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Jan Awrejcewicz Technical University of Lodz, Poland
Mathematical modeling and simulation of the transitional wobblestone dynamics
It is known that certain bodies of mass centre not coincident with their centroid and of principal centroidal axes of inertia not coincident with their geometric axes exhibit certain interesting dynamical behaviors. An example of such a rigid body is the so-called a rattle back top or wobblestone (a half-ellipsoid solid), which lied on a flat horizontal surface and set in rotational motion about the vertical axis (spin) can rotate in only one direction. The imposition of an initial velocity in the opposite direction leads to rapid cessation of rotation in this direction, and subsequently the stone starts its transverse vibrations and rotation in the opposite direction.
The integral model of dry friction components is built with assumption of classical Coulomb friction law and with specially developed model of normal stress distribution coupled with rolling resistance for elliptic contact shape. In order to avoid a necessity of numerical integration over the contact area at each the numerical simulation step, few versions of approximate model are developed and then tested numerically. In the numerical experiments the simulation results of the Celtic stone with the friction forces modelled by the use of approximants of different complexity (from no coupling between friction force and torque to the second order Pad approximation) are compared to results obtained from model with friction approximated in the form of piecewise polynomial functions based on the Taylor series with Hertzian stress distribution. The coefficients of the corresponding approximate models are found by the use of optimization methods as in identification process using the real experiment data.
We show how in the beginning wobbling without slipping (non-holonomic constraints) and without energy loss have been assumed in a process of modelling and dynamic analysis of the Celtic stone. Then we explore a few paradigm shifts in physics theory aimed at understanding and reliable modelling of the Celtic stone bifurcation and chaotic dynamics: (i) various linearization procedures keeping linear dependence of friction and velocity of a contact point between two bodies; (ii) research through the theory of bifurcation and chaos to explain stability and scenarios of transition between regular and chaotic transitional stone dynamics; (iii) application of the perturbation/asymptotic approaches to study local stone dynamics; (iv) key role of coupling between friction forces and torques assuming a circle contact including both aerodynamic damping and rolling friction factors.
Developments and trends of the state-of-the art of the performed literature survey exhibit importance of the coupling between friction force and torque, which play an essential role in the dynamics of some mechanical systems. If the contact between two bodies is very small (the point contact) the sliding friction force opposes the sliding relative velocity and can be successfully modelled by the use of classical one-dimensional Coulomb friction law. In this case the friction torque (drilling friction) and its influence on sliding friction force can be neglected (since the contact point cannot transmit a torque). The so far mentioned spurious effects observed during the Celtic stone dynamics are also exhibited by a billiard ball, Thompson top, and electric polishing machine and they cannot be mathematically modelled or explained by the use of the assumption of one-dimensional dry friction model.
It is known that certain bodies of mass centre not coincident with their centroid and of principal centroidal axes of inertia not coincident with their geometric axes exhibit certain interesting dynamical behaviors. An example of such a rigid body is the so-called a rattle back top or wobblestone (a half-ellipsoid solid), which lied on a flat horizontal surface and set in rotational motion about the vertical axis (spin) can rotate in only one direction. The imposition of an initial velocity in the opposite direction leads to rapid cessation of rotation in this direction, and subsequently the stone starts its transverse vibrations and rotation in the opposite direction.
The integral model of dry friction components is built with assumption of classical Coulomb friction law and with specially developed model of normal stress distribution coupled with rolling resistance for elliptic contact shape. In order to avoid a necessity of numerical integration over the contact area at each the numerical simulation step, few versions of approximate model are developed and then tested numerically. In the numerical experiments the simulation results of the Celtic stone with the friction forces modelled by the use of approximants of different complexity (from no coupling between friction force and torque to the second order Pad approximation) are compared to results obtained from model with friction approximated in the form of piecewise polynomial functions based on the Taylor series with Hertzian stress distribution. The coefficients of the corresponding approximate models are found by the use of optimization methods as in identification process using the real experiment data.
We show how in the beginning wobbling without slipping (non-holonomic constraints) and without energy loss have been assumed in a process of modelling and dynamic analysis of the Celtic stone. Then we explore a few paradigm shifts in physics theory aimed at understanding and reliable modelling of the Celtic stone bifurcation and chaotic dynamics: (i) various linearization procedures keeping linear dependence of friction and velocity of a contact point between two bodies; (ii) research through the theory of bifurcation and chaos to explain stability and scenarios of transition between regular and chaotic transitional stone dynamics; (iii) application of the perturbation/asymptotic approaches to study local stone dynamics; (iv) key role of coupling between friction forces and torques assuming a circle contact including both aerodynamic damping and rolling friction factors.
Developments and trends of the state-of-the art of the performed literature survey exhibit importance of the coupling between friction force and torque, which play an essential role in the dynamics of some mechanical systems. If the contact between two bodies is very small (the point contact) the sliding friction force opposes the sliding relative velocity and can be successfully modelled by the use of classical one-dimensional Coulomb friction law. In this case the friction torque (drilling friction) and its influence on sliding friction force can be neglected (since the contact point cannot transmit a torque). The so far mentioned spurious effects observed during the Celtic stone dynamics are also exhibited by a billiard ball, Thompson top, and electric polishing machine and they cannot be mathematically modelled or explained by the use of the assumption of one-dimensional dry friction model.
Dumitru Baleanu Cankaya University, Turkey
Advances in fractional dynamics:theory and applications
Fractional calculus is crossing a period of confrontation to the experimental confirmation of its huge merits. During the last decades, in many disciplines of science and engineering the non-locality was better described by using the fractional calculus methods and techniques. In this talk the recent developments of the fractional dynamics will be presented.
Fractional calculus is crossing a period of confrontation to the experimental confirmation of its huge merits. During the last decades, in many disciplines of science and engineering the non-locality was better described by using the fractional calculus methods and techniques. In this talk the recent developments of the fractional dynamics will be presented.
Mike Field Rice University, USA
Asynchronous Networks, Adaptation and the Visualization of Complex Dynamics
Much contemporary mathematical research in dynamics is still motivated and strongly influenced by relatively old problems originating in classical mechanics. However, contemporary challenges in dynamics are of a quite different character. For example, in complex distributed systems, components may run on different clocks, connectivity may vary in time, components may stop and then restart, and the mathematical models may be at best piecewise continuous and piecewise smooth. We discuss some of the mathematical challenges in analyzing these types of asynchronous network as well illustrate with some mathematically tractable, but complex, examples. We conclude with a description of schemes for visualizing dynamics in complex systems. Several of the methods we present are strongly influenced by ideas from neuroscience, most notably Spike-Timing Dependent Plasticity.
Much contemporary mathematical research in dynamics is still motivated and strongly influenced by relatively old problems originating in classical mechanics. However, contemporary challenges in dynamics are of a quite different character. For example, in complex distributed systems, components may run on different clocks, connectivity may vary in time, components may stop and then restart, and the mathematical models may be at best piecewise continuous and piecewise smooth. We discuss some of the mathematical challenges in analyzing these types of asynchronous network as well illustrate with some mathematically tractable, but complex, examples. We conclude with a description of schemes for visualizing dynamics in complex systems. Several of the methods we present are strongly influenced by ideas from neuroscience, most notably Spike-Timing Dependent Plasticity.
Raoul Nigmatullin Kazan Federal University
Self-similar structure of the measured data and "universal" fitting function
A new general fitting function and method of its calculation based on the Self-Similar (SS) organization of random sequences is presented. The proposed analytical function helps to fit the response of many complex systems when their recorded data form a self-similar curve. The verified SS principle opens new possibilities for the fitting of economical, meteorological and other complex data when the mathematical model is absent but the reduced description in terms of some universal set of the fitting parameters is necessary. This fitting function is verified on economical (price of a commodity versus time) and weather (the Earth's mean temperature surface data versus time) data. Other available data are also analyzed in the frame of this new approach. For these nontrivial cases it becomes possible to receive a very good fit of initial data set. The general conditions of application of this fitting method describing the response of many complex systems and the forecast possibilities are discussed.
A new general fitting function and method of its calculation based on the Self-Similar (SS) organization of random sequences is presented. The proposed analytical function helps to fit the response of many complex systems when their recorded data form a self-similar curve. The verified SS principle opens new possibilities for the fitting of economical, meteorological and other complex data when the mathematical model is absent but the reduced description in terms of some universal set of the fitting parameters is necessary. This fitting function is verified on economical (price of a commodity versus time) and weather (the Earth's mean temperature surface data versus time) data. Other available data are also analyzed in the frame of this new approach. For these nontrivial cases it becomes possible to receive a very good fit of initial data set. The general conditions of application of this fitting method describing the response of many complex systems and the forecast possibilities are discussed.
Manuel Ortigueira UNINOVA, Portugal
A new look into the discrete-time fractional calculus
A derivative based discrete-time signal processing is presented. Both nabla (forward) and delta (backward) derivatives are studied and generalized including the fractional case. The corresponding exponentials are introduced as eigenfunctions of such derivatives. These lead to discrete-time Laplace transforms that are used to define and study the linear discrete-time derivatives.
A derivative based discrete-time signal processing is presented. Both nabla (forward) and delta (backward) derivatives are studied and generalized including the fractional case. The corresponding exponentials are introduced as eigenfunctions of such derivatives. These lead to discrete-time Laplace transforms that are used to define and study the linear discrete-time derivatives.