Dynamics of a nonlinear periodic structure with cyclic symmetry

Modal Analysis of a Nonlinear Periodic Structure with Cyclic Symmetry

“…This resonance scenario results in NNM motions which take the form of traveling waves and which are represented by ellipses in the configuration space. A detailed analytical study of these modes is given in reference [6]. Due to the existence of a phase difference between the coordinates, a different phase condition is considered for the NNM computation: only one initial velocity is set to zero, which is compatible with a traveling-wave motion.…”

“…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].…”

Reduced-order modelling using nonlinear modes and triple nonlinear modal synthesis

“…The key feature of this method is the use of so-called nonlinear modes to capture the nonlinearities in the reduction basis. First initiated by Rosenberg [7][8][9] in the 1960s, the concept of nonlinear mode, which can be seen as an extension of linear modes to nonlinear systems, has been broadened by the contributions of many authors since then [10][11][12][13][14][15][16][17][18][19][20]. In addition to the literature that provides today researchers and engineers with numerous analytical and numerical techniques to calculate nonlinear modes, thoroughly reviewed by Renson et al in [21], the improving hardware and software capabilities provided by modern scientific computing make it possible nowadays to apply it to industrial structures.…”