Nonlinear structural vibrations by the linear acceleration method

Numerical integration method for calculating dynamic response of nonlinear elastic structure

Tuning of the Active Hair Bundle

The organs of the inner ear rely upon a population of several thousand sensory hair cells to amplify and transduce acoustic, seismic, and kinesthetic signals. Each hair cell detects mechanical disturbances by means of its hair bundle, a motile organelle consisting of actin-filled, villous projections (called stereocilia) endowed with assemblies (called adaptation motors) of mechano-sensitive ion channels and myosin molecules that power both spontaneous and evoked movements. Active hair-bundle motility serves two functions: it mechanically amplifies sensory stimuli; and it regulates their transduction into electrical signals that drive the hair-cell synapse. To characterize these two functions, we consider here a model of the mechanical and electrical dynamics of the hair bundle of the bullfrog sacculus. Under simplifying assumptions, we reduce this model of a muscle fiber, and outline a procedure for estimating its parameters from experiment. We delineate the bifurcation structure of this simplified model, and analyse by perturbation methods its behavior in various dynamical regimes, notably in the relaxation-oscillation regime that displays prominently the hair bundle\u27s active process; and in the near-Hopf-bifurcation regime at which auditory hair cells are thought to operate in vivo. We find close similarities between the dynamics of the active hair bundle and those of simplified models of a spiking neuron. In light of this analysis, we offer an account of the biophysical mechanisms underlying the spontaneous oscillations, frequency specificity, nonlinear gain, and self-tuning predicted for auditory hair bundles poised near a Hopf bifurcation

ANALYSIS OF THE NONLINEAR VIBRATIONS OF ELECTROSTATICALLY ACTUATED MICRO-CANTILEVERS IN HARMONIC DETECTION OF RESONANCE (HDR)

Micro- and nano-cantilevers have the potential to revolutionize physical, chemical, and biological sensing; however, an accurate and scalable detection method is required. In this work, a fully electrical actuation and detection method is presented, known as the Harmonic Detection of Resonance (HDR). In HDR, harmonic components of the current are measured to determine the cantilever\u27s resonance frequency. These harmonics exist as a result of nonlinearities in the system, principally in the electrostatic actuation force. In order to better understand this rich harmonic structure, a theoretical investigation of the micro-cantilever is undertaken. Both a lumped parameter model and a more accurate continuum model are used to derive the governing nonlinear modal equations of motion (EOM) of the cantilever. Various approximate solution methods applicable to nonlinear equations are then discussed including numerical integration, perturbation, and averaging. An averaging method known as the method of harmonic balance is then used to obtain steady state solutions to the micro-cantilever EOM. Low-order closed-form harmonic balance solutions are derived which explain many of the important features of the HDR results, such as the presence of parasitic capacitance in the first harmonic and super-harmonic resonance peaks in higher harmonics. Finally, higher-order computer generated harmonic balance solutions are presented which show good agreement with the experimental HDR results, validating both the modeling and the solution methods used

Approximate Periodic Solutions for Oscillatory Phenomena Modelled by Nonlinear Differential Equations

We apply the Fourier-least squares method (FLSM) which allows us to find approximate periodic solutions for a very general class of nonlinear differential equations modelling oscillatory phenomena. We illustrate the accuracy of the method by using several significant examples of nonlinear problems including the cubic Duffing oscillator, the Van der Pol oscillator, and the Jerk equations. The results are compared to those obtained by other methods

Mixed Boundary Value Problem for Nonlinear Fractional Volterra Integral Equation

In this paper we present the existence of solutions for a nonlinear fractional integral equation of Volterra type with mixed boundary conditions, some necessary hypotheses have been developed to prove the existence of solutions to the proposed equation. Krasnoselskii Theorem, Banach Contraction principle, and Leray-Schauder degree theory are the basic theorems used here to find the results. A simple example of the application of the main result is presented

Bifurcations, Chaos, Controlling and Synchronization of Certain Nonlinear Oscillators

In this set of lectures, we review briefly some of the recent developments in the study of the chaotic dynamics of nonlinear oscillators, particularly of damped and driven type. By taking a representative set of examples such as the Duffing, Bonhoeffer-van der Pol and MLC circuit oscillators, we briefly explain the various bifurcations and chaos phenomena associated with these systems. We use numerical and analytical as well as analogue simulation methods to study these systems. Then we point out how controlling of chaotic motions can be effected by algorithmic procedures requiring minimal perturbations. Finally we briefly discuss how synchronization of identically evolving chaotic systems can be achieved and how they can be used in secure communications.Comment: 31 pages (24 figures) LaTeX. To appear Springer Lecture Notes in Physics Please Lakshmanan for figures (e-mail: [email protected]

A New Method to Predict Vessel Capsizing in a Realistic Seaway

A recently developed approach, in the area of nonlinear oscillations, is used to analyze the single degree of freedom equation of motion of a oating unit (such as a ship) about a critical axis (such as roll). This method makes use of a closed form analytic solution, exact upto the rst order, and takes into account the the complete unperturbed (no damping or forcing) dynamics. Using this method very-large-amplitude nonlinear vessel motion in a random seaway can be analysed with techniques similar to those used to analyse nonlinear vessel motions in a regular (periodic) or random seaway. The practical result being that dynamic capsizing studies can be undertaken considering the shortterm irregularity of the design seaway. The capsize risk associated with operation in a given sea state can be evaluated during the design stage or when an operating area change is being considered. Moreover, this technique can also be used to guide physical model tests or computer simulation studies to focus on critical vessel and environmental conditions which may result in dangerously large motion amplitudes. Extensive comparitive results are included to demonstrate the practical usefulness of this approach. The results are in the form of solution orbits which lie in the stable or unstable manifolds and are then projected onto the phase plane

Lagrangian Descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems

In this paper we develop new techniques for revealing geometrical structures in phase space that are valid for aperiodically time dependent dynamical systems, which we refer to as Lagrangian descriptors. These quantities are based on the integration, for a finite time, along trajectories of an intrinsic bounded, positive geometrical and/or physical property of the trajectory itself. We discuss a general methodology for constructing Lagrangian descriptors, and we discuss a "heuristic argument" that explains why this method is successful for revealing geometrical structures in the phase space of a dynamical system. We support this argument by explicit calculations on a benchmark problem having a hyperbolic fixed point with stable and unstable manifolds that are known analytically. Several other benchmark examples are considered that allow us the assess the performance of Lagrangian descriptors in revealing invariant tori and regions of shear. Throughout the paper "side-by-side" comparisons of the performance of Lagrangian descriptors with both finite time Lyapunov exponents (FTLEs) and finite time averages of certain components of the vector field ("time averages") are carried out and discussed. In all cases Lagrangian descriptors are shown to be both more accurate and computationally efficient than these methods. We also perform computations for an explicitly three dimensional, aperiodically time-dependent vector field and an aperiodically time dependent vector field defined as a data set. Comparisons with FTLEs and time averages for these examples are also carried out, with similar conclusions as for the benchmark examples.Comment: 52 pages, 25 figure

Accurate approximations for nonlinear vibrations

As global issues such as climate change and overpopulation continue to grow, the role of the engineer is forced to adapt. The general population now places an emphasis not only on the performance of a mechanical system, but also the efficiency with which this can be achieved. In pushing these structures to their maximum efficiency, a number of design and engineering challenges can arise. In particular, the occurrence of geometric nonlinearities can lead to failures in the linear modelling techniques that have traditionally been used. The aim of this thesis is to increase the understanding of a number of widely-used, nonlinear methods, so that they may eventually be used with the same ease and confidence as traditional linear techniques. A key theme throughout this work is the notion that nonlinear behaviour is typically approximated in some way, rather than finding exact solutions. This is not to say that exact solutions cannot be found, but rather that the process of doing so, or the solutions themselves, can be prohibitively complicated. Across the techniques considered, there is a desire to accurately predict the frequency-amplitude relationship, whether this be for the free or forced response of the system. Analytical techniques can be used to produce insight that may be inaccessible through the use of numerical methods, though they require assumptions to be made about the structure. In this thesis, a number of these methods are compared in terms of their accuracy and their usability, so that the influence of the aforementioned assumptions can be understood. Frequency tuning is then used to bring the solutions from three prominent methods in line with one another. The Galerkin method is used to project a continuous beam model into a discrete set of modal equations, as is the traditional method for treating such a system. Motivated by microscale beam structures, an updated approach for incorporating nonlinear boundary conditions is developed. This methodology is then applied to two example structures to demonstrate the importance of this procedure in developing accurate solutions. The discussion is expanded to consider non-intrusive reduced-order modelling techniques, which are typically applied to systems developed with commercial finite element software. By instead applying these methods to an analytical nonlinear system, it is possible to compare the approximated results with exact analytical solutions. This allows a number of observations to be made regarding their application to real structures, noting a number of situations in which the static cases applied or the software itself may influence the solution accuracy

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