Effect of uncertainties and noise on the nonlinear vibrations of a slender beam

Resumo

Uncertainty analysis is fundamental in structural engineering in order to incorporate imprecise
information, errors, and limitations into theoretical models. Various techniques have been developed over the years
to access the uncertainty, with advances regarding semilinear boundary value problems to correctly represent the
probability spaces, namely spectral decomposition and collocation. However, the analysis is continuously
improving for initial value problems, and nonlinear problems in general since the methods developed to decompose
the probability spaces do not behave well in such cases. When uncertainties are given in spaces of distributions,
such as white noise processes, the applicability of those techniques is even more limited, leaving the Monte Carlo
sampling strategies as the final and, in various cases, the only choice of analysis. We consider a planar nonlinear
beam equation under an additive white noise excitation and imperfections to exemplify these limitations. To obtain
the dynamics of such a system, we spatially discretize the problem using the first linear vibration mode and analyze
the resulting differential equation by varying the uncertainty parameters. A stochastic differential equation of Itô
type is derived to verify the white noise excitation. We demonstrate the global dynamics through the generalized
cell mapping, which is used to construct both the Perron-Frobenius and the Koopman operator of the differential
equations. This example shows the effects of noise on the resonant solutions of nonlinear dynamic problems,
stabilizing the nonresonant solution. Also, the mutual impact of noise with imperfections demonstrates the
correlation of parametric uncertainty with random vibrations.