The University of Southampton
University of Southampton Institutional Repository

Dynamics of convecting elastic solids

Dynamics of convecting elastic solids
Dynamics of convecting elastic solids
The dynamics of a class of convecting elastic media is considered. On the basis of an appropriate variational principle, the general field equation governing small oscillations is derived. The variational formulation demands (i) conservations of mass, (ii) conservation of energy, and (iii) conservation of the identity of particles. Of these, conservation of mass needs to be satisfied explicitly as a constraint. This is achieved by constraining the classical mechanical Lagrangian using a Lagrange multiplier with the continuity equation. Hamilton's principle modified for a control volume in this way then leads to the equation of motion for small oscillations of convecting gyroelastic solids. The mathematical structure of the field equation thus derived is examined. The origins of the 'gyroscopic' and the 'centrifugal' effects are traced. These can be associated with various terms in the expression for the Lagrangian density. In particular, terms in the kinetic energy density that are independent the velocity field, those that are linear in the velocity field and those that are quadratic in the velocity field are associated with the centrifugal, gyroscopic, and inertia terms in the equation of motion respectively. A close mathematical analogy between the dynamics of this class of continua and the dynamics of discrete gyroscopic-centrifugal systems having fixed material particles is noted. The free vibration problem is posed in its generality. An appropriate Rayleigh quotient is defined. The stationarity associated with the quotient can potentially be used for computational work. Illustrative examples and applications are discussed.
Bhaskar, A
d4122e7c-5bf3-415f-9846-5b0fed645f3e
Bhaskar, A
d4122e7c-5bf3-415f-9846-5b0fed645f3e

Bhaskar, A (2010) Dynamics of convecting elastic solids. Days on Diffraction 2010 - International Conference, Russian Federation. 08 - 11 Jun 2010.

Record type: Conference or Workshop Item (Paper)

Abstract

The dynamics of a class of convecting elastic media is considered. On the basis of an appropriate variational principle, the general field equation governing small oscillations is derived. The variational formulation demands (i) conservations of mass, (ii) conservation of energy, and (iii) conservation of the identity of particles. Of these, conservation of mass needs to be satisfied explicitly as a constraint. This is achieved by constraining the classical mechanical Lagrangian using a Lagrange multiplier with the continuity equation. Hamilton's principle modified for a control volume in this way then leads to the equation of motion for small oscillations of convecting gyroelastic solids. The mathematical structure of the field equation thus derived is examined. The origins of the 'gyroscopic' and the 'centrifugal' effects are traced. These can be associated with various terms in the expression for the Lagrangian density. In particular, terms in the kinetic energy density that are independent the velocity field, those that are linear in the velocity field and those that are quadratic in the velocity field are associated with the centrifugal, gyroscopic, and inertia terms in the equation of motion respectively. A close mathematical analogy between the dynamics of this class of continua and the dynamics of discrete gyroscopic-centrifugal systems having fixed material particles is noted. The free vibration problem is posed in its generality. An appropriate Rayleigh quotient is defined. The stationarity associated with the quotient can potentially be used for computational work. Illustrative examples and applications are discussed.

Full text not available from this repository.

More information

Published date: June 2010
Venue - Dates: Days on Diffraction 2010 - International Conference, Russian Federation, 2010-06-08 - 2010-06-11

Identifiers

Local EPrints ID: 155855
URI: https://eprints.soton.ac.uk/id/eprint/155855
PURE UUID: 5540f127-3904-47bb-b7ed-781ae2739d7a

Catalogue record

Date deposited: 02 Jun 2010 08:56
Last modified: 18 Feb 2019 17:31

Export record

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of https://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×