Lagrangians for classically damped linear multi-degrees-of-freedom dynamical systems are obtained using simple and elementary methods. Such dynamical systems are very widely used to model and analyze small amplitude vibrations in numerous naturally occurring and engineered systems. An invariant of the motion is also obtained.
Issue Section:
Technical Brief
References
1.
Helmholtz
, H. V.
, 1887
, “Uber die Physikalische Bedeutung des Princips der Kleinsten Wirkung
,” J. Reine Angew. Math.
, 100
, pp. 137
–166
.2.
Santilli
, R. M.
, 1987
, Foundations of Theoretical Mechanics I: The Inverse Problem in Newtonian Mechanics
, Springer-Verlag
, New York
, pp. 110
–111
, 131–132.3.
Udwadia
, F. E.
, and Cho
, H.
, 2013
, “Lagrangians for Damped Linear Multi-Degree-of-Freedom Systems
,” ASME J. Appl. Mech.
, 80
(4
), p. 041023
.4.
Douglas
, J.
, 1941
, “Solution of the Inverse Problem of the Calculus of Variations
,” Trans. Am. Math. Soc.
, 50
(1
), pp. 71
–128
.5.
Hojman
, S.
, and Ramos
, S.
, 1982
, “Two-Dimensional s-Equivalent Lagrangians and Separability
,” J. Phys. A
, 15
(11
), pp. 3475
–3480
.6.
Mesdag
, T.
, Sarlet
, W.
, and Crampin
, M.
, 2011
, “The Inverse Problem for Lagrangian Systems With Certain Non-Conservative Forces
,” Differ. Geom. Appl.
, 29
(1
), pp. 55
–72
.7.
Leitmann
, G.
, 1963
, “Some Remarks on Hamilton's Principle
,” ASME J. Appl. Mech.
, 30
(4
), pp. 623
–625
.8.
Udwadia
, F. E.
, Leitmann
, G.
, and Cho
, H.
, 2011
, “Some Further Remarks on Hamilton's Principle
,” ASME J. Appl. Mech.
, 78
(1
), p. 011014
.9.
Junkins
, J. L.
, and Kim
, Y.
, 1993
, Introduction to Dynamics and Control of Flexible Structures
(AIAA Education Series), American Institute of Aeronautics and Astronautics
, Washington, DC
.10.
Meirovitch
, L.
, 1997
, Principles and Techniques of Vibrations
, Prentice-Hall
, Upper Saddle River, NJ
.11.
Kelly
, G.
, 2006
, Advanced Vibration Analysis
, CRC Press
, Boca Raton, FL
.12.
Carinena
, J. F.
, and Ranada
, M. F.
, 2005
, “Lagrangian Formulation of Nonlinear Riccati Systems: One Dimensional Integrability and Two-Dimensional Superintegrability
,” J. Math. Phys.
, 46
(6
), p. 062703
.13.
Caughey
, T. K.
, and O'Kelley
, M. E. J.
, 1965
, “Classical Normal Models in Damped Linear Dynamics Systems
,” ASME J. Appl. Mech.
, 32
(3
), pp. 583
–588
.14.
Horn
, R. A.
, and Johnson
, C. R.
, 1990
, Matrix Analysis
, Cambridge University Press
, New York
.15.
Pars
, L. A.
, 1972
, A Treatise on Analytical Dynamics
, Oxbow Press
, Woodbridge, CT
.Copyright © 2016 by ASME
You do not currently have access to this content.
