Numerical and Experimental Study of Nonlinear Localization in a Flexible Structure with Vibro‐Impacts

Abstract

In this work a numerical and experimental study of nonlinear motion confinement phenomena in a nonlinear flexible assembly with vibro‐impacts is carried out. The assembly consists of two coupled cantilever beams whose motion is constrained by rigid barriers. In the theoretical model the impact nonlinearities are simulated by clearance nonlinearities with steep stiffness characteristics; special care is taken to model energy dissipation due to inelastic impacts. The theoretical results confirm that the vibro‐impact system possess motion confinement properties. Both transient and steady state motions are studied, and it is shown that under certain conditions the vibrational energy of the system is passively confined to only one of the two beams. These essentially nonlinear responses exist inspite of direct coupling between the two beams, and have no analogs in linear theory. The experimental results confirm the theoretical predictions both in the transient and steady state regimes. The passive nonlinear motion confinement phenomena reported in this and previous works can be utilized to enhance the controllability of flexible structures with symmetries, and to improve current shock and vibration isolation designs of such systems.

Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures

“…Axially vibrating beams with or without distributed elastic constraints have been considered in [22,23] by following a stochastic or deterministic approach, respectively. Nonlinear problems have been studied only recently [24,25]; in particular the existence of localized nonlinear normal modes has been proved in [26][27][28]. In [26] it has been observed that the origin of nonlinear mode localization in lumped-mass periodic systems is the dependence of the frequencies of substructures on their vibration amplitude; thus nonlinearities cause mistuning and localization takes place even in perfectly periodic systems.…”

Proper Orthogonal Decomposition (Pod) of a Class of Vibroimpact Oscillations

“…The parameters of the system are shown in Table 1. These parameters are the same as those of the beam used to experimentally study localization in a coupled system in a previous work [18]. The #exural rigidity is EI, the linear mass density m and the length is¸.…”

“…It is advisable to employ double-precision computations throughout to reduce the numerical inaccuracies as much as possible. The number of K}L modes used (p) should be in accordance with equation (18). For the want of enhanced accuracy, the higher K}L modes should not be carelessly used as some of them are spurious and would lead to numerical instabilities.…”

“…The #exural rigidity is EI, the linear mass density m and the length is¸. The natural frequencies ( G ) and the modal damping values (d G ), of the cantilever beam without the non-linearity, were obtained experimentally, in reference [18]. The damping values (d G ), given in Table 1, will be used for all the simulations.…”

On measures of nonlinearity effects for uncertain dynamical systems—Application to a vibro-impact system

“…(4)- (6). All the random variables are defined in a probability space (Θ, F , P) (A) Parametric probabilistic model of data uncertainties for the non-linear term.…”