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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.

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 [

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. [

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 [

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

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

Calculation of the coefficients using (

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

The first three statistical moments of the first nine measured natural frequencies; mean value,

| | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|

| 115 | 145 | 275 | 396 | 504 | 530 | 556 | 704 | 781 |

| 4.60 | 5.90 | 8.93 | 15.22 | 11.92 | 15.92 | 15.35 | 17.98 | 25.95 |

| 0.485 | 0.366 | 0.895 | 0.292 | 0.166 | 1.086 | 1.312 | 0.622 | 1.419 |

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

| | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|

| 115 | 145 | 275 | 396 | 504 | 530 | 566 | 704 | 781 |

| 4.53 | 5.87 | 8.72 | 15.18 | 11.90 | 15.36 | 14.53 | 17.78 | 24.32 |

| 0.37 | 0.36 | 1.35 | 0.74 | 0.33 | 2.95 | 3.48 | 1.88 | 6.40 |

Experiment setup for modal analysis of FRC plates.

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

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.

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

The authors declare that they have no conflicts of interest.

This work was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) under project no. DFG-KR 1713/18-1 and the Technical University of Munich within the funding programme Open Access Publishing. The support is gratefully acknowledged.