Enhancing the dynamic range of targeted energy transfer in acoustics using several nonlinear membrane absorbers

In order to enhance the robustness and the energy range of efficiency of targeted energy transfer (TET) phenomena in acoustics, we discuss in this paper about the use of multiple nonlinear membrane absorbers (called nonlinear energy sinks or NES) placed in parallel. We show this way, mainly thanks to an experimental set-up with two membranes, that the different absorbers have additional effects that extend the efficiency and the possibilities of observation of TET. More precisely, we present the different behavior of the system under sinusoidal forcing and free oscillations, characterizing the phenomena for all input energies. The frequency responses are also presented, showing successive clipping of the original resonance peak of the system, and strongly modulated regimes (SMR). A model is finally used to generalize these results to more than two NES and to simulate the case of several very similar membranes in parallel which shows how to extend the existence zone of TET.     

Experimental evidence of energy pumping in acousticsMise en évidence expérimentale de pompage énergétique en acoustique

This Note presents an experimental vibro-acoustic set-up that aims to reproduce the energy pumping phenomenon between an acoustic medium and an essentially nonlinear oscillator. It shows a one-way irreversible transfer of energy between the first acoustic mode in a tube and a thin visco-elastic membrane. To cite this article: B. Cochelin et al., C. R. Mecanique 334 (2006).     

Backbone curves,Neimark-Sacker boundaries and appearance of quasi-periodicity in nonlinear oscillators: application to 1:2 internal resonance and frequency combs in MEMS

Meccanica - Quasi-periodic solutions can arise in assemblies of nonlinear oscillators as a consequence of Neimark-Sacker bifurcations. In this work, the appearance of Neimark-Sacker bifurcations is...     

A parallel computer implementation of the Asymptotic Numerical Method to study thermal convection instabilities

We have developed a numerical model to efficiently compute steady-state combined buoyancy and thermocapillary convection solutions. It features a parallel computer implementation of an Asymptotic Numerical Method to perform steady-state path-following and locate bifurcation points in problems involving large size algebraic systems, up to few million degrees of freedom. The model has been first validated on a problem for which a reference solution exists and then is used to analyse the influence of the container size and shape on Rayleigh–Bénard–Marangoni convection and its related cellular pattern.     

Assessment of the harmonic balance method on a self-oscillating one-degree-of-freedom system with regularized friction

Time-periodic solutions of dynamical systems can be looked for using a discretization method. This paper tests the harmonic balance method (HBM) on a one-degree-of-freedom system (mass, damper, spring, belt) with a regularized friction law. Its relative error is computed with respect to the number of discretization unknowns. Despite the widespread idea that frequency methods are hardly applicable to friction problems, the HBM compares well with a classical time-domain method for this nonlinear system. The main conclusion of this article is that the HBM, without any specific optimization, is well suited for regularized friction.     

A comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems

The high-order purely frequency-based harmonic balance method (HBM) presented by Cochelin and Vergez (2009) [1] and extended by Karkar et al. (2013) [2] now allows to follow the periodic solutions of regularized non-smooth systems (stiff systems). This paper compares its convergence property to a reference method in applied mathematics: orthogonal collocation with piecewise polynomials. A first test is conducted on a nonlinear smooth 2 degree-of-freedom spring mass system, showing better convergence of the HBM. The second test is conducted on a one degree-of-freedom vibro-impact system with a very stiff regularization of the impact law. The HBM continuation of the nonlinear mode was found to be very robust, even with a very large number of harmonics. Surprisingly, the HBM was found to have a better convergence than the collocation method for this vibro-impact system.     

A planar rod model with flexible thin-walled cross-sections. Application to the folding of tape springs

This paper is focused on the modeling of rod-like elastic bodies that have an initially curved and thin-walled cross-section and that undergo important localized changes of the cross-section shape. The typical example is the folding of a carpenter’s tape measure for which the folds are caused by the flattening of the cross-section in some localized areas. In this context, we propose a planar rod model that accounts for large displacements and large rotations in dynamics. Starting from a classical shell model, the main additional assumption consists in introducing an elastica kinematics to describe the large changes of the cross-section shape with very few parameters. The expressions of the strain and kinetic energies are derived by performing an analytical integration over the section. The Hamilton principle is directly introduced in a suitable finite element software to solve the problem. The folding, coiling and deployment of a tape spring is studied to demonstrate the ability of the model to account for several phenomena: creation of a single fold and associated snap-through behavior, splitting of a fold into two, motion of a fold along the tape during a dynamic deployment, scenarios of coiling and uncoiling of a bistable tape spring. This 1D model may also be relevant for future applications in biomechanics, biophysics and nanomechanics.     

Analysis of Laminated Rubber Bearings with a Numerical Reduction Model Method

In this paper, we present a reduction method for modeling slender laminated elastomeric structures, which is developed in the context of nearly incompressible hyperelasticity. This method, based on a finite element formulation, consists in projecting the unknown fields onto a polynomial basis in order to reduce the dimension of the problem and the model size. Two types of finite elements are used, one for plane-strain and the other for 3D structures. Comparisons with classical finite element models on single layers show the reliability of the present method. The method proposed successfully predicts the global and local behavior and since it reduces both the model size and the computing time. It can be used to model slender bearing consisting of several layers.     

Modelling Pressurized Water Reactor cores in terms of porous media

The aim of this study is to develop a tractable model of a nuclear reactor core taking the complexity of the structure (including its nonlinear behaviour) and fluid flow coupling into account. The mechanical behaviour modelling includes the dynamics of both the fuel assemblies and the fluid. Each rod bundle is modelled in the form of a deformable porous medium; then, the velocity field of the fluid and the displacement field of the structure are defined over the whole domain. The fluid and the structure are first modelled separately, before being linked together. The equations of motion for the structure are obtained using a Lagrangian approach and, to be able to link up the fluid and the structure, the equations of motion for the fluid are obtained using an arbitrary Lagragian Eulerian approach. The finite element method is applied to spatially discretize the equations. Simulations are performed to analyse the effects of the characteristics of the fluid and of the structure. Finally, the model is validated with a test involving two fuel assemblies, showing good agreement with the experimental data.     

High order harmonic balance formulation of free and encapsulated microbubbles

The radial responses of free and encapsulated microbubbles excited by an ultrasonic plane wave with a large wavelength in comparison with the bubble size are governed by NonLinear Ordinary Differential Equations (NL-ODEs). The nonlinear frequency response gives the harmonic content of the time response and constitutes the expected outcome of a high order harmonic analysis. In this paper, high order harmonic balance analysis of modified “RPNNP” (bubble), Hoff and Marmottant (contrast agents) models is performed with an open-source software program. For this purpose, the original NL-ODEs are recast into nonlinear systems in which the nonlinearities are at most quadratic. In the spectral domain, this recast provides close form and aliasing-free solutions of arbitrarily large numbers of harmonics. Relevant quantities such as primary and secondary resonances and the nonlinear amplitude threshold of the excitation wave are evaluated. The frequency curves drawn up characterize the bending and quantify the jump frequencies and amplitudes of each harmonic component. The results obtained with this predictive method confirm that it should provide a useful tool for nonlinear bubble detection and sizing and for contrast agent designing.