Output-Only Modal Identification of a NonuniformBeam by using Decomposition Methods

Rickey A. Caldwell, Jr.Department of Mechanical Engineering

Michigan State UniversityEast Lansing, MI, USAcaldwe20@msu.edu

517-355-8310

Brian F. Feeny!Department of Mechanical Engineering

Michigan State UniversityEast Lansing, MI, USAfeeny@egr.msu.edu517-353-9451

Reduced-order mass weighted proper orthogonal decompo-sition (RMPOD), smooth orthogonal decomposition (SOD),and state variable modal decomposition (SVMD) are usedto extract modal parameters from a nonuniform experimen-tal beam. The beam was sensed by accelerometers. Ac-celerometer signals were integrated and passed through ahigh-pass filter to obtain velocities and displacements, allof which were used to build the necessary ensembles for thedecomposition matrices. Each of these decomposition meth-ods was used to extract mode shapes and modal coordinates.RMPOD can directly quantify modal energy, while SOD andSVMD directly produce estimates of modal frequencies. Theextracted mode shapes and modal frequencies were com-pared to an analytical approximation of these quantities, andto frequencies estimated by applying the fast Fourier trans-form to accelerometer data. SVMD is also applied to esti-mate modal damping, which was compared to that estimatedby logarithmic decrement applied to modal coordinate sig-nals, with varying degrees of success.

1 IntroductionOutput-only modal analysis uses displacements, ac-

celerations, or velocities to determine modal information.Output-only modal analysis is useful when inputs cannotbe recreated or are unknown. Other benefits include theavoidance of frequency response functions (FRFs) andFRF matrices and related testing procedures [1] whichreduces test time and amount of data needed. Output-onlymodal analysis can be done in time domain or frequencydomain. Some examples of time domain methods include

!Address all correspondence to this author.

the eigensystem realization algorithm [2], Ibrahim timedomain method [3], independent component analysis [4, 5]and the polyreference method [6]. Examples of frequencydomain methods are orthogonal polynomial methods [7, 8],complex mode indicator function [9], and frequency domaindecomposition [10]. The methods that will be exploredin this work are in the time domain and are extensions ofproper orthogonal decomposition (POD). These includemass-weighted proper decomposition, smooth orthogonaldecomposition, and state variable modal decomposition. Theoutputs used are displacements, velocities, and accelerations.

In POD a structure is sensed with M sensors, whosesignals are processed as needed to generate displacementtime signals. An M " N ensemble matrix X is createdsuch that each row corresponds to a sensor and eachcolumn is a time step. That is, X = [x1 x2 · · · xM]T , wherexi = [xi(0) xi(ΔT ) xi(2ΔT ) · · · xi(NΔT )], and N is thenumber of time samples. This ensemble matrix is used inboth POD and the other decomposition methods.

In POD, the “correlation matrix” R is formed using the

ensemble matrix, such that R =XXT

N. Then the eigenvalue

problem (EVP) Rψψψ = λψψψ is solved. If the mass is uniform,and the system is lightly damped, POD produces estimatesof the mode shapes from free vibration responses [11–13]and random responses [11]. The former case has beenverified in experiments [14–16].

The decomposition methods which are generalizationsof POD reported here are reduced-order mass-weighted POD

1

(RMPOD), state-variable modal decomposition (SVMD),and smooth orthogonal decomposition (SOD). These meth-ods are applied to a nonuniform beam experiment. RMPODhas been applied successfully in simulations [8]. SVMDhas been verified for a uniform beam [17]. SOD has beenapplied in simulations [18] so an experimental applicationsare of interest.

In the next two sections, the beam experiment is de-scribed, and an approximate analytical description of themodes is outlined. Then RMPOD, SOD, and SVMD are ap-plied to the data from an experimental beam. In each section,the application of the method is introduced, and the resultsfrom the beam experiment are presented.

EXPERIMENTAL SETUPA thin nonuniform lightly damped cantilevered beam, a

Buck Bros. tapered saw blade, shown in Fig. 1, was sensedwith eleven accelerometers (PCB model 352B10) whichhave a sensitivity of 10±0.5mVg . The beam was clamped ina fixture such that the length was 11.5 inches (0.2921 m).The width was 3.5 inches (0.0889 m) at the clamped end,tapering from 3.5 inches (0.0889 m) at a location of 1.78inches (0.0452 m) from the clamp, to 0.80 inches (0.02032m) at the free end. The beam was clamped such that themidline of the taper was horizontal, and the flexure of thebeam was in the horizontal plane. Parameters of the beamare listed in Table 1. The beam material was unknown. Thedensity was measured, and Young’s modulus was assumedto be that of stainless steel.

The accelerometers were placed at one inch intervalsstarting one inch from the clamped end and progressing tothe free end. Each accelerometer was attached to the beamusing wax. The resulting signals from the accelerometerswere connected through a PCB Model 481 signal condi-tioner which also amplified the signal with a gain of 10. Af-ter the signals were passed through the signal conditionerthey were sent to the TEAC GX-1 data recorder. The sig-nals were sampled at a rate of 5000 Hz and sent through alow-pass filter with a cutoff frequency of 2000 Hz. This sat-isfied the Nyquist criterion and prevented aliasing. A fastFourier transform (FFT) of any of the acceleration signalsduring free vibration revealed the following natural frequen-cies: 8.45 Hz, 40.28 Hz, 107.4 Hz, 205.1 Hz, 498 Hz, and677 (Table 2).

In the free-vibration experiments the beam was strucktwo inches from the clamped end with an impact hammer.The resulting accelerations were recorded and importedinto MathWorks’ MATLAB. In MATLAB further signalprocessing was performed. Since the accelerometers hadphase distortions near 8 Hz, a high-pass filter was used witha cutoff frequency of 20 Hz. This attenuated the first modeand consequently removed the first mode from the resultsof the decomposition methods. Within MATLAB the meanof each signal was subtracted out and linear trends were

Fig. 1: Experimental beam

removed using the detrend command. The signal waspassed through a second-order high-pass filter twice: onceforward and the second time backwards, which correctedthe resulting phase shift caused by the filter. The signal wasthen integrated using the MATLAB function, cumtrapz,which approximates the integral using the trapezoid rule.One iteration of this process yielded velocities. The meansubtraction, detrending, filtering, and integration processeswere repeated again to produce displacements. Acceler-ations, velocities, and displacements were used to createensemble matrices for RMPOD, SOD, and SVMD. Themodal assurance criterion (MAC) [19] was used to evaluatethe quality of the extracted mode shapes as referenced to thediscretized analytical mode shapes of the nonuniform beammodel, which is discussed in the next section.

The impact tests were conducted with a variety of im-pulses to gain insight into their effects on modal identifica-tion. The beam was struck with an impact hammer at sensorlocation of 2 in (0.05 m), 6 in (0.15 m), and 11 in (0.28 m)from the clamped end. Two impulse magnitudes were used,the first, a small amplitude impulse such that all resultingfree vibration accelerations were less than 10g’s. The sec-ond, a large amplitude impulse where the free vibration ac-celerations were greater than 20g’s and less than 50g’s. Fi-nally, the signal of resulted free vibrations were divided intofour equal time bins. The results presented in this paperused the following parameters: a small amplitude impulsetwo inches from the clamped end and the second time bin,t # [1.3406,1.4594] seconds.

ANALYTICAL APPROXIMATION

In order to evaluate the experimental results an analyt-ical approximation to the nonuniform Euler-Bernoulli equa-tion was formulated. The general equation describing aclamped-free beam with variable cross-section is

m(x)y(x, t)+d2

dx2

!EI(x)

d2

dx2y(x, t)

"= 0 (1)

2

with boundary conditions

y(0, t) = 0ddxy(0, t) = 0

d2

dx2[EI(x)y(L, t)] = 0

d3

dx3[EI(x)y(L, t)] = 0

where y(x, t) is the transverse displacement at location x ofthe beam, m(x) is the mass per unit length, E is Young’smodulus, and I is the cross-sectional area moment of interia.

We discretized this equation by approximating y(x, t) $=M

∑i=1

qi(t)ui(x), where ui(x) are assumed modes and qi(t) are

the assumed modal coordinates. Inserting into Eqn. (1), mul-tiplying by u j(x), and integrating (the second term by partstwice) yields

Mq+Kq= 0 (2)

where the elements mi j ofM and ki j of K are

mi j =! L

0m(x)ui(x)u j(x)dx

(2.5)

ki j =! L

0EI(x)u%%i (x)u%%j (x)dx

The spatial discretization of y(x, t) can be expressed asy$= Uq, where U is a modal matrix made of column vectorsthat are spatial descretizations of the ui(x). As such, the ele-

ments of y are yi = y(xi, t) $=M

∑j=1

u j(xi)q j(t). We can assume

that there exists a discretized system of equations

#My+ #Ky= 0 (3)

that faithfully discretizes the original system Eqn. (1), suchthat the system matrices are related by M = UT #MU andK= UT #KU.

Assuming synchronous motion, such that q(t) = pr(t),Eqn. (2) leads to the eignevalue problem (EVP)

λMp=Kp. (4)

Material Property Value

Young’s Modulus (E) 190"109 Pascals

*Density (ρ) 7035 kg/m3

Height (h) 0.00066 m

Width (w)0.08787 m 0& 0.0456 m

'0.27409x+0.1004

0.0457 m& x& 0.2921 m

Length (L) 0.2921 m

Table 1: Material properties for the beam. *Estimated den-sity includes the mass of the accelerometers

Solving this EVP leads to estimates λi $= ω2i of the modalfrequencies of the beam model, and a modal matrix P for thesystem of equations (2). Applying the transformation q= Prdiagonalizes Eqn. (2). As such, composing q = Pr andy= Uq, we find that y$= UPr transforms system Eqn. (3) inoriginal coordinates, to the diagonal system in r. Then thediscretized mode shapes are approximated by the columnsof the composite modal matrix UP.

In application, a matrix U is created such thatU = [u1 u2 · · · uM] where ui’s are the discretized assumedmodal functions. We obtain these from the true modalfunctions of the damped-free uniform Euler-Bernoulliequation. We then build the associated mass and stiffnessmatrices M and K using Eqn. (2.5). Matrix P is createdsuch that P = [p1 p2 · · · pM] where pi’s are from theresulting eigensystem in Eqn. (4). Then, the LNMs forthe nonuniform beam are approximated as columns of UP.These modes will be compared to experimentally estimatedmodes in the following section.

When this model was applied to the experimental sawblade, the estimated modal frequencies were 9.02 Hz, 43.51Hz, 112.00 Hz, 214.46 Hz, 350.50 Hz, 520.93 Hz, and725.83 Hz, as listed in Table 2.

SOURCES OF ERRORSources of error for the frequency estimation are dis-

cussed here. The ratios between analytical and experimentalfrequencies, shown in Table 2, are uniformly between 1.04and 1.08. This could be caused by an error in the effectiveparameter group, for example due to errors in E and h,which would systematically scale the estimated frequencies.The error due to measured dimensions and density are likelyto be small. Discretized models sometimes are slightlystiffened by the “constraint” associated with discretization.

Another source of error may be in the assumptionsembedded in the beam model. Since the beam is wide, there

3

Mode no. AnalyticalApprox.

Beam FFT Ratio

1 9.02 Hz 8.55 Hz 1.06

2 43.51 Hz 40.28 Hz 1.08

3 112.00 Hz 107.40 Hz 1.04

4 214.46 Hz 205.10 Hz 1.05

5 350.50 Hz not seen inFFT

N/A

6 520.93 Hz 498.00 Hz 1.05

7 725.83 Hz 677.50 Hz 1.07

Table 2: Second column: Frequencies predicted by the ana-lytical approximation. Third column: Frequencies estimatedfrom FFTs of experimental beam accelerations

.

may be some influence of plate characteristics. The infinite

uniform 1-D plate equation is∂2w∂t2

+Dw%%%% = 0, while that

of the uniform beam is∂2w∂t2

+EImu%%%% = 0. The ratio between

parameter groups isEImD

= 1' ν2, where ν is Poisson’sratio, which bounds the deviation between the infinite 1-Dplate and the Euler-Bernoulli beam. Using ν = 0.3, thisleads to an decrease of up to about 5%, in the analyticallyestimated frequencies, which would increase the differencebetween the frequencies of the approximated model and theexperiments.

REDUCED-ORDER MASS-WEIGHTED PROPERORTHOGONAL DECOMPOSITION

It was noted above that POD produces estimates ofmode shapes when the mass distribution is uniform. Whenthe mass distribution is nonuniform and known, the weightedEVP RMψψψ= λψψψ produces estimates of the normal modes.

Mass-weighted POD is straight forward when themass matrix is of the same dimension as the correlationmatrix. The challenge addressed by RMPOD is when themass distribution is not dimensionally compatible with thecorrelation matrix R. A common reason for the mass matrixto be larger than R is that the number of available sensors,M, may be limited. For example the mass matrix producedby a finite-element program is easily greater than 100"100.Thus by mathematical necessity in order to create a cor-relation matrix whose dimensions match the mass matrixthe experimenter needs at least 100 sensors. Likewise,a continuous structure will always have more degrees offreedom than sensors. RMPOD uses a reduced-order massmatrix Mr of dimension M " M, such that the matrixmultiplication RMr is possible. Then the following problem

is solved: RMrψψψ = λψψψ. The eigenvectors ψψψ correspond toLNMs and the eigenvalues are mass-weighted mean squaredvalues of the modal coordinates [20].

When the dimension ofM is larger than that of R, an in-terpolation scheme can be used to reduce the effective orderof the system to that of the number of sensors used in build-ing R [20]. For the case of a one-dimensional distributedparameter system, using linear interpolation represented byinterpolation functions ηi(x) leads to a reduced mass matrixMr whose elements are

Mi j =Mji =! L

0m(x)ηi(x)η j(x)dx. (5)

We used ηi(x) and η j(x) as tent-shaped interpolation func-tions of the form ηi(x) = 1

h (x'(i'1)h) for (i'1)h& x< ih,ηi(x) = 1

h (x' (i+ i)h) for ih & x < (i+ 1)h, and ηi(x) = 0otherwise, where h is the spatial interval of the sensors onthe beam.

In this work, we inserted an expression representing themass distribution m(x), as defined by the taper in the sawblade, into Eqn. (5) to obtain the Mr matrix for weightingthe RMPOD process.

As such, RMPOD was applied. A permutation of inputparameters and ensembles was used in the RMPOD decom-position to gain some experience regarding their effects onthe modal decomposition results. Results were evaluatedusing the modal assurance criterion (MAC) [19] values withdiscretized analytical approximation mode shapes as thereference. In this case, the RMPOD based on accelerationensembles, with small impulses located 2 inches (0.0508 m)from the clamped end gave the best overall performance,although only marginally better than other permutations ofthe testing input parameters. Figures 2, 3, and 4 show theplots of the extracted modes, modal coordinate accelerationsof the extracted modes, and the magnitudes of the FFTs ofthe modal coordinate accelerations for the second, third, andfourth mode respectively.

We used the modal coordinates to further evaluate thedecompositions. The modal coordinates are defined throughthe transformation X = ΨΨΨQ, where the jth column of themodal matrix ΨΨΨ is ψ, from the EVP and Q is an ensem-ble of modal coordinate time histories. Each row is a sam-pled time history corresponding to the associated column inΨΨΨ. Then the modal coordinate accelerations are given byQa =ΨΨΨ'1A (hereA is the acceleration ensemble). The mag-nitude of the FFT of the modal coordinate accelerations forthe second mode showed a single peak at 39.14 Hz. Thethird modal coordinate acceleration had a maximum peak at107.6 Hz and smaller peak at 39.14 Hz. This shows somepollution [21] from the second mode into the third modal. Asimilar phenomenon occurs for the fourth modal coordinateacceleration, which had a maximum peak at 205.5 Hz, fol-

4

lowed by 39.14 Hz, and finally, 107.6 Hz. Despite this modalpollution, the extracted mode shapes were strong approxi-mations to linear normal modes, which is evident by MACvalues close to unity for these mode shapes. Those valuesare 0.986, 0.852, and 0.912 for the second, third, and fourthmode, respectively. The extracted modes shapes are shownin comparison to the discretized analytically approximatedmode shapes in Figs. 2, 3, and 4. Figure 5 was included asan example of a poor extraction.

Fig. 2: Top: Second mode shape extracted by RMPOD (()plotted with the analytical approximation’s discretized modeshape. Middle: Second mode, modal acceleration coordinatefrom RMPOD. Bottom: FFT of the modal coordinate accel-eration

SMOOTH ORTHOGONAL DECOMPOSITIONIn the case of SOD two correlation matrices are created.

One is the displacement correlation matrix R, such that

Fig. 3: Top: Third mode shape extracted by RMPOD (()plotted with the analytical approximation’s discretized modeshape. Middle: Third mode, modal acceleration coordinatefrom the RMPOD. Bottom: FFT of the modal coordinateacceleration

R =XXT

Nand the other is the velocity correlation matrix

S =VVTN

, where V is an ensemble of velocity measure-ments. R and S must be of the same dimensions. Next,R and S are used in the generalized eigenvalue problemdescribed by λRφφφ = Sφφφ. The natural modal frequenciesare estimated as ωi =

)λi, i = 1, · · · M, and LMNs are

approximated by columns of ΨΨΨ=ΦΦΦ'T , where ΦΦΦ is a matrixcontaining the eigenvectors of the generalized eigenvalueproblem. Derivations of the above ideas can be foundin [17, 18]. SOD has been shown to extract approximationsto LNMs and natural frequencies from simulated discreteand continuous systems [18].

Chelidze et al. did extensive simulations comparing

5

Fig. 4: Top: Fourth mode shape extracted by RMPOD (()plotted with the analytical approximation’s discretized modeshape. Middle: Fourth mode, modal acceleration coordinateform RMPOD. Bottom: FFT of modal the acceleration coor-dinates

SOD to POD [18]. Our work contributes to the field by firstusing experimental data, and second by using a nonuniformbeam.

The results of applying SOD to the free vibrations of thenonuniform beam of this study are included next. The natu-ral frequencies estimated by SODwere 43.72 Hz, 107.77 Hz,and 203.53 Hz. The extracted mode shapes had MAC valuesof 0.999, 0.820, and 0.937. From these it is suggested thatthe SOD can extract the lower modes of a lightly dampednonuniform beam. Figures 6, 7, and 8 show the SOD ex-tracted modes for the 2nd, 3rd, and 4th modes respectively.These modes are plotted with the analytical approximationsof a nonuniform Euler-Bernoulli beam. This research showsthat SOD can be used to extract modal information froma lightly damped freely vibrating nonuniform cantilevered

Fig. 5: Seventh mode shape extracted by RMPOD of instruc-tive purpose as an example of poor extraction

beam.

STATE VARIABLE MODAL DECOMPOSITIONIn SVMD [22] the outputs of the freely vibrating, lightly

damped beam were used to estimate the mode shapes, nat-ural frequencies, and in some cases modal damping ofthe beam. When applying SVMD one must first create astate-variable ensemble matrix, Y = [VT XT]T, where Vis the velocity ensemble matrix and Xis the displacementensemble matrix [22]. As such Y = [y(t1) y(t2) · · · y(tN)],where y(t) = [x1(t), · · · xM(t); x1(t), · · · xM(t)]T . A

2M" 2M correlation matrix is created such that R =YYT

N,

and unique to SVMD, a second 2M " 2M nonsymmetric

correlation matrix is created, N, such that N=YWT

N, where

W= [AT VT ]T = [y(t1) · · · y(tN)].

6

Fig. 6: Top: Second mode shape extracted by SOD (()plotted with the analytical approximation’s discretized modeshape. Middle: Second mode, modal acceleration coordinateform RMPOD. Bottom: FFTs of modal coordinates

Once the two correlation matrices are computed thenan eigenvalue problem is cast as λRφφφ = Nφφφ. This problemcan be solved for 2M eigenvalues λ and eigenvectors φφφ.If this eigensystem is solved in MATLAB using the eigcommand, it produces two matrices ΛΛΛ and ΦΦΦ in matrix formsatisfying, RΦΦΦΛΛΛ= SΦΦΦ. The eigenvalue matrix ΛΛΛ is diagonaland contains information about the natural frequencies and,in theory, modal damping. The real part of eigenvalue λiindicates the exponential decay rate of the mode, while theimaginary part represents the damped modal frequency.

The eigenvector matrix ΦΦΦ contains modal informationbut the inverse transpose of this matrix must be taken toextract the mode shapes [22]. So the matrix of eigenvectorsis ΨΨΨ = ΦΦΦ'T and each 2M " 1 column of ΨΨΨ containsinformation about the mode shapes of the beam; the bottomM"1 rows will contain the mode shapes from the displace-ments and approximate LNMs. These mode shapes maybe complex. If damping is approximately Caughey [23]

Fig. 7: Top: Third mode shape extracted by SOD (() plottedwith the analytical approximation’s discretized mode shape.Middle: Third mode, modal acceleration coordinate formRMPOD. Bottom: FFTs of modal coordinates

or Rayleigh (proportional) then the real parts dominate thecomplex modes and approximately correspond to the LNMs.

Using SVMD on the experimental beam, approxima-tions to LNMs were extracted as as shown in Figs 9, 10, and11 below. The extracted damped modal frequencies from theimaginary parts of the eigenvalues were 40.08 Hz, 106.42Hz, and 205.08 Hz for the second, third, and forth mode,respectively. The MAC values for these modes when com-pared to the analytical approximation were 0.9921, 0.9729,and 0.9865 for the second, third, and fourth mode respec-tively.

DAMPING RATIO ESTIMATIONIn the limit of modal (Caughey) damping, the eigenval-

ues associated with the underdamped modes are expectedto be complex in the form λi = 'ζiωi±ωd ı. Where ζi isthe damping ratio, ωi is the circular frequency, and ωdi =ωi

$1'ζ2i is the damped circular frequency. Noting that

7

Fig. 8: Top: Fourth mode shape extracted by SOD (() plottedwith the analytical approximation’s discretized mode shape.Middle: Fourth mode, modal acceleration coordinate formRMPOD. Bottom: FFTs of modal coordinates

Mode ζSVMD ζlogdec

2 0.0130 0.0126

3 0.0036 0.0116

4 0.0106 0.0064

5 -0.0022 0.003

6 0.0141 0.013

Table 3: SVMD extracted damping ratios and damping ra-tios computed from the log decrement of the SVMD modalcoordinates, using all peaks in the modal coordinates

|λi| =%λiλi = ωi and λi + λi = 2ζiωi, where the overbar

indicates a complex conjugate, then

ζi =λi+λi2|λi|

(6)

The modal damping ratios extracted by SVMD are in

Fig. 9: Top: Second mode shape extracted by SVMD (()plotted with the analytical approximation’s discretized modeshape (-). Middle: Second modal coordinate displacementfrom SVMD. Bottom: FFT of modal coordinate displace-ment

Table 3 for the second through six modes. The second col-umn shows the SVMD extracted damping ratio and the thirdcolumn shows the damping ratio computed from the SVMDmodal coordinates using logarithmic decrement on all thelocal maxima of the modal coordinate. The second modeshows great agreement between the SVMD extracted damp-ing and the logarithmic decrement. The agreement betweenthe two values deteriorate after the second mode. A simi-lar trend was mentioned in [17], and based on insight gainedfrom this paper attempts were made to extract better dampingestimates.

IMPROVING DAMPING ESTIMATESIGNAL SEGMENTS

In some situations, when damped structures undergofree vibrations higher frequency modes decay faster thanlower frequency modes. If a time segment of the free vibra-tion signal was chosen based on the settling time, Ts, of thehighest frequency of interest, for example the fourth mode,it maybe possible to improve the extraction of modal damp-ing using SVMD. The damping estimates in Table 3 usedthe time segment t # [1.1142 2.2000] seconds and all theincluded peaks in modal coordinates were used to compute

8

Fig. 10: Top: Third mode shape extracted by SVMD (()plotted with the analytical approximation’s discretized modeshape (-). Middle: Third modal coordinate displacementform SVMD. Bottom: Fast Fourier Transform of modal co-ordinate displacement

the logarithmic decrement. Figure 12 shows the modal co-ordinate in this time window. The settling time computedusing Ts = 4/ζω where ζ and ω and are taken from theinitial SVMD estimates. In this calculation ζ4 = 0.01 andω4 = 1275.66 rad/s which results in Ts4 = 0.2929 seconds.Using this as a guide for capturing enough cycles of the lowermodes for SVMD a time range of t # [1.1142 1.5330] sec-onds was used.

Upon close inspection of the modal coordinates it wasnoticed that the first few peaks showed some transient distor-tions, which may be a result of the filtering. As such the firstcouple of peaks are excluded in future logarithmic decrementcalculations. In the long time window, t # [1.1142 2.2000]seconds, peaks 3 and 40 were used in computing the loga-rithmic decrement of the second mode, peaks 3 and 40, and3 and 60 were used for the third and fourth modes respec-tively. The reason for the difference is that the second modehas a lower frequency and therefore fewer cycles this canbe seen in Fig. 12 and the resulting logarithmic decrementcalculations can be seen in Table 4 for t # [1.1142 2.2000]seconds and Table 5 for t # [1.1142 1.5330] seconds.

The shorter time segment showed improved damping ra-tio extraction. Next, proportional modal damping in addition

Fig. 11: Top: Fourth mode shape extracted by SVMD (()plotted with the analytical approximation’s discretized modeshape (-). Middle: Fourth modal coordinate displacementfrom SVMD. Bottom: FFT of modal coordinate displace-ment

to the shorter time segment was considered as a means forbetter modal damping extraction.

PROPORTIONAL DAMPINGIf the damping is assumed to be proportional then the

damping matrix is CCC = αMMM+ βKKK. Then the modal damp-ing coefficients are 2ζiωi = α+ βΛΛΛ. Three cases were ex-amined: β = 0, α = 0 and least squares solution of α andβ using SVMD extracted data from Table 3. The result-ing modal damping extractions are shown in Table 4 fort # [1.1142 2.2000] seconds, Table 5 for t # [1.1142 1.5330]seconds and Figs. 13, 14 and 15.

CONCLUSIONWe applied three output-only modal analysis decom-

position methods to a nonuniform beam. The fast Fouriertransform was applied to the beam’s accelerations to identitythe modal frequencies. The first mode was filtered outsince it was below the range of reliable accelerometerperformance. The beam was modeled as a nonuniformEuler-Bernoulli beam. An analytical approximation of the

9

Fig. 12: SVMD modal coordinates plotted with maximumpeaks ((), and boundaries for log decrement calculations(• *). Time segment 1.1142 & t & 2.2000 Top: Secondmode, Middle: Third mode, Bottom: Fourth mode

1.1142& t & 2.2000

Mode ωSVMD ζSVMD Log Dec (n)

2 251.45 0.0130 0.0122 (20)

3 644.96 0.0036 0.0126 (40)

4 1275.66 0.0106 0.0063 (60)

Table 4: SVMD modal damping long time segment esti-mates. 2nd column: Log decrement was computed usingpeaks [3,(n)].

beam was created and its mode shapes were determined andused to predict the natural frequencies of experimental beamand showed agreement with the FFT frequencies.

The reduced-order mass-weighted POD was appliedusing a permutation of conditions involving impulse lo-cation, impulse amplitudes, and ensembles created fromdisplacement, velocity and acceleration signals. RMPOD

Fig. 13: Top: SVMD modal coordinates for the secondmode 1.1142 & t & 2.2000 (solid curve), standard SVMDdamping(!), logarithmic damping (() for peak 3 to peak20, least square solution to 2ωiζi = α+ βΛΛΛ damping (*)Bottom: SVMD modal coordinates for the second mode1.1142 & t & 1.5330 (solid curve), standard SVMD damp-ing (!), logarithmic damping (() for peak 3 to peak 14, leastsquare solution to 2ωiζi = αIII+βΛΛΛ damping (*)

Fig. 14: Top: SVMD modal coordinates for the third mode1.1142 & t & 2.2000 (solid curve), standard SVMD damp-ing (!), logarithmic damping (() for peak 3 to peak 60, leastsquare solution to 2ωiζi = αI+ βΛΛΛ damping (*), Bottom:SVMD modal coordinates for the third mode 1.1142 & t &1.5330 (solid curve), standard SVMD damping (!), logarith-mic damping (() for peak 3 to peak 40, least square solutionto 2ωiζi = αIII+βΛΛΛ damping (*)

10

Fig. 15: Top: SVMD modal coordinates for the fourth mode1.1142 & t & 2.2000 (solid curve), standard SVMD damp-ing (!), logarithmic damping (() for peak 3 to peak 60, leastsquare solution to 2ωiζi = αI+ βΛΛΛ damping (*) Bottom:SVMD modal coordinates for the fourth mode 1.1142& t &1.5330 (solid curve), standard SVMD damping (!), logarith-mic damping (() for peak 3 to peak 40, least square solutionto 2ωiζi = αIII+βΛΛΛ damping (*)

1.1142& t & 1.5330

Mode ωSVMD ζSVMD Log Dec(n)

2 253.34 0.0058 0.0123 (14)

3 663.17 0.0097 0.0113 (40)

4 1275.53 0.0030 0.0037 (40)

Table 5: SVMD modal damping short time segment esti-mates. Logarithmic decrement was computed using peaks[3,(n)].

extracted approximations to the second, third, and fourthLNMs suggested by MAC values of 0.986, 0.852, and0.912 between extracted, modes and analytical modes ofthe second, third, and fourth modes, respectively. Furtherconfirmation on the quality of the modes was provided fromcomputing the modal coordinates and taking their FFTs. Thepeak frequency for the lowest extracted mode was dominant.For increasingly higher modal coordinates, frequencies ofother modes leaked in. Apparently, the pollution of thesemodes did not greatly affect the approximation to the LNMs.

SOD and SVMD were also used on the same data toextract the modal frequencies and approximations to theLNMs. The SOD extracted frequencies agreed with theFFT of the experimental beam. SOD extracted mode shapesagreed with the analytical approximation. Similar agreement

1.1142& t & 2.2000 α= 2.53 β= 1.45"10'6

Mode SVMDζ2 0.0053

3 0.0025

4 0.0019

Table 6: SVMD modal damping long time least square pro-portional estimates

1.1142& t & 1.5330 α= 6.946 β= 1.23"10'6

Mode SVMDζ2 0.0139

3 0.0056

4 0.0035

Table 7: SVMD modal damping short time least square pro-portional estimates

was found when SVMD was applied.

SVMD was able to extract the modal damping ratiodirectly without the use of modal coordinates. SVMDextraction of the lower modes’ modal damping was betterwith longer time signals. This could be a result of thehigher modes decay faster and as time increases the signal isdominated by the lower modes. Likewise with short signalsegment near the time of impact SVMD’s ability to extracthigh modes improved. With similar reasoning at the shortersignal near the impact are not as dominated by the lowerfrequencies modes Assuming proportional damping shouldbest performance with taking using α and β determinedfrom the least squares solution.

These tests suggest that RMPOD, SOD, and SVMDcan be reliable methods of modal identification, at least forthe lower modes of a structure and are easy to implement.The only signal processing needed is in integrating the ac-celerometer signals into the desired states (displacement, ve-locity, or acceleration), and high pass filtering used to pre-vent integrator drift. The relative performance of these threemethods cannot determined from the results presented in thispaper. Several impulses was used to excited the beam at avariety of locations and only one permutation of those inputsare reported. The data set reported in this paper optimizedthe results of RMPOD only.When these excitation parame-ters are applied to other data sets, the methods seem to beequally reliable and in the absence of an analytical approxi-mation, the methods could be used in concert to cross checkresults. Table 8 summarizes some of the benefits and draw-backs of each method. However, the drawbacks of each de-composition method are not serious.

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PROS and CONS

RMPOD

can estimate mode shapes frequencies not directly es-timated (need QQQ)

RMPOVs estimate modalstrength

need to compute the re-duced mass matrix

requires single RRR

requires XXX only

input signal not needed

SVMD

can estimate mode shapes no modal strength, exceptby QQQ

estimate modal frequenciesdirectly

need XXX , VVV , and AAA

possibility of modal damp-ing directly

mass not required

input signal not needed

SOD

can estimate mode shapes no modal strength exceptby QQQ

estimate modal frequencydirectly

need XXX and VVV

mass not required

input signal not needed

Table 8: Pros and cons of each decomposition method

ACKNOWLEDGMENTSThis work was supported by the National Science

Foundation grant numbers CMMI-0943219 and CMMI-0727838. Any opinions, findings, and conclusions orrecommendations are those of the authors and do not nec-essarily reflect the views of the National Science Foundation.

Additional support was received from the Diversity Pro-grams Office and the College of Engineering at MichiganState University.

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