1,721,099 research outputs found
The finite element method for fractional non-local thermal energy transfer in non-homogeneous rigid conductors
Full text linkIn a non-local fractional-order model of thermal energy transport recently introduced by the authors, it is assumed that local and non-local contributions coexist at a given observation scale: while the first is described by the classical Fourier transport law, the second involves couples of adjacent and non-adjacent elementary volumes, and is taken as proportional to the product of the masses of the interacting volumes and their relative temperature, through a material-dependent, distance-decaying power-law function. As a result, a fractional-order heat conduction equation is derived. This paper presents a pertinent finite element method for the solution of the proposed fractional-order heat conduction equation. Homogenous and non-homogeneous rigid bodies are considered. Numerical applications are carried out on 1D and 2D bodies, including a standard finite difference solution for validation
An interval framework for uncertain frequency response of multi-cracked beams with application to vibration reduction via tuned mass dampers
No full textThe paper addresses the frequency response of beams in presence of open cracks with interval parameters. On adopting the standard Euler–Bernoulli beam theory, every crack is modelled as a linearly-elastic rotational spring whose stiffness and position are treated as uncertain-but-bounded parameters. A two-step method is proposed to calculate the bounds of all response variables. First, the sensitivity functions of the response are calculated as every uncertain parameter varies within the respective interval. Next, the bounds of the response are computed by either a sensitivity-based method or a global optimization technique, the former if the response is monotonic with respect to all uncertain parameters and the latter if the response is non-monotonic with respect to even one parameter only. The method relies on analytical forms for all response variables and the associated sensitivity functions. The applications focus on the frequency response of multi-cracked beams equipped with tuned mass dampers, showing potential and accuracy of the method
On the Stochastic Response of a Fractionally-damped Duffing Oscillator
Full text linkA numerical method is presented to compute the response of a viscoelastic Duffing oscillator
with fractional derivative damping, subjected to a stochastic input. The key idea
involves an appropriate discretization of the fractional derivative, based on a preliminary
change of variable, that allows to approximate the original system by an equivalent system
with additional degrees of freedom, the number of which depends on the discretization of
the fractional derivative. Unlike the original system that, due to the presence of the fractional
derivative, is governed by non-ordinary differential equations, the equivalent system
is governed by ordinary differential equations that can be readily handled by standard integration
methods such as the Runge–Kutta method. In this manner, a significant reduction
of computational effort is achieved with respect to the classical solution methods, where
the fractional derivative is reverted to a Grunwald–Letnikov series expansion and numerical
integration methods are applied in incremental form. The method applies for fractional
damping of arbitrary order a (0 < a < 1) and yields very satisfactory results. With respect to
its applications, it is worth remarking that the method may be considered for evaluating
the dynamic response of a structural system under stochastic excitations such as earthquake
and wind, or of a motorcycle equipped with viscoelastic devices on a stochastic road
ground profile
Stochastic response of offshore structures by a new approach to statistical cubicization
No full textThis study presents a new statistical cubicization approach for predicting the stochastic response of offshore platforms subjected to a Morison-type nonlinear drag loading. Statistics of the original system are obtained from an equivalent nonlinear system, which is constructed by replacing the Morison drag force by a cubic polynomial function of the relative fluid-structure velocity, up to cubic order: A Volterra series expansion with a finite Fourier series representation is used to approximate the response of the equivalent system. Exact solutions are developed to express the Fourier coefficients of the second and third-order response as functions of the Fourier coefficients of the first-order relative fluid-structure velocity. Response statistics are then computed by ensemble averaging over a suitable number of realizations of the first-order Gaussian response. Response statistics up to the sixth order, computed for a variety of sea conditions, show accuracy and efficiency of the proposed method as compared to digital simulation
A correction method for dynamic analysis of linear systems
No full textThis paper proposes an analytical method to improve the accuracy of the dynamic response of classically damped linear systems, as given by a standard truncated modal analysis. Upon computing the first m undamped modes of a n-degree-of-freedom system, two sets of equations in the Rn nodal space are built, which are uncoupled and govern the contribution to the response of the m computed modes and the remaining (n−m) unknown modes, respectively. The first set is solved in the Rm modal space by using the m available modes; the second set is solved in a reduced R(n−m) nodal space, without computing additional modes. Specifically, it is shown that the particular solution of the second set of equations may be obtained by a series expansion involving repetitive time derivatives of the first-order static solution. The convergence conditions of such a series are discussed and proved on a rigorous basis. Numerical applications are also presented to demonstrate the effectiveness of the proposed method
Stochastic response of linear and non-linear systems to α-stable Lévy white noises
No full textThe stochastic response of linear and non-linear systems to external α-stable Lévy white noises is investigated. In the literature, a differential equation in the characteristic function (CF) of the response has been recently derived for scalar systems only, within the theory of the so-called fractional Einstein-Smoluchowsky equations (FESEs). Herein, it is shown that the same equation may be built by rules of stochastic differential calculus, previously applied by one of the authors to systems driven by arbitrary delta-correlated processes. In this context, a straightforward formulation for multi-degree-of-freedom (MDOF) systems is also developed. Approximate CF solutions to the derived equation are sought for polynomial non-linearities, in stationary conditions. To this aim a wavelet representation is used, in conjunction with a weighted residual method. Numerical results prove in excellent agreement with exact solutions, when available, and digital simulation dat
Solution strategies for 1D elastic continuum with long-range interactions: Smooth and fractional decay
Full text linkAn elastic continuum model with long-range forces is addressed in this study within the context of approximate analytical methods Such a model stems from a mechanically-based approach to non-local theory where long-range central forces are introduced between non-adjacent volume elements Specifically, long-range forces depend on the relative displacement on the volume product between interacting elements and they are proportional to a proper, material-dependent, distance-decaying function Smooth-decay functions lead to integro-differential governing equations whereas hypersingular, fractional-decay functions lead to a fractional differential governing equation of Marchaud type In this paper the Galerkin and the Rayleigh-Ritz method are used to build approximate solutions to the integro-differential and the fractional differential governing equations Numerical applications show the accuracy of the proposed approximate solutions as compared to the finite difference approximation and to the fractional finite difference approximatio
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