Analysis and identification of linear and nonlinear normal modes in vibrating systems

• Main Author: Vakakis, Alexander F.
• Format: Others
• Published: 1991
• Online Access: https://thesis.library.caltech.edu/3200/1/Vakakis_af_1991.pdf; Vakakis, Alexander F. (1991) Analysis and identification of linear and nonlinear normal modes in vibrating systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/60CA-1Q64. https://resolver.caltech.edu/CaltechETD:etd-08232004-105610 <https://resolver.caltech.edu/CaltechETD:etd-08232004-105610>

In the first part of this work, the free and forced oscillations of a class of strongly nonlinear, undamped, discrete oscillators, are studied. The free motions are examined by using the notion of "nonlinear normal mode," originally introduced by Rosenberg. Analytical methods for computing similar and nonsimilar normal modes are presented, and the mode stability is analyzed. Normal mode bifurcations are found to exist in these systems, increasing in complexity as the degree of nonlinearity increases. A specific application is given with a two degree of freedom, hamiltonian oscillator with cubic nonlinearity. The low energy motions are analyzed by means of Poincare' maps and an approximate averaging technique. When the energy is increased, chaotic motions are observed in the Poincare' maps, resulting from the transverse intersections of the stable and unstable manifolds of an unstable normal mode. Moreover, the generation of subharmonic orbits resulting from the breakdown of invariant KAM Tori is examined by using Subharmonic Melnikov analysis. The similar and nonsimilar forced steady state motions are examined by considering special (nonharmonic) periodic excitations. For the case of cubic nonlinearity, a theorem on the necessary and sufficient conditions that a force should satisfy in order to lead to an exact steady state is given. In the second part of the work, techniques for identifying systems with closely spaced modes and weak nonlinearities are developed. Modal interference in the Complex plane is modeled by expanding the Frequency Response Function of the "perturbing mode" in Taylor series, and retaining only the two first terms. The distorted Nyquist plots of systems with stiffness and/or damping nonlinearities are analytically studied, by using the concept of "equivalent linearization." Based on the analytical results, refined identification algorithms are proposed, and their applicability is tested by analyzing theoretical and experimental data.