On the Stability of the Linearly Related Modes of Certain Nonlinear Two-Degree-of-Freedom Systems

[+] Author and Article Information
C. P. Atkinson

University of California, Berkeley, Calif.

J. Appl. Mech 28(1), 71-77 (Mar 01, 1961) (7 pages) doi:10.1115/1.3640469 History: Received March 14, 1960; Online September 16, 2011


This paper presents a method for analyzing a pair of coupled nonlinear differential equations of the Duffing type in order to determine whether linearly related modal oscillations of the system are possible. The system has two masses, a coupling spring and two anchor springs. For the systems studied, the anchor springs are symmetric but the masses are not. The method requires the solution of a polynomial of fourth degree which reduces to a quadratic because of the symmetric springs. The roots are a function of the spring constants. When a particular set of spring constants is chosen, roots can be found which are then used to set the necessary mass ratio for linear modal oscillations. Limits on the ranges of spring-constant ratios for real roots and positive-mass ratios are given. A general stability analysis is presented with expressions for the stability in terms of the spring constants and masses. Two specific examples are given.

Copyright © 1961 by ASME
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