Abstract
A new formulation is presented here for the existence and calculation of nonlinear normal modes in undamped nonlinear autonomous mechanical systems. As in the linear case an expression is developed for the mode in terms of the amplitude, mode shape and frequency, with the distinctive feature that the last two quantities are amplitude and total phase dependent. The dynamic of the periodic response is defined by a one-dimensional nonlinear differential equation governing the total phase motion. The period of the oscillations, depending only on the amplitude, is easily deduced. It is established that the frequency and the mode shape provide the solution to a 2 π -periodic nonlinear eigenvalue problem, from which a numerical Galerkin procedure is developed for approximating the nonlinear modes. The procedure is applied to various mechanical systems with two degrees of freedom.
Highlights
Extending the concept of normal modes of vibrating systems to the case where the restoring forces contain nonlinear terms has been a challenge to many authors, mainly because the principle of linear superposition does not hold for nonlinear systems
Normal modes approach to be extended to nonlinear theory, and it is concluded that nonlinear normal modes (NNMs) may provide a valuable theoretical tool for understanding some peculiarities of nonlinear systems such as mode bifurcations and nonlinear mode localization
Following the pioneer work by Rosenberg [2] on conservative systems, several attempts have been made to develop methods of calculating NNMs. These include the harmonic method developed by Szemplinska-Stupnicka [3], the normal form theory [4,5], the invariant manifold method [6,7], the perturbation method [8], the balance harmonic procedure [9], the method of multiple scales [10], and various combinations
Summary
Normal modes approach to be extended to nonlinear theory, and it is concluded that nonlinear normal modes (NNMs) may provide a valuable theoretical tool for understanding some peculiarities of nonlinear systems such as mode bifurcations and nonlinear mode localization. A new formulation is presented for the existence and calculation of the synchronous periodic oscillation of an undamped autonomous nonlinear mechanical system. The modal vector and the frequency provide the solution to a nonlinear eigenvector–eigenvalue 2p-periodic problem, which makes it possible to calculate these quantities using the classical Galerkin procedure in the space of 2p-periodic functions [10]. The Galerkin computational procedure is applied to two-dimensional mechanical systems in the odd case as well as in the general case. It seems that the presence of internal resonances in the underlying linear system does not require particular attention in these calculations
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
