Lagrangians for Damped Linear Multi-Degree-of-Freedom Systems

[en] This paper deals with finding Lagrangians for damped, linear multi-degree-of-freedom systems. New results for such systems are obtained using extensions of the results for single and two degree-of-freedom systems. The solution to the inverse problem for an n-degree-of-freedom linear gyroscopic system is obtained as a special case. Multi-degree-of-freedom systems that commonly arise in linear vibration theory with symmetric mass, damping, and stiffness matrices are similarly handled in a simple manner. Conservation laws for these damped multi-degree-of-freedom systems are found using the Lagrangians obtained and several examples are provided.

Disciplines :

Mechanical engineering

Author, co-author :

Udwadia, Firdaus; University of Southern California > Department of Aerospace and Mechanical Engineering

Cho, Hancheol ; Université de Liège > Département d'aérospatiale et mécanique > Laboratoire de structures et systèmes spatiaux

Language :

English

Title :

Lagrangians for Damped Linear Multi-Degree-of-Freedom Systems

Publication date :

2013

Journal title :

Journal of Applied Mechanics

ISSN :

0021-8936

eISSN :

1528-9036

Publisher :

American Society of Mechanical Engineers, New York, United States - New York

Volume :

Issue :

Pages :

041023-1 – 041023-10

Peer reviewed :

Peer Reviewed verified by ORBi

Available on ORBi :

since 20 January 2016

Statistics

Number of views

49 (0 by ULiège)

Number of downloads

1008 (0 by ULiège)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

OpenAlex citations

Bibliography

Bolza, O., 1931, Vorlesungen über Varationsrechnung, Koehler und Amelang, Leipzig, Germany.
Leitmann, G., 1963, "Some Remarks on Hamilton's Principle," ASME J. Appl. Mech., 30, pp. 623-625.
Udwadia, F. E., Leitmann, G., and Cho, H., 2011, "Some Further Remarks on Hamilton's Principle," ASME J. Appl. Mech., 78, p. 011014.
He, J.-H., 2004, "Variational Principles for Some Nonlinear Partial Differential Equations With Variable Coefficients," Chaos, Solitons Fractals, 19, pp. 847-851.
Darboux, G., 1894, Leçons sur la Théorie Générale des Surfaces, Vol. 3, Gauthier-Villars, Paris.
Helmholtz, H. v., 1887, "Uber die physikalische Bedeutung des Princips der kleinsten Wirkung," J. Reine Angew. Math., 100, pp. 137-166.
Santilli, R. M., 1978, Foundations of Theoretical Mechanics I: The Inverse Problem in Newtonian Mechanics, Springer-Verlag, New York, pp. 110-111 and 131-132.
Douglas, J., 1941, "Solution of the Inverse Problem of the Calculus of Variations," Trans. Am. Math. Soc., 50, pp. 71-128.
Hojman, S. and Ramos, S., 1982, "Two-Dimensional s-Equivalent Lagrangians and Separability," J. Phys. A, 15, pp. 3475-3480.
Mestdag, T., Sarlet, W., and Crampin, M., 2011, "The Inverse Problem for Lagrangian Systems With Certain Non-Conservative Forces," Diff. Geom. Applic., 29, 2011, pp. 55-72.
Ray, J. R., 1979, "Lagrangians and Systems They Describe - How Not to Treat Dissipation in Quantum Mechanics," Am. J. Phys., 47, pp. 626-629.
Pars, L. A., 1972, A Treatise on Analytical Dynamics, Ox Bow, Woodbridge, CT, p. 82.