Normal vibrations of a general class of conservative oscillators

International audienceThis paper considers normal vibrations with curvilinear trajectories in a configuration space of systems which are close to systems permitting rectilinear normal modes of vibration. Analysis of trajectories of normal vibrations in the configuration space is used

Matching of local expansions in the theory of non-linear vibrations

International audienceNormal vibrations in non-linear systems are a generalization of normal (principal) vibrations of linear systems [1–3]. In this case all position coordinates can be defined well from any one of them. R. M. Rosenberg is credited with being the first to introduce broad classes of conservative systems allowing normal vibrations with rectilinear trajectories in a configurational space. In systems of a more general type, trajectories of normal vibrations are curvilinear. Assume that in a conservative system the potential energy is a positively definite polynomial in the coordinates. At small amplitudes a linear system is to be selected as the initial one, while at large amplitudes a homogeneous non-linear system allows normal vibrations with rectilinear trajectories. In the vicinity of a linear system, trajectories of normal vibrations can be determined as power series in the amplitude; while in the vicinity of a homogeneous non-linear system, they can be determined as power series in the inverse amplitude. In order to join together local expansions and to investigate the behavior of normal vibration trajectories at arbitrary amplitude values, fractional rational diagonal Padeápproximants are used. Necessary conditions for the convergence of a succession of Padeápproximants have been obtained, and that allows one to establish relations between quasi-linear and essentially non-linear expansions: that is, to decide which of them correspond to the same solution and which to different ones. Additional modes of vibrations exist only in a non-linear systems; as the amplitude decreases, they vanish at a certain limiting point

Approximate 3-Dimensional Electrical Impedance Imaging

We discuss a new approach to three-dimensional electrical impedance imaging based on a reduction of the information to be demanded from a reconstruction algorithm. Images are obtained from a single measurement by suitably simplifying the geometry of the measuring chamber and by restricting the nature of the object to be imaged and the information required from the image. In particular we seek to establish the existence or non-existence of a single object (or a small number of objects) in a homogeneous background and the location of the former in the (x,y)-plane defined by the measuring electrodes. Given in addition the conductivity of the object rough estimates of its position along the z-axis may be obtained. The approach may have practical applications.Comment: 12 pages, 4 figures, LaTeX, Appendix added and other minor change

Nonlinear normal vibration modes and their applications in some applied problems

International audienceNonlinear normal vibration modes (NNMs) are a generalization of the normal vibrations in the linear systems. In conception of NNMs by Kauderer-Rosenberg all position coordinates can be defined from any one of them. In conception of NNMs by Lyapunov-Shaw-Pierre all phase coordinates can be defined from two selected ones. Curvilinear trajectories of NNMs in a configuration space, or in a phase space, can be obtained as power series. The NNMs theory is used to study vibrations of some linear structure connected with the single-DOF nonlinear absorber. An essentially nonlinear oscillator, a snap-through truss with three equilibrium positions, and a vibro-impact oscillator are chosen as absorbers. Construction and stability analysis of the NNMs are presented. If the localized mode is stable, and the non-localized vibration mode is unstable, the vibration energy is concentrated in the absorber. Free damped oscillations of the double tracked road vehicle with a nonlinear response of the suspension can be considered by the NNMs theory too. The 7-DOF nonlinear model is used to analyze the suspension dynamics with smooth characteristics. The quarter-car model is considered for a case of the non-smooth characteristic of the shock absorber

Review of Applications of Nonlinear Normal Modes for Vibrating Mechanical Systems

International audienceThis paper is an extension of the previous review Nonlinear Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments done by the authors, and it is devoted to applications of nonlinear normal modes (NNMs) theory. NNMs are typical regimes of motions in wide classes of nonlinear mechanical systems. The significance of NNMs for mechanical engineering is determined by several important properties of these motions. Forced resonances motions of nonlinear systems occur close to NNMs. Nonlinear phenomena, such as nonlinear localization and transfer of energy, can be analyzed using NNMs. The NNMs analysis is an important step to study more complicated behavior of nonlinear mechanical systems. This review focuses on applications of Kauderer–Rosenberg and Shaw–Pierre concepts of nonlinear normal modes. The Kauderer–Rosenberg NNMs are applied for analysis of large amplitude dynamics of finite-degree-of-freedom nonlinear mechanical systems. Systems with cyclic symmetry, impact systems, mechanical systems with essentially nonlinear absorbers, and systems with nonlinear vibration isolation are studied using this concept. Applications of the Kauderer–Rosenberg NNMs for discretized structures are also discussed. The Shaw–Pierre NNMs are applied to analyze dynamics of finite-degree-of-freedom mechanical systems, such as floating offshore platforms, rotors, piece-wise linear systems. Studies of the Shaw–Pierre NNMs of beams, plates, and shallow shells are reviewed, too. Applications of Shaw–Pierre and King–Vakakis continuous nonlinear modes for beam structures are considered. Target energy transfer and localization of structures motions in light of NNMs theory are treated. Application of different asymptotic methods for NNMs analysis and NNMs based model reduction are reviewed

Construction of homoclinic and heteroclinic trajectories in mechanical systems with several equilibrium positions

International audienceA new approach for a construction of homo and heteroclinic trajectories of some principal non-linear dynamical systems is utilized here, namely the non-linear Schrodinger equation, non-autonomous Duffing equation and the equation of a parametrically excited damped pendulum are considered. PadeÕ and quasi-PadeÕ approximants and a convergence condition used earlier in the theory of non-linear normal vibration modes made possible to solve a boundary-value problems formulated for the orbits and to determine initial amplitude values of the trajectories with admissible precision. The approach proposed here is more exact than the generally accepted one because it is not necessary to use here separatrix trajectories of the corresponding autonomous equations

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

Snap-through truss as an absorber of forced oscillations

International audienceThe possibility of absorption of linear elastic system forced oscillations by means of an essentially nonlinear absorber (snap-through truss), which is attached to the linear subsystem, is analyzed. The simplest mass–spring linear subsystem is chosen to study the problem of forced oscillations absorption. The combination of the nonlinear normal vibrations modes method, the Rauscher approach and the asymptotic analysis is used to study forced oscillations of the two-dof system. The absorption vibrations mode in the form of nonlinear normal vibrations mode with small oscillations amplitudes of the linear subsystem and large amplitudes of the snap-through truss is studied. It is shown that this mode is stable over the wide range of the excitation frequency

Nonlinear Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments

International audienceTwo principal concepts of nonlinear normal vibrations modes (NNMs), namely the Kauderer–Rosenberg and Shaw–Pierre concepts, are analyzed. Properties of the NNMs and methods of their analysis are presented. NNMs stability and bifurcations are discussed. Combined application of the NNMs and the Rauscher method to analyze forced and parametric vibrations is discussed. Generalization of the NNMs to continuous systems dynamics is also described

An application of the ince algebraization to the stability of non-linear normal vibration modes

International audienceA normal vibration mode stability in conservative non-linear systems is investigated. The algebraization by !nee (transition from linear equations with periodic coefficients to equations with singular points) is used. The normal mode stability in homogeneous systems, whose potential is an even homogeneous function of the variables and systems close to the homogeneous one, is investigated. Eigenvalues and eigenfunctions are obtained. Conditions when a number of instability zones in a non-linear system parameters space are finite (finite zoning or finite-gap conditions) are also obtained