Backbone curves of coupled cubic oscillators in one-to-one internal resonance: bifurcation scenario, measurements and pa
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Backbone curves of coupled cubic oscillators in one-to-one internal resonance: bifurcation scenario, measurements and parameter identification Arthur Givois
. Jin-Jack Tan
. Cyril Touze´
. Olivier Thomas
Received: 7 August 2019 / Accepted: 31 January 2020 Ó Springer Nature B.V. 2020
Abstract A system composed of two cubic nonlinear oscillators with close natural frequencies, and thus displaying a 1:1 internal resonance, is studied both theoretically and experimentally, with a special emphasis on the free oscillations and the backbone curves. The instability regions of uncoupled solutions are derived and the bifurcation scenario as a function of the parameters of the problem is established, showing in an exhaustive manner all possible solutions. The backbone curves are then experimentally measured on a circular plate, where the asymmetric modes are known to display companion configurations with close eigenfrequencies. A control system based on a Phase-Locked Loop (PLL) is used to measure the backbone curves and also the frequency response
A. Givois (&) O. Thomas Arts et Metiers Institute of Technology, LISPEN, HESAM Universite´, F-59000 Lille, France e-mail: [email protected] A. Givois Conservatoire National des Arts et Me´tiers, LMSSC EA 3196, 292 rue Saint-Martin, 75141 Paris Cedex 03, France J.-J. Tan C. Touze´ IMSIA, CNRS, CEA, ENSTA Paris, Institut Polytechnique de Paris, 828 Boulevard des Mare´chaux, 91762 Palaiseau Cedex, France J.-J. Tan PO Box 513, 5600 MB, Eindhoven, The Netherlands
function in the forced and damped case, including unstable branches. The model is used for a complete identification of the unknown parameters and an excellent comparison is drawn out between theoretical prediction and measurements. Keywords Nonlinear vibrations Backbone curve Bifurcations 1:1 Resonance Stability Measurements Model identification
1 Introduction Nonlinear system displaying internal resonance has been the subject of a number of studies as a strong nonlinear coupling could lead to solutions that are completely different from linear predictions [4, 12, 24, 25, 27, 40, 42]. Internal resonance is closely related to the normal form theory and Poincare´’s theorem where the specific resonance relationship between eigenfrequencies is linked to a resonant monom that cannot be cancelled through a nearidentity transform [15, 31, 44, 46]. In vibration theory, these systems are usually denoted in series of numbers, e.g. 1:2 and 1:1:2, which refers to the relationship between the eigenfrequencies of the system. For instance, a 1:2 resonant system has eigenfrequencies related by x2 ’ 2x1 while a 1:1:2 system exhibits x2 ’ x1 and x3 ’ 2x1 .
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Meccanica
Among all possible internal resonances, 1:1 resonance is described by two oscillators having close eigenfrequencies and may appear as the simplest one and the first to be studied. It occurs in numerous mechanical systems having known symmetries such as strings, where the two polarizations of a same m
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