Mode Localization in a Class of Multidegree-of-Freedom Nonlinear Systems with Cyclic Symmetry

Abstract

The free oscillations of n-degree-of-freedom (DOF) nonlinear systems with cyclic symmetry and weak coupling between substructures are examined. An asymptotic methodology is used to detect localized nonsimilar normal modes, i.e., free periodic motions spatially confined to only a limited number of substructures of the cyclic system. It is shown that nonlinear mode localization occurs in the perfectly symmetric, weakly coupled structure, in contrast to linear mode localization, which exists only in the presence of substructure "mistuning." In addition to the localized modes, nonlocalized modes are also found in the weakly coupled system. The stability of the identified modes is investigated by means of an approximate two-timing averaging methodology, and the general theory is applied to the case of a cyclic system with three-DOF. The theoretical results are then verified by direct numerical integrations of the equations of motion.

Modal Analysis of a Nonlinear Periodic Structure with Cyclic Symmetry

“…The prediction is corrected by a shooting procedure in order to solve (7) in which the variations of the initial conditions and the period are forced to be orthogonal to the predictor step. At iteration k, the corrections…”

“…Wei and Pierre examined the effects of dry friction on a nearly cyclic structure using the harmonic balance method [5]. In a series of papers, Vakakis and co-workers demonstrated that, in contrast to the findings of linear theory, nonlinear mode localization may occur in perfectly cyclic nonlinear systems [6,7,8,9]. Other studies dealing with mode localization in nonlinear cyclic systems are [10,11,12].…”

“…In the present study, we revisit the free dynamics of such systems using the numerical algorithm for NNM computation proposed in [13,14]. This computational framework allows relaxing the assumptions of weak nonlinearity and/or weak coupling used in [6,7,8,9]. In addition, it is shown that modal interactions can take place in nonlinear systems without necessarily having commensurate natural frequencies in the underlying linear system.…”

“…NNMs and asymptotic analysis were used in [6,7,8,9] to examine the free and forced vibrations of nonlinear periodic systems with cyclic symmetry. In the present study, we revisit the free dynamics of such systems using the numerical algorithm for NNM computation proposed in [13,14].…”

Localization in Multiple Nearly-Identical Tuned Vibration Absorbers

“…Localization in both linear and non linear dynamical systems has been studied by many researchers. Samples can be found in [3][4][5][6][7][8]. In a previous study of localization in systems of multiple vibration absorbers, Alsuwaiyan and Shaw [9] investigated the existence of localization in free vibration modes for both translational and torsional vibration absorber systems.…”