Chapter 15 Resonant Characterization of Nonlinear Structures in the Co-existence of Multiple Resonant Components Nidish Narayanaa Balaji, Matthew R. W. Brake, D. Dane Quinn, and Malte Krack Abstrac t The study of nonlinear normal modes has become a very popular subdiscipline in the structural dynamics community. This is principally due to the fact that they allow for a practical generalization of the concept of spectral invariant manifolds in linear dynamics. There have been a lot of analytical successes achieved in this area through the application of the method of multiple scales which decomposes the response of a nonlinear oscillator into fast-changing dynamics occurring on a slowly varying manifold. Numerical calculations have also been carried out using a periodic ansatz (with resonance-based constraints). Most of such investigations in the past have focused on cases where a single nonlinear mode is studied in isolation. Although this alone provides very interesting results, such as invariant manifold characterization and internal resonance detection, these are usually not sufficient to study the coupling between multiple nonlinear modes. Recent experimental studies have shown that such coupling can lead to very nontrivial trends in the resonant characteristics, making it difficult to correlate computational realizations with experimental measurements. The present paper takes a computational approach frequency-domain numerical simulations conducted using quasi-periodic harmonic balance to obtain numerical insights into the steady-state multi-resonant behavior. These results are compared with signal processing conducted on transient ringdown responses to determine the ramifications of commonly employed signal processing techniques like frequency-domain filtering for mode isolation. Analytical results are also computed using multiple scales for comparisons. Results are presented for a simplified system with two kinds of nonlinearities: geometric and dry friction. Keyword s Nonlinear resonance · Quasi-periodic oscillations · Modal coupling · Method of multiple scales · Nonlinear structural dynamics 15.1 Introduction Resonance of nonlinear structures is an area that has been undergoing increasing interest in the structural dynamics community over the last few decades [1]. Nonlinear normal modes (NNMs) generalize the concept of resonance in linear dynamics to nonlinear systems. Unlike linear systems, in which the resonance is an invariant feature of the dynamics irrespective of the amplitude of response (or excitation), the presence of nonlinearity implies a strong amplitude dependence of the resonance characteristics (resonant frequency, damping, mode shapes, etc.). Consequently, there have been several computational as well as analytical techniques that have been successfully applied for the estimation of these NNMs in terms of amplitude-dependent resonant characterization (see, for instance, [2–4]). Apart from the theoretical interest in the NNMs, the practical interest stems from the fact that, like modal representations in linear vibrations, these NNMs allow for the construction of reduced-order nonlinear modal models. Such models provide a computationally very efficient representation of a potentially much larger system. Stemming from the single nonlinear mode N. N. Balaji ( ) · M. R. W. Brake Department of Mechanical Engineering, Rice University, Houston, TX, USA e-mail: nidish.balaji@ila.uni-stuttgart.de; brake@rice.edu D. Dane Quinn Department of Mechanical Engineering, The University of Akron, Akron, OH, USA e-mail: quinn@uakron.edu M . Krac k Institute of Aircraft Propulsion Systems, University of Stuttgart, Stuttgart, Germany e-mail: malte.krack@ila.uni-stuttgart.de © The Society for Experimental Mechanics, Inc. 2024 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1 , Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-36999-5_15 111
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