Spectral Representation of Uncertainty in Experimental Vibration Modal Data

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 withmajor 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.


Introduction
Uncertainty quantification concerns representation and solution of simulation models, e.g., a differential equation or a finite element model, when some levels of modeling such as input parameters are not exactly known.In such conditions, the model is said to be stochastic, i.e., it exhibits some degree of uncertainty.Probabilistic structural dynamics, in particular, endeavor to take into account uncertainties relating various aspects of real structures such as material and geometric parameters, loading terms, and initial/boundary conditions and exploring related impacts on the structure responses.To improve the performance, durability, and efficiency of structures, an exact knowledge of geometrical and material parameters is required.Characterization of the stochastic response due to these uncertainties by stochastic methods has gained interest among researchers in past decades.Stochastic methods in conjunction with finite element method (FEM) have been widely used to quantify uncertainty in structural responses [1][2][3][4][5][6][7][8][9].Two major issues have to be addressed regarding stochastic analysis of such structures under uncertainty: first, how the uncertainties can be efficiently identified and modeled in numerical simulations, particularly in finite element models and, secondly, how uncertainties affect the behavior of aforesaid structures.
The former issue requires quantifying the randomness in uncertain parameters.This can be efficiently characterized by the statistical properties of the parameters, e.g., probability density function (PDF).However, identification of the appropriate PDF type characterizing the parameter uncertainties demands to know a priori information which may be collected from experimental tests.Various methods have been developed in past decades for PDF identification from experimental data (cf.[10][11][12][13]).The latter issue depends on the availability of an exact model relating inputs to outputs.Construction of such model is not possible due to many common assumptions in modeling of structural dynamics.For that reason, experimental methods are still the most reliable approach for investigation of the uncertainties.

Advances in Acoustics and Vibration
In this paper, uncertainties relating to the experimentally identified natural frequencies of composite plates are investigated.Uncertainties in such materials may have different sources, e.g., manufacturing tolerances, fiber orientations, or physical properties of individual components.To this end, experimental modal analysis using the noncontact method by employing Laser-Vibrometer is conducted on 100 samples of plates having identical nominal topology.The statistical properties of the identified natural frequencies of the plates, in particular, are discussed in detail.The random frequencies then are represented employing generalized Polynomial Chaos (gPC) expansion [14][15][16][17].This provides us with major advantage to approximate the experimental data using closed form functions combining deterministic coefficients and random orthogonal basis.The coefficients then are estimated employing optimization procedure comparing theoretical and experimental values of statistical moments.
This paper is organized as follows: the basic formulation of the spectral-based representation of random parameters is given in the next section.The numerical-experimental simulations are presented in Section 3 and the last section denotes the conclusion.

Spectral-Based Representation of Random Parameters
The spectral discretization methods are the key advantage for the efficient stochastic reduced basis representation of uncertain parameters in finite element modeling.This is because these methods provide a similar application of the deterministic Galerkin projection and collocation methods to reduce the order of complex systems.In this way, it is common to employ a truncated expansion to discretize the input random quantities of the structure and system responses.The unknown coefficients of the expansions then can be calculated based on the FE model outputs.Let us consider the uncertain parameter () where  ∈ Ω is the vector random variable characterizing the uncertainty in the parameter and Ω denotes the random space.Under the limited variance, i.e.,  2 < ∞, the parameter can be approximated by which is known as the truncated generalized Polynomial Chaos (gPC) expansion of the parameter.The deterministic coefficients   are calculated employing the stochastic Galerkin projection as [17] where  is the joint probability density function (PDF) of random vector  which for independent random variables   can be written as the multiplication of the individual PDF for each variable   (  ); i.e., and ℎ  denotes the norm of polynomials defined as For the sake of simplicity, we will focus on monodimensional random input in this work; i.e.,  = {}.

Estimation of the gPC Coefficients from Experimental
Data.Calculation of the coefficients using (2) requires prior information on the PDF of uncertain parameters which may not be available.Statistical moments, in contrast, exhibit adequate implicit information on the probability properties of random quantities.Once the experimental data on the uncertain parameters are available, the estimation of the gPC coefficients from statistical moments [11,17] is possible.
Here, the statistical moments derived analytically from the gPC expansion are compared to those calculated from the data.For the truncated gPC expansion, only the first few orders of moments are required to calculate the coefficients.
The major benefit is that information on the probability distribution of uncertain parameters does not have to be known a priori.The coefficients are then estimated comparing statistical moments constructed from the gPC expansion and experimental data via an optimization procedure.The th-oder statistical moment,   , of uncertain  having the gPC expansion given in ( 1) is calculated as with  0 = 1 and  1 =  0 .In this paper, we use the probabilistic orthonormal Hermite polynomials for Ψ  () and, accordingly, () = (1/ √ 2)exp(− 2 /2).The th-order central statistical moment is then calculated as This leads imminently to the following expression for the second central statistical moment (variance)  2 of the uncertain parameter  as in which ℎ 2  = ! for the Hermite polynomials.Similar expressions can be derived for the higher order moments as functions of the gPC coefficients.The calculated moments form the gPC expansion can be compared to the corresponding values obtained from experimental data for an uncertain parameter.In such a way, one can attempt to estimate the unknown coefficients from the available experimental data.That is, for a given set of experimental data { 1 ,  2 , . . .,   } , are calculated as in which E() = (1/) ∑  =1   is the mean value of samples.An error function based on the least-square criterion corresponding to the difference between the moments derived from ( 5) and ( 8) can be used to estimate the optimal coefficients   .This leads to a minimization problem as follows: minimize The first condition of the process denotes that the expected value of the data represents the first coefficient of the gPC expansion.Since the calculated moments for the gPC expansion are nonlinear functions of the coefficients, one has to employ nonlinear optimization procedure.The optimization leads to unique solution under the convergence condition for coefficients of one-dimensional gPC; i.e., ‖ +1 ‖ < ‖  ‖.

Experimental and Numerical Study
As a case study, in this section, the natural frequencies of fiber-reinforced composite (FRC) plates are represented as random parameters.The experimental modal analysis has been performed on 100 sample plates with nominal identical topology of  = 250 mm,  = 125 mm, and thickness of 2 mm; see Figure 1.The plates were suspended by very thin elastic bands to simulate free boundary conditions.The Laser-Vibrometer has been employed to collect the vibration responses due to the excitation force from impulse hammer with tip force transducer at some predefined points of the plates.The average of 5 impacts at each point was recorded.
The sample test has returned to rest before the next impact is taken.A standard data acquisition facility along with a postprocessing modal analysis software has been used to 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 frequencies.Spectral representation of the measured random frequencies requires estimating the statistical moments.The second-order gPC expansion is employed to approximate the frequencies.This leads to three unknown coefficients and, consequently, estimation of the first three statistical moments as given in Table 1.The third statistical moment is given by the skewness,  3  , defined as  expri 3 / 3   .The random natural frequencies are represented using second-order gPC expansions having random Hermite polynomials () as basis; i.e., The unknown deterministic coefficients    are calculated by equality of the statistical moments of the gPC expansions given in (6) and the experimental estimations given in Table 1.The optimization Application optimtool in MATLAB5 for constrained nonlinear problem is employed for this purpose.This leads immediately to the following expressions for optimization problem defined in (9): The nonlinear optimization problem is then solved to estimate    as given in   that the second-order gPC expansion has enough accuracy to represent uncertainty in the random frequencies.Once the unknown coefficients are known, the statistical properties of the measured data can be calculated using the constructed gPC expansions.

Conclusion
Natural frequencies of composite plates have been considered as random parameters.The generalized Polynomial Chaos expansions has been employed to approximate the uncertainty in the measured natural frequencies.The method offers the major advantage that the unknown deterministic coefficients of the expansions have to be calculated instead of random parameters.This has been performed by comparing the statistical moments of the experimental results for 100 identical plates and from the expansions via the minimization of least-square based error.The results have been given for the first nine natural frequencies using second-order expansions.

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

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

Table 1 :
The first three statistical moments of the first nine measured natural frequencies; mean value,   , the standard deviation   , and the skewness  3 .

Table 2
. The second-order coefficients are very small compared to the first two coefficients.This denotes Advances in Acoustics and Vibration

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