By Alexander F. Vakakis
The papers during this quantity tackle complex nonlinear themes within the common parts of vibration mitigation and approach id, akin to, tools of research of strongly nonlinear dyanmical platforms; options and methodologies for reading complicated, multi-frequency transitions in damped nonlinear responses; new ways for passive vibration mitigation in line with nonlinear specified strength move (TET) and the linked idea of nonlinear strength sink (NES); and an summary and review of present nonlinear process identity ideas.
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Additional resources for Advanced Nonlinear Strategies for Vibration Mitigation and System Identification
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 ) . Moreover, it is clear that we seek periodic solutions of (1) with dominant frequencies identical to the frequency of the external periodic force, and approximately equal to the eigenfrequency of the linear oscillator (that is, the frequency detuning HV provides a slight frequency mismatch).
At points of bifurcation, both the polynomial (7) and its derivative with respect to Z should be equal to zero: 3D 3 Z 2 2D 2 Z D1 0 (9) It follows that to compute the bifurcation points we need to satisfy simultaneously the set of equations (7) and (9); this yields the bifurcation curve in parameter space ( A, O , V ) where SN bifurcations occur.
Hence, we will be interested in fundamental nonlinear resonances of system (1). Targeted Energy Transfer in Systems with Periodic Excitations 55 After substitution of (2) and (3) into (1) and subsequent averaging aver the fast oscillations of frequency unity, we obtain the following slow–flow (complex modulation) equations: M1 jHV M1 HM2 H A jH M1 M2 2(1 H ) 2(1 H ) 2 M2 O(1 H ) M2 2 jHV M1 HM2 j(1 H ) 2 HA j M2 M2 M2 M1 2(1 H ) 2(1 H ) 2 2 (4) The system of equations (4) has a complicated structure and cannot be solved analytically.
Advanced Nonlinear Strategies for Vibration Mitigation and System Identification by Alexander F. Vakakis