On the existence of normal mode of vibrations of nonlinear systems with two degrees of freedom

Rosenberg, R. M.

doi:10.1090/qam/99954

Author: R. M. Rosenberg
Journal: Quart. Appl. Math. 22 (1964), 217-234
DOI: https://doi.org/10.1090/qam/99954
MathSciNet review: QAM99954
Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: A nonlinear, conservative system having two degrees of freedom is considered. The potential energy is subject to certain restrictions which are consistent with the strain energy of springs. Then, the notion of transversals; i.e., of curves orthogonal to equipotential lines, is introduced in order to prove the existence of an in-phase and of an out-of-phase normal mode of vibration, no matter how nonlinear the springs. It is shown, that the transversals are also very helpful in the discussion of general motions of the system.

R. M. Rosenberg, Normal modes of nonlinear dualmode systems, J. Appl. Mech. 27 (1960), 263–268. MR 0111194
R. M. Rosenberg, On normal vibrations of a general class of nonlinear dual-mode systems, Trans. ASME Ser. E. J. Appl. Mech. 28 (1961), 275–283. MR 120796
R. M. Rosenberg and C. S. Hsu, On the geometrization of normal vibrations of nonlinear systems having many degrees of freedom, Analytic methods in the theory of non-linear vibrations (Proc. Internat. Sympos. Non-linear Vibrations, Vol. I, 1961) Izdat. Akad. Nauk Ukrain. SSR, Kiev, 1963, pp. 380–416 (English, with Russian summary). MR 0159084
R. M. Rosenberg, The normal modes of nonlinear $n$-degree-of-freedom systems, Trans. ASME Ser. E. J. Appl. Mech. 29 (1962), 7–14. MR 137340

R. M. Rosenberg, and J. K. Kuo, Non-similar normal mode vibrations of nonlinear systems having two degrees of freedom, to be published in J. Appl. Mech.

See for instance N. Minorski, Nonlinear oscillations, D. Van Nostrand Company, Inc. 1962, pp. 9–26

Hans Kauderer, Nichtlineare Mechanik, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958 (German). MR 0145709