Nonlinear vibration absorber optimal design via asymptotic approach

This paper tackles the classical problem of Vibration Absorbers (VAs) operating in the nonlinear dynamic regime. Since traditional linear VAs suffer from the drawback of a narrow bandwith and numerous structures exhibit nonlinear behavior, nonlinear absorbers are of practical interest. The resonant dynamic behavior of a nonlinear hysteretic VA attached to a damped nonlinear structure is investigated analytically via asymptotics and numerically via path following. The response of the reduced-order model, obtained by projecting the dynamics of the primary structure onto the mode to control, is evaluated using the method of multiple scales up to the first nonlinear order beyond the resonance. Here, the asymptotic response of the two-degree-of-freedom system with a 1:1 internal resonance is shown to be in very close agreement with the results of path following analyses. The asymptotic solution lends itself to a versatile optimization based on differential evolutionary

Dynamics of a linear oscillator connected to a small strongly non-linear hysteretic absorber

The present investigation deals with the dynamics of a two-degrees-of-freedom system which consists of a main linear oscillator and a strongly nonlinear absorber with small mass. The nonlinear oscillator has a softening hysteretic characteristic represented by a Bouc-Wen model. The periodic solutions of this system are studied and their calcu- lation is performed through an averaging procedure. The study of nonlinear modes and their stability shows, under specific conditions, the existence of localization which is responsible for a passive irreversible energy transfer from the linear oscillator to the nonlinear one. The dissipative effect of the nonlinearity appears to play an important role in the energy transfer phenomenon and some design criteria can be drawn regarding this parameter among others to optimize this energy transfer. The free transient response is investigated and it is shown that the energy transfer appears when the energy input is sufficient in accordance with the predictions from the nonlinear modes. Finally, the steady-state forced response of the system is investigated. When the input of energy is sufficient, the resonant response (close to nonlinear modes) experiences localization of the vibrations in the nonlinear absorber and jump phenomena

7th International Conference on Nonlinear Vibrations, Localization and Energy Transfer: Extended Abstracts

International audienceThe purpose of our conference is more than ever to promote exchange and discussions between scientists from all around the world about the latest research developments in the area of nonlinear vibrations, with a particular emphasis on the concept of nonlinear normal modes and targeted energytransfer

Design Procedure of a Nonlinear Vibration Absorber Using Bifurcation Analysis

A nonlinear energy sink (NES) is characterized by its ability to passively realize targeted energy transfer as well as multimodal damping. This latter feature seems to make this device very well suited for reducing the vibration level of MDOF linear structures. The perspective of dealing with MDOF linear primary structures requires the development of an efficient NES design procedure. This paper poses the basis of such a procedure based upon the bifurcation analysis of a system composed of a linear oscillator coupled to a NES, using the software MatCont

Dynamical interaction of an elastic system and essentially nonlinear absorber

International audienceThe nonlinear two-degrees-of-freedom system under consideration consists of a linear oscillator with a relatively big mass which is an approximation of some continuous elastic system, and an essentially nonlinear oscillator with a relatively small mass which is an absorber of the main linear system vibrations. Free and forced vibrations of the system are investigated. Analysis of nonlinear normal vibration modes shows that a stable localized vibration mode, which provides the vibration regime appropriate for an absorption, exists in a large region of the system parameters.In this regime amplitudes of vibrations of the main elastic system are small; simultaneously vibrations of the absorber are significant.Frequency response of the system under external periodic force is obtained. The dynamical interaction of elastic string under impact impulse and the essentially nonlinear absorber is considered too. Absorption of a longitudinal traveling wave in the system is analyzed

Exploring the limitations of linear and nonlinear vibration absorbers

Een veel toegepaste manier om overmatige trillingsamplitudes te beperken is het lokaal aanhechten van een licht massa-veer-demper systeem, beter bekend als een dynamische trillingsdemper (dynamic vibration absorber). Aangezien de trillende hoofdstructuur zelf niet gewijzigd moet worden en men typisch met een zeer klein dynamisch element sterke trillingsreducties kan bekomen, wordt dit element zeer veel gebruikt in tal van ingenieurstoepassingen zoals onder meer bij gebouwen (aardbeving,wind), bruggen (verkeer), transportvoertuigen (auto, helikopter) en machines. Sinds de uitvinding hiervan door Frahm meer dan honderd jaar geleden, hebben onnoemelijk veel onderzoekers nieuwe types van trillingsdempers ontwikkeld in de hoop de klassieke dynamische trillingsdemper te verbeteren. Een zeer recent voorbeeld hiervan is het gebruik van sterk niet-lineaire veren die erin slagen om de trillingen in een veel ruimer frequentiegebied te onderdrukken. In dit eindwerk wordt zowel ingegaan op het klassiek lineaire element als op zijn niet-lineaire tegenhanger. Hoewel reeds zeer veel onderzoek is gewijd aan beide elementen, zijn we erin geslaagd om het inzicht in verschillende aspecten sterk te verruimen wat vooral bij het praktisch implementeren van cruciaal belang is

Nonlinear resonance and excitability in interconnected systems

Engineering design amounts to develop components and interconnect them to obtain a desired behaviour. While in the context of equilibrium dynamics there is a well-developed theory that can account for robustness and optimality in this process, we still lack a corresponding methodology for nonequilibrium dynamics and in particular oscillatory behaviours. With the aim of fostering such a theory, this thesis studies two basic interconnections in the contexts of nonlinear resonance and excitability, two phenomena with the potential of encompassing a large number of applications. The first interconnection is considered in the context of vibration absorption. It corresponds to coupling two Duffing oscillators, the prototypical example of nonlinear resonator. Of primary interest is the frequency response of the system, which quantifies the behaviour in presence of harmonic forces. The analysis focuses on how isolated families of solutions appear and merge with a main one. Using singularity theory it is possible to organise these solutions in the space of parameters and delimit their presence through numerical methods. The second interconnection studied in this dissertation appears in the context of excitable circuits. Combining a fast excitable system and a slower oscillatory system that share a similar structure naturally leads to bursting. The resulting system has a slow-fast structure that can be leveraged in the analysis. The first step of this analysis is a novel slow-fast model of bistability between a rest state and a spiking attractor. Following this, the analysis moves to the complete interconnection, and in particular on how it can generate different patterns of bursting activity

Space-time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables

This paper presents space-time numerical simulation and validation of analytical predictions for the finite-amplitude forced dynamics of suspended cables. The main goal is to complement analytical and numerical solutions, accomplishing overall quantitative/qualitative comparisons of nonlinear response characteristics. By relying on an approximate, kinematically non-condensed, planar modeling, a simply supported horizontal cable subject to a primary external resonance and a 1:1, or 1:1 vs. 2:1, internal resonance is analyzed. To obtain analytical solution, a second-order multiple scales approach is applied to a complete eigenfunction-based series of nonlinear ordinary-differential equations of cable damped forced motion. Accounting for both quadratic/cubic geometric nonlinearities and multiple modal contributions, local scenarios of cable uncoupled/coupled responses and associated stability are predicted, based on chosen reduced-order models. As a cross-checking tool, numerical simulation of the associated nonlinear partial-differential equations describing the dynamics of the actual infinite-dimensional system is carried out using a finite difference technique employing a hybrid explicit-implicit integration scheme. Based on system control parameters and initial conditions, cable amplitude, displacement and tension responses are numerically assessed, thoroughly validating the analytically predicted solutions as regards the actual existence, the meaningful role and the predominating internal resonance of coexisting/competing dynamics. Some methodological aspects are noticed, along with a discussion on the kinematically approximate versus exact, as well as planar versus non-planar, cable modeling

Bistable nonlinear damper based on a buckled beam configuration

International audienceThis article addresses a particular realization of a compact bistable nonlinear absorber based on the concept of Nonlinear Energy Sink. The article presents both a detailed description of the absorber mechanics and an illustration of the targeted energy transfer between the absorber and a linear system. The experimental results are accompanied with the numerical simulations. Beside practical improvements linked to the features of absorber design, the obtained results stay in line with those found for simpler realizations of a bistable Nonlinear Energy Sinks

Dynamic modeling and stability analysis of a nonlinear system with primary resonance

In recent years, there has been growing interest in the study of nonlinear phenomena. This is due to the modernization of structures related to the need of using lighter, more resistant and flexible materials. Thus, this work aims to study the behavior of a mechanical system with two degrees of freedom with nonlinear characteristics in primary resonance. The structure consists of the main system connected to a secondary system to act as a Nonlinear Dynamic Vibration Absorber, which partially or fully absorbs the vibrational energy of the system. The numerical solutions of the problem are obtained using the Runge-Kutta methods of the 4th order and approximate analytical solutions are obtained using the Multiple Scales Method. Then, the approximation error between the two solutions is analyzed. Using the aforementioned perturbation method, the responses for the ordinary differential equations of the first order can be determined, which describe the modulation amplitudes and phases. Thus, the solution in steady state and the stability are studied using the frequency response. Furthermore, the behavior of the main system and the absorber are investigated through numerical simulations, such as responses in the time domain, phase planes and Poincaré map; which shows that the system displays periodic, quasi-periodic and chaotic movements. The dynamic behavior of the system is analyzed using the Lyapunov exponent and the bifurcation diagram is presented to better summarize all the possible behaviors as the force amplitude varies. In general, the main characteristics of a dynamic system that experiences the chaotic response will be identified