CN1033480A - Use the modal analysis method of microcomputer - Google Patents

Abstract

A kind ofly works is made method of mode analysis by microcomputer, by being carried out simulation test, works obtains force signal and vibration response signal, through gathering sampling, be converted to digital signal, calculate the amplitude-frequency response of each measuring point by microcomputer, and it is average to do lump, is the estimated value of model frequency with the Frequency point on its each peak, finds out and the corresponding model frequency of objective function, the damping vibration shape when by optimization again.Can save expensive signal processor and analog magnetic tape recorder, low price, precision height.

Description

Use the modal analysis method of microcomputer

The present invention relates to a kind of modal analysis method.Natural frequency, the vibration shape, the modal damping that relates more specifically to a kind of definite vibrational system such as compares at the modal analysis method of the characteristic parameter of mode.

So-called model analysis is exactly the process by vibration test identification modal parameter.So-called modal parameter is promptly represented the characteristic parameter of mode of oscillation.The characteristic of the vibrational system of a complexity can be decomposed into one group of independently stack of natural mode of vibration separately.Wherein each mode can be described with following parameters, that is:

Model frequency is a natural frequency;

The modal damping ratio; With

Mode Shape.

When wherein Mode Shape represents that works is made pure modal vibration under this frequency, the vibration shape of each point on the works.The vibration shape can be with an array representation.Each element is corresponding to the relative amplitude of vibrating on this coordinate in the array.The vibration shape is only represented the form vibrated, and the amplitude of the actual vibration of it and works is distinguishing.A continuous structure has a unlimited vibration shape.Each vibration shape correspondence single order mode.The mode that frequency is low is called lower mode, and the mode that frequency is high is called high order mode.Limited interior mode of interested frequency range is only studied in general restriction because of the capacity that is subjected to test apparatus and computing machine.

Model analysis can be used to 1, determines that the resonant frequency of structure reaches the distortion of object when resonance, thereby is used for the fault diagnosis and the quality control of machine; 2, revise limited element calculation model, and be used for the calculating (being works vibrates course under certain load calculating) of vibratory response; 3, making structural dynamic revises, promptly answer " if after certain parts of works have been done certain and revised design; how mode of oscillation will change ", or further answer: " if make the frequency of vibration or the vibration shape do certain variation, will do the modification how to design " structure.So model analysis is one of optimal design key.Be core with the computing machine, the computer-aided design (CAD) that the CAT system with data acquisition, signal analysis and model analysis function can be engineering goods provides experimental basis.It is the important means of dynamic design, fault diagnosis and the quality control of complex product.

Referring to Fig. 3, Figure 3 shows that the external existing block diagram that carries out the device of model analysis.Act on the works certain any force signal (producing) and the works on a certain measuring point by the force transducer 1 that is placed on this some place to the vibration response signal (producing) of this dynamic exciting power by the vibration transducer 2 that is placed on this measuring point place respectively through prime amplifier 12,13 amplify, be recorded in after the amplification on the magnetic tape recorder 5, making quick Fourier transform (FFT) and calculate transport function or frequency response function through the anti-signal processor 8 of delivering to after mixing low-pass filter 6,7 filtering respectively again.All the transport function of measuring point is by the serial or parallel interface input computing machine 9 of signal processor 8.Simulate the modal parameter of works through computing machine 9 again,, thereby set up the mathematical model of works by printer 10 or plotting apparatus 11 outputs.

Above method need be stored the signal of input with magnetic tape recorder, and need make quick Fourier transform to input signal earlier with signal processor and handle, and this signal processor is had relatively high expectations on making, and import charges is big.Adopt in addition that each peak point is the model frequency estimated value on the amplitude-versus-frequency curve on indivedual measuring points, the shortcoming of existence is because of vibration measuring point as is positioned on the nodel line of certain single order Mode Shape, do not occur the peak value of this order frequency on the amplitude frequency curve of this point; Otherwise a certain peak on the amplitude-versus-frequency curve also may be to measure or analyze noise causes rather than mode.The real modal identification method of existing optimization requires internal memory bigger again, often is to carry out on bigger robot calculator.

The purpose of this invention is to provide a kind of method in order to overcome above shortcoming, its signal processor of making quick Fourier transform is to realize according to the method for process flow diagram with software on microcomputer; And it is average that the amplitude-frequency response of the transport function of each measuring point is done lump, and be the estimated value of model frequency with the Frequency point on each peak; And, discern the modal damping ratio, thereby calculate the identification Mode Shape by finding out with optimization and the corresponding identification model frequency of the minimum value of objective function E.

Another object of the present invention provides a kind of method of obtaining objective function E and promptly forms matrix H according to the test transfer function data earlier, again according to following formula 10 compute matrix T, then according to following formula 11 compute matrix A, again according to following formula 5 compute matrix , again according to following formula 14 calculating target functions

Another purpose of the present invention provides a kind of optimization and surveys the suitable high and steep ü  of the late at night   nurse green pepper  speech nautilus L  of Dong moral and pound joyous ǖ and try to gain that craftsman そ  is emerging to castrate that the nautilus L of saddle cloth mould  Dong mire tenon stone mountain benevolence is timid not to have flat sparrow hawk to refer to that net for catching beasts page or leaf bucktooth imitates

Modal analysis method provided by the present invention the steps include: that (a) treats the geodesic structure thing and carry out pulse or stable state modal test at random, obtains force signal by force transducer, and obtains relevant vibration response signal on each measuring point by vibration transducer; (b) simultaneously the vibration response signal on force signal and the measuring point is done the parallel acquisition sampling and is converted to digital signal, deposit in the computing machine; (c) calculate the real part and the imaginary part of the transport function of each measuring point respectively, and obtain its amplitude-frequency response; (d) it is average the amplitude-frequency response of the transport function of each measuring point to be done lump, and is the estimated value of model frequency with the Frequency point on its each peak; (e) find out and the corresponding identification model frequency of the minimum value of objective function E by optimization, identification modal damping ratio, thus and calculate the identification Mode Shape.

The analysis of transport function is the basis of Modal Parameter Identification, as if the x(t that is input as of system), be output as y(t).Their Fourier transform is X(f) and Y(f), then system transter is defined as

H（f）＝Y（f）/X（f） （1）

In order to suppress the input end The noise, the foundation of calculation of transfer function of the present invention is

$\mathrm{H(f)=}\frac{X^{*}\mathrm{(\; f\; )Y(\; f\; )}}{X^{*}\mathrm{(\; f\; )X(\; f\; )}}=\frac{G_{\mathrm{x\; y}}\mathrm{(\; f\; )}}{G_X\mathrm{(\; f\; )}}\mathrm{(2)}$

The conjugation of " * " expression plural number in the formula.G _Xy(f) be called the cross-power spectrum that output is imported, Gx(f) be called the auto-power spectrum of input.

H(f) be that a plural number has real part and imaginary part, the quadratic sum of its real part and imaginary part evolution again is amplitude, also is the function of frequency.

The real mode identification step of optimization provided by the present invention is:

According to the real modal theory of proportional damping, the excitation of P point, the acceleration transport function of l point measurement is

$H_{\mathrm{l\; p}}\mathrm{(\; \omega \; )=}\underset{\mathrm{i\; =\; 1}}{\overset{N}{\mathrm{\&Sigma;}}}\frac{{\mathrm{-\phi }}_{\mathrm{l\; i}}{\mathrm{<figure-callout\; id="2"\; label="\phi \; pi\; \omega "\; filenames="871075687\_DRIMG12.png"\; state="\{\{state\}\}">\phi \; </figure-callout>}}_{\mathrm{pi}}{\omega }^{}{}^2}{{\mathrm{mi(\; \omega }}^2{}_i{\mathrm{-<figure-callout\; id="2"\; label="\omega "\; filenames="871075687\_DRIMG12.png"\; state="\{\{state\}\}">\omega </figure-callout>}}^2{\mathrm{+j\; 2S}}_i{\omega }_i\mathrm{\omega \; )}}\mathrm{(4)}$

In the formula

The ω excited frequency

ω _lNatural frequency i=1,2 ..., N

The exponent number of N system in the frequency-of-interest scope

m _iI rank modal mass

ζ _iI rank modal damping ratio

φ _LiI rank Mode Shape i=1,2 ..., N; L=1,2 ... L.

This recognition methods, at first supposition only utilizes imaginary frequency characteristic to discern, and by (4) complex function is expanded into real part and imaginary part,

Its imaginary part is:

$H^I{}_{\mathrm{l\; p\; \prime }}\mathrm{(\omega \; )=}\underset{\mathrm{i\; =\; 1}}{\overset{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="871075687\_DRIMG12.png"\; state="\{\{state\}\}">N</figure-callout>}}{\mathrm{\&Sigma;}}}\frac{{\mathrm{2\; <figure-callout\; id="3"\; label="\omega "\; filenames="871075687\_DRIMG13.png"\; state="\{\{state\}\}">\omega </figure-callout>}}^3{\omega }_i{S}_i{A}_{\mathrm{il}}}{{\mathrm{(\omega }}^2{}_i{\mathrm{-\omega }}^2{)}^2{\mathrm{+4S}}^2{}_i{\omega }^2{}_i{\omega }^2}\mathrm{(5)}$

In the formula $A_{\mathrm{i\; l}}=\frac{{\phi }_{\mathrm{l\; i}}{\phi }_{\mathrm{p\; i}}}{m_i}\mathrm{(6)}$

A _IlBe directly proportional with the i first order mode.

By fft analysis gained test transport function, when the fft block size is 1024,512 complex values are arranged.When system damping hour, have only the point of near-resonance to be only effectively.Modal identification method of the present invention only utilizes a near-resonance S test figure.S can be taken as 7.If exponent number is N, the test figure of using altogether is m=N * S, and this m frequency is designated as Ω ₁, Ω ₂... Ω _mForm m * L rank transport function imaginary-part matrix for whole measuring points

Also can form vibration shape matrix to be established by (6)

By real mode formula (5), can write out matrix equation

H＝TA （9）

When excitation frequency is relevant with model frequency, damping for matrix T in the formula, and is irrelevant with the vibration shape.

Suppose that in certain iteration searching process, model frequency and damping ratio are known, can calculate matrix T by (10), and (7) matrix H are known test figures.So can solve vibration shape matrix A from equation (9).Therefore but matrix T is not a square formation, and line number is much larger than columns, use least square solution:

A＝（T ^TT） ^-1T ^TH＝T ^TH （11）

Subscript T is a transposition in the formula, and subscript-1 is for inverting.T ^IBe also referred to as the generalized inverse or the pseudoinverse of matrix T.

Set up the objective function E of an optimization then.

If on a certain step of iteration, model frequency, the damping when vibration shape are all calculated, and Dai Huishi mode fundamental formular (5) can construct the transport function imaginary-part matrix of mathematical model again

＝TA （12）

Be called transport function or mode transport function after the match, it also is a matrix with the difference of test transfer function matrix

E＝H～

（13）

Make the Frobenious norm of error matrix E

As the objective function of optimizing identification.

Optimize the identification model frequency then, identification modal damping ratio, thus and obtain discerning Mode Shape.

The invention has the advantages that and on microcomputer, to realize model analysis on computers especially, thereby enlarged the use of microcomputer, owing to saved expensive signal processor and analog magnetic recording view, so than wanting considerably cheaper on the import instrument price, and the result who measures is consistent with the result who calculates gained with finite element theory, and precision is higher.

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Sensor arrangement synoptic diagram when Fig. 1 is test; Wherein Fig. 1 (a) is a pulsed modal test synoptic diagram, and Fig. 1 (b) is a stable state modal test synoptic diagram;

Fig. 2 is the input/output relation figure of works (system);

Fig. 3 is existing model analysis device synoptic diagram;

Fig. 4 is the block schematic diagram of signals collecting part;

Fig. 5 is a model analysis general flow chart of the present invention;

Fig. 6 carries out the process flow diagram of transfer function analysis for the present invention;

Fig. 7 is for optimizing the schematic flow diagram of Modal Parameter Identification;

Fig. 8 is the schematic flow diagram of calculation optimization objective function E;

Fig. 9 is for representing the curve relevant with model analysis or the figure of figure; Wherein Fig. 9 (a) is the stable state oscillogram of force signal at random, Fig. 9 (b) is the vibration response signal oscillogram of a measuring point, Fig. 9 (c) is the real part oscillogram of the transport function of this measuring point, Fig. 9 (d) is the imaginary part oscillogram of the transport function of this measuring point, Fig. 9 (e) is the amplitude-frequency response oscillogram of this measuring point, Fig. 9 (f) is the average amplitude-frequency response oscillogram of the lump of amplitude-frequency response of 16 measuring points after average, and Fig. 9 (g)～Fig. 9 (k) is respectively 5 Mode Shape figure on the identification frequency.

Referring to Fig. 1 (a), Fig. 1 (b), Fig. 2,2 is works among Fig. 1 (a), and 1 for being contained in the force transducer that knocks on the hammer 111, and 3 for being arranged in the acceleration transducer on the measuring point.When knocking the vibration response signal that on sensor 3, can obtain measuring point when certain is some on the works 2 with knocking hammer 111, also can knock hammer 111 and knock, and the signal of as Fig. 1 (b) shown in stable state random signal generator 5 being exported is delivered to vibrator 4 and carried out exciting as pulsed.Coupled force transducer 1 is placed on certain on the works 2, and Fig. 2 represents the input/output relation of works (system), and X is excitation (power) signal of input, Y _lBe the vibration response signal of output, l=1,2 ... L, always total L measuring point, Y ₁, Y ₂Y _lBe respectively L the vibration response signal on the measuring point, treating as a stranger the geodesic structure thing except that works shown in Figure 1 also can be very complicated structure.

Referring to Fig. 3, Fig. 3 is existing model analysis schematic representation of apparatus, is described in the explanation prior art part in front to this part.

Referring to Fig. 4, Fig. 4 is a signals collecting part calcspar, wherein the force signal that produced of force transducer 1 is through prime amplifier 121, the anti-low-pass filter 6 that mixes, sampling holder 15, modulus, (A/D) converter is converted to digital signal with sampled signal, be stored in the computing machine then, another road, resists and mixes low-pass filter 7, sampling holder 16 also through prime amplifier 131 from the vibration response signal of acceleration transducer 3 simultaneously, analog to digital converter is transformed to digital signal with the vibration response signal of sampling and is sent to computing machine and stores.Can import the vibration response signal of force signal and this measuring point to different measuring points simultaneously.So that calculate the transport function of this measuring point.

Referring to Fig. 5, Fig. 5 is a model analysis general flow chart of the present invention.Step wherein has: 25 are pulse or stable state modal test at random.26 for carrying out the double-channel signal collection.27 for carrying out transfer function analysis.28 is that the lump of amplitude-frequency response is average, and the amplitude versus frequency characte that is about to each measuring point averages so that obtain the lump mean value of amplitude-frequency response.29 is the real modal analysis that is optimized.

Referring to Fig. 6, Fig. 6 is the process flow diagram of calculation of transfer function.In step 40 at first to each input digit force signal, and digital vibration signal is set up offset binary code time data file, and in step 41, make i=1, then in step 42, from the time data file, take out force signal and vibration signal each one (1024 points) successively, and convert ASCII character to, follow computation of mean values in step 43, and removal DC component, then in step 44, carry out the quick Fourier transform fft analysis of binary channels, calculate input auto-power spectrum Gx, output auto-power spectrum Gy and cross-power spectrum Gxy, then carry out step 45, whether make i＜NAVE time judgement, then enter step 46 less than NAVE as i, make i=i+1, carry out step 42 once more after returning, 43,44 handle next piece power and vibration signal, till carrying out NAVE number repeatedly, the average time that adopts when NAVE is transfer function analysis.Its size determines by signal statistics, and is relatively poor as the signal stationarity, then NAVE should obtain bigger, desirable generally speaking 16～64 times.Gx in step 47 then to calculating for NAVE time, Gy, Gxy averages, and exists then

And imaginary part calculation of transfer function real part Hr=(Re(Gxy) in the step 48)/(Gx)

Hi＝ (Im（Gxy）)/(Gx)

Referring to Fig. 7, Fig. 8.Preferential Modal Parameter Identification step has: step 51 is ω=ω for setting initial estimate ^o _i, ζ i=ζ ^o _i=0.008(i=1,2 ..., N), ω wherein ^o _iBe the angular frequency value (angular frequency estimated value) that i the peak of lump after average located, ζ ^o _i=0.008 for using the golden section point of Fibonacci method (i.e. 0.616 method) between 0.0001 to 0.0020, because objective function E is subjected to the influence of dampingratio i less, and be subjected to the influence of frequency bigger, so optimize the identification frequency in the present invention earlier, and then optimize the identification damping ratio, in step 52 according to the test transfer function data through by formula (7) calculate to form matrix H, a certain the impulsing a little of P representative in the matrix, L represents the measuring point number, owing to get S data (for example 7 data) near a resonance point, the test figure of using altogether for the N rank is m=N * S (as 7N).All measuring point is formed m * L rank transport function imaginary-part matrix.Then entering step 53 is i=1, promptly asks near i.e. identification model frequency and the identification modal damping ratio of first resonance point in first rank earlier.Then in program 55, make ω _i=ω ^o _iAnd then in step, calculate the objective function E of optimization, and in step 61, make Ec=E then, make ω i=ω in step 54 again ^o _i-△ ω promptly strides a step-length left, and this step-length △ ω=△ F/4(wherein △ F is the angular frequency resolution of transfer function analysis, and the objective function E of calculation optimization in step 57 makes E again in step 60 then _L=E.In step 56, make ω=ω again ^o _i+ △ ω (being that a step-length is striden on the right side), and the desirable for example △ of △ ω F/4, its size is decided by the desired precision of model frequency.The calculating of the objective function E that is optimized in step 59 again makes E again in step 62 _R=E.In step 63, judge E again _CWhether than E _L, E _R, all little, if like this then flow process enters the optimization that step 67 is carried out the modal damping ratio, as E _CNot to compare E _L, E _RAll little, then enter and whether carry out E in the step 64 _L＜E _RJudgement, in this way, then enter step 65, and make E _R=E _C, E _C=E _LAnd enter step 54 once more, and stride a step-length left, calculating target function E in step 57 again, and in step 60, make E _L=E, and then in step 63, make E _CWhether than E _L, E _RAll little judgement is judged E in step 64 _LBe not＜E _R, then enter step 66 and make E _L=E _C, E _C=E _R, and enter step 56, and stride a step-length to the right, make ω i=ω ^o _i+ △ ω, calculating target function E in step 59 makes E again in step 62 again _R=E carries out E then in step 63 _C＜E _LAnd E _C＜E _RJudgement, in this way, then finished the optimization of model frequency, as not, then continue to enter step 64, repeat above-mentioned relevant step then.After the optimization of model frequency finishes, promptly enter step 67, make ζ a=0.001 therein, ζ _b=0.02, Ex=E then enters step 69, makes ζ _i=ζ _y+ ζ _a+ (ζ _b-ζ _x), calculating target function in step 70 makes Ey=E again in step 73 again, enters step 74 again and judges whether obviously 0.0002 also optional other values herein of ζ y-ζ x＜0.0002, and this is more relevant than precision of optimizing with modal damping.As for being, optimize and finish, desirable ζ _xWith ζ _yMean value be the modal damping ratio.As for not, then enter step 75 and whether judge Ey＞Ex, as for being then to enter step 76 and make ζ _b=ζ y, ζ y=ζ x, Ey=Ex then for step 69, makes ζ i=ζ x+(ζ _b-ζ y), carries out the calculating of priority target function E again in step 71, in next step 72, make Ex=E again, carry out the judgement of step 74 then, look judged result or enter step 79, or still enter the judgement of step 75.In in step 75, judge Ey＜Ex, then enter step 77, make ζ a=ζ x, ζ x=ζ y and Ex=Ey, then again through step 68,70, and carry out step 74 again, 75 judgement after 73, until ζ y-ζ x＜till 0.0002 o'clock, thereby the mean value of the ζ x of desirable this moment and ζ y is the modal damping ratio.Then enter step 79 and whether make i＞judgement of N, as denying, then enter step 78, make i=i+1 return again, be optimized the second rank model frequency, reach damping ratio, follow the 3rd rank ... till the N rank are optimized model frequency and damping ratio and are obtained, at last will be, print analysis result in modal parameter deposits such as the Mode Shape in the objective function E calculation procedure (A matrix) and each rank model frequency, modal damping compare.

Referring to Fig. 8, Fig. 8 is concrete steps of relevant calculation optimization aim function among Fig. 7, at first in step 78 by one group of ω i and ζ i(i=1,2 ...,, then in step 79, calculate the generalized inverse matrix T of T matrix N) according to formula 10 calculating and composition T matrix ^I=(T ^TT) ^-1T ^T, then in step 80, calculate corresponding to this group ω i, the vibration shape matrix A=T under the ζ i ^IH, then compute mode transfer function matrix in step 81 =TA then calculates the optimization aim function in step 82

Referring to Fig. 9, below be example with a rectangular parallelepiped member, list the result of its model analysis, thereby practicality of the present invention is described.Shown in Fig. 1 (b), this rectangular parallelepiped member is at place random signal exciting, and survey its vibratory response at 16 measuring point places, Fig. 9 (a) is the oscillogram of force signal, and transverse axis is time t, (b) is the oscillogram of the vibration response signal of a measuring point wherein, Fig. 9 (c) then is the real part curve of the transport function of this measuring point of calculating, abscissa is a frequency, and Fig. 9 (d) then is the imaginary part curve of transport function on this measuring point, and its abscissa is a frequency.Fig. 9 (c) is the amplitude-frequency response of this measuring point, as can be seen from the figure has five resonant frequencies in 250 hertz scope.Fig. 9 (f) is the lump mean value curve of the amplitude-frequency response of 16 measuring points, therefrom can find out has five resonant frequencies to can be used as estimated frequency, promptly the 1st rank estimate that model frequency is 20.00 hertz, the 2nd rank estimate that model frequency is 50.50 hertz, the 3rd rank estimate that model frequency is 58.00 hertz, the 4th rank estimate that model frequency is 107.00 hertz, and the 5th rank estimate that model frequency is 227.00 hertz.Resulting the 1st rank identification model frequency is 20.00 hertz after optimizing identification, and identification modal damping ratio is 0.0182; The 2nd rank identification model frequency is 51.00 hertz, and identification modal damping ratio is 0.0073; The 3rd rank identification model frequency is 58.50 hertz, and identification modal damping ratio is 0.0092; The 4th rank identification model frequency is 107.50 hertz, and identification modal damping ratio is 0.0045; The 5th rank identification model frequency is 227.50 hertz, identification modal damping ratio is that 0.0035 Fig. 9 (g) is the former figure of rectangular parallelepiped member, the vibration shape animation display of making according to the identification Mode Shape of calculating when Fig. 9 (h)～Fig. 9 (l) is respectively 5 rank model frequencies therefrom can be found out in exciting situation lower member vibration situation.

The present invention exists in the disk by flow process of the present invention is compiled into computer program, can cooperate with IBMPC/XT and compatible etc. thereof to realize real modal analysis.

The modal parameter of trying to achieve with this method is all titled with " identification " two words in this manual.

Claims (3)

1, a kind ofly works is made method of mode analysis, it is characterized in that comprising the following steps: by microcomputer

A, works is carried out pulse or stable state stochastic simulation, obtain force signal, and obtain vibration response signal on each measuring point by vibration transducer by force transducer;

B, the vibration response signal on force signal and the measuring point is done parallel acquisition sampling, and be converted to digital signal, deposit in the microcomputer;

C, calculate the real part and the imaginary part of the transport function of each measuring point respectively by microcomputer, and obtain its amplitude-frequency response;

D, that the amplitude-frequency response of the transport function of each measuring point is done lump is average, and is the estimated value of model frequency with the Frequency point on its each peak;

E, find out and the corresponding model frequency of the minimum value of objective function E by optimization, thus modal damping than and calculate Mode Shape.

2, modal analysis method as claimed in claim 1 is characterized in that obtaining above-mentioned objective function Method be according to formula 10 compute matrix T, then according to formula 11 compute matrix A, again according to formula 5 compute matrix

, again according to formula 14 calculating target functions, wherein

Formula 5 is

Ail=(φ li φ pi)/(mi) wherein

ω in the formula: excited frequency ω i=natural frequency i=1,2 ... N

N: the exponent number of system in the frequency-of-interest scope

Mi: i rank modal mass

Si: i rank modal damping ratio

φ li i rank Mode Shape i=1,2 ... N; 1=1,2 ... L.

Formula 10 is

Formula 11 is A=(T ^TT) ^-1T ^TH-T ^TH;

Formula 14 is an objective function

And

＝TA

3, modal analysis method as claimed in claim 1 is characterized in that above-mentioned model frequency is by searching for certain step-length, and the damping ratio of above-mentioned mode is found out with Fibonacci method.

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