represent RHmkDirectional derivative on normal direction n；
By the time t in formula (1) it is discrete be ti=t0+ i Δ t, wherein Δ t are sampling time interval, i=1,2 ..., I, I
For total number of sample points, t0For initial time, then delay time THmkCan using discrete representation as：
Unified new time delay axle τ is established according to formula (3), and makes time delayFall into new time delay axle τ：
τj=τ0+jΔt (3)
In formula (3), j=1,2 ..., J, τ0=t0-RHmin/ c, RHminRepresent all RHmkIn minimum value；
According to formula (4) and (5), to time-domain equivalent, majorant for integral and its derivative introduce Lagrange interpolating functions in a steady stream respectively：
In formula (4) and (5),For τjK-th of equivalent source source strength integration of moment, Φj(τ) is Lagrange interpolation letters
Number；
Step 3, formula (4) and (5) are substituted into formula (1), structure tiThe time domain normal direction of upper m-th of the measurement point of moment holographic facet H
Particle vibration velocity vHm(t) shown in interpolation formula such as formula (6)：
In formula (6),
The summation form of formula (6) is converted into matrix form, builds tiThe method of all M measurement points on moment holographic facet H
To shown in the matrix representation forms such as formula (7) of particle vibration velocity：
In formula (7),
Wherein, the transposition of " T " representing matrix；
Step 4, in formula (7), as i=1, formula (7) is changed into：
In formula (11),WithIt is known.Pseudoinverse is asked to obtain least square solution by formula (12) singular value decomposition
Ψ1：
In formula (12), the pseudoinverse of "+" representing matrix；
As i=2, formula (7) is changed into：
In formula (13),WithIt is known, Ψ1It is to solve to obtain by formula (12).It is strange by formula (14)
Different value, which is decomposed, asks pseudoinverse to obtain least square solution Ψ2：
By that analogy, as i=I, formula (7) is changed into：
Pseudoinverse is asked to obtain least square solution Ψ by formula (16) singular value decompositionI：
Tikhonov regularization in standard regularization and generalized crossover are used in the solution formula (12), (14) and (16) at each moment
Verification technique solves suitable time-domain equivalent majorant for integral Ψ in a steady streami；
Step 5, the equivalent source source strength for solving each moment integrate ΨiIn substitution formula (17), institute on body structure surface S is calculated
There is the normal direction vibration velocity of all L reconstruction points constantly：
In formula (17),
Wherein,τSlkRepresent
Time delay between upper l-th of the reconstruction point of body structure surface S and k-th of time-domain equivalent source, RSlkRepresent body structure surface S upper l-th
The distance between reconstruction point and k-th time-domain equivalent source,Represent RSlkDirectional derivative on normal direction n.
Wherein, the time domain normal direction particle vibration velocity of each examination network point is to use vibration velocity sensor array on the holographic facet H
Once snapshot measurement obtains on holographic facet H.
Wherein, the structural vibration has time-varying characteristics, and the reconstruction of surface normal vibration velocity is carried out in time domain.
Compared with the prior art, the present invention has the beneficial effect that：
1st, the inventive method is using time domain particle vibration velocity as input, and higher weight is provided to rebuild body structure surface transient state vibration velocity
Build precision.
2nd, the inventive method can rebuild random appearance body structure surface transient state vibration velocity, therefore be more suitable in Practical Project complicated
The analysis of structure transient oscillation characteristic, important input quantity can be provided for the fault diagnosis of labyrinth.
Brief description of the drawings
Fig. 1 is the body structure surface transient state vibration velocity method for reconstructing schematic diagram that the present invention uses particle plane vibration speed measurement；
Fig. 2 (a) is that the inventive method tests vibrational structure, holographic facet and equivalent source face location map；
Fig. 2 (b) is that the inventive method tests mesh point division schematic diagram；
Fig. 3 (a) is the time domain normal direction vibration velocity signal graph at the inventive method experimental configuration surface A；
Fig. 3 (b) is the time domain normal direction vibration velocity signal graph at the inventive method experimental configuration surface B；
Fig. 3 (c) is the time domain normal direction vibration velocity signal graph at the inventive method experimental configuration surface C；
Fig. 3 (d) is the time domain normal direction vibration velocity signal graph at the inventive method experimental configuration surface D；
The phase valuation factor E that Fig. 4 (a) is obtained by the inventive method experimentpNumeric distribution figure；
The amplitude evaluation points E that Fig. 4 (b) is obtained by the inventive method experimentaNumeric distribution figure；
Fig. 5 (a) is the measurement normal direction vibration velocity Modal Space distribution on the inventive method t=3.50ms moment body structure surfaces S；
Fig. 5 (b) is the measurement normal direction vibration velocity Modal Space distribution on the inventive method t=5.31ms moment body structure surfaces S；
Fig. 5 (c) is the measurement normal direction vibration velocity Modal Space distribution on the inventive method t=6.78ms moment body structure surfaces S；
Fig. 5 (d) is the normal direction vibration velocity Modal Space distribution rebuild using the inventive method at the t=3.50ms moment；
Fig. 5 (e) is the normal direction vibration velocity Modal Space distribution rebuild using the inventive method at the t=5.31ms moment；
Fig. 5 (f) is the normal direction vibration velocity Modal Space distribution rebuild using the inventive method at the t=6.78ms moment.
Embodiment
Referring to Fig. 1, the present embodiment uses the body structure surface transient state vibration velocity method for reconstructing of particle plane vibration speed measurement, as follows
Rebuild body structure surface time domain normal direction vibration velocity：
Step 1, all types of sound sources can regard the combination of monopole as, therefore choose a series of monopole conducts
Equivalent source, which is arranged in inside vibrational structure, carrys out the approximate representation structure；As shown in figure 1, it is respectively arranged entirely in structural outer and inside
Breath face H and equivalent source face E, and holographic facet H and equivalent source face E shape and body structure surface S are conformal；In the holographic facet H, equivalent
M measurement point, K time-domain equivalent source and L normal direction vibration velocity reconstruction point are respectively uniformly distributed on source face E and body structure surface S, it is holographic
The equivalent source sizing grid in examination network size, equivalent source face E on the H of face and the reconstruction sizing grid phase on body structure surface S
Together, the equivalent source position on examination network point position, equivalent source face E and on holographic facet H and the reconstruction point position on body structure surface S
Put identical；All equivalent source source strengths and corresponding Green functions can be expressed as according to the vibration velocity of any instant any point in sound field
Convolution superposition, so as to obtain the time domain normal direction particle vibration velocity v of upper m-th of the measurement point of holographic facet HHmAnd body structure surface S upper the (t)
The time domain normal direction vibration velocity v of l reconstruction pointSl(t) respectively with time-domain equivalent strong q in a steady streamk(t) relation between：
In formula (20) and (21), " * " represents the convolution of two time-domain functions, and n is body structure surface normal unit vector, and ρ is
Media density, c be sound spread speed, δ (t-RHmk/ c) and δ (t-RSlk/ c) it is Dirac functions, H (t-RHmk/ c) and H (t-
RSlk/ c) it is unit jump function, RHmkThe distance between upper m-th of the measurement point of holographic facet H and k-th time-domain equivalent source are represented,
RSlkThe distance between upper l-th of the reconstruction point of body structure surface S and k-th time-domain equivalent source are represented,WithRepresent respectively
RHmkAnd RSlkDirectional derivative on normal direction n；
Definition
τHmk=t-RHmk/c (23)
τSlk=t-RSlk/c (24)
In formula (22)-(24),Represent k-th of time-domain equivalent majorant for integral in a steady stream, τHmkRepresent upper m-th of the survey of holographic facet H
Time delay between amount point and k-th of time-domain equivalent source, τSlkRepresent upper l-th of the reconstruction point of body structure surface S and k-th of time domain
Time delay between equivalent source；
By formula (22)-(24) substitute into formula (20) and (21), respectively structure t under upper m-th of the measurement point of holographic facet H when
Domain method is to particle vibration velocity vHm(t) and upper l-th of the reconstruction point of body structure surface S time domain normal direction vibration velocity vSl(t) with time-domain equivalent in a steady stream
Majorant for integralAnd time-domain equivalent majorant for integral derivative in a steady streamBetween relation such as formula (25) and (26) shown in：
Step 2, according to formula (27) to time t carry out it is discrete：
ti=t0+iΔt (27)
In formula (27), i=1,2 ..., I, I be total time step number, Δ t is time step, t0For initial time.Then postpone
Time τHmkAnd τSlkCan using discrete representation as：
Time domain normal direction particle vibration velocity v on collection holographic facet HHm(ti) after, the time-domain equivalent solved in formula (25) is in a steady stream
Majorant for integralAnd its derivativeThe time domain normal direction calculated in formula (26) on body structure surface S can not be directly substituted into shake
Speed, because the time-domain equivalent majorant for integral in a steady stream required in formula (26)And its derivativeIt may be not comprised in
The time-domain equivalent that formula (25) solves majorant for integral in a steady streamAnd its derivativeIn, therefore in order to realize transient state vibration velocity weight
Build, Lagrange interpolating functions are introduced into time-domain equivalent in a steady stream majorant for integral and establish unified time shaft；
Unified new time delay axle τ is established according to formula (30), and makes time delayWithFall into new time delay
In axle τ：
τj=τ0+jΔt (30)
In formula (30), j=1,2 ..., J, τ0=t0-RHmin/ c, RHminRepresent all RHmkIn minimum value.Then tiMoment
Corresponding time delayFor：
According to formula (32) and (33) respectively to time-domain equivalent majorant for integral in a steady streamAnd its derivativeIntroduce Lagrange
Interpolating function：
In formula (32) and (33),For τjK-th of equivalent source source strength integration of moment, Φj(τ) andRespectively
Lagrange interpolating functions and its derivative, its expression formula are respectively：
Step 3, formula (32) and (33) are substituted into formula (25), structure tiThe time domain method of upper m-th of the measurement point of moment holographic facet H
To particle vibration velocity vHm(t) shown in interpolation formula such as formula (36)：
In formula (36),
Formula (32) and (33) are substituted into formula (26), build tiThe time domain normal direction of upper l-th of the reconstruction point of moment body structure surface S is shaken
Fast vSl(t) shown in interpolation formula such as formula (37)：
In formula (37),
The summation form of formula (36) and (37) is converted into matrix form, builds t respectivelyiAll M on moment holographic facet H
The matrix representation forms such as formula of the normal direction vibration velocity of all L reconstruction points in the normal direction particle vibration velocity and body structure surface S of measurement point
(38) and shown in (39)：
In formula (38) and (39),
Wherein, the transposition of " T " representing matrix；
Step 4, in formula (38), as i=1, formula (38) is changed into：
In formula (45),WithIt is known.Pseudoinverse is asked to obtain least square solution by formula (46) singular value decomposition
Ψ1：
In formula (46), the pseudoinverse of "+" representing matrix；
As i=2, formula (38) is changed into：
In formula (47),WithIt is known, Ψ1It is to solve to obtain by formula (46).It is strange by formula (48)
Different value, which is decomposed, asks pseudoinverse to obtain least square solution Ψ2：
By that analogy, as i=I, formula (38) is changed into：
Pseudoinverse is asked to obtain least square solution Ψ by formula (50) singular value decompositionI：
Due to solving time-domain equivalent majorant for integral Ψ in a steady streamiProcess be usually morbid state, therefore during in order to obtain suitable
Domain equivalent source source strength integration, needs using Tikhonov regularization in standard just in the solution formula (46), (48) and (50) at each moment
Then change technology, wherein regularization parameter can be chosen by Generalized Cross Validation method；
Step 5, the equivalent source source strength for solving each moment integrate ΨiIn substitution formula (39), structure can be calculated
The normal direction vibration velocity of all moment all reconstruction points on the S of surface；
The time domain normal direction particle vibration velocity of each examination network point is to be listed in holographic facet H using vibration velocity sensor array on holographic facet H
Last snapshot measurement obtains.Structural vibration has time-varying characteristics, and the reconstruction of surface normal vibration velocity is carried out in time domain.
The inspection of method：
Outwardly and inwardly distinguish distribution holographic face H and equivalent source face E, and holographic facet H and equivalent source face E in body structure surface S
Shape and body structure surface S it is conformal.The inventive method rebuilds structure by measuring the time domain normal direction particle vibration velocity on holographic facet
The time domain normal direction vibration velocity on surface, and compared with the time domain normal direction vibration velocity of laser vibration measurer measurement.
In this experiment, vibrational structure is the hollow iron plate of one piece of semi-cylinder, and its external diameter, height and thickness is respectively
0.25m, 0.6m and 5.5mm.Radius 10mm steel ball is made into a single pendulum, hits the clamped iron plate back side, produces transient oscillation.
The position relationship of iron plate surface S, holographic facet H and equivalent source face E under cylindrical coordinate system o (r, θ, z) is shown in Fig. 2 (a), coordinate original
For point positioned at the center of semicylinder, iron plate surface S, holographic facet H and equivalent source face E radius are respectively rs=0.25m, rh=0.28m
And reThe level angle θ scopes of=0.245m, iron plate surface S, holographic facet H and equivalent source face E places cylinder are all from π/6rad
To 5 π/6rad.5 × 5 measurement points are all distributed on iron plate surface S and holographic facet H respectively, are distributed 5 × 5 etc. on the E of equivalent source face
On effect source, iron plate surface S, holographic facet H and equivalent source face E interval of the consecutive points on horizontal angle θ and z directions be all respectively π/
6rad and 0.1m, referring to Fig. 2 (b).Time-domain signal sample frequency is 51.2kHz, sampling number 447.
To examine normal direction vibration velocity of the inventive method in time domain to rebuild effect, four spaces are chosen on the S of iron plate surface
Point A (0.25m, 2 π/3rad, 0.1m), B (0.25m, 2 π/3rad, 0m), C (0.25m, pi/2 rad, -0.1m) and D (0.25m, π/
2rad, -0.2m) come compare the normal direction vibration velocity time domain waveform measured on the S of iron plate surface and rebuild normal direction vibration velocity time domain waveform.Figure
3 (a), Fig. 3 (b), Fig. 3 (c) and Fig. 3 (d) correspond to reconstruction point A, reconstruction point B, reconstruction point C and reconstruction point D respectively, solid line table in figure
Show the iron plate surface time domain normal direction vibration velocity of laser vibration measurer measurement, figure dotted line represents the iron plate table rebuild using the inventive method
Face time domain normal direction vibration velocity.Structure can effectively be reconstructed using the inventive method by comparing solid line and dotted line in figure and can be seen that
The time domain normal direction vibration velocity on surface.
For the reconstruction effect of quantitative evaluation the inventive method, phase valuation factor EpWith amplitude evaluation points EaIt is introduced into, its
It is respectively defined as：
In formula (51) and (52), v (rl, t) and represent the time domain normal direction vibration velocity that laser vibration measurer measures, vc(rl, t) and represent this
The time domain normal direction vibration velocity that inventive method is rebuild, ＜ ＞ represent to average.Evaluation points EpIt is to be surveyed for weighing laser vibration measurer
Phase error between the time domain normal direction vibration velocity that the time domain normal direction vibration velocity and the inventive method of amount are rebuild, works as EpValue closer to 1
When, phase error is smaller.Evaluation points EaIt is the time domain normal direction vibration velocity and the inventive method for weighing laser vibration measurer measurement
Amplitude error between the time domain normal direction vibration velocity of reconstruction, works as EaValue closer to 0 when, amplitude error is smaller.With formula (51) and
(52) E on iron plate surface S at each point is calculated respectivelypAnd EaValue.Fig. 4 (a) represent E a littlepIt is worth, it is most of contour in figure
Line value is more than 0.88, and contour minimum value is 0.8.Fig. 4 (b) represent E a littleaIt is worth, most of value of contours are less than in figure
0.12, contour maximum is 0.15.Referring to Fig. 4 (a) and Fig. 4 (b), it can be seen that measurement normal direction vibration velocity value and reconstruction normal direction are shaken
Speed value is all fine in consistent manner, and phase and the amplitude of reconstruction can effectively represent their measured value.
For examine the inventive method spatial domain vibration velocity rebuild effect, have chosen three moment 3.50ms, 5.31ms and
6.78ms.Fig. 5 (a), Fig. 5 (b) and Fig. 5 (c) are respectively the iron plate surface normal measured at 3.50ms, 5.31ms and 6.78ms moment
Vibration velocity Modal Space is distributed, and Fig. 5 (d), Fig. 5 (e) and Fig. 5 (f) are respectively to use this hair at 3.50ms, 5.31ms and 6.78ms moment
The iron plate surface normal vibration velocity Modal Space distribution that bright method is rebuild.Compare Fig. 5 (a) and Fig. 5 (d), Fig. 5 (b) and Fig. 5 (e), figure
5 (c) and Fig. 5 (f) are as can be seen that the normal direction vibration velocity for the distribution of normal direction vibration velocity Modal Space and measurement rebuild using the inventive method
Modal Space distribution is almost identical.
Above-mentioned experiment shows, complex appearance body structure surface transient state vibration velocity can be reconstructed well using the inventive method.
Claims (3)
1. a kind of body structure surface transient state vibration velocity method for reconstructing using particle plane vibration speed measurement, it is characterized in that：Step is as follows：
Step 1, in vibrational structure outwardly and inwardly it is respectively arranged holographic facet H and equivalent source face E, and holographic facet H and equivalent source face E
Shape and body structure surface S it is conformal；M survey is respectively uniformly distributed on described holographic facet H, equivalent source face E and body structure surface S
Measure point, K time-domain equivalent source and L normal direction vibration velocity reconstruction point；In examination network size, equivalent source face E on holographic facet H etc.
Effect source sizing grid is identical with the reconstruction sizing grid on body structure surface S, and examination network point position on holographic facet H, equivalent
Equivalent source position on the E of source face is identical with the reconstruction point position on body structure surface S；Gather upper m-th of the measurement point of holographic facet H when
Domain method is to particle vibration velocity vHm(t)；
Step 2, the time domain normal direction particle vibration velocity v according to formula (1) structure upper m-th of the measurement point of t holographic facet HHmAnd time domain (t)
Equivalent source source strength integratesAnd time-domain equivalent majorant for integral derivative in a steady streamBetween relation：
In formula (1), τHmk=t-RHmk/ c, τHmkRepresent prolonging between upper m-th of the measurement point of holographic facet H and k-th of time-domain equivalent source
Slow time, n are body structure surface normal unit vector, and ρ is Media density, and c is the spread speed of sound, RHmkRepresent on holographic facet H
The distance between m-th measurement point and k-th time-domain equivalent source,Represent RHmkDirectional derivative on normal direction n；
By the time t in formula (1) it is discrete be ti=t0+ i Δ t, wherein Δ t are sampling time interval, i=1,2 ..., I, I to adopt
Total sample, t0For initial time, then delay time THmkCan using discrete representation as：
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In formula (3), j=1,2 ..., J, τ0=t0-RHmin/ c, RHminRepresent all RHmkIn minimum value；
According to formula (4) and (5), to time-domain equivalent, majorant for integral and its derivative introduce Lagrange interpolating functions in a steady stream respectively：
In formula (4) and (5),For τjK-th of equivalent source source strength integration of moment, Φj(τ) is Lagrange interpolating functions；
Step 3, formula (4) and (5) are substituted into formula (1), structure tiThe time domain normal direction particle of upper m-th of the measurement point of moment holographic facet H shakes
Fast vHm(t) shown in interpolation formula such as formula (6)：
In formula (6),
The summation form of formula (6) is converted into matrix form, builds tiThe normal direction particle of all M measurement points on moment holographic facet H
Shown in the matrix representation forms of vibration velocity such as formula (7)：
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</mrow>
</mrow>
In formula (7),
<mrow>
<msubsup>
<mi>V</mi>
<mi>H</mi>
<mi>i</mi>
</msubsup>
<mo>=</mo>
<msup>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>v</mi>
<mrow>
<mi>H</mi>
<mn>1</mn>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>v</mi>
<mrow>
<mi>H</mi>
<mn>2</mn>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msub>
<mi>v</mi>
<mrow>
<mi>H</mi>
<mi>M</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mi>T</mi>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msubsup>
<mi>G</mi>
<mrow>
<mi>H</mi>
<mi>v</mi>
</mrow>
<mrow>
<mi>j</mi>
<mi>i</mi>
</mrow>
</msubsup>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>H</mi>
<mn>11</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>H</mi>
<mn>12</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>H</mi>
<mn>1</mn>
<mi>K</mi>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>H</mi>
<mn>21</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>H</mi>
<mn>22</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>H</mi>
<mn>2</mn>
<mi>K</mi>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>H</mi>
<mi>M</mi>
<mn>1</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>H</mi>
<mi>M</mi>
<mn>2</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>H</mi>
<mi>M</mi>
<mi>K</mi>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, the transposition of " T " representing matrix；
Step 4, in formula (7), as i=1, formula (7) is changed into：
<mrow>
<msubsup>
<mi>V</mi>
<mi>H</mi>
<mn>1</mn>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>H</mi>
<mi>v</mi>
</mrow>
<mn>11</mn>
</msubsup>
<msup>
<mi>&Psi;</mi>
<mn>1</mn>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula (11),WithIt is known.Pseudoinverse is asked to obtain least square solution Ψ by formula (12) singular value decomposition1：
<mrow>
<msup>
<mi>&Psi;</mi>
<mn>1</mn>
</msup>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>H</mi>
<mi>v</mi>
</mrow>
<mn>11</mn>
</msubsup>
<mo>&rsqb;</mo>
</mrow>
<mo>+</mo>
</msup>
<msubsup>
<mi>V</mi>
<mi>H</mi>
<mn>1</mn>
</msubsup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>12</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula (12), the pseudoinverse of "+" representing matrix；
As i=2, formula (7) is changed into：
<mrow>
<msubsup>
<mi>V</mi>
<mi>H</mi>
<mn>2</mn>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>H</mi>
<mi>v</mi>
</mrow>
<mn>12</mn>
</msubsup>
<msup>
<mi>&Psi;</mi>
<mn>1</mn>
</msup>
<mo>+</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>H</mi>
<mi>v</mi>
</mrow>
<mn>22</mn>
</msubsup>
<msup>
<mi>&Psi;</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>13</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula (13),WithIt is known, Ψ1It is to solve to obtain by formula (12).Pass through formula (14) singular value point
Solution asks pseudoinverse to obtain least square solution Ψ2：
<mrow>
<msup>
<mi>&Psi;</mi>
<mn>2</mn>
</msup>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>H</mi>
<mi>v</mi>
</mrow>
<mn>22</mn>
</msubsup>
<mo>&rsqb;</mo>
</mrow>
<mo>+</mo>
</msup>
<mo>&lsqb;</mo>
<msubsup>
<mi>V</mi>
<mi>H</mi>
<mn>2</mn>
</msubsup>
<mo>-</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>H</mi>
<mi>v</mi>
</mrow>
<mn>12</mn>
</msubsup>
<msup>
<mi>&Psi;</mi>
<mn>1</mn>
</msup>
<mo>&rsqb;</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>14</mn>
<mo>)</mo>
</mrow>
</mrow>
By that analogy, as i=I, formula (7) is changed into：
<mrow>
<msubsup>
<mi>V</mi>
<mi>H</mi>
<mi>I</mi>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>H</mi>
<mi>v</mi>
</mrow>
<mrow>
<mn>1</mn>
<mi>I</mi>
</mrow>
</msubsup>
<msup>
<mi>&Psi;</mi>
<mn>1</mn>
</msup>
<mo>+</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>H</mi>
<mi>v</mi>
</mrow>
<mrow>
<mn>2</mn>
<mi>I</mi>
</mrow>
</msubsup>
<msup>
<mi>&Psi;</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<mo>...</mo>
<mo>+</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>H</mi>
<mi>v</mi>
</mrow>
<mrow>
<mi>I</mi>
<mi>I</mi>
</mrow>
</msubsup>
<msup>
<mi>&Psi;</mi>
<mi>I</mi>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>15</mn>
<mo>)</mo>
</mrow>
</mrow>
Pseudoinverse is asked to obtain least square solution Ψ by formula (16) singular value decompositionI：
<mrow>
<msup>
<mi>&Psi;</mi>
<mi>I</mi>
</msup>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>H</mi>
<mi>v</mi>
</mrow>
<mrow>
<mi>I</mi>
<mi>I</mi>
</mrow>
</msubsup>
<mo>&rsqb;</mo>
</mrow>
<mo>+</mo>
</msup>
<mo>&lsqb;</mo>
<msubsup>
<mi>V</mi>
<mi>H</mi>
<mi>I</mi>
</msubsup>
<mo>-</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>H</mi>
<mi>v</mi>
</mrow>
<mrow>
<mn>1</mn>
<mi>I</mi>
</mrow>
</msubsup>
<msup>
<mi>&Psi;</mi>
<mn>1</mn>
</msup>
<mo>-</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>H</mi>
<mi>v</mi>
</mrow>
<mrow>
<mn>2</mn>
<mi>I</mi>
</mrow>
</msubsup>
<msup>
<mi>&Psi;</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<mo>...</mo>
<mo>-</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>H</mi>
<mi>v</mi>
</mrow>
<mrow>
<mo>(</mo>
<mi>I</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
<mi>I</mi>
</mrow>
</msubsup>
<msup>
<mi>&Psi;</mi>
<mrow>
<mo>(</mo>
<mi>I</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</msup>
<mo>&rsqb;</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>16</mn>
<mo>)</mo>
</mrow>
</mrow>
Tikhonov regularization in standard regularization and Generalized Cross Validation are used in the solution formula (12), (14) and (16) at each moment
Technology solves suitable time-domain equivalent majorant for integral Ψ in a steady streami；
Step 5, the equivalent source source strength for solving each moment integrate ΨiIn substitution formula (17), institute is calculated on body structure surface S sometimes
Carve the normal direction vibration velocity of all L reconstruction points：
<mrow>
<msubsup>
<mi>V</mi>
<mi>S</mi>
<mi>i</mi>
</msubsup>
<mo>=</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>S</mi>
<mi>v</mi>
</mrow>
<mrow>
<mn>1</mn>
<mi>i</mi>
</mrow>
</msubsup>
<msup>
<mi>&Psi;</mi>
<mn>1</mn>
</msup>
<mo>+</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>S</mi>
<mi>v</mi>
</mrow>
<mrow>
<mn>2</mn>
<mi>i</mi>
</mrow>
</msubsup>
<msup>
<mi>&Psi;</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<mo>...</mo>
<mo>+</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>S</mi>
<mi>v</mi>
</mrow>
<mrow>
<mi>j</mi>
<mi>i</mi>
</mrow>
</msubsup>
<msup>
<mi>&Psi;</mi>
<mi>j</mi>
</msup>
<mo>+</mo>
<mo>...</mo>
<mo>+</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>S</mi>
<mi>v</mi>
</mrow>
<mrow>
<mi>i</mi>
<mi>i</mi>
</mrow>
</msubsup>
<msup>
<mi>&Psi;</mi>
<mi>i</mi>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>17</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula (17),
<mrow>
<msubsup>
<mi>V</mi>
<mi>S</mi>
<mi>i</mi>
</msubsup>
<mo>=</mo>
<msup>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>v</mi>
<mrow>
<mi>S</mi>
<mn>1</mn>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>v</mi>
<mrow>
<mi>S</mi>
<mn>2</mn>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msub>
<mi>v</mi>
<mrow>
<mi>S</mi>
<mi>L</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>t</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mi>T</mi>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>18</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msubsup>
<mi>G</mi>
<mrow>
<mi>S</mi>
<mi>v</mi>
</mrow>
<mrow>
<mi>j</mi>
<mi>i</mi>
</mrow>
</msubsup>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>S</mi>
<mn>11</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>S</mi>
<mn>12</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>S</mi>
<mn>1</mn>
<mi>K</mi>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>S</mi>
<mn>21</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>S</mi>
<mn>22</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>S</mi>
<mn>2</mn>
<mi>K</mi>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>S</mi>
<mi>L</mi>
<mn>1</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>S</mi>
<mi>L</mi>
<mn>2</mn>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msup>
<mi>g</mi>
<mi>j</mi>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&tau;</mi>
<mrow>
<mi>S</mi>
<mi>L</mi>
<mi>K</mi>
</mrow>
<mi>i</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>19</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein,τSlkRepresent structure
Time delay between upper l-th of the reconstruction point of surface S and k-th of time-domain equivalent source, RSlkRepresent upper l-th of the reconstruction of body structure surface S
The distance between point and k-th time-domain equivalent source,Represent RSlkDirectional derivative on normal direction n.
2. the body structure surface transient state vibration velocity method for reconstructing according to claim 1 using particle plane vibration speed measurement, it is characterized in that：
The time domain normal direction particle vibration velocity of each examination network point is to be listed in holographic facet H upper one using vibration velocity sensor array on the holographic facet H
Secondary snapshot measurement obtains.
3. the body structure surface transient state vibration velocity method for reconstructing according to claim 1 using particle plane vibration speed measurement, it is characterized in that：
The structural vibration has time-varying characteristics, and the reconstruction of surface normal vibration velocity is carried out in time domain.
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