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On normal mode vibrations

Published online by Cambridge University Press:  24 October 2008

R. M. Rosenberg
R. M. Rosenberg
Affiliation:
University of California, Berkeley
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Extract

1. Introduction. In linear systems, the concept of ‘free vibrations in normal modes’ is well defined and fully understood. The meaning of this phrase is far less clear when it is applied to non-linear systems. It is the purpose here to define and examine the free vibrations in normal modes (and their stability) in certain non-linear systems composed of masses and springs and having a finite number of degrees of freedom. Of necessity, such a paper is in some degree conceptual in nature.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 3 , July 1964 , pp. 595 - 611
Copyright
Copyright © Cambridge Philosophical Society 1964

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References

REFERENCES

(1) Bradistolov, G. Über periodische und asymptotische Lösungen beim n-fachen Pendel. Math. Ann. 116 (1936), 181.CrossRefGoogle Scholar
(2) Bradistolov, G. Über periodische Bewegungen des n−fachen Pendels in der Ebene. Math. Ann. 116 (1936), 602.CrossRefGoogle Scholar
(3) Darboux, G. Leçons sur la théorie générale des surfaces, vol. II (Gauthier-Villars; 1889).Google Scholar
(4) Hsu, C. S. On a restricted class of coupled Hill's equations and some applications. J. Appl. Mech. 28 (1961), 551.CrossRefGoogle Scholar
(5) Hsu, C. S. On the parametric excitation of a dynamic system having multiple degrees of freedom. J. Appl. Mech. (to appear).Google Scholar
(6) Kauderer, H. Nichtlineare Mechanik (Springer; Berlin, 1958).CrossRefGoogle Scholar
(7) Mettler, E. Stabilitätsfragen bei freien Schwingungen mechanischer Systeme. Ing.-Arch. 28 (1959), 213.CrossRefGoogle Scholar
(8) Pöschl, Th. Über Hauptschwingungen mit endlichen Schlagweiten, I. Ing.-Arch. 20 (1952), 189.CrossRefGoogle Scholar
(9) Pöschl, Th. Über Hauptschwingungen mit endlichen Schlagweiten, II. Ing.-Arch. 21 (1953), p. 396CrossRefGoogle Scholar
Pöschl, Th. corrigendum, Ing.-Arch. 22 (1954), 294.CrossRefGoogle Scholar
(10) Rosenberg, R. M. Normal modes of nonlinear dual-mode systems. J. Appl. Mech. 27 (1960), 263.CrossRefGoogle Scholar
(11) Rosenberg, R. M. On normal vibrations of a general class of nonlinear dual-mode systems. J. Appl. Mech. 28 (1961), 275.CrossRefGoogle Scholar
(12) Rosenberg, R. M. The natural modes of nonlinear n−degree of freedom systems. J. Appl. Mech. 29 (1962), 7.CrossRefGoogle Scholar
(13) Rosenberg, R. M. The Ateb(h) functions and their properties. Quart. Appl. Math. 20 (1963), 37.CrossRefGoogle Scholar
(14) Rosenberg, R. M. and Atkinson, C. P. On the natural modes and their stability in non-linear two-degree-of-freedom systems. J. Appl Mech. 26 (1959), 377.CrossRefGoogle Scholar
(15) Rosenberg, R. M. and Hsu, C. S. On the geometrization of normal vibrations of non-linear systems having many degrees of freedom. Proc. Internal. IUTAM Symposium on Nonlinear Vibrations;Kiev 1961.Google Scholar
(16) Rosenberg, R. M. and Kuo, J. K. Nonsimilar normal mode vibrations of nonlinear-systems having two degrees of freedom. To be published in J. Appl. Mech.Google Scholar
(17) Schmieden, C. Nichtlineare Schwingungen bei zwei Freiheitsgraden, I. Ing.-Arch. 25 (1957), 292.CrossRefGoogle Scholar
(18) Schmieden, C. Nichtlineare Schwingungen bei zwei Freiheitsgraden, II. Ing.-Arch. 26 (1958), 110.CrossRefGoogle Scholar
(19) Synge, J. L. On the geometry of dynamics. Philos. Trans. Roy. Soc. London, Ser. A, 222 (1926), 31.Google Scholar
(20) Valeev, K. G. On the solutions and characteristic exponents of some systems of linear-differential equations with periodic coefficients. Prikl. Mat. Meh. (English translation), 24 (1960), 877.Google Scholar
(21) Valeev, K. G. On Hill's method in the theory of linear differential equations with periodic-coefficients. Prikl. Mat. Meh. (English translation), 24 (1960), 1475.Google Scholar