Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-18T04:25:34.427Z Has data issue: false hasContentIssue false
  • English
  • Français

Lattice vibrations with Rayleigh dissipation

Published online by Cambridge University Press:  17 February 2009

J. N. Boyd  and
P. N. Raychowdhury
J. N. Boyd
Affiliation:
Mathematical Sciences Department, Virginia Commonwealth University, Richmond, Virginia 23284-2014, USA.
P. N. Raychowdhury
Affiliation:
Mathematical Sciences Department, Virginia Commonwealth University, Richmond, Virginia 23284-2014, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We approximate a linear array of coupled harmonic oscillators as a symmetric circular array of identical masses and springs. The springs are taken to possess mass distributed along their lengths. We give a Lagrangian formulation of the problem of finding the natural frequencies of oscillation for the array. Damping terms are included by means of the Rayleigh dissipation function. A transformation to symmetry coordinates as determined by the group of rotations of the circle uncouples the equations of motion.

Type
Research Article
Information
The ANZIAM Journal , Volume 42 , Issue 2 , October 2000 , pp. 244 - 253
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Boyd, J. N. and Raychowdhury, P. N., “A one-dimensional crystal with nearest neighbors coupled through their velocities”, J. Dynamic Systems, Measurement and Control 103 (1982) 293–296.Google Scholar
[2]Boyd, J. N. and Raychowdhury, P. N., “An application of projection operators to a one-dimensional crystal”, Bull. of the Institute of Math., Academia Sinica 7 (1979) 133–134.Google Scholar
[3]Boyd, J. N. and Raychowdhury, P. N., “Representation theory of finite Abelian groups applied to a linear diatomic crystal”, Internat. J. Math. and Math. Sci. 3 (1980) 559–574.CrossRefGoogle Scholar
[4]Boyd, J. N. and Raychowdhury, P. N., “A group theoretic approach to generalized harmonic vibrations in a one-dimensional lattice”, Internat. J. Math. and Math. Sci. 9 (1986) 131–136.CrossRefGoogle Scholar
[5]Boyd, J. N. and Raychowdhury, P. N., “A double chain of coupled circuits in analogy with mechanical lattices”, Internat. J. Math. and Math. Sci. 14 (1991) 403–406.CrossRefGoogle Scholar
[6]Boyd, J. N. and Raychowdhury, P. N., “A geometrical approach to maximizing a variance”, Appl. Math. Modelling 18 (1994) 697–700.CrossRefGoogle Scholar
[7]Goldstein, H., Classical Mechanics (Addison-Wesley, Reading, MA, 1965) 21–22.Google Scholar
[8]Hamermesh, M., Group Theory and Its Application to Physical Problems (Addison-Wesley, Reading, MA, 1962).CrossRefGoogle Scholar
[9]Nussbaum, A., “Group theory and normal modes”, Amer. J. Physics 36 (1968) 529–539.CrossRefGoogle Scholar