Tailoring the resonances of nonlinear mechanical systems
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ORIGINAL PAPER
Tailoring the resonances of nonlinear mechanical systems Thibaut Detroux Gaetan Kerschen
· Jean-Philippe Noël ·
Received: 5 March 2020 / Accepted: 5 October 2020 © Springer Nature B.V. 2020
Abstract The objective of this paper is the realization of a desired frequency–amplitude dependence of a specific vibration mode of a nonlinear system. To this end, we introduce an additional nonlinearity with an a priori arbitrary mathematical form in the considered system and synthesize its restoring force through optimization. To greatly facilitate the optimization process, the restoring force is optimized piece by piece from low to large motion amplitudes. The suppression of a modal interaction and the enforcement of isochronicity are considered to demonstrate the proposed procedure. Keywords Modal design · Nonlinear resonance · Nonlinear normal mode · Nonlinearity synthesis · Restoring force 1 Introduction The 20th century witnessed extraordinary advances in nonlinear systems theory. Today, engineers in academia and research centers are exploiting this theory to Thibaut Detroux (B) · Gaetan Kerschen Space Structures and Systems Laboratory, Department of Aerospace and Mechanical Engineering, University of Liège, Quartier Polytech 1, Allée de la Découverte 9 (B52/3), 4000 Liège, Belgium e-mail: [email protected] Jean-Philippe Noël Control Systems Technology Group, Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, Netherlands
develop modeling, identification, simulation and control techniques. Even though important progress is yet to be achieved toward more effective and robust nonlinear methods, the next great challenge is to provide practitioners with techniques to design with and for nonlinearity. This is the challenge the present paper attempts to address. Since the 2000s, the benefits of nonlinearity have been more and more systematically explored in engineering. Among a variety of potential examples, the nonlinear implementation of signal processing operations is worthy of mention. In [1], tunable rectification was achieved using a granular crystal with bifurcating dynamics leading to quasiperiodic and chaotic states. By coupling nonlinear modes through internal resonances, Ref. [2] proposed a frequency stabilization mechanism for micromechanical resonators. In [3], a chain of nonlinear resonators involving a cascade of parametric resonances was also used to passively divide frequencies. Some further examples of advanced nonlinear design studies are to be found in the literature about wave propagation in smart materials. As an example, in [4], an array of spherical particles realized a nonlinear lens to transform an incident sound wave into a compact pulse of high acoustic energy, with applications in biomedical imaging and damage detection. In the field of mechanical vibrations, intentionally utilizing nonlinearity has similarly gathered attention over the past few years. For instance, harvesting energy from ambient vibrations is a typical subject of research
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