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On the Relation of the Principle of Maximum Dissipation to the Principles of Gauss and Jourdain

Kerim Yunt(kerimyunt***at***web.de)

Abstract: The aim of this work is to establish the relation between three well-known principles of dynamics for finite-dimensional Lagrangian systems subject to non-impulsive dissipative force laws. These principles are the principle of maximum dissipation (PMD), the principle of Jourdain and Gauss’ principle. Among the principles of mechanics, the principle of Jourdain, which is also known as the principle of virtual power, is a formulation of the evolution conditions of a Lagrangian system on velocity level and constitutes the natural connection to relate mechanical variational principles to dissipativity. In this work, a dissipation based definition of the principle of Jourdain is presented for rheonomic (explicitly time dependent) mechanical systems, which evolve under the influence of convex dissipation potentials. Further, it is shown, that the variational condition of the dissipative principle of Jourdain is the necessary condition for the maximization of the total dissipated power with respect to generalized velocities. The principle of maximum dissipation is related to the dissipative principle of Jourdain. For dynamics, where the evolution requires the determination of the accelerations of the system, it is shown that in the presence of dissipative force laws, that the augmentation of the optimization problem of least constraints by the time derivative of total dissipation potential for fixed generalized positions and velocities is required. A dissipative principle of Gauss is formulated by making use of the subdifferential of the total time derivative of the total dissipation potential. Further, it is shown, that the principle of maximum dissipation is extended to dynamics in the form of a principle, that requires the maximization of the time derivative of the total dissipation potential for fixed generalized positions and velocities with respect to passive dissipative forces. The dual problem of least constraints is derived from the maximization of the total time derivative of the total dissipation with respect to passive dissipative forces.

Keywords: Dissipative Systems, Nonsmooth Analysis, Principle of Jourdain, Gauss’ Principle

Category 1: Applications -- Science and Engineering (Basic Sciences Applications )

Category 2: Complementarity and Variational Inequalities

Category 3: Convex and Nonsmooth Optimization

Citation: submitted to ZAMM (Zeitschrift für Angewandte Mechanik und Mathematik)

Download: [PDF]

Entry Submitted: 01/23/2013
Entry Accepted: 01/23/2013
Entry Last Modified: 01/23/2013

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