Perturbation methods in evolutionary spectral analysis for linear dynamics and equivalent statistical linearization

Abstract

a b s t r a c tThe analysis of large-scale structures subject to transient random loads, coherent in space and time, is a classic problem encountered in earthquake and wind engineering. The simulation-based framework is usually seen as the most convenient approach for both linear and nonlinear dynamics. However, the generation of statistically consistent samples of an excitation field remains a heavy computational task. In light of this, perturbation techniques are applied to develop and improve evolutionary spectral analysis.Advantageously performed in a standard modal basis, this evolutionary spectral analysis for linear structures requires the computation of the modal impulse response matrix. However, this matrix has no general closed-form expression in the presence of modal coupling. We propose therefore to model it by an asymptotic approximation, obtained by the inverse Fourier transform of an asymptotic expansion of the modal transfer matrix of the structure. This latter expansion considers the modal coupling as a perturbation of a main decoupled system. This strategy leads to an expansion known in a closed-form. Finally, the semi-group property allows the use of an efficient recurrence relation to approximate the modal evolutionary transfer matrix, i.e. the evolutionary extension of the transfer matrix.The asymptotic expansion-based method and the recurrence relation are then applied to nonlinear transient dynamics by using Gaussian equivalent linearization. This extension is formalized by a multiple timescales approach, allowing to consider a linearized structure, namely a time variant system, as piecewise linear time invariant depending on a statistical timescale. The proposed developments are finally illustrated on realistic civil engineering applications.

Second-order moment of the first passage time of a quasi-Hamiltonian oscillator with stochastic parametric and forcing excitations

“…1, are known to be versatile and accurate, although highly time consuming. Other approaches are based on the solution of the (generalized) Pontryagin equation(s) [12,[15][16][17][18], sometimes with the finite difference method [19,20]; alternative methods to determine the transient evolution of joint probability density functions include path integral approaches [21][22][23][24], smooth particle hydrodynamics or other Lagrangian methods [25], semi-analytical methods such as the Galerkin projection scheme [26,27], the Poisson distribution based assumptions [28] or other applications of the perturbation method in evolutionary spectral analysis [29].…”

First passage time as an analysis tool in experimental wind engineering

“…Beside, the analysis of non-stationary problems can be done in several different ways, (i) either through Monte Carlo simulations (Kloeden and Platen, 1992;Primo zi c, 2011;Giles, 2008;Vanvinckenroye and Deno€ el, 2015) providing for instance the time evolution of the joint probability density function in transient and, eventually, stationary regimes, (ii) either through a more theoretical context, where the state-variable probability density function and the first passage time are obtained as the solutions of the Fokker-Planck-Kolmogorov and generalized Pontryagin equations. These equations can be solved numerically through a path integration method (Kougioumtzoglou and Spanos, 2014a), the perturbation method (Canor et al, 2016), the smooth particle hydrodynamics method (Canor and Deno€ el, 2013), high dimensional finite elements Kr al, 2014, 2017), or other approximate techniques. Comparisons of approached and numerical solutions for the first passage times and the associated, so-called, survival probability, are widely available (Kougioumtzoglou and Spanos, 2014b;Kougioumtzoglou, 2014a, 2014b;Palleschi and Torquati, 1989).…”

Closed-form Response of a Linear Fractional Visco-elastic Oscillator under Arbitrary Stationary Input

“…The MTSA also provides a very simple way to manage non proportional damping in a deterministic or stochastic context [31]. It has also given very concise and accurate results in the context of non Gaussian loading [40], nonstationary loading [41] slightly nonlinear systems [36], and oscillators featuring some visco-elastic components [31]. It is not limited to the variance of the response.…”