Research ArticleSpectral Representation of Uncertainty inExperimental Vibration Modal Data

K. Sepahvand ,1 Ch. A. Geweth,1 F. Saati,1 M. Klaerner,2 L. Kroll,2 and S. Marburg1

1Chair of Vibroacoustics of Vehicles and Machines, Department of Mechanical Engineering, Technical University of Munich,85748 Garching b. Munich, Germany2Department of Lightweight Structures and Polymer Technology, Faculty ofMechanical Engineering, Technical University of Chemnitz,09107 Chemnitz, Germany

Correspondence should be addressed to K. Sepahvand; k.sepahvand@tum.de

Received 21 April 2018; Accepted 19 June 2018; Published 9 July 2018

Academic Editor: Akira Ikuta

Copyright © 2018 K. Sepahvand et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

It is well known that structures exhibit uncertainty due to various sources, such as manufacturing tolerances and variations inphysical properties of individual components. Modeling and accurate representation of these uncertainties are desirable in manypractical applications. In this paper, spectral-based method is employed to represent uncertainty in the natural frequencies offiber-reinforced composite plates. For that, experimental modal analysis using noncontact method employing Laser-Vibrometer isconducted on 100 samples of plates having identical nominal topology. The random frequencies then are represented employinggeneralized Polynomial Chaos (gPC) expansions having unknown deterministic coefficients.This provides us withmajor advantageto approximate the random experimental data using closed form functions combining deterministic coefficients and randomorthogonal basis. Knowing the orthogonal basis, the statistical moments of the data are used to estimate the unknown coefficients.

1. Introduction

Uncertainty quantification concerns representation and solu-tion of simulation models, e.g., a differential equation or afinite element model, when some levels of modeling such asinput parameters are not exactly known. In such conditions,the model is said to be stochastic, i.e., it exhibits some degreeof uncertainty. Probabilistic structural dynamics, in particu-lar, endeavor to take into account uncertainties relating vari-ous aspects of real structures such as material and geometricparameters, loading terms, and initial/boundary conditionsand exploring related impacts on the structure responses.To improve the performance, durability, and efficiency ofstructures, an exact knowledge of geometrical and materialparameters is required. Characterization of the stochasticresponse due to these uncertainties by stochasticmethods hasgained interest among researchers in past decades. Stochasticmethods in conjunction with finite element method (FEM)have been widely used to quantify uncertainty in structuralresponses [1–9]. Two major issues have to be addressed

regarding stochastic analysis of such structures under uncer-tainty: first, how the uncertainties can be efficiently identifiedand modeled in numerical simulations, particularly in finiteelement models and, secondly, how uncertainties affect thebehavior of aforesaid structures.

The former issue requires quantifying the randomnessin uncertain parameters. This can be efficiently character-ized by the statistical properties of the parameters, e.g.,probability density function (PDF). However, identificationof the appropriate PDF type characterizing the parameteruncertainties demands to know a priori information whichmay be collected from experimental tests. Various methodshave been developed in past decades for PDF identificationfrom experimental data (cf. [10–13]).The latter issue dependson the availability of an exact model relating inputs tooutputs. Construction of such model is not possible dueto many common assumptions in modeling of structuraldynamics. For that reason, experimental methods are stillthe most reliable approach for investigation of the uncertain-ties.

HindawiAdvances in Acoustics and VibrationVolume 2018, Article ID 9695357, 5 pageshttps://doi.org/10.1155/2018/9695357

2 Advances in Acoustics and Vibration

In this paper, uncertainties relating to the experimentallyidentified natural frequencies of composite plates are inves-tigated. Uncertainties in such materials may have differentsources, e.g., manufacturing tolerances, fiber orientations, orphysical properties of individual components. To this end,experimental modal analysis using the noncontact methodby employing Laser-Vibrometer is conducted on 100 samplesof plates having identical nominal topology. The statisticalproperties of the identified natural frequencies of the plates,in particular, are discussed in detail. The random frequen-cies then are represented employing generalized PolynomialChaos (gPC) expansion [14–17]. This provides us with majoradvantage to approximate the experimental data using closedform functions combining deterministic coefficients andrandom orthogonal basis. The coefficients then are estimatedemploying optimization procedure comparing theoreticaland experimental values of statistical moments.

This paper is organized as follows: the basic formulationof the spectral-based representation of random parametersis given in the next section. The numerical-experimentalsimulations are presented in Section 3 and the last sectiondenotes the conclusion.

2. Spectral-Based Representation ofRandom Parameters

The spectral discretization methods are the key advantagefor the efficient stochastic reduced basis representation ofuncertain parameters in finite element modeling. This isbecause these methods provide a similar application of thedeterministic Galerkin projection and collocation methodsto reduce the order of complex systems. In this way, itis common to employ a truncated expansion to discretizethe input random quantities of the structure and systemresponses. The unknown coefficients of the expansions thencan be calculated based on the FE model outputs. Let usconsider the uncertain parameter 𝑃(𝜉) where 𝜉 ∈ Ω isthe vector random variable characterizing the uncertaintyin the parameter and Ω denotes the random space. Underthe limited variance, i.e., 𝜎2 < ∞, the parameter can beapproximated by

𝑃 (𝜉) ≈ 𝑁∑𝑖=0

𝑝𝑖Ψ𝑖 (𝜉) (1)

which is known as the truncated generalized PolynomialChaos (gPC) expansion of the parameter. The determinis-tic coefficients 𝑝𝑖 are calculated employing the stochasticGalerkin projection as [17]

𝑝𝑖 = 1ℎ2𝑖 ∫Ω 𝑃 (𝜉) Ψ𝑖 (𝜉) 𝑓 (𝜉) d𝜉 (2)

where 𝑓 is the joint probability density function (PDF) ofrandom vector 𝜉 which for independent random variables 𝜉𝑖can be written as the multiplication of the individual PDF foreach variable 𝑓𝑖(𝜉𝑖); i.e.,

𝑓 (𝜉) 𝑑𝜉 = 𝑓1 (𝜉1) 𝑓2 (𝜉2) ⋅ ⋅ ⋅ 𝑓𝑛 (𝜉𝑛) d𝜉1d𝜉2 ⋅ ⋅ ⋅ d𝜉𝑛 (3)

and ℎ𝑖 denotes the norm of polynomials defined as

ℎ2𝑖 = ⟨Ψ𝑖 (𝜉) , Ψ𝑖 (𝜉)⟩ = ∫ΩΨ2𝑖 (𝜉) 𝑓 (𝜉) d𝜉 (4)

For the sake of simplicity, we will focus onmonodimensionalrandom input in this work; i.e., 𝜉 = {𝜉}.2.1. Estimation of the gPC Coefficients from ExperimentalData. Calculation of the coefficients using (2) requires priorinformation on the PDF of uncertain parameters which maynot be available. Statistical moments, in contrast, exhibitadequate implicit information on the probability propertiesof random quantities. Once the experimental data on theuncertain parameters are available, the estimation of thegPC coefficients from statistical moments [11, 17] is possible.Here, the statistical moments derived analytically from thegPC expansion are compared to those calculated from thedata. For the truncated gPC expansion, only the first feworders of moments are required to calculate the coefficients.The major benefit is that information on the probabilitydistribution of uncertain parameters does not have to beknown a priori. The coefficients are then estimated compar-ing statistical moments constructed from the gPC expansionand experimental data via an optimization procedure. The𝑘th-oder statistical moment, 𝑚𝑘, of uncertain 𝑃 having thegPC expansion given in (1) is calculated as

𝑚𝑘 = E [𝑃𝑘] = ∫Ω[ 𝑁∑𝑖=0

𝑝𝑖Ψ𝑖 (𝜉)]𝑘

𝑓 (𝜉) d𝜉,𝑘 = 0, 1, . . .

(5)

with 𝑚0 = 1 and 𝑚1 = 𝑝0. In this paper, we use theprobabilistic orthonormal Hermite polynomials for Ψ𝑖(𝜉)and, accordingly, 𝑓(𝜉) = (1/√2𝜋)exp(−𝜉2/2). The 𝑘th-ordercentral statistical moment is then calculated as

𝜇𝑘 = E [(𝑃 − E [𝑃])𝑘] = 𝑘∑𝑖=0

(𝑘𝑖) (−1)𝑘−𝑖𝑚𝑖𝑚𝑘−𝑖1 ,𝑘 = 2, 3, . . .

(6)

This leads imminently to the following expression for the sec-ond central statistical moment (variance) 𝜇2 of the uncertainparameter 𝑃 as

𝜇2 = 𝑚2 − 𝑚21 =𝑁∑𝑖=1

𝑝2𝑖 ℎ2𝑖 (7)

in which ℎ2𝑖 = 𝑖! for the Hermite polynomials. Similarexpressions can be derived for the higher order moments asfunctions of the gPC coefficients. The calculated momentsform the gPC expansion can be compared to the correspond-ing values obtained from experimental data for an uncertainparameter. In such a way, one can attempt to estimate theunknown coefficients from the available experimental data.That is, for a given set of experimental data {𝑃1, 𝑃2, . . . , 𝑃𝑀}

Advances in Acoustics and Vibration 3

Table 1:The first three statistical moments of the first nine measured natural frequencies; mean value, 𝜇𝑓, the standard deviation 𝜎𝑓, and theskewness 𝛾3.

𝑓1 𝑓2 𝑓3 𝑓4 𝑓5 𝑓6 𝑓7 𝑓8 𝑓9𝜇𝑓 [Hz] 115 145 275 396 504 530 556 704 781𝜎𝑓 [Hz] 4.60 5.90 8.93 15.22 11.92 15.92 15.35 17.98 25.95𝛾3𝑓 [–] 0.485 0.366 0.895 0.292 0.166 1.086 1.312 0.622 1.419

on𝑃, the experimental central moments, 𝜇expri𝑘

, are calculatedas

𝜇expri𝑘

= 1𝑀𝑀∑𝑗=1

[𝑃𝑗 − E (𝑃)]𝑘 , 𝑘 = 2, 3, . . . (8)

in which E(𝑃) = (1/𝑀)∑𝑀𝑗=1 𝑃𝑗 is the mean value ofsamples. An error function based on the least-square crite-rion corresponding to the difference between the momentsderived from (5) and (8) can be used to estimate the optimalcoefficients 𝑝𝑖. This leads to a minimization problem asfollows:

minimize𝑝𝑖

𝑘∑𝑛=1

𝑓2𝑛 (𝑝𝑖)s.t. 𝑓0 (𝑝𝑖) = 𝜇expri1 − 𝜇1 = 0

𝑓𝑛 (𝑝𝑖) = 𝜇expri𝑘 − 𝜇𝑘, 𝑛 ≥ 2(9)

The first condition of the process denotes that the expectedvalue of the data represents the first coefficient of the gPCexpansion. Since the calculated moments for the gPC expan-sion are nonlinear functions of the coefficients, one has toemploy nonlinear optimization procedure. The optimizationleads to unique solution under the convergence condition forcoefficients of one-dimensional gPC; i.e., ‖𝑝𝑖+1‖ < ‖𝑝𝑖‖.3. Experimental and Numerical Study

As a case study, in this section, the natural frequencies offiber-reinforced composite (FRC) plates are represented asrandom parameters. The experimental modal analysis hasbeen performed on 100 sample plates with nominal identicaltopology of 𝑎 = 250 mm, 𝑏 = 125 mm, and thicknessof 2 mm; see Figure 1. The plates were suspended by verythin elastic bands to simulate free boundary conditions. TheLaser-Vibrometer has been employed to collect the vibrationresponses due to the excitation force from impulse hammerwith tip force transducer at some predefined points of theplates. The average of 5 impacts at each point was recorded.The sample test has returned to rest before the next impactis taken. A standard data acquisition facility along with apostprocessing modal analysis software has been used toextract modal data, e.g., natural frequencies. The measuredfirst nine natural frequencies of plates are given in Figure 2.As shown, a considerable range of uncertainty is observed infrequencies. Spectral representation of the measured randomfrequencies requires estimating the statistical moments. The

FRC

plat

e

LSVHammer

Spectrumanalysis

Vibrationmodal data

Elasticbands

Figure 1: Experiment setup for modal analysis of FRC plates.

second-order gPC expansion is employed to approximatethe frequencies. This leads to three unknown coefficientsand, consequently, estimation of the first three statisticalmoments as given in Table 1. The third statistical moment isgiven by the skewness, 𝛾3𝑓 , defined as 𝜇expri3 /𝜎3𝑓. The randomnatural frequencies are represented using second-order gPCexpansions having random Hermite polynomials 𝐻(𝜉) asbasis; i.e.,

𝑓𝑛 = 2∑𝑖=0

𝑝𝑛𝑖𝐻𝑖 (𝜉) = 𝑝𝑛0 + 𝑝𝑛1𝜉 + 𝑝𝑛2 (𝜉2 − 1) ,𝑛 = 1, 2, . . . , 9

(10)

The unknown deterministic coefficients 𝑝𝑛𝑖 are calculated byequality of the statistical moments of the gPC expansionsgiven in (6) and the experimental estimations given in Table 1.The optimization Application optimtool in MATLAB� forconstrained nonlinear problem is employed for this purpose.This leads immediately to the following expressions foroptimization problem defined in (9):

𝑓1 (𝑝𝑛0) = 𝜇𝑓 − 𝑝𝑛0 = 0𝑓2 (𝑝𝑛1 , 𝑝𝑛2) = 𝜎2𝑓 − 𝑝2𝑛1 + 2𝑝2𝑛2𝑓3 (𝑝𝑛1 , 𝑝𝑛2) = 𝛾3𝑓 − 1𝜎3

𝑓

(6𝑝2𝑛1𝑝𝑛2 + 8𝑝3𝑛2)(11)

The nonlinear optimization problem is then solved to esti-mate 𝑝𝑛𝑖 as given in Table 2.The second-order coefficients arevery small compared to the first two coefficients.This denotes

4 Advances in Acoustics and Vibration

20 40 60 80 100105

110

115

120

125

20 40 60 80 100

130

140

150

160

20 40 60 80 100

260

270

280

290

300

20 40 60 80 100340

360

380

400

420

440

20 40 60 80 100

480

490

500

510

520

20 40 60 80 100500

520

540

560

20 40 60 80 100520

540

560

580

20 40 60 80 100660

680

700

720

740

20 40 60 80 100

750

800

850

f1

f2

f3

f4

f5

f6

f7

f8

f9

Figure 2: Measured first nine natural frequencies of 100 FRC plates (all in Hz).

Table 2: The coefficients of the gPC expansions approximating the measured uncertain natural frequencies.

𝑓1 𝑓2 𝑓3 𝑓4 𝑓5 𝑓6 𝑓7 𝑓8 𝑓9𝑝𝑛0 115 145 275 396 504 530 566 704 781𝑝𝑛1 4.53 5.87 8.72 15.18 11.90 15.36 14.53 17.78 24.32𝑝𝑛2 0.37 0.36 1.35 0.74 0.33 2.95 3.48 1.88 6.40

that the second-order gPC expansion has enough accuracy torepresent uncertainty in the random frequencies. Once theunknown coefficients are known, the statistical properties ofthe measured data can be calculated using the constructedgPC expansions.

4. Conclusion

Natural frequencies of composite plates have been consideredas random parameters. The generalized Polynomial Chaosexpansions has been employed to approximate the uncer-tainty in the measured natural frequencies. The methodoffers the major advantage that the unknown deterministiccoefficients of the expansions have to be calculated instead ofrandom parameters. This has been performed by comparingthe statistical moments of the experimental results for 100identical plates and from the expansions via theminimizationof least-square based error. The results have been givenfor the first nine natural frequencies using second-orderexpansions.

Data Availability

The measured data used to support this study were suppliedfrom project no. DFG-KR 1713/18-1 funded by the GermanResearch Foundation (Deutsche Forschungsgemeinschaft-DFG). There is no restrictions to use the officially publisheddata.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the German Research Founda-tion (Deutsche Forschungsgemeinschaft,DFG) under projectno. DFG-KR 1713/18-1 and the Technical University ofMunich within the funding programme Open Access Pub-lishing. The support is gratefully acknowledged.

Advances in Acoustics and Vibration 5

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