T. K. Caughey1 has shown that a necessary and sufficient condition that a damped, linear, n-degree-of-freedom system possess classical linear normal modes is that the damping matrix be diagonalized by the same transformation which uncouples the undamped system. Rosenberg2 has defined normal modes for nonlinear n-degree-of-freedom undamped systems and has shown the existence of such modes for various classes of nonlinear systems. In linear systems, the frequency is independent of the amplitude and, if a set of masses is vibrating in unison, it is not surprising that in some cases they continue to do so as the motion damps out. In nonlinear vibrations, however, frequency depends upon amplitude so that a series of masses vibrating at different amplitudes in a Rosenberg normal mode might generally be expected to lose synchronization as their amplitudes damp out. Two classes of systems are discussed here in which normal modes are preserved under damping, and several examples are given.
Skip Nav Destination
Normal Modal Vibrations for Some Damped n-Degree-of-Freedom Nonlinear Systems
George W. Morgenthaler
Electronics and Mathematics Research Department, The Martin Company, Denver, Colo.
- Views Icon Views
- Share Icon Share
- Search Site
Morgenthaler, G. W. (December 1, 1966). "Normal Modal Vibrations for Some Damped n-Degree-of-Freedom Nonlinear Systems." ASME. J. Appl. Mech. December 1966; 33(4): 877–880. https://doi.org/10.1115/1.3625196
Download citation file:
- Ris (Zotero)
- Reference Manager