Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results

2009 ◽  
Vol 63 (1) ◽  
Author(s):  
Yuriy A. Rossikhin ◽  
Marina V. Shitikova
Keyword(s):  
Fractional Calculus ◽  
Degrees Of Freedom ◽  
Solid Mechanics ◽  
State Of The Art ◽  
Dynamic Problems ◽  
Engineering Problems ◽  
Multilayered Systems ◽  
Linear And Nonlinear ◽  
Linear And Nonlinear Systems

The present state-of-the-art article is devoted to the analysis of new trends and recent results carried out during the last 10years in the field of fractional calculus application to dynamic problems of solid mechanics. This review involves the papers dealing with study of dynamic behavior of linear and nonlinear 1DOF systems, systems with two and more DOFs, as well as linear and nonlinear systems with an infinite number of degrees of freedom: vibrations of rods, beams, plates, shells, suspension combined systems, and multilayered systems. Impact response of viscoelastic rods and plates is considered as well. The results obtained in the field are critically estimated in the light of the present view of the place and role of the fractional calculus in engineering problems and practice. This articles reviews 337 papers and involves 27 figures.

Is there Chaos on the German Labor Market? / Ist der deutsche Arbeitsmarkt chaotisch?

1999 ◽  
Vol 218 (5-6) ◽  
Author(s):  
Michael Neugart
Keyword(s):  
Time Series ◽  
Labor Market ◽  
Chaotic Dynamics ◽  
Alternative Hypothesis ◽  
The Past ◽  
Labor Market Dynamics ◽  
Linear And Nonlinear ◽  
Bds Test ◽  
Linear And Nonlinear Systems

SummaryEvidence on the role of chaotic and nonlinear dynamics on labor markets is mixed. It is unclear whether nonlinear relationships are responsible for the dynamic patterns observed in Europe during the past decades. In this paper, we test German labor market data for the null hypothesis of an i.i.d. process with the BDS test. As several processes including chaotic, nonlinear deterministic, and stochastic linear and nonlinear systems are nested within the alternative hypothesis, time series are whitened with linear and nonlinear filters. Lyapunov exponents and correlation dimensions are applied to the residuals of the filtered time series to test for chaotic dynamics. There seems to be a nonlinear deterministic core to German labor market dynamics. Chaos does not occur.

scholarly journals Numerical Bifurcation Methods applied to Climate Models: Analysis beyond Simulation

2019 ◽  
Author(s):  
Henk A. Dijkstra
Keyword(s):  
Degrees Of Freedom ◽  
Climate Models ◽  
State Of The Art ◽  
Climate System ◽  
Special Issue ◽  
Personal View ◽  
Spatially Extended ◽  
System Variability ◽  
Numerical Bifurcation

Abstract. In this special issue contribution, I provide a personal view on the role of bifurcation analysis of climate models in the development of a theory of climate system variability. The state-of-the-art of the methodology is shortly outlined and the main part of the paper deals with examples of what has been done and what has been learned. In addressing these issues, I will discuss the role of a hierarchy of climate models, concentrate on results for spatially extended (stochastic) models (having many degrees of freedom) and evaluate the importance of these results for a theory of climate system variability.

scholarly journals Quantum optics with one or two photons

2015 ◽  
Vol 471 (2180) ◽  
pp. 20150208 ◽  
Author(s):  
G. J. Milburn ◽  
S. Basiri-Esfahani
Keyword(s):  
Gravitational Field ◽  
Nonlinear Systems ◽  
Degrees Of Freedom ◽  
Single Photon ◽  
Optical Cavity ◽  
Photon State ◽  
Single Photon State ◽  
Linear And Nonlinear ◽  
Photon States ◽  
Linear And Nonlinear Systems

We discuss the concept of a single-photon state together with how they are generated, measured and interact with linear and nonlinear systems. In particular, we consider how a single-photon state interacts with an opto-mechanical system: an optical cavity with a moving mirror and how such states can be used as a measurement probe for the mechanical degrees of freedom. We conclude with a discussion of how single-photon states are modified in a gravitational field due to the red-shift.

scholarly journals Numerical bifurcation methods applied to climate models: analysis beyond simulation

2019 ◽  
Vol 26 (4) ◽  
pp. 359-369 ◽  
Author(s):  
Henk A. Dijkstra
Keyword(s):  
Degrees Of Freedom ◽  
Climate Models ◽  
State Of The Art ◽  
Climate System ◽  
Special Issue ◽  
Personal View ◽  
Spatially Extended ◽  
System Variability ◽  
Numerical Bifurcation

Abstract. In this special issue contribution, I provide a personal view on the role of bifurcation analysis of climate models in the development of a theory of climate system variability. The state of the art of the methodology is shortly outlined, and the main part of the paper deals with examples of what has been done and what has been learned. In addressing these issues, I will discuss the role of a hierarchy of climate models, concentrate on results for spatially extended (stochastic) models (having many degrees of freedom) and evaluate the importance of these results for a theory of climate system variability.

Energy Dissipation in Dynamical Systems Through Sequential Application and Removal of Constraints

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Jimmy Issa ◽  
Ranjan Mukherjee ◽  
Alejandro R. Diaz
Keyword(s):  
Energy Dissipation ◽  
Nonlinear Systems ◽  
Kinetic Energy ◽  
Degrees Of Freedom ◽  
General Application ◽  
Finite Dimensional ◽  
Elastic Systems ◽  
Random Switching ◽  
Linear And Nonlinear ◽  
Linear And Nonlinear Systems

A strategy to remove energy from finite-dimensional elastic systems is presented. The strategy is based on the cyclic application and removal of constraints that effectively remove and restore degrees of freedom of the system. In general, application of a constraint removes kinetic energy from the system, while removal of the constraint resets the system for a new cycle of constraint application. Conditions that lead to a net loss in kinetic energy per cycle and bounds on the amount of energy removed are presented. In linear systems, these bounds are related to the modes of the system in its two states, namely, with and without constraints. It is shown that energy removal is always possible, even using a random switching schedule, except in one scenario, when energy is trapped in modes that span an invariant subspace with special orthogonality properties. Applications to nonlinear systems are discussed. Examples illustrate the process of energy removal in both linear and nonlinear systems.

Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids

1997 ◽  
Vol 50 (1) ◽  
pp. 15-67 ◽  
Author(s):  
Yuriy A. Rossikhin ◽  
Marina V. Shitikova
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Review Article ◽  
Mechanics Of Solids ◽  
Dynamic Problems ◽  
Weakly Singular Kernels ◽  
Viscoelastic Models ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Linear And Nonlinear

The aim of this review article is to collect together separated results of research in the application of fractional derivatives and other fractional operators to problems connected with vibrations and waves in solids having hereditarily elastic properties, to make critical evaluations, and thereby to help mechanical engineers who use fractional derivative models of solids in their work. Since the fractional derivatives used in the simplest viscoelastic models (Kelvin-Voigt, Maxwell, and standard linear solid) are equivalent to the weakly singular kernels of the hereditary theory of elasticity, then the papers wherein the hereditary operators with weakly singular kernels are harnessed in dynamic problems are also included in the review. Merits and demerits of the simplest fractional calculus viscoelastic models, which manifest themselves during application of such models in the problems of forced and damped vibrations of linear and nonlinear hereditarily elastic bodies, propagation of stationary and transient waves in such bodies, as well as in other dynamic problems, are demonstrated with numerous examples. As this takes place, a comparison between the results obtained and the results found for the similar problems using viscoelastic models with integer derivatives is carried out. The methods of Laplace, Fourier and other integral transforms, the approximate methods based on the perturbation technique, as well as numerical methods are used as the methods of solution of the enumerated problems. This review article includes 174 references.

On the Use of Quantum Thermal Bath in Unimolecular Fragmentation Simulations

2019 ◽  
Author(s):  
Riccardo Spezia ◽  
Hichem Dammak
Keyword(s):  
Degrees Of Freedom ◽  
Molecular Simulations ◽  
Zero Point ◽  
Thermal Bath ◽  
Unimolecular Dissociation ◽  
Point Energy ◽  
Rotational Degrees Of Freedom ◽  
The Difference ◽  
Nuclear Effects

<div> <div> <div> <p>In the present work we have investigated the possibility of using the Quantum Thermal Bath (QTB) method in molecular simulations of unimolecular dissociation processes. Notably, QTB is aimed in introducing quantum nuclear effects with a com- putational time which is basically the same as in newtonian simulations. At this end we have considered the model fragmentation of CH4 for which an analytical function is present in the literature. Moreover, based on the same model a microcanonical algorithm which monitor zero-point energy of products, and eventually modifies tra- jectories, was recently proposed. We have thus compared classical and quantum rate constant with these different models. QTB seems to correctly reproduce some quantum features, in particular the difference between classical and quantum activation energies, making it a promising method to study unimolecular fragmentation of much complex systems with molecular simulations. The role of QTB thermostat on rotational degrees of freedom is also analyzed and discussed. </p> </div> </div> </div>

THE ROLE OF FINITE ELEMENT METHOD IN CIVIL ENGINEERING

2018 ◽  
Vol 5 (02) ◽  
Author(s):  
Er. Hardik Dhull
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Approximate Solution ◽  
Analytical Method ◽  
Civil Engineering ◽  
The Finite Element Method ◽  
Engineering Problems ◽  
Building Components ◽  
Element Method

The finite element method is a numerical method that is used to find solution of mathematical and engineering problems. It basically deals with partial differential equations. It is very complex for civil engineers to study various structures by using analytical method,so they prefer finite element methods over the analytical methods. As it is an approximate solution, therefore several limitationsare associated in the applicationsin civil engineering due to misinterpretationof analyst. Hence, the main aim of the paper is to study the finite element method in details along with the benefits and limitations of using this method in analysis of building components like beams, frames, trusses, slabs etc.

Mathematical modeling of conversion of non-Gaussian random processes, signals and noise in linear and nonlinear systems

2018 ◽  
pp. 66-78
Author(s):  
V. M. Artyushenko ◽  
V. I. Volovach
Keyword(s):  
Mathematical Modeling ◽  
Nonlinear Systems ◽  
Random Processes ◽  
Mathematical Transformation ◽  
Positive And Negative Feedback ◽  
Non Linear Systems ◽  
Linear And Nonlinear ◽  
Non Gaussian ◽  
Linear And Nonlinear Systems ◽  
Inertial Systems

The questions connected with mathematical modeling of transformation of non-Gaussian random processes, signals and noise in linear and nonlinear systems are considered and analyzed. The mathematical transformation of random processes in linear inertial systems consisting of both series and parallel connected links, as well as positive and negative feedback is analyzed. The mathematical transformation of random processes with polygamous density of probability distribution during their passage through such systems is considered. Nonlinear inertial and non-linear systems are analyzed.

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