0
Research Papers

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

[+] Author and Article Information
Firdaus E. Udwadia

Professor
Departments of Aerospace and Mechanical Engineering,
Civil Engineering,
Mathematics,
Systems Architecture Engineering,
and Information and Operations Management,
University of Southern California,
430 K Olin Hall,
Los Angeles, CA 90089

Hancheol Cho

Graduate Student
Department of Aerospace and Mechanical Engineering,
University of Southern California,
Los Angeles, CA 90089

Since we cannot differentiate the function L(t,q,q·), with respect to a dependent variable, say q·i, by L(t,q,q·)/q·i we mean L(t,s,r)/ri|s=qr=q·, where t, s, and r are considered independent variables; similarly, by L(t,q,q·)/qi, we mean L(t,s,r)/si|s=qr=q·.

There is a slight abuse of notation here, since in Eqs. (11)–(14) the variables t, q, and q· are considered to be independent, while in the equations of motion, x and x· (see Eq. (8)) are considered to be functions of time t.

Manuscript received March 1, 2012; final manuscript received October 27, 2012; accepted manuscript posted November 19, 2012; published online May 23, 2013. Assoc. Editor: Wei-Chau Xie.

J. Appl. Mech 80(4), 041023 (May 23, 2013) (10 pages) Paper No: JAM-12-1087; doi: 10.1115/1.4023019 History: Received March 01, 2012; Revised October 27, 2012; Accepted November 19, 2012

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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

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. [CrossRef]
Udwadia, F. E., Leitmann, G., and Cho, H., 2011, “Some Further Remarks on Hamilton's Principle,” ASME J. Appl. Mech., 78, p. 011014. [CrossRef]
He, J.-H., 2004, “Variational Principles for Some Nonlinear Partial Differential Equations With Variable Coefficients,” Chaos, Solitons Fractals, 19, pp. 847–851. [CrossRef]
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. [CrossRef]
Hojman, S. and Ramos, S., 1982, “Two-Dimensional s-Equivalent Lagrangians and Separability,” J. Phys. A, 15, pp. 3475–3480. [CrossRef]
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. [CrossRef]
Ray, J. R., 1979, “Lagrangians and Systems They Describe—How Not to Treat Dissipation in Quantum Mechanics,” Am. J. Phys., 47, pp. 626–629. [CrossRef]
Pars, L. A., 1972, A Treatise on Analytical Dynamics, Ox Bow, Woodbridge, CT, p. 82.

Figures

Grahic Jump Location
Fig. 1

Linear 2-DOF mass-spring-damper system with b = 2 m

Grahic Jump Location
Fig. 2

General 2-DOF mass-spring-damper system

Grahic Jump Location
Fig. 3

General 3-DOF mass-spring-damper system

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In