Hamilton’s Principle as Variational Inequality for Mechanical Systems with Impact

The classical form of Hamilton’s principle holds for conservative systems with perfect bilateral constraints. Several attempts have been made in literature to generalise Hamilton’s principle for mechanical systems with perfect unilateral constraints involving impulsive motion. This has led to a number of different variants of Hamilton’s principle, some expressed as variational inequalities. Up to now, the connection between these different principles has been missing. The aim of this paper is to put these different principles of Hamilton in a unified framework by using the concept of weak and strong extrema. The difference between weak and strong variations of the motion is explained in detail. Each type of variation leads to a variant of the principle of Hamilton in the form of a variational inequality. The conclusion of the paper is that each type of variation leads to different necessary and sufficient conditions on the impact law. The principle of Hamilton with strong variations is valid for perfect unilateral constraints with a completely elastic impact law, whereas the weak form of Hamilton’s principle only requires perfect unilateral constraints and no condition on the energy.     

Discontinuous fold bifurcations in mechanical systems

Summary  This paper treats discontinuous fold bifurcations of periodic solutions of discontinuous systems. It is shown how jumps in the fundamental solution matrix lead to jumps of the Floquet multipliers of periodic solutions. A Floquet multiplier of a discontinuous system can jump through the unit circle, causing a discontinuous bifurcation. Numerical examples are treated, which show discontinuous fold bifurcations. A discontinuous fold bifurcation can connect stable branches to branches with infinitely unstable solutions. Received 20 September 2000; accepted for publication 26 June 2001     

The historical development of classical stability concepts: Lagrange, Poisson and Lyapunov stability

A brief historical overview is given which discusses the development of classical stability concepts, starting in the seventeenth century and finally leading to the concept of Lyapunov stability at the beginning of the twentieth century. The aim of the paper is to find out how various scientists thought about stability and to which extent their work is related to the stability concepts bearing their names, i.e. Lagrange, Poisson and Lyapunov stability. To this end, excerpts of original texts are discussed in detail. Furthermore, the relationship between the various works is addressed.     

Bifurcations in Nonlinear Discontinuous Systems

This paper treats bifurcations of periodic solutions indiscontinuous systems of the Filippov type. Furthermore, bifurcations offixed points in non-smooth continuous systems are addressed. Filippov'stheory for the definition of solutions of discontinuous systems issurveyed and jumps in fundamental solution matrices are discussed. It isshown how jumps in the fundamental solution matrix lead to jumps of theFloquet multipliers of periodic solutions. The Floquet multipliers canjump through the unit circle causing discontinuous bifurcations.Numerical examples are treated which show various discontinuousbifurcations. Also infinitely unstable periodic solutions are addressed.     

Attractivity of Equilibrium Sets of Systems with Dry Friction

The dynamics of mechanical systems with dry friction elements, modelledby set-valued force laws, can be described by differential inclusions.An equilibrium set of such a differential inclusion corresponds to astationary mode for which the friction elements are sticking. Theattractivity properties of the equilibrium set are of major importancefor the overall dynamic behaviour of this type of systems. Conditionsfor the attractivity of the equilibrium set of MDOF mechanical systemswith multiple friction elements are presented. These results areobtained by application of a generalisation of LaSalle's principle fordifferential inclusions of Filippov-type. Besides passive systems, alsosystems with negative viscous damping are considered. For such systems,only local attractivity of the equilibrium set can be assured undercertain conditions. Moreover, an estimate for the region of attractionis given for these cases. The effectiveness of the results isillustrated by means of both 1DOF and MDOF examples.     

Dynamics of a Rolling Disk in the Presence of Dry Friction

In this paper we are interested in the dynamics and numerical treatment of a rolling disk on a flat support. The objective of the paper is to develop a numerical model which is able to simulate the dynamics of a rolling disk taking into account various kinds a friction models (resistance against sliding, pivoting and rolling). A mechanical model of a rolling disk is presented in the framework of Non-smooth Dynamics and Convex Analysis. In an analytical study, approximations are derived for the energy decay of the system during the final stage of the motion for various kinds of frictional dissipation models. Finally, the numerical and analytical results are discussed and compared with experimental results available in literature.     

The demicellization of alkyltrimethylammonium bromides in 0.1 M sodium chloride solution studied by isothermal titration calorimetry

The demicellization of the cationic detergents dodecyltrimethylammonium bromide, tetradecyltrimetylammonium bromide, and cetyltrimethylammonium bromide was studied at temperatures between 20 and 60 degrees C in 0.1 M NaCl (pH 6.4) using isothermal titration calorimetry (ITC). We determined the critical micellization concentration (cmc) of the cationic detergents which show a minimum at temperatures between 20 and 34 degrees C. In accordance with the lengthening of the hydrophobic tail of the detergents the cmc decreases with increasing alkyl chain length. The thermodynamic parameters describing the changes of enthalpy (DeltaH(demic)), the changes of entropy (DeltaS(demic)) and the Gibbs free energy change (DeltaG(demic)) for demicellization were first obtained using the pseudophase-separation model. The aggregation number n at the cmc as well as the demicellization enthalpy, entropy and Gibbs free energy change were also calculated using a simulation based on the mass-action model. Furthermore, we investigated the demicellization of CTAB in deionized water in comparison to demicellization in sodium chloride solution to determine the influence of counter ion binding on the demicellization.     

Simulation of femtosecond pulse propagation in air

We present numerical simulations of the propagation of intense spatio-temporal fields propagating in air. Characteristic parameters of the propagation like peak intensities, plasma densities and diameters of the plasma channels are obtained. The evolution of the spectral content of the pulses is investigated and, in agreement with recent experiments, white-light supercontinuum generation is observed. We show in which areas of the pulse the supercontinuum is generated and conclude that self-phase modulation is responsible for the spectral broadening. Furthermore we investigate azimuthal instabilities of intense rotationally symmetric pulsed beams propagating in air. Although the spatial-temporal evolution of the field is strongly influenced by the onset of plasma generation, the instabilities are basically caused by the Kerr effect. We conclude that calculations assuming rotational symmetry become unrealistic due to the fast growth of azimuthal instabilities in the focal region.     

The Euler-Maupertuis principle of least action as variational inequality

Starting from Hamilton's Principle, the current paper discusses how we can derive the Euler-Maupertuis Principle of Least Action in the context of non-smooth dynamics. This variational principle allows us to directly obtain the space curve y (x) of a point-mass in a potential field V (x, y) without referring to the temporal dynamics. This paper generalises the Euler-Maupertuis Principle of Least Action to systems with impact by formulating the principle as a variational inequality. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The direct method of Lyapunov for nonlinear dynamical systems with fractional damping

Nonlinear Dynamics - In this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the...