Extended complexification method to study nonlinear passive control

The present research work aims to design a passive vibration control based on nonlinear energy pumping. An extended asymptotic approach is introduced based on the invariant manifold approach for the case of 1:1 resonance. It consists in introducing an extended form of Manevitch's complex variables, taking into consideration higher harmonics, enabling the detection of the invariant manifold of the system at fast timescale. At the slow timescale, equilibrium points and singularities are identified analytically in order to predict periodic regimes and strongly modulated responses. The example of a passive shunt loudspeaker using a nonlinear absorber is studied. Unlike classical investigations, the first and third harmonics are taken into consideration. It is demonstrated that the presence of the third harmonic improves the approximations of the results. Different cases are considered, where the obtained analytical results are in good agreement with those obtained via direct numerical integration of the principal system of equations.