Nonlinear Complex Modes Research Articles

Precession and nutation dynamics of nonlinearly coupled non-coaxial three-dimensional matter wave vortices.

Nonlinearity is the driving force for numerous important effects in nature typically showing transitions between different regimes, regular, chaotic or catastrophic behavior. Localized nonlinear modes have been the focus of intense research in areas such as fluid and gas dynamics, photonics, atomic and solid state physics etc. Due to the richness of the behavior of nonlinear systems and due to the severe numerical demands of accurate three-dimensional (3D) numerical simulations presently only little knowledge is available on the dynamics of complex nonlinear modes in 3D. Here, we investigate the dynamics of 3D non-coaxial matter wave vortices that are trapped in a parabolic potential and interact via a repulsive nonlinearity. Our numerical simulations demonstrate the existence of an unexpected and fascinating nonlinear regime that starts immediately when the nonlinearity is switched-on and is characterized by a smooth dynamics representing torque-free precession with nutations. The reported motion is proven to be robust regarding various effects such as the number of particles, dissipation and trap deformations and thus should be observable in suitably designed experiments. Since our theoretical approach, i.e., coupled nonlinear Schrödinger equations, is quite generic, we expect that the obtained novel dynamical behavior should also exist in other nonlinear systems.

Nonlinear Modal Analysis of Mistuned Periodic Structures Subjected to Dry Friction

This paper deals with the dynamics of a cyclic system, representative of a bladed disk subjected to dry friction forces, and exhibits structural mistuning. The nonlinear complex modes are computed by solving the eigenproblem associated to the free response of the whole structure and are then used to better understand the forced response to a traveling wave excitation. Similarly to the underlying linear system, the tuned model possesses pairs of modes that can be linearly combined to form traveling waves, unlike those of the mistuned structure. However, due to the nonlinearity, the modal properties are not constant but vary with the vibration amplitude in both cases. A qualitative analysis is also performed to assess the impact of the mistuning magnitude on the response and suggests that further statistical investigations could be of great interest for the design of bladed-disks, in terms of vibration mitigation and robustness.

Complex modes in plasmonic nonlinear slot waveguides

We analyze linear and nonlinear modes of metal–dielectric–metal slot waveguides taking into account the effect of losses at the metallic cladding. First, we study a linear core and then assume it to be nonlinear with a Kerr-type nonlinear response. In the linear case, we summarize the earlier results of the frequency dispersion analysis showing the mode transformation when losses are increasing. For the nonlinear case, after a brief review of the earlier results on the lossless plasmonic slot waveguides, we demonstrate that losses lead to the existence of complex nonlinear modes which we describe through the dependence of mode power on the propagation constant. We show that losses bring many novel features into this nonlinear system resulting in novel bifurcations and novel types of nonlinear guided modes, as well as the existence of limiting values for the propagation constant and the modes with an unlimited power. We also study the mode stability by means of numerical simulation using the finite difference time domain method.

Complex non-linear modal analysis for mechanical systems: Application to turbomachinery bladings with friction interfaces

A method for modal analysis of non-linear and non-conservative mechanical systems is proposed. In particular, dry-friction nonlinearities are considered although the method is not restricted to these. Based on the concept of complex nonlinear modes, eigensolutions are written as generalized Fourier series and the eigenproblem is then formulated in the frequency-domain. An alternating frequency-time domain method is used for the calculation of implicit nonlinear forces. A two degrees-of-freedom example featuring dry-friction illustrates the method and highlights the effects of dissipation on modal parameters. The stabilizing effects of friction in presence of negative damping in the system are also addressed.Then an application on a large-scale nonlinear system consisting of a turbomachinery blade, with dry-friction interfaces is proposed. In the latter, an original framework for the description of two-dimensional frictional motions by complex variables is proposed and applied, in particular, to a Dahl model. Effects of friction parameters and models on the blade's modal characteristics are investigated.

A numerical method for nonlinear complex modes with application to active–passive damped sandwich structures

In this paper, a numerical method is proposed for determining complex vibrations modes of sandwich structures with piezoelectric and viscoelastic layers. Based on homotopy and asymptotic numerical techniques, this method leads to the damping properties calculation (loss factor and natural frequency per mode) of the hybrid sandwich structure. The numerical results of the loss factor and natural frequency are compared to those obtained from analytical beam model and from numerical studies the modal strain energy method.

Application of a Nonlinear Modal Instability Approach to Brake Systems

A method, called the Complex NonLinear Modal Analysis (CNLMA), is proposed for the calculation of the periodic solutions of nonlinear mechanical systems with continued nonlinearities. The equivalent linearization concept and the notion of nonlinear complex modes are applied in order to analyze the steady-state responses of autonomous nonlinear systems. The CNLM-Analysis appears very interesting in regard to computational time; it also necessitates very few computer resources. This method was applied to study instability phenomena in a nonlinear model with a constant brake friction coefficient.

A note on the number of periodic orbits near a resonant equilibrium point

The author derives a formula for the number M(n, d) of complex nonlinear normal modes in the (n+1) degrees of freedom system with Hamiltonian H2+H, where H2 describes a system of (n+1) uncoupled oscillators with the same frequency (the so-called (1:1:1:.:1) resonance) and H is a generic homogeneous perturbation of degree 2d which Poisson commutes with H2.