Preprint Article Version 1 This version is not peer-reviewed

An Analysis of Dynamic Vibrations: The Invariant Normal Curvature Wave Equation

Version 1 : Received: 25 September 2017 / Approved: 25 September 2017 / Online: 25 September 2017 (16:23:04 CEST)

How to cite: Afas, K.C.; Moschandreou, T. An Analysis of Dynamic Vibrations: The Invariant Normal Curvature Wave Equation. Preprints 2017, 2017090122 (doi: 10.20944/preprints201709.0122.v1). Afas, K.C.; Moschandreou, T. An Analysis of Dynamic Vibrations: The Invariant Normal Curvature Wave Equation. Preprints 2017, 2017090122 (doi: 10.20944/preprints201709.0122.v1).

Abstract

This paper attempts to address the phenomenon of normal vibrations, (referred to as dynamic vibrations) occurring on a surface which is already in vibrational motion due to other kinematic phenomena. Such a surface will have a metric tensor, normal, and ambient velocity which diverges from the surface’s original various dynamic tensorial descriptors. This paper formulates the wave equation defined in a coordinate space, and extends the equation to observe vibrations on a surface with the use of the Laplace-Beltrami operator in a tensorial fashion drawing on conventions from the newly established Calculus of Moving Surfaces (CMS). The Paper then identifies the way which these normal vibrations will manifest within ambient space. Finally, a counter-intuitive relation between the magnitude of such dynamic vibrations and dynamic surface’s time-dependent mean curvature presents itself, for which dynamic vibrations superimposed on original dynamic motion will eliminate the other, and the surface remains static under an arbitrary initial motion. From this condition, resubstitution within the wave equation yields a novel coupled PDE system which within contains the time-evolution of the surface’s mean curvature and its dynamic vibrations. This analysis can have application for algorithms designed for mechanical stabilization during seismic activity, as well as analyzing stabilization algorithms for other various applications and can also have application with respect to biotechnological innovations for analyzing properties defined on the surface of cell-Like biological entities such as metabolism, lipid content, and actin dynamics.

Subject Areas

vibrations

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