Abstract. We study numerically and analytically the dynamics of passive energy transfer from a damped linear oscillator to an essentially nonlinear end attachment. This transfer is caused by either fundamental or subharmonic resonance capture, and in some cases is initiated by nonlinear beat phenomena. It is shown that, due to the essential nonlinearity, the end attachment is capable of passively absorbing broadband energy at both high and low frequencies, acting, in essence, as a passive broadband boundary controller. Complicated transitions in the damped dynamics can be interpreted based on the topological structure and bifurcations of the periodic solutions of the underlying undamped system. Moreover, complex resonance capture cascades are numerically encountered when we increase the number of degrees of freedom of the system. The ungrounded essentially nonlinear end attachment discussed in this work can find application in numerous practical settings, including vibration and shock isolation of structures, seismic isolation, flutter suppression, and packaging. 1. Introduction. We study passive and irreversible energy transfer from a linear oscillator to an essentially nonlinear attachment, which, in essence, acts as a nonlinear energy sink (NES); such energy transfer we refer to as nonlinear energy pumping. In previous works (Vakakis and Gendelman (2001), Vakakis et al. (2003)) grounded and relatively heavy nonlinear attachments were considered, a feature that limits their attractiveness in practical applications. To eliminate these restrictions, an ungrounded and light nonlinear attachment is considered in this work, which, in addition, possesses the feature of modularity. As shown in Lee et al. (2005), even though the system considered has a simple configuration, it possesses a very complicated structure of undamped periodic orbits, which, in turn, give rise to a complicated
We study passive and nonlinear targeted energy transfers induced by resonant interactions between a singledegree-of-freedom nonlinear energy sink (NES) and a 2-DOF in-flow rigid wing model. We show that it is feasible to partially or even completely suppress aeroelastic instability by passively transferring vibration energy from the wing to the NES in a one-way irreversible fashion. Moreover, this instability suppression is performed by partially or completely eliminating its triggering mechanism. Numerical parametric studies identify three main mechanisms for suppressing aeroelastic instability: recurring burstout and suppression, intermediate suppression, and complete elimination. We investigate these mechanisms both numerically by the Hilbert-Huang transform and analytically by a complexification-averaging technique. Each suppression mechanism involves strong 1:1 resonance capture during which the NES absorbs and dissipates a significant portion of energy fed from the flow to the wing. Failure of suppression is associated with restoring the underlying triggering mechanism of instability, which is a series of superharmonic resonance captures followed by escapes from resonance. Finally, using a numerical continuation technique, we perform a bifurcation analysis to examine sensitive dependence on initial conditions and thus robustness of instability suppression. Nomenclature b = semichord length, c=2, where c is a chord length C = nondimensional coefficient for essentially nonlinear coupling stiffness, b 2 k s =m! 2 C L; = lift curve slope, @C L =@j 0 c 1 , c 2 = nonlinear heave and pitch stiffness factors d; = offset attachment of the NES to the wing, measured from and positive ahead of the elastic axis (e.a.); nondimensional offset, d=b e; = location of the aerodynamic center (a.c.) measured from the e.a. (positive ahead of e.a.); its nondimensional parameter, e=b h, , z = heave (positive downward), pitch (positive clockwise), NES (positive downward) degrees of freedom K h , K = coefficients of linear heave and pitch stiffnesses L, M = lift and aerodynamic moment acting at the a.c., respectively; the equivalent aerodynamic forces at the e.a. are L ea L and M ea M eL eL for small angles m, I = mass of the airfoil and its mass moment of inertia with respect to the e.a.m s , k s , c s = mass, essentially nonlinear stiffness, and damping in the NES q = dynamic pressure, 1 2 1 U 2 , where 1 is the density of the flow r = radius of gyration of the cross section of the wing, I =mb 2 p S = planform area of the wing S ; x = mass unbalance in the airfoil, mx cg ; its nondimensional parameter, S =mb, x cg =b t,= physical and nondimensional times ( ! t) U = constant and uniform flow speed around the wing x cg = location of the center of gravity (c.g.) measured from the e.a. (positive aft of the e.a.) y, v = nondimensional heave (y h=b) and NES (v z=b) modes y , = steady-state amplitude ratio in the heave and pitch modes = mass ratio between the NES and the wing, m s =m = reduced speed of the flow, U=b! = nondimensional lin...
In the field of seismic protection of structures, it is crucial to be able to diminish 'as much as possible' and dissipate 'as fast as possible' the load induced by seismic (vibration-shock) energy imparted to a structure by an earthquake. In this context, the concept of passive nonlinear energy pumping appears to be natural for application to seismic mitigation. Hence, the overall problem discussed in this paper can be formulated as follows: Design a set of nonlinear energy sinks (NESs) that are locally attached to a main structure, with the purpose of passively absorbing a significant part of the applied seismic energy, locally confining it and then dissipating it in the smallest possible time. Alternatively, the overall goal will be to demonstrate that it is feasible to passively divert the applied seismic energy from the main structure (to be protected) to a set of preferential nonlinear substructures (the set of NESs), where this energy is locally dissipated at a time scale fast enough to be of practical use for seismic mitigation. It is the aim of this work to show that the concept of nonlinear energy pumping is feasible for seismic mitigation. We consider a two degree-of-freedom (DOF) primary linear system (the structure to be protected) and study seismic-induced vibration control through the use of Vibro-Impact NESs (VI NESs). Also, we account for the possibility of attaching to the primary structure additional alternative NES configurations possessing essential but smooth nonlinearities (e.g., with no discontinuities). We study the performance of the NESs through a set of evaluation criteria. The damped nonlinear transitions that occur during the operation of the VI NESs are then studied by superimposing wavelet spectra of the nonlinear responses to appropriately defined frequency -energy plots (FEPs) of branches of periodic orbits of underlying Conservative systems.
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