A system of n masses, equal or not, interconnected by nonlinear “symmetric” springs, and having n degrees of freedom is examined. The concept of normal modes is rigorously defined and the problem of finding them is reduced to a geometrical maximum-minimum problem in an n-space of known metric. The solution of the geometrical problem reduces the coupled equations of motion to n uncoupled equations whose natural frequencies can always be found by a single quadrature. An infinite class of systems, of which the linear system is a member, has been isolated for which the frequency amplitude can be found in closed form.
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The Normal Modes of Nonlinear n-Degree-of-Freedom Systems
R. M. Rosenberg
University of California, Berkeley, Calif.
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Rosenberg, R. M. (March 1, 1962). "The Normal Modes of Nonlinear n-Degree-of-Freedom Systems." ASME. J. Appl. Mech. March 1962; 29(1): 7–14. https://doi.org/10.1115/1.3636501
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