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Spectral Representation of Uncertainty in Experimental Vibration Modal Data

Spectral Representation of Uncertainty in Experimental Vibration Modal Data Hindawi Advances in Acoustics and Vibration Volume 2018, Article ID 9695357, 5 pages https://doi.org/10.1155/2018/9695357 Research Article Spectral Representation of Uncertainty in Experimental Vibration Modal Data 1 1 1 2 2 1 K. Sepahvand , Ch.A.Geweth, F. Saati, M. Klaerner, L. Kroll, and S. Marburg Chair of Vibroacoustics of Vehicles and Machines, Department of Mechanical Engineering, Technical University of Munich, 85748 Garching b. Munich, Germany Department of Lightweight Structures and Polymer Technology, Faculty of Mechanical 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 in physical properties of individual components. Modeling and accurate representation of these uncertainties are desirable in many practical applications. In this paper, spectral-based method is employed to represent uncertainty in the natural frequencies of fiber-reinforced composite plates. For that, experimental modal analysis using noncontact method employing Laser-Vibrometer is conducted on 100 samples of plates having identical nominal topology. The random frequencies then are represented employing generalized Polynomial Chaos (gPC) expansions having unknown deterministic coefficients. This provides us with major advantage to approximate the random experimental data using closed form functions combining deterministic coefficients and random orthogonal basis. Knowing the orthogonal basis, the statistical moments of the data are used to estimate the unknown coefficients. 1. Introduction regarding stochastic analysis of such structures under uncer- tainty: first, how the uncertainties can be efficiently identified Uncertainty quantification concerns representation and solu- and modeled in numerical simulations, particularly in ni fi te tion of simulation models, e.g., a dieff rential equation or a element models and, secondly, how uncertainties aeff ct the finite element model, when some levels of modeling such as behavior of aforesaid structures. input parameters are not exactly known. In such conditions, The former issue requires quantifying the randomness the model is said to be stochastic, i.e., it exhibits some degree in uncertain parameters. This can be efficiently character- of uncertainty. Probabilistic structural dynamics, in particu- ized by the statistical properties of the parameters, e.g., lar, endeavor to take into account uncertainties relating vari- probability density function (PDF). However, identification ous aspects of real structures such as material and geometric of the appropriate PDF type characterizing the parameter parameters, loading terms, and initial/boundary conditions uncertainties demands to know a priori information which and exploring related impacts on the structure responses. may be collected from experimental tests. Various methods To improve the performance, durability, and efficiency of have been developed in past decades for PDF identification structures, an exact knowledge of geometrical and material from experimental data (cf. [10–13]). The latter issue depends parameters is required. Characterization of the stochastic ontheavailabilityofanexact modelrelatinginputsto response due to these uncertainties by stochastic methods has outputs. Construction of such model is not possible due gained interest among researchers in past decades. Stochastic to many common assumptions in modeling of structural methods in conjunction with finite element method (FEM) dynamics. For that reason, experimental methods are still have been widely used to quantify uncertainty in structural the most reliable approach for investigation of the uncertain- responses [1–9]. Two major issues have to be addressed ties. 2 Advances in Acoustics and Vibration In this paper, uncertainties relating to the experimentally and ℎ denotes the norm of polynomials defined as identified natural frequencies of composite plates are inves- tigated. Uncertainties in such materials may have different 2 2 ℎ = ⟨Ψ 𝜉 ,Ψ 𝜉 ⟩ = ∫ Ψ 𝜉 𝑓 𝜉 d𝜉 ( ) ( ) ( ) ( ) (4) 𝑖 𝑖 𝑖 𝑖 sources, e.g., manufacturing tolerances, b fi er orientations, or Ω physical properties of individual components. To this end, For the sake of simplicity, we will focus on monodimensional experimental modal analysis using the noncontact method random input in this work; i.e., 𝜉 ={𝜉} . by employing Laser-Vibrometer is conducted on 100 samples of plates having identical nominal topology. eTh statistical properties of the identified natural frequencies of the plates, 2.1. Estimation of the gPC Coefficients from Experimental in particular, are discussed in detail. The random frequen- Data. Calculation of the coefficients using (2) requires prior cies then are represented employing generalized Polynomial information on the PDF of uncertain parameters which may not be available. Statistical moments, in contrast, exhibit Chaos (gPC) expansion [14–17]. This provides us with major advantage to approximate the experimental data using closed adequate implicit information on the probability properties form functions combining deterministic coefficients and of random quantities. Once the experimental data on the random orthogonal basis. The coefficients then are estimated uncertain parameters are available, the estimation of the employing optimization procedure comparing theoretical gPC coefficients from statistical moments [11, 17] is possible. and experimental values of statistical moments. Here, the statistical moments derived analytically from the Thispaper isorganizedasfollows: thebasic formulation gPC expansion are compared to those calculated from the of the spectral-based representation of random parameters data. For the truncated gPC expansion, only the rfi st few is given in the next section. The numerical-experimental orders of moments are required to calculate the coefficients. simulations are presented in Section 3 and the last section eTh major benefit is that information on the probability denotes the conclusion. distribution of uncertain parameters does not have to be known apriori.Thecoecffi ientsare thenestimatedcompar- ing statistical moments constructed from the gPC expansion 2. Spectral-Based Representation of and experimental data via an optimization procedure. The Random Parameters 𝑘th-oder statistical moment, 𝑚 ,ofuncertain 𝑃 having the gPC expansion given in (1) is calculated as eTh spectral discretization methods are the key advantage for the efficient stochastic reduced basis representation of uncertain parameters in finite element modeling. This is 𝜉 ] 𝑚 = E[𝑃 ]= ∫ [ ∑ 𝑝 Ψ ( ) 𝑓 (𝜉 )d𝜉, 𝑘 𝑖 𝑖 because these methods provide a similar application of the Ω (5) 𝑖=0 deterministic Galerkin projection and collocation methods to reduce the order of complex systems. In this way, it 𝑘 = 0,1,... is common to employ a truncated expansion to discretize the input random quantities of the structure and system with 𝑚 =1 and 𝑚 =𝑝 .Inthispaper,weusethe 0 1 0 responses. The unknown coefficients of the expansions then probabilistic orthonormal Hermite polynomials for Ψ(𝜉) canbecalculatedbasedon theFEmodel outputs. Letus and, accordingly, 𝑓(𝜉) = (1/ 2𝜋)exp (−𝜉 /2).The 𝑘th-order consider the uncertain parameter 𝑃(𝜉) where 𝜉 ∈Ω is central statistical moment is then calculated as the vector random variable characterizing the uncertainty in the parameter and Ω denotes the random space. Under 𝑘 𝑘 𝑘−𝑖 𝑘−𝑖 𝜇 = E[(𝑃− E [𝑃 ]) ]= ∑ ( ) (−1) 𝑚 𝑚 , 𝑘 𝑖 1 thelimitedvariance,i.e., 𝜎 <∞,the parametercan be (6) 𝑖=0 approximated by 𝑘 = 2,3,... 𝑃 (𝜉 ) ≈ ∑ 𝑝 Ψ (𝜉 ) (1) 𝑖 𝑖 This leads imminently to the following expression for the sec- 𝑖=0 ond central statistical moment (variance) 𝜇 of the uncertain which is known as the truncated generalized Polynomial parameter 𝑃 as Chaos (gPC) expansion of the parameter. eTh determinis- tic coecffi ients 𝑝 arecalculatedemployingthestochastic 2 2 2 𝜇 =𝑚 −𝑚 = ∑ 𝑝 ℎ (7) 2 2 1 𝑖 𝑖 Galerkin projection as [17] 𝑖=1 𝑝 = ∫ 𝑃 (𝜉 )Ψ (𝜉 )𝑓 (𝜉 )d𝜉 (2) 𝑖 𝑖 in which ℎ =𝑖! for the Hermite polynomials. Similar expressions can be derived for the higher order moments as functions of the gPC coefficients. The calculated moments where 𝑓 is the joint probability density function (PDF) of form the gPC expansion can be compared to the correspond- random vector 𝜉 which for independent random variables 𝜉 ing values obtained from experimental data for an uncertain can be written as the multiplication of the individual PDF for parameter. In such a way, one can attempt to estimate the each variable 𝑓 (𝜉 ); i.e., 𝑖 𝑖 unknown coefficients from the available experimental data. 𝑓 (𝜉 )𝑑𝜉 =𝑓 (𝜉 )𝑓 (𝜉 )⋅⋅⋅𝑓 (𝜉 )d𝜉 d𝜉 ⋅⋅⋅ d𝜉 (3) That is, for a given set of experimental data {𝑃 ,𝑃 ,...,𝑃 } 1 1 2 2 𝑛 𝑛 1 2 𝑛 1 2 𝑀 Advances in Acoustics and Vibration 3 Table 1: eTh first three statistical moments of the first nine measured natural frequencies; mean value, 𝜇 , the standard deviation 𝜎 ,and the 𝑓 𝑓 skewness 𝛾 . 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 1 2 3 4 5 6 7 8 9 𝜇 [Hz] 115 145 275 396 504 530 556 704 781 𝜎 [Hz] 4.60 5.90 8.93 15.22 11.92 15.92 15.35 17.98 25.95 𝛾 [–] 0.485 0.366 0.895 0.292 0.166 1.086 1.312 0.622 1.419 expri on 𝑃, the experimental central moments, 𝜇 ,are calculated Elastic as bands Vibration expri 𝜇 = ∑ [𝑃 − E 𝑃 ] , 𝑘 = 2,3,... ( ) (8) modal data LSV Hammer 𝑗=1 in which E(𝑃) = (1/𝑀)∑ 𝑃 is the mean value of 𝑗=1 samples. An error function based on the least-square crite- rion corresponding to the difference between the moments derived from (5) and (8) can be used to estimate the optimal Spectrum coefficients 𝑝 . This leads to a minimization problem as 𝑖 analysis follows: Figure 1: Experiment setup for modal analysis of FRC plates. minimize ∑ 𝑓 (𝑝 ) 𝑛 𝑖 𝑛=1 (9) expri second-order gPC expansion is employed to approximate s.t.𝑓 (𝑝 )=𝜇 −𝜇 =0 0 𝑖 1 the frequencies. This leads to three unknown coefficients expri and, consequently, estimation of the rst fi three statistical 𝑓 (𝑝 )=𝜇 −𝜇 ,𝑛≥2 𝑛 𝑖 𝑘 moments as given in Table 1. eTh third statistical moment is expri given by the skewness, 𝛾 , defined as 𝜇 /𝜎 .Therandom The first condition of the process denotes that the expected 3 3 𝑓 value of the data represents the rfi st coecffi ient of the gPC natural frequencies are represented using second-order gPC expansion. Since the calculated moments for the gPC expan- expansions having random Hermite polynomials 𝐻(𝜉) as sion are nonlinear functions of the coecffi ients, one has to basis; i.e., employ nonlinear optimization procedure. eTh optimization leads to unique solution under the convergence condition for 𝑓 = ∑ 𝑝 𝐻 (𝜉 )=𝑝 +𝑝 𝜉+𝑝 (𝜉 −1), coefficients of one-dimensional gPC; i.e., ‖𝑝 ‖<‖𝑝 ‖. 𝑛 𝑛 𝑖 𝑛 𝑛 𝑛 𝑖+1 𝑖 𝑖 0 1 2 𝑖=0 (10) 3. Experimental and Numerical Study 𝑛 = 1,2,...,9 As a case study, in this section, the natural frequencies of The unknown deterministic coefficients 𝑝 are calculated by fiber-reinforced composite (FRC) plates are represented as equality of the statistical moments of the gPC expansions random parameters. eTh experimental modal analysis has given in (6) and the experimental estimations given in Table 1. been performed on 100 sample plates with nominal identical eTh optimization Application optimtool in MATLAB for topology of 𝑎 = 250 mm, 𝑏 = 125 mm, and thickness constrained nonlinear problem is employed for this purpose. of 2mm; seeFigure1.Theplatesweresuspended byvery This leads immediately to the following expressions for thin elastic bands to simulate free boundary conditions. eTh optimization problem den fi ed in (9): Laser-Vibrometer has been employed to collect the vibration responses due to the excitation force from impulse hammer 𝑓 (𝑝 )=𝜇 −𝑝 =0 1 𝑛 𝑓 𝑛 with tip force transducer at some predefined points of the 0 0 plates. eTh average of 5 impacts at each point was recorded. 2 2 2 𝑓 (𝑝 ,𝑝 )=𝜎 −𝑝 +2𝑝 2 𝑛 𝑛 The sample test has returned to rest before the next impact 𝑓 𝑛 𝑛 1 2 1 2 (11) is taken. A standard data acquisition facility along with a 2 3 postprocessing modal analysis software has been used to 𝑓 (𝑝 ,𝑝 )=𝛾 − (6𝑝 𝑝 +8𝑝 ) 3 𝑛 𝑛 3 𝑛 𝑛 𝑛 1 2 𝑓 3 2 1 2 extract modal data, e.g., natural frequencies. The measured first nine natural frequencies of plates are given in Figure 2. As shown, a considerable range of uncertainty is observed in The nonlinear optimization problem is then solved to esti- frequencies. Spectral representation of the measured random mate 𝑝 as given in Table 2. The second-order coefficients are frequencies requires estimating the statistical moments. eTh very small compared to the first two coefficients. This denotes FRC plate 4 Advances in Acoustics and Vibration 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 490 520 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 740 850 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 Figure 2: Measured first nine natural frequencies of 100 FRC plates (all in Hz). Table 2: eTh coefficients of the gPC expansions approximating the measured uncertain natural frequencies. 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 1 2 3 4 5 6 7 8 9 𝑝 115 145 275 396 504 530 566 704 781 𝑝 4.53 5.87 8.72 15.18 11.90 15.36 14.53 17.78 24.32 𝑝 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 to Data Availability represent uncertainty in the random frequencies. Once the eTh measured data used to support this study were supplied unknown coefficients are known, the statistical properties of from project no. DFG-KR 1713/18-1 funded by the German the measured data can be calculated using the constructed Research Foundation (Deutsche Forschungsgemeinschaft- gPC expansions. DFG). er Th e is no restrictions to use the officially published data. 4. Conclusion Natural frequencies of composite plates have been considered Conflicts of Interest as random parameters. eTh generalized Polynomial Chaos expansions has been employed to approximate the uncer- eTh authors declare that they have no conflicts of interest. tainty in the measured natural frequencies. The method oer ff s the major advantage that the unknown deterministic coefficients of the expansions have to be calculated instead of Acknowledgments random parameters. This has been performed by comparing the statistical moments of the experimental results for 100 This work was supported by the German Research Founda- identical plates and from the expansions via the minimization tion (Deutsche Forschungsgemeinschaft, DFG) under project of least-square based error. eTh results have been given no. DFG-KR 1713/18-1 and the Technical University of for the first nine natural frequencies using second-order Munich within the funding programme Open Access Pub- expansions. lishing. The support is gratefully acknowledged. f f f 7 4 1 f f f 8 5 2 f f f 9 6 3 Advances in Acoustics and Vibration 5 References [1] M. Hanss, “Applied fuzzy arithmetic: An introduction with engineering applications,” Applied Fuzzy Arithmetic: An Intro- duction with Engineering Applications, pp. 1–256, 2005. [2] M.KleiberandT.D.Hien, eTh Stochastic Finite Element Method: Basic Perturbation Technique and Computer Implementation, JohnWiley&Sons,New York,NY, USA, 1992. [3] R. Ghanem, “Hybrid stochastic finite elements and generalized monte carlo simulation,” Journal of Applied Mechanics,vol.65, no. 4, pp. 1004–1009, 1998. [4] I. Elishako,ff Probabilistic eTh ory of Structures ,2nd,Dover,NY, [5] K.Sepahvand,S.Marburg,and H.-J. Hardtke,“Stochastic structural modal analysis involving uncertain parameters using generalized polynomial chaos expansion,” International Journal of Applied Mechanics,vol.3,no.3, pp.587–606,2011. [6] Z.Zhang,T.A.El-Moselhy,I. M. Elfadel,andL.Daniel, “Calculation of generalized polynomial-chaos basis functions and gauss quadrature rules in hierarchical uncertainty quan- tification,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems,vol.33,no.5,pp. 728–740, 2014. [7] K. Sepahvand and S. Marburg, “Stochastic dynamic analysis of structures with spatially uncertain material parameters,” International Journal of Structural Stability and Dynamics,vol. 14, no. 8, 2014. [8] K. Sepahvand and S. Marburg, Random and Stochastic Struc- tural Acoustic Analysis,JohnWiley Sons,New York,2016. [9] K. Sepahvand, “Spectral stochastic finite element vibration analysis of b fi er-reinforced composites with random b fi er ori- entation,” Composite Structures, vol. 145, pp. 119–128, 2016. [10] K. Sepahvand, S. Marburg, and H.-J. Hardtke, “Stochastic free vibration of orthotropic plates using generalized polynomial chaos expansion,” JournalofSound andVibration,vol.331,no. 1, pp.167–179,2012. [11] K. Sepahvand and S. Marburg, “On construction of uncertain material parameter using generalized polynomial chaos expan- sion from experimental data,” Procedia IUTAM,vol.6,pp.4–17, [12] K. Sepahvand, M. Scheffler, and S. Marburg, “Uncertainty quan- tification in natural frequencies and radiated acoustic power of composite plates: analytical and experimental investigation,” Applied Acoustics, vol. 87, pp. 23–29, 2015. [13] K. Sepahvand, “Stochastic finite element method for random harmonic analysis of composite plates with uncertain modal damping parameters,” Journal of Sound and Vibration,vol.400, pp. 1–12, 2017. [14] N. Wiener, “The homogeneous chaos,” American Journal of Mathematics,vol.60, no.4,pp.897–936, 1938. [15] R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer, New York, NY, USA, 1991. [16] D. Xiu and G. E. Karniadakis, “eTh wiener-askey polynomial chaos for stochastic differential equations,” SIAM Journal on Scienticfi Computing ,vol.24,no.2,pp. 619–644,2002. [17] K. Sepahvand, S. Marburg, and H.-J. Hardtke, “Uncertainty quantification in stochastic systems using polynomial chaos expansion,” International Journal of Applied Mechanics,vol.2, no. 2, pp. 305–353, 2010. 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Spectral Representation of Uncertainty in Experimental Vibration Modal Data

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Hindawi Advances in Acoustics and Vibration Volume 2018, Article ID 9695357, 5 pages https://doi.org/10.1155/2018/9695357 Research Article Spectral Representation of Uncertainty in Experimental Vibration Modal Data 1 1 1 2 2 1 K. Sepahvand , Ch.A.Geweth, F. Saati, M. Klaerner, L. Kroll, and S. Marburg Chair of Vibroacoustics of Vehicles and Machines, Department of Mechanical Engineering, Technical University of Munich, 85748 Garching b. Munich, Germany Department of Lightweight Structures and Polymer Technology, Faculty of Mechanical 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 in physical properties of individual components. Modeling and accurate representation of these uncertainties are desirable in many practical applications. In this paper, spectral-based method is employed to represent uncertainty in the natural frequencies of fiber-reinforced composite plates. For that, experimental modal analysis using noncontact method employing Laser-Vibrometer is conducted on 100 samples of plates having identical nominal topology. The random frequencies then are represented employing generalized Polynomial Chaos (gPC) expansions having unknown deterministic coefficients. This provides us with major advantage to approximate the random experimental data using closed form functions combining deterministic coefficients and random orthogonal basis. Knowing the orthogonal basis, the statistical moments of the data are used to estimate the unknown coefficients. 1. Introduction regarding stochastic analysis of such structures under uncer- tainty: first, how the uncertainties can be efficiently identified Uncertainty quantification concerns representation and solu- and modeled in numerical simulations, particularly in ni fi te tion of simulation models, e.g., a dieff rential equation or a element models and, secondly, how uncertainties aeff ct the finite element model, when some levels of modeling such as behavior of aforesaid structures. input parameters are not exactly known. In such conditions, The former issue requires quantifying the randomness the model is said to be stochastic, i.e., it exhibits some degree in uncertain parameters. This can be efficiently character- of uncertainty. Probabilistic structural dynamics, in particu- ized by the statistical properties of the parameters, e.g., lar, endeavor to take into account uncertainties relating vari- probability density function (PDF). However, identification ous aspects of real structures such as material and geometric of the appropriate PDF type characterizing the parameter parameters, loading terms, and initial/boundary conditions uncertainties demands to know a priori information which and exploring related impacts on the structure responses. may be collected from experimental tests. Various methods To improve the performance, durability, and efficiency of have been developed in past decades for PDF identification structures, an exact knowledge of geometrical and material from experimental data (cf. [10–13]). The latter issue depends parameters is required. Characterization of the stochastic ontheavailabilityofanexact modelrelatinginputsto response due to these uncertainties by stochastic methods has outputs. Construction of such model is not possible due gained interest among researchers in past decades. Stochastic to many common assumptions in modeling of structural methods in conjunction with finite element method (FEM) dynamics. For that reason, experimental methods are still have been widely used to quantify uncertainty in structural the most reliable approach for investigation of the uncertain- responses [1–9]. Two major issues have to be addressed ties. 2 Advances in Acoustics and Vibration In this paper, uncertainties relating to the experimentally and ℎ denotes the norm of polynomials defined as identified natural frequencies of composite plates are inves- tigated. Uncertainties in such materials may have different 2 2 ℎ = ⟨Ψ 𝜉 ,Ψ 𝜉 ⟩ = ∫ Ψ 𝜉 𝑓 𝜉 d𝜉 ( ) ( ) ( ) ( ) (4) 𝑖 𝑖 𝑖 𝑖 sources, e.g., manufacturing tolerances, b fi er orientations, or Ω physical properties of individual components. To this end, For the sake of simplicity, we will focus on monodimensional experimental modal analysis using the noncontact method random input in this work; i.e., 𝜉 ={𝜉} . by employing Laser-Vibrometer is conducted on 100 samples of plates having identical nominal topology. eTh statistical properties of the identified natural frequencies of the plates, 2.1. Estimation of the gPC Coefficients from Experimental in particular, are discussed in detail. The random frequen- Data. Calculation of the coefficients using (2) requires prior cies then are represented employing generalized Polynomial information on the PDF of uncertain parameters which may not be available. Statistical moments, in contrast, exhibit Chaos (gPC) expansion [14–17]. This provides us with major advantage to approximate the experimental data using closed adequate implicit information on the probability properties form functions combining deterministic coefficients and of random quantities. Once the experimental data on the random orthogonal basis. The coefficients then are estimated uncertain parameters are available, the estimation of the employing optimization procedure comparing theoretical gPC coefficients from statistical moments [11, 17] is possible. and experimental values of statistical moments. Here, the statistical moments derived analytically from the Thispaper isorganizedasfollows: thebasic formulation gPC expansion are compared to those calculated from the of the spectral-based representation of random parameters data. For the truncated gPC expansion, only the rfi st few is given in the next section. The numerical-experimental orders of moments are required to calculate the coefficients. simulations are presented in Section 3 and the last section eTh major benefit is that information on the probability denotes the conclusion. distribution of uncertain parameters does not have to be known apriori.Thecoecffi ientsare thenestimatedcompar- ing statistical moments constructed from the gPC expansion 2. Spectral-Based Representation of and experimental data via an optimization procedure. The Random Parameters 𝑘th-oder statistical moment, 𝑚 ,ofuncertain 𝑃 having the gPC expansion given in (1) is calculated as eTh spectral discretization methods are the key advantage for the efficient stochastic reduced basis representation of uncertain parameters in finite element modeling. This is 𝜉 ] 𝑚 = E[𝑃 ]= ∫ [ ∑ 𝑝 Ψ ( ) 𝑓 (𝜉 )d𝜉, 𝑘 𝑖 𝑖 because these methods provide a similar application of the Ω (5) 𝑖=0 deterministic Galerkin projection and collocation methods to reduce the order of complex systems. In this way, it 𝑘 = 0,1,... is common to employ a truncated expansion to discretize the input random quantities of the structure and system with 𝑚 =1 and 𝑚 =𝑝 .Inthispaper,weusethe 0 1 0 responses. The unknown coefficients of the expansions then probabilistic orthonormal Hermite polynomials for Ψ(𝜉) canbecalculatedbasedon theFEmodel outputs. Letus and, accordingly, 𝑓(𝜉) = (1/ 2𝜋)exp (−𝜉 /2).The 𝑘th-order consider the uncertain parameter 𝑃(𝜉) where 𝜉 ∈Ω is central statistical moment is then calculated as the vector random variable characterizing the uncertainty in the parameter and Ω denotes the random space. Under 𝑘 𝑘 𝑘−𝑖 𝑘−𝑖 𝜇 = E[(𝑃− E [𝑃 ]) ]= ∑ ( ) (−1) 𝑚 𝑚 , 𝑘 𝑖 1 thelimitedvariance,i.e., 𝜎 <∞,the parametercan be (6) 𝑖=0 approximated by 𝑘 = 2,3,... 𝑃 (𝜉 ) ≈ ∑ 𝑝 Ψ (𝜉 ) (1) 𝑖 𝑖 This leads imminently to the following expression for the sec- 𝑖=0 ond central statistical moment (variance) 𝜇 of the uncertain which is known as the truncated generalized Polynomial parameter 𝑃 as Chaos (gPC) expansion of the parameter. eTh determinis- tic coecffi ients 𝑝 arecalculatedemployingthestochastic 2 2 2 𝜇 =𝑚 −𝑚 = ∑ 𝑝 ℎ (7) 2 2 1 𝑖 𝑖 Galerkin projection as [17] 𝑖=1 𝑝 = ∫ 𝑃 (𝜉 )Ψ (𝜉 )𝑓 (𝜉 )d𝜉 (2) 𝑖 𝑖 in which ℎ =𝑖! for the Hermite polynomials. Similar expressions can be derived for the higher order moments as functions of the gPC coefficients. The calculated moments where 𝑓 is the joint probability density function (PDF) of form the gPC expansion can be compared to the correspond- random vector 𝜉 which for independent random variables 𝜉 ing values obtained from experimental data for an uncertain can be written as the multiplication of the individual PDF for parameter. In such a way, one can attempt to estimate the each variable 𝑓 (𝜉 ); i.e., 𝑖 𝑖 unknown coefficients from the available experimental data. 𝑓 (𝜉 )𝑑𝜉 =𝑓 (𝜉 )𝑓 (𝜉 )⋅⋅⋅𝑓 (𝜉 )d𝜉 d𝜉 ⋅⋅⋅ d𝜉 (3) That is, for a given set of experimental data {𝑃 ,𝑃 ,...,𝑃 } 1 1 2 2 𝑛 𝑛 1 2 𝑛 1 2 𝑀 Advances in Acoustics and Vibration 3 Table 1: eTh first three statistical moments of the first nine measured natural frequencies; mean value, 𝜇 , the standard deviation 𝜎 ,and the 𝑓 𝑓 skewness 𝛾 . 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 1 2 3 4 5 6 7 8 9 𝜇 [Hz] 115 145 275 396 504 530 556 704 781 𝜎 [Hz] 4.60 5.90 8.93 15.22 11.92 15.92 15.35 17.98 25.95 𝛾 [–] 0.485 0.366 0.895 0.292 0.166 1.086 1.312 0.622 1.419 expri on 𝑃, the experimental central moments, 𝜇 ,are calculated Elastic as bands Vibration expri 𝜇 = ∑ [𝑃 − E 𝑃 ] , 𝑘 = 2,3,... ( ) (8) modal data LSV Hammer 𝑗=1 in which E(𝑃) = (1/𝑀)∑ 𝑃 is the mean value of 𝑗=1 samples. An error function based on the least-square crite- rion corresponding to the difference between the moments derived from (5) and (8) can be used to estimate the optimal Spectrum coefficients 𝑝 . This leads to a minimization problem as 𝑖 analysis follows: Figure 1: Experiment setup for modal analysis of FRC plates. minimize ∑ 𝑓 (𝑝 ) 𝑛 𝑖 𝑛=1 (9) expri second-order gPC expansion is employed to approximate s.t.𝑓 (𝑝 )=𝜇 −𝜇 =0 0 𝑖 1 the frequencies. This leads to three unknown coefficients expri and, consequently, estimation of the rst fi three statistical 𝑓 (𝑝 )=𝜇 −𝜇 ,𝑛≥2 𝑛 𝑖 𝑘 moments as given in Table 1. eTh third statistical moment is expri given by the skewness, 𝛾 , defined as 𝜇 /𝜎 .Therandom The first condition of the process denotes that the expected 3 3 𝑓 value of the data represents the rfi st coecffi ient of the gPC natural frequencies are represented using second-order gPC expansion. Since the calculated moments for the gPC expan- expansions having random Hermite polynomials 𝐻(𝜉) as sion are nonlinear functions of the coecffi ients, one has to basis; i.e., employ nonlinear optimization procedure. eTh optimization leads to unique solution under the convergence condition for 𝑓 = ∑ 𝑝 𝐻 (𝜉 )=𝑝 +𝑝 𝜉+𝑝 (𝜉 −1), coefficients of one-dimensional gPC; i.e., ‖𝑝 ‖<‖𝑝 ‖. 𝑛 𝑛 𝑖 𝑛 𝑛 𝑛 𝑖+1 𝑖 𝑖 0 1 2 𝑖=0 (10) 3. Experimental and Numerical Study 𝑛 = 1,2,...,9 As a case study, in this section, the natural frequencies of The unknown deterministic coefficients 𝑝 are calculated by fiber-reinforced composite (FRC) plates are represented as equality of the statistical moments of the gPC expansions random parameters. eTh experimental modal analysis has given in (6) and the experimental estimations given in Table 1. been performed on 100 sample plates with nominal identical eTh optimization Application optimtool in MATLAB for topology of 𝑎 = 250 mm, 𝑏 = 125 mm, and thickness constrained nonlinear problem is employed for this purpose. of 2mm; seeFigure1.Theplatesweresuspended byvery This leads immediately to the following expressions for thin elastic bands to simulate free boundary conditions. eTh optimization problem den fi ed in (9): Laser-Vibrometer has been employed to collect the vibration responses due to the excitation force from impulse hammer 𝑓 (𝑝 )=𝜇 −𝑝 =0 1 𝑛 𝑓 𝑛 with tip force transducer at some predefined points of the 0 0 plates. eTh average of 5 impacts at each point was recorded. 2 2 2 𝑓 (𝑝 ,𝑝 )=𝜎 −𝑝 +2𝑝 2 𝑛 𝑛 The sample test has returned to rest before the next impact 𝑓 𝑛 𝑛 1 2 1 2 (11) is taken. A standard data acquisition facility along with a 2 3 postprocessing modal analysis software has been used to 𝑓 (𝑝 ,𝑝 )=𝛾 − (6𝑝 𝑝 +8𝑝 ) 3 𝑛 𝑛 3 𝑛 𝑛 𝑛 1 2 𝑓 3 2 1 2 extract modal data, e.g., natural frequencies. The measured first nine natural frequencies of plates are given in Figure 2. As shown, a considerable range of uncertainty is observed in The nonlinear optimization problem is then solved to esti- frequencies. Spectral representation of the measured random mate 𝑝 as given in Table 2. The second-order coefficients are frequencies requires estimating the statistical moments. eTh very small compared to the first two coefficients. This denotes FRC plate 4 Advances in Acoustics and Vibration 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 490 520 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 740 850 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 Figure 2: Measured first nine natural frequencies of 100 FRC plates (all in Hz). Table 2: eTh coefficients of the gPC expansions approximating the measured uncertain natural frequencies. 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 𝑓 1 2 3 4 5 6 7 8 9 𝑝 115 145 275 396 504 530 566 704 781 𝑝 4.53 5.87 8.72 15.18 11.90 15.36 14.53 17.78 24.32 𝑝 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 to Data Availability represent uncertainty in the random frequencies. Once the eTh measured data used to support this study were supplied unknown coefficients are known, the statistical properties of from project no. DFG-KR 1713/18-1 funded by the German the measured data can be calculated using the constructed Research Foundation (Deutsche Forschungsgemeinschaft- gPC expansions. DFG). er Th e is no restrictions to use the officially published data. 4. Conclusion Natural frequencies of composite plates have been considered Conflicts of Interest as random parameters. eTh generalized Polynomial Chaos expansions has been employed to approximate the uncer- eTh authors declare that they have no conflicts of interest. tainty in the measured natural frequencies. The method oer ff s the major advantage that the unknown deterministic coefficients of the expansions have to be calculated instead of Acknowledgments random parameters. This has been performed by comparing the statistical moments of the experimental results for 100 This work was supported by the German Research Founda- identical plates and from the expansions via the minimization tion (Deutsche Forschungsgemeinschaft, DFG) under project of least-square based error. eTh results have been given no. DFG-KR 1713/18-1 and the Technical University of for the first nine natural frequencies using second-order Munich within the funding programme Open Access Pub- expansions. lishing. The support is gratefully acknowledged. f f f 7 4 1 f f f 8 5 2 f f f 9 6 3 Advances in Acoustics and Vibration 5 References [1] M. Hanss, “Applied fuzzy arithmetic: An introduction with engineering applications,” Applied Fuzzy Arithmetic: An Intro- duction with Engineering Applications, pp. 1–256, 2005. [2] M.KleiberandT.D.Hien, eTh Stochastic Finite Element Method: Basic Perturbation Technique and Computer Implementation, JohnWiley&Sons,New York,NY, USA, 1992. [3] R. Ghanem, “Hybrid stochastic finite elements and generalized monte carlo simulation,” Journal of Applied Mechanics,vol.65, no. 4, pp. 1004–1009, 1998. [4] I. Elishako,ff Probabilistic eTh ory of Structures ,2nd,Dover,NY, [5] K.Sepahvand,S.Marburg,and H.-J. Hardtke,“Stochastic structural modal analysis involving uncertain parameters using generalized polynomial chaos expansion,” International Journal of Applied Mechanics,vol.3,no.3, pp.587–606,2011. [6] Z.Zhang,T.A.El-Moselhy,I. M. 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