By Alexander F. Vakakis

The papers during this quantity deal with complicated nonlinear issues within the normal components of vibration mitigation and approach identity, comparable to, equipment of research of strongly nonlinear dyanmical structures; innovations and methodologies for reading complicated, multi-frequency transitions in damped nonlinear responses; new methods for passive vibration mitigation in line with nonlinear special power move (TET) and the linked proposal of nonlinear strength sink (NES); and an summary and review of present nonlinear approach id thoughts.

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2008; Starosvetsky and Gendelman, 2008). This system is described by the following set of equations, y1 HO ( y1 y2 ) (1 HV ) y1 (4H / 3)( y1 y2 )3 H A cos t H y2 HO ( y2 y1 ) (4H / 3)( y2 y1 )3 0 (1) 54 Oleg V. Gendelman and Y. Starosvetsky where y1 and y2 are the displacements of the linear oscillator and the attachment respectively, HO the damping coefficient, H A the amplitude of the external force, and HV a frequency detuning parameter. The parameter H 1 is the small parameter of the problem which scales the coupling between the two oscillators, the damping forces, the amplitude of the external force, the detuning parameter, and the mass of the NES.

The coefficients A, O , V are adopted to be of O(1) . The coefficient of the nonlinear term can be modified by proper rescaling of the dependent variables and the forcing amplitude; the value (4H / 3) is chosen for the sake of convenience. We will analyze the steady state periodic responses of (1) by the complexification – averaging (CX-A) methodology. To this end, we apply the following coordinate transformations, denoting the center-of-mass and relative displacements of the system, v y1 H y2 w y1 y2 (2) and then switch the analysis to complex variables: M1 exp( jt ) v jv M2 exp( jt ) w jw (3) By (3) we partition the dynamics into slow and fast components, and make the additional ansatz that the sought steady state responses are in the form of fast oscillations exp( jt ) modulated by slowly-varying complex amplitudes Mi (t ) .

Gendelman and Y. Starosvetsky where y1 and y2 are the displacements of the linear oscillator and the attachment respectively, HO the damping coefficient, H A the amplitude of the external force, and HV a frequency detuning parameter. The parameter H 1 is the small parameter of the problem which scales the coupling between the two oscillators, the damping forces, the amplitude of the external force, the detuning parameter, and the mass of the NES. The coefficients A, O , V are adopted to be of O(1) .