On the Stochastic Response of a Fractionally-damped Duffing Oscillator

A 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

The finite element method for fractional non-local thermal energy transfer in non-homogeneous rigid conductors

In 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

Solution strategies for 1D elastic continuum with long-range interactions: Smooth and fractional decay

An 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

A new displacement-based framework for non-local Timoshenko beams

In this paper, a new theoretical framework is presented for modeling non-locality in shear deformable beams. The driving idea is to represent non-local effects as long-range volume forces and moments, exchanged by non-adjacent beam segments as a result of their relative motion described in terms of pure deformation modes of the beam. The use of these generalized measures of relative motion allows constructing an equivalent mechanical model of non-local effects. Specifically, long-range volume forces and moments are associated with three spring-like connections acting in parallel between couples of non-adjacent beam segments, and separately accounting for pure axial, pure bending and pure shear deformation modes. The variational consistency of the proposed non-local beam model is demonstrated by minimization of an appropriate total potential energy functional. Numerical results concerning the static behavior for different boundary and loading conditions are presented. It is shown that the proposed non-local beam model is able to capture experimental data on the static deflection of micro-beams, available in the literature

Stationary and non-stationary stochastic response of linear fractional viscoelastic systems

A method is presented to compute the stochastic response of single-degree-of-freedom (SDOF) structural systems with fractional derivative damping, subjected to stationary and non-stationary inputs. Based on a few manipulations involving an appropriate change of variable and a discretization of the fractional derivative operator, the equation of motion is reverted to a set of coupled linear equations involving additional degrees of freedom, the number of which depends on the discretization of the fractional derivative operator. As a result of the proposed variable transformation and discretization, the stochastic analysis becomes very straightforward and simple since, based on standard rules of stochastic calculus, it is possible to handle a system featuring Markov response processes of first order and not of infinite order like the original one. Specifically, for inputs of most relevant engineering interest, it is seen that the response second-order statistics can be readily obtained in a closed form, to be implemented in any symbolic package. The method applies for fractional damping of arbitrary order α (0 ≤ α ≤ 1). The results are compared to Monte Carlo simulation data

Non-local stiffness and damping models for shear-deformable beams

This paper presents the dynamics of a non-local Timoshenko beam. The key assumption involves modeling non-local effects as long-range volume forces and moments mutually exerted by non-adjacent beam segments, that contribute to the equilibrium of any beam segment along with the classical local stress resultants. Elastic and viscous long-range volume forces/moments are endowed in the model. They are built as linearly depending on the product of the volumes of the interacting beam segments and on generalized measures of their relative motion, based on the pure deformation modes of the beam. Attenuation functions governing the space decay of the non-local effects are introduced. Numerical results are presented for a variety of non-local parameter

The mechanically-based approach to 3D non-local linear elasticity theory: Long-range central interactions

This paper presents the generalization to a three-dimensional (3D) case of a mechanically-based approach to non-local elasticity theory, recently proposed by the authors in a one-dimensional (1D) case. The proposed model assumes that the equilibrium of a volume element is attained by contact forces between adjacent elements and by long-range forces exerted by non-adjacent elements. Specifically, the long-range forces are modelled as central body forces depending on the relative displacement between the centroids of the volume elements, measured along the line connecting the centroids. Further, the long-range forces are assumed to be proportional to a proper, material-dependent, distance-decaying function and to the products of the interacting volumes. Consistently with the modelling of the long-range forces as central body forces, the static boundary conditions enforced on the free surface of the solid involve only local stress due to contact forces. The proposed 3D formulation is developed both in a mechanical and in a variational context. For this the elastic energy functionals of the solid with long-range interactions are introduced, based on the principle of virtual work to set the proper correspondence between the mechanical and the kinematic variables of the model. Numerical applications are reported for 2D solids under plane stress condition

analysis of dynamic instabilities in bridges under wind action through a simple friction based mechanical model

Abstract In the field of stability of structures under nonconservative loads, the concept of follower force has long been debated by scientists due to the lack of actual experimental evidence. Bigoni and Noselli's work [2] aimed to investigate flutter and divergence instability phenomena through a purely mechanical model with Coulomb friction represents a praiseworthy attempt to shed light on this issue. A two-degree-of-freedom (DOF) system, conceived as a variant of the Ziegler column, was set up experimentally. The follower load was induced by a frictional force acting on a wheel mounted at the column end, so that the rolling friction vanishes and the sliding frictional force keeps always coaxial to the column, thus representing a tangential follower force. Along this research line, in this contribution a model is elaborated that stems from the analysis of an elastically supported rigid plate that represents the behaviour of a bridge deck suspended on springs and subjected to a wind-induced force. The wind force has been simulated by a Coulomb friction force acting on a wheel mounted on the plate aerodynamic centre, so that the sliding friction force keeps perpendicular to the plate axis throughout the system motion, thus representing a follower force. To properly reproduce the wind force, the friction force is applied to the wheel by a lever mechanism wherein one of the two lever arms involves the plate rotation via a particular circular guide. The corresponding equations of motion of the bridge deck are derived in a completely dimensionless form. Depending on the mechanical characteristics of the plate and the magnitude of the friction force, stability, flutter or divergence phenomena may occur. The occurrence of these phenomena is numerically investigated by integration of the equations of motion. The development of an experimental framework of the model to corroborate these intuitions is the object of an ongoing research

Potassium partitioning between leaves and clusters: Role of rootstock

Different scion/rootstock combinations in grapevine (Vitis vinifera/Vitis spp.) were tested for nutritional properties and juice composition. Chardonnay and Cabernet Sauvignon each grafted on 22 rootstock varieties (10 new crosses and 12 already used in viticulture) were grown in outdoor pots containing very poor nutritional substrate. Crop load had a strong effect on juice composition and potassium nutrition. Results indicate that under our experimental conditions rootstock can have an effect on potassium partitioning between leaves and cluster 9 out of 22 rootstocks that we tried were able to improve leaf potassium content without inducing a significant increase in juice potassium conten

Preliminary study of Armenian grapevines phenolic contents

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