Multiple timescale spectral analysis

Probabilistic Engineering Mechanics

2016

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.

Section: Has Beenmentioning

confidence: 99%

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

Canor

Caracoglia

Probabilistic Engineering Mechanics

2016

“…This latter number relating thus the characteristic frequency of turbulence U/L to the characteristic frequency ω 0 of the structure is typically small in wind engineering applications, in the range [10 −3 ; 10 −1 ], and oers the possible use of multiple timescales approaches such as the well-known background/resonant decomposition [Davenport, 1961] or its higher-order adaptation [Denoël, 2011[Denoël, , 2015. The normalization property of the PSD translates into…”

Section: Dimensionless Formulationmentioning

confidence: 99%

“…Second, as the timescales related to the wind uctuations and the structure are well separated, the use of a perturbation approach is welcome. With this respect, the recent Multiple Timescale Spectral Analysis [Denoël, 2015] is substantially used in the formulation. The combination of these two arguments are validated through an extensive Monte Carlo simulation spanning a wide parameter space.…”

mentioning

confidence: 99%

“…no integration instead of integration in a 1-D space, formally. The idea was extended to similar higher-order linear applications [Denoël, 2009, Denoël, 2011 and recently generalized to multi-degree-of-freedom and nonlinear systems [Denoël, 2015]. This technique identies the dierent contributions to the integral, then sequentially focuses on each of them with a game of stretch-and-shrink operations.…”

mentioning

confidence: 99%

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Response of an oscillator to a random quadratic velocity-feedback loading

Journal of Wind Engineering and Industrial Aerodynamics

Carassale

2015

The Aerodynamic force acting on compact structures is often modeled as a quadratic function of the wind velocity, which uctuates randomly due to the atmospheric turbulence. When a exible structure is considered and the quasi-steady assumption is applied, the wind velocity is substituted by the wind-structure relative velocity and, even in case of linear structures, the composite aerodynamic-mechanical system is governed by a nonlinear dierential equation characterized by a quadratic feedback term. This class of dynamical systems has been deeply investigated (with dierent levels of simplications) applying several alternative mathematical approaches. In this paper we dene a system approximation based on a 2nd-order Volterra series and obtain its statistical response in terms of cumulants. The response cumulants are calculated applying the Multiple Scale Spectral Analysis leading to analytic or semi-analytic expressions. All the approximations are validated through Monte Carlo simulation within a wide parameter space. Then, the analytical structure of the obtained expressions is used to discuss, from a qualitative point of view, the behavior of the considered dynamical system. IntroductionIn several engineering problems, the driving force is expressed as a nonlinear transformation of the input, the response and its time derivatives. In structural engineering, this is the case in vibrations of base-excited structures, where the internal forces depend on the (nonlinear transformation of) relative displacements and velocities between the structure and the base motion [Constantinou and Papageorgiou, 1990]. Other examples include the aerodynamic or hydrodynamic loads on exible structures, where the eective loads might be expressed, in a quasi-steady framework, as a memoryless nonlinear transformation of some relative velocities (e.g. Kareem, 1987). The common feature shared by these models with nonlinear feedback is the presence of random parametric excitation terms. The nonlinearity associated with these terms, as well as other more specic terms (like the quadratic structural velocity term considered in this paper) make the development of exact analytical solutions rather challenging. In this paper we consider the problem of a single degree-of-freedom linear oscillator, subject to a quasi-steady aerodynamic loading. The aerodynamic force is dened as the square of the wind-structure relative velocity and results from the sum of ve terms: a constant term, terms proportional to the turbulence uctuation and its square, terms proportional to the structure velocity and its square and a term proportional to the product of wind velocity uctuation and structure velocity. Another specicity of this problem concerns the stochasticity of the wind velocity uctuation, which is usually modeled as a zero-mean, stationary, Gaussian random process. The need to regard this problem from a probabilistic viewpoint makes it even more dicult to develop closed-form solutions. Of course Monte Carlo simulations are able to deal with the m...

“…This observation has triggered the development of a unified approach, the Multiple Timescale Spectral Analysis (Denoël 2015), which is based on the sole assumption of separation of timescales. In the particular case of second order statistics (variances and covariances) in linear elasto-dynamics, it degenerates into the well-known background/resonant decomposition, and in other cases, it allows the derivation of closed form asymptotic approximations of the statistical properties of the structural responses.…”

Section: Introductionmentioning

confidence: 99%

Applications of the multiple timescale spectral analysis in wind engineering

Insights and Innovations in Structural Engineering, Mechanics and Computation

2016

The random response of civil engineering structures to the buffeting action of wind loads is typically composed of several components, usually referred to as the background component, in the low frequency zone and the resonant component(s) in the neighborhood of modal natural frequencies. It has become customary to study separately and add the contributions of these components to the total response, at least as far as the second order response (variance of structural responses) is concerned. Such a decomposition exists but is less usual for the computation of covariances of modal coordinates or of structural displacements, which are in turn necessary for the determination of internal stresses. The question of such a decomposition also holds for nonlinear systems, or even for the higher statistical moments of a linear structural system, should the response be non Gaussian. With very wide ranges of applicability, the Multiple Timescale Spectral Analysis summarizes under a unified framework recent works aiming at the development of such decompositions. This paper briefly pictures this particular theory based on perturbation methods, and provides illustrations of its applicability to the problems cited above.