CN108228958A - Oscillator field monitoring optimizing method based on modal intensity - Google Patents
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Abstract
The invention discloses a kind of oscillator field monitoring optimizing method based on modal intensity, for this method first by rejecting the low mode of modal intensity, retaining high energy mode, reduction is used for the mode quantity of calculating；It is based on mode independency principle again, from being evenly arranged in required all the sensors position, it rejects and smaller position is contributed to mode independence, only retain and ensure the effective position of mode independence, one position is rejected by multiple loop iteration, each iteration, the final reduction for realizing measuring point quantity and point position are preferred.Mode superposition method is then based on, only with the measurement data of the preferably rear sensor group in reduced number, position, reconstructs the vibratory response field of total, you can obtain the vibratory response at each position of total.The present invention solves the problems, such as that scale is excessive when large-scale structures make vibration monitoring to ship, establishes the preferred method of vibration measurement sensor quantity and position, and simplicity, the accurate method of monitoring layout are provided for vibration monitor system.
Description
Technical field
The present invention relates to ship monitoring technology more particularly to a kind of oscillator field monitoring optimizing methods based on modal intensity.
Background technology
The vibration monitoring of the large-scale structures such as Ship Structure, which exists, is difficult to the contradiction overcome：Measurement sensor quantity is with measuring
Precision is difficult to satisfactory to both parties.High-acruracy survey needs big quantity sensor to be evenly arranged in structure, and number of sensors is very few to be caused to survey
Accuracy of measurement is insufficient.But ship load ability and space are limited, reduce the adjuncts such as number of sensors and its signal wire to the greatest extent, it can
To reduce the weight occupied and space, while construction and maintenance cost can be saved, there is important value to Ship Design.
Invention content
The technical problem to be solved in the present invention is for the defects in the prior art, to provide a kind of based on modal intensity
Oscillator field monitoring optimizing method can save space, weight and cost by reducing number of sensors, and significantly reduce measurement analysis
Workload solves vibration measurement sensor quantity and the optimal selection problem of position.
The technical solution adopted by the present invention to solve the technical problems is：A kind of oscillator field monitoring based on modal intensity is excellent
Change method, includes the following steps：
1) vibration acceleration sensor is evenly arranged in large scale shell structure surface；
2) vibration data of each vibratory response point is tested into row energization to housing；Each vibration acceleration sensor is
One vibratory response point；
3) model analysis is carried out to test data, identifies each mode, and obtained each Mode Shape；
Define modal intensityIt is represented as the numerical value for characterizing the i-th rank mode degree of strength,
Wherein, m is mode sum,Represent the Mode Shape value at the position of the i-th rank mode excitation at j-th, ωi
For the mode circular frequency；
4) the normalization modal intensity of each mode is calculated, it is then preferred according to result of calculation progress mode, it preferably accounts for total strong
Spend the larger N number of high energy mode of the intensity of setting ratio；
5) to initial vibration response point, the KEM method based on modal intensity, from the s being largely evenly arranged in structure
Portion is rejected in a Sensor position, only retains E Sensor position, while ensures to know by E measuring point
The mode not gone out keeps Line independent, and is complete；Wherein, E<S and E are not less than structural modal truncation number；
6) to the measurement sensor position preferably obtained and quantity, after these position measurement vibratory responses, to whole knot
Structure oscillator field is reconstructed, and obtains the vibratory response of all sites in structure, realizes the vibration monitoring to structure.
By said program, overall strength setting ratio is 80% to 90% in the step 4).
By said program, the elimination method of Sensor position is as follows in the step 5)：
5.1) according to modal superposition principle, the response of system is expressed as：
Wherein, { u } ∈ Rs×1, represent physical coordinates：φsFor the i-th rank modal vector, φs∈Rs×m, represent modal vector square
Battle array；qiFor the i-th rank modal coordinate, { q } ∈ Rm×1, the sum of behalf measurement sensor, m represents rank number of mode；
5.2) structural matrix E：
Due to E2=E, therefore E is idempotent matrix, characteristic value l either 0, the mark of idempotent matrix is equal to order,
I-th of element representation, i-th of degree of freedom or test point on its diagonal is to matrix φsOrder contribution, that is, the line to A
The contribution of property independence；
Write the diagonal entry of matrix E as a column vector：
EI=[E11,E22,…,Ess]T
If Eii=0, then it represents that None- identified mode of concern, E on corresponding i-th measuring pointii=1 represents corresponding
Measuring point is key point, should retain in testing and cannot remove, accordingly select EI matrixes in close to 1 element corresponding to sensing
Device point position as the monitoring position of vibratory response, realizes sensors optimum placement.
By said program, total oscillator field is reconstructed using oscillator field reconstructing method in the step 6), is had
Body is as follows：
According to vibration superposition theory, have
V=Φ A
In formula：V is the normal direction vibration velocity column vector of s measuring point of body structure surface；Φ is Produced by Modal normal component matrix；
A is modal coordinate column vector；
Vibration velocity at E point position of body structure surface is：
VE=ΦEA
In formula：VE=[v1,v2,…,vE]TTo take out the vibratory response shape at E measuring point from surface vibration velocity vector V
Into measuring point response vector；ΦEFor the corresponding mode of oscillation matrix of E point position, A is modal coordinate column vector；Then：
Above formula is the calculation formula of oscillator field reconstruct, it shows that can measure part-structure response comes reconfigured geometry any position
It responds (assuming that known to all Mode Shapes), MIEI method of these sensing stations as described in upper step preferably obtains.
The beneficial effect comprise that：The present invention solves to ship the scale when large-scale structures make vibration monitoring
Excessive problem establishes the preferred method of vibration measurement sensor quantity and position, can substitute traditional uniform cloth
The method for testing vibration of big quantity sensor is put, so as to save space, weight and cost, and significantly reduces measurement analysis workload,
Simplicity, the accurate method of monitoring layout are provided for vibration monitor system, it can also be used to the common of large scale structure vibratory response
It measures, it is contemplated that play weight in ship acoustics status assessment, vibration monitoring of engineering structure, radiated noise estimation and Noise Sources Identification etc.
It acts on, potential application foreground is wide.
Description of the drawings
Below in conjunction with accompanying drawings and embodiments, the invention will be further described, in attached drawing：
Fig. 1 is the structural modal vibration shape example schematic of the embodiment of the present invention；
Fig. 2 is the normalization modal intensity schematic diagram of the embodiment of the present invention；
Fig. 3 be the embodiment of the present invention MIEI method iterative process in remaining measuring point modal matrix conditional number variation diagram；
Fig. 4 is the typical structural vibration response reconstruction result of the embodiment of the present invention and measured value correlation curve；
Fig. 5 is the reconstruction result of the typical frequencies lower housing vibration field distribution of the embodiment of the present invention and actual measurement response comparison.
Specific embodiment
In order to make the purpose , technical scheme and advantage of the present invention be clearer, with reference to embodiments, to the present invention
It is further elaborated.It should be appreciated that specific embodiment described herein is not used to limit only to explain the present invention
The fixed present invention.
As shown in Figure 1, the present invention realizes at least to pass for solving the problems, such as that Ship Vibration monitors sensors optimum placement
Sensor quantity and preferred position measurement structure vibratory response.It is general invention defines modal intensity to solve the problems, such as this
It reads, and number of sensors is reduced in terms of Modal Space and geometric space two based on modal intensity, step is as follows：First by picking
Except the low mode of modal intensity, retain high energy mode, reduction is used for the mode quantity of calculating；It is based on mode independency principle again,
It from being evenly arranged in required all the sensors position, rejects and smaller position is contributed to mode independence, only retain and ensure mould
A position, the final contracting for realizing measuring point quantity are rejected in the effective position of state independence by multiple loop iteration, each iteration
Subtract preferred with point position.Mode superposition method is then based on, only with the measurement number of the preferably rear sensor group in reduced number, position
According to reconstructing the vibratory response field of total, you can obtain the vibratory response at each position of total.
It is specific as follows：
1st, modal intensity defines
Structural modal is by its vibrational energy and influences that high energy mode and weak mode can be divided into, they are in structural vibration radiation
Effect played in noise is different：High energy mode coverage in structural vibration is big, radiated is high, and weak Effect of Mode
Range small (local mode), radiated are low.Structural modal vibration shape example is as shown in Figure 1, the second-order modal vibration shape is significantly higher than
Other mode are typical high energy mode, and generally measure can be relatively low for high order mode in addition to indivedual mode.
The isostructural vibration monitoring of hull should more focus on the vibration of high energy mode initiation, and sensor arrangement should preferably reflect height
The point position of energy mode rejects and reflects the position of weak mode to reduce measurement sensor sum.Otherwise, the ginseng of excessive weak mode
The position for reflecting weak modal information with final measuring point position is caused to tend to, can not reflect true structural vibration and its radiation
Noise states do not meet the demand of structural vibration noise control.
To retain high energy mode, rejecting weak mode, requirement defines modal intensity, is only adopted in point position preferred process
It is participated in high energy mode, inhibits the point position for reflecting weak mode, make vibration monitoring of engineering structure result that can more reflect with radiating under water
The contact of noise, and can further reduce number of sensors, improve engineering adaptability.
Quantization defines the degree of mode power, can to avoid measuring point it is preferred when a large amount of mode disorderings participations, can not be directed to
Property test and inhibit modality-specific.This is that solution point position layout result is average to all mode, can not reflect structure high energy
The key of modal vibration problem.
According to modal theory, the i-th rank mode is to the contribution of system vibration response gross energy：
In formula, [P]m×rTo act on the excitation matrix of m response point, { Φ by r vibrational excitationiFor the i-th first order mode to
Amount, ωiFor the mode circular frequency.
With PjRepresent j-th of vibrational excitation,Represent the Mode Shape at the position of the i-th rank mode excitation at j-th
Value, then above formula is reduced to：
Assuming that vibrational excitation is uniformly distributed equivalent excitationTime variable t is not considered, then above formula
For：
In formula | P | it is a constant, represents the amplitude of m uniformly distributed equivalent excitations,J-th yuan for the i-th rank modal vector
Element,Represent the phase difference between each position vibrational excitation, k is to represent meeting for imaginary number.
The phase difference of structure each vibrational excitation only in very low frequencies rigid motionIn resonance band and high frequency,
Phase differenceIt is equally distributed stochastic variable (same to m orthogonal arbitrary excitation).Therefore above formula can be changed
For：
It is square directly proportional due to vibrational energy and oscillation intensity, then from the above equation, we can see that oscillation intensity with
It is directly proportional.
Therefore quantization defines modal intensityNumerical tabular as the i-th rank mode degree of strength of characterization
Show.Normalization mode can also be used, enableWherein n is the mode quantity for participating in calculating, and defined mode is strong
Spend δiNumerical value facilitates and is used as weighted parameter between 0 and 1.
2nd, mode is preferred
According to normalization modal intensity definition, take and account for a certain proportion of N number of mode participation subsequent sensor position of overall strength
It is preferred that.This ratio can be determined according to engineering experience, be recommended using 80% ratio, i.e., the intensity for preferably accounting for overall strength 80% is larger
N number of high energy mode, give up the weak mode for only accounting for overall strength 20%, should " sixteen " ratio meet Pareto Principle.It is examined in engineering
Consider the influence of various disturbance factors, what can also be guarded takes 90% ratio.
It for surveying mode shown in Fig. 1, is calculated by the definition of normalization modal intensity, has obtained whole 12 ranks
Corresponding normalization modal intensity (as shown in Figure 2) data of mode, it is found that wherein only the 2nd~7 rank and this 7 mode of the 10th rank are strong
It has been more than just 90% to spend accounting, thus preferable 2nd~7 rank and the 10th rank this 7 high energy mode participate in it is follow-up calculate, it is so far real
It is preferred the mode based on modal intensity has been showed.
3rd, the preferred MIEI methods of sensing station
Establish KEM method (the Modal intensity-based effective based on modal intensity
Independence method or MIEI), realize Sensor position and quantity optimization, target is from being largely evenly arranged
Portion is rejected in the s Sensor position in structure, only retains E (E<S and not less than structural modal truncation number)
A Sensor position, while the mode for ensureing to identify by E measuring point still maintains Line independent, and be complete
's.It is complete to refer to that the measurement data using this E optimization position reconstructs entire vibratory response field, so as to grasp shaking for total
Dynamic response state.
According to modal superposition principle, the response of system can be expressed as：
Wherein, { u } ∈ Rs×1, represent physical coordinates：φsFor the i-th rank modal vector, φs∈Rs×m, represent modal vector square
Battle array；qiFor the i-th rank modal coordinate, { q } ∈ Rm×1, the sum of behalf measurement sensor, m represents rank number of mode.The minimum of above formula
Two, which multiply solution, is
If it is considered that measurement noise, then：
{ u }=φs{q}+{v}
Wherein { v } represents variance as σ2Gaussian distribution white noise.Corresponding true generalized coordinatesIt certainly exists
One deviation, it is assumed that this process is that a unbiased is effectively estimated, thenThe covariance matrix P of { q } is：
E represents mathematic expectaion in above formula, and matrix [Q] is Fisher information matrix.Assuming that measurement noise independently of each other and
Identical to the statistical property of each sensor measurement signal, then [Q] can be reduced to：
It is also Fisher information matrix by the visible A matrixes of above formula.Good estimation in order to obtain, it is necessary to make certain of Information Matrix
Standard (determinant, norm, mark etc.) obtains maximization, and the covariance that can thus obtain parameter { q } estimated bias is minimum, from
And obtain parameterBest estimate.Here common 2- norms are selected | | A | |2And the conditional number cond (A) of A is controlled
The quality of estimation.If conditional number is bigger, illustrate that the condition of matrix is bad, if smaller, illustrate that the condition of matrix is good
It is good.It is not difficult to obtain by norm theory：
As it can be seen that the requirement to A matrix conditions and norm can be indirectly by φsSelection realize.
The mark of matrix A can be write as：
Wherein AiRepresent contribution of i-th of degree of freedom to the mark of matrix A.Those do not interfere with or are not significantly affected by matrix
The mark of A or the A of determinanti, it is all extra to represent those degree of freedom, can be removed.
The characteristic equation of dematrix A：
(A- λ I) ψ=0
ψTA ψ=λ；ψTψ=I
Obviously have：
A-1=ψTλ-1ψ
Structural matrix E：
It it is easy to show that, E2=E.Therefore E is idempotent matrix, characteristic value l or 0.Mark of idempotent matrix etc.
Yu Zhi, i-th of element representation, i-th of degree of freedom or test point on diagonal are to matrix φsOrder contribution, it is that is, right
The contribution of the Line independent of A.Write the diagonal entry of matrix E as a column vector (effective Independent Vector)：
EI=[E11,E22,…,Ess]T
If Eii=0, then it represents that None- identified mode of concern, E on corresponding i-th measuring pointii=1 represents corresponding
Measuring point is key point, should retain in testing and cannot remove.Accordingly select EI matrixes in close to 1 element corresponding to sensing
Device point position as the monitoring position of vibratory response, realizes sensors optimum placement.
With reference to above-mentioned mode preferred process, the mode for first carrying out Modal Space is preferred, then several by the aforementioned progress of this step
The point position in what space is preferred, collectively forms the preferred MIEI methods of sensing station based on modal intensity.To Modal Space
The preferably upper section of mode described in, and it is preferred to the point position of geometric space when, usually represent that candidate is surveyed with the diagonal element of matrix E
Contribution of the point to modal matrix linear independence degree (independence).By E matrix diagonals element size come to each candidate measuring point
Priority is ranked up, and excludes the measuring point of corresponding diagonal element minimum every time with iterative algorithm, then carries out next iteration, directly
To obtaining satisfied measuring point number, the Line independent of modal matrix and Fisher information battle array after reduction are finally maintained as far as possible
Determinant maximizes, so as to obtain the best estimate of vibratory response from test data.
MIEI methods utilize the idempotence of composite mode matrix E, gradually delete the minimum degree of freedom of the order contribution to E, thus
Optimize Fisher information matrix, make based on the modal vector that modal intensity preferably goes out in the case of minimum measuring point, as far as possible
Linear independence is kept, so as to which maximum modal response information can be collected in testing.This method is used to vibrate point layout
The problems such as measure-point amount is excessive, scale is excessive, the selection of crucial measuring point can be solved, is that feasible method can be used in engineering.
4th, oscillator field reconstructing method
According to vibration superposition theory, have
V=Φ A
In formula：V is the normal direction vibration velocity column vector of s measuring point of body structure surface；Φ is Produced by Modal normal component matrix；
A is modal coordinate column vector.
Body structure surface E (E<S) vibration velocity at a point position is：
VE=ΦEIn A formulas：VE=[v1,v2,…,vE]TTo take out shaking at E measuring point from surface vibration velocity vector V
Dynamic response forms measuring point response vector；ΦEFor the corresponding mode of oscillation matrix of E point position, A is modal coordinate column vector.
Then：
Above formula is the calculation formula of oscillator field reconstruct, it shows that can measure part-structure response comes reconfigured geometry any position
It responds (assuming that known to all Mode Shapes), MIEI method of these sensing stations as described in upper section preferably obtains.
So far, it using the preferred measurement sensor position of the method for step 1 to 3 and quantity, vibrates and rings in these position measurements
Ying Hou is reconstructed total oscillator field using method in step 4, you can the vibration for obtaining all sites in structure is rung
Should, the vibration monitoring to structure is only realized with a small amount of sensor.
One specific embodiment：
This example carries out individual layer cylindrical shell (closed at both ends) mould measurement and vibratory response test under water, then by MIEI
Method and oscillator field reconstructing method carry out data processing, it is final reconstructed after Shell model vibratory response data, and with actual measurement
Date comprision, the accuracy of verification method achievement.
There is the sealing of arch hush panel at tested cylinder shell model axial direction both ends, axial 1.8 meters long, 1.2 meters of diameter, and coverboard is thick
0.8 centimetre, model is 6 meters of underwater during test.Experimental provision and acceleration transducer point layout schematic diagram as shown in figure 3,
200 vibration acceleration sensors are evenly distributed on cylindrical shell inner surface, share 10 circles, often enclose 20 sensors.Vibration exciter
Be resiliently mounted on pedestal to housing into row energization, in test system excitation point to each vibratory response point transfer admittance, and
Test the vibration data of each vibratory response point.Utilize B＆K companies 3560D type multichannel analysis systems and PULSE software systems
Data sampling and analysis are carried out, vibrating sensor uses the accelerometer of B＆K companies production, and test data is input to self-editing
MATLAB programs are calculated.
(a) MIEI methods result
After the completion of experiment test, model analysis is carried out to test data with Reflex softwares, the cylindrical shell of identification is underwater
Below 200Hz mode totally 12 (being shown in Table 1), and obtained each Mode Shape.
Mode that 1 experimental test and analysis of table obtains and preferably after modal frequency
By the definition in step 1, it is as shown in Figure 2 to calculate each mode normalization modal intensity；It is carried out by the method in step 2
Mode is preferred, and the mode for choosing accounting 90% participates in follow-up calculating, and the mode preferably gone out is shown in Table 1.Then the sensing as described in step 3
The preferred MIEI methods in device position, the 200 response measuring points arranged using cylindrical shell in experiment are transported as starting point by successive ignition
Calculate the measuring point for gradually reducing reservation.In iterative process, the conditional number for the modal matrix that remaining measuring point (sensing station) is formed is not
Disconnected to reduce (as shown in Figure 3), when iteration 186 times, remaining measuring point only remain 14, the conditional number of remaining measuring point modal matrix takes
Obtain minimum.Modal matrix conditional number is smaller, then calculating error when follow-up modal matrix is inverted is smaller.
By matrix theory, if it must be full rank (sequency spectrum) that matrix, which has inverse matrix, it is equivalent to of equation in equation group
Number has to be larger than the number of unknown number, and otherwise equation is without determining solution.These mathematical theories are in mould measurement and oscillator field reconstruct side
The reflection in face is exactly that the measurement sensor quantity finally retained cannot be less than mode number.This experiment uses preceding 12 rank mode, then
Effective measure-point amount cannot be less than 12, and since inevitably there are various disturbing factors and error, works in experiment
Effective measure-point amount should also increased journey in practice.Changed by the conditional number of Fig. 3, it may be determined that preferred sensing station
Number is 14, and the oscillator field reconstruct of next step is carried out with these preferred measuring points.The Position Number of the preferred measuring point of MIEI methods is shown in
Table 2.
The preferred sensor arrangement Position Number of 2 MIEI methods of table
| 61 | 71 | 75 | 77 | 86 | 94 | 98 |
| 106 | 114 | 118 | 121 | 131 | 135 | 137 |
(b) oscillator field reconstruction result
With the vibratory response that sensor at preferred 14 positions of MIEI methods measures, oscillator field reconstructs according to step 4
Method calculates the vibratory response of whole 200 response points of cylindrical shell, obtains the entire hull vibration field of reconstruct.
Fig. 4 is typical structural vibration response reconstruction result and measured value correlation curve.Mark DOF57 in figure represents the
No. 57 Sensors, other and so on.Solid line is reconstruct as a result, dotted line is measured value in figure.
Fig. 5 shows the reconstruction result of entire hull vibration field distribution and actual measurement response comparison under typical frequencies.Wherein left figure
The response of total for reconstruct, right figure are actual measurement response distribution.As seen from Figure 5, the oscillator field of reconstruct and actual measurement oscillator field
It coincide fine.
The overall average of error is weighed on all measuring points of error, all frequencies after reconstruct, the results showed that mean error
Only 2.7dB, and number of sensors is reduced to 14 by 200, economy has reached 93%.
The above results show that this intellectual achievement the method has enough accuracy and feasibility, are suitble in engineering
It promotes and applies.
It should be understood that for those of ordinary skills, can be improved or converted according to the above description,
And all these modifications and variations should all belong to the protection domain of appended claims of the present invention.
Claims (4)
1. a kind of oscillator field monitoring optimizing method based on modal intensity, includes the following steps：
1) vibration acceleration sensor is evenly arranged in large scale shell structure surface；
2) vibration data of each vibratory response point is tested into row energization to housing；Each vibration acceleration sensor is one
Vibratory response point；
3) model analysis is carried out to test data, identifies each mode, and obtained each Mode Shape；
Define modal intensityIt is represented as the numerical value for characterizing the i-th rank mode degree of strength,
Wherein, m is mode sum,Represent the Mode Shape value at the position of the i-th rank mode excitation at j-th, ωiFor this
Mode circular frequency；
4) the normalization modal intensity of each mode is calculated, it is then preferred according to result of calculation progress mode, it preferably accounts for overall strength and sets
The larger N number of high energy mode of the intensity of certainty ratio；
5) to initial vibration response point, the KEM method based on modal intensity, from s biography being largely evenly arranged in structure
Portion is rejected in sensor point position, only retains E Sensor position, while ensures to identify by E measuring point
Mode keep Line independent, and be complete；Wherein, E<S and E are not less than structural modal truncation number；
6) it to the measurement sensor position preferably obtained and quantity, after these position measurement vibratory responses, shakes to total
Dynamic field is reconstructed, and obtains the vibratory response of all sites in structure, realizes the vibration monitoring to structure.
2. the oscillator field monitoring optimizing method according to claim 1 based on modal intensity, which is characterized in that the step
4) overall strength setting ratio is 80% to 90% in.
3. the oscillator field monitoring optimizing method according to claim 1 based on modal intensity, which is characterized in that the step
5) elimination method of Sensor position is as follows in：
5.1) according to modal superposition principle, the response of system is expressed as：
Wherein, { u } ∈ Rs×1, represent physical coordinates：φsFor the i-th rank modal vector, φs∈Rs×m, represent modal vector matrix；qi
For the i-th rank modal coordinate, { q } ∈ Rm×1, the sum of behalf measurement sensor, m represents rank number of mode；
5.2) structural matrix E：
Due to E2=E, therefore E is idempotent matrix, characteristic value l either 0, the mark of idempotent matrix is equal to order, diagonal
I-th of element representation, i-th of degree of freedom or test point on line is to matrix φsOrder contribution, that is, the Line independent to A
Contribution；
Write the diagonal entry of matrix E as a column vector：
EI=[E11,E22,…,Ess]T
If Eii=0, then it represents that None- identified mode of concern, E on corresponding i-th measuring pointii=1 represents that corresponding measuring point is
Key point should retain and cannot remove in testing, accordingly select EI matrixes in close to 1 element corresponding to Sensor
Sensors optimum placement as the monitoring position of vibratory response, is realized in position.
4. the oscillator field monitoring optimizing method according to claim 1 based on modal intensity, which is characterized in that the step
6) total oscillator field is reconstructed using oscillator field reconstructing method in, it is specific as follows：
According to vibration superposition theory, have
V=Φ A
In formula：V is the normal direction vibration velocity column vector of s measuring point of body structure surface；Φ is Produced by Modal normal component matrix；A is
Modal coordinate column vector；
Vibration velocity at E point position of body structure surface is：
VE=ΦEA
In formula：VE=[v1,v2,…,vE]TSurvey is formed to take out the vibratory response at E measuring point from surface vibration velocity vector V
Point response vector；ΦEFor the corresponding mode of oscillation matrix of E point position, A is modal coordinate column vector；Then：
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| CN111460576A (en) * | 2020-03-19 | 2020-07-28 | 东南大学 | Characteristic value tracking method for flutter analysis of wings |
| CN113358308A (en) * | 2021-06-03 | 2021-09-07 | 哈尔滨工业大学 | Combined structure transverse displacement determination method based on limited measuring points and global mode |
| CN113465734A (en) * | 2021-09-02 | 2021-10-01 | 清华大学 | Real-time estimation method for structural vibration |
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