Nonlinear normal modes of a two degree of freedom oscillator with a bilateral elastic stop

A study of the non linear modes of a two degree of freedom mechanical system with bilateral elastic stop is considered. The issue related to the non-smoothness of the impact force is handled through a regularization technique. In order to obtain the Nonlinear Normal Mode (NNM), the harmonic balance method with a large number of harmonics, combined with the asymptotic numerical method, is used to solve the regularized problem. These methods are present in the software "package" MANLAB. The results are validated from periodic orbits obtained analytically in the time domain by direct integration of the non regular problem. The two NNMs starting respectively from the two linear normal modes of the associated underlying linear system are discussed. The energy-frequency plot is used to present a global vision of the behavior of the modes. The dynamics of the modes are also analyzed comparing each periodic orbits and modal lines. The first NNM shows an elaborate dynamics with the occurrence of multiple impacts per period. On the other hand, the second NNM presents a more simple dynamics with a localization of the displacement on the first mass

Radial Dependence of the Pattern Speed of M51

The grand-design spiral galaxy M51 has long been a crucial target for theories of spiral structure. Studies of this iconic spiral can address the question of whether strong spiral structure is transient (e.g. interaction-driven) or long-lasting. As a clue to the origin of the structure in M51, we investigate evidence for radial variation in the spiral pattern speed using the radial Tremaine-Weinberg (TWR) method. We implement the method on CO observations tracing the ISM-dominant molecular component. Results from the method's numerical implementation--combined with regularization, which smooths intrinsically noisy solutions--indicate two distinct patterns speeds inside 4 kpc at our derived major axis PA=170 deg., both ending at corotation and both significantly higher than the conventionally adopted global value. Inspection of the rotation curve suggests that the pattern speed interior to 2 kpc lacks an ILR, consistent with the leading structure seen in HST near-IR observations. We also find tentative evidence for a lower pattern speed between 4 and 5.3 kpc measured by extending the regularized zone. As with the original TW method, uncertainty in major axis position angle (PA) is the largest source of error in the calculation; in this study, where \delta PA=+/-5 deg. a ~20% error is introduced to the parameters of the speeds at PA=170 deg. Accessory to this standard uncertainty, solutions with PA=175 deg. (also admitted by the data) exhibit only one pattern speed inside 4 kpc, and we consider this circumstance under the semblance of a radially varying PA.Comment: 14 pages in emulateapj format, 12 figures, accepted for publication in Ap

Aggregated motion estimation for real-time MRI reconstruction

Real-time magnetic resonance imaging (MRI) methods generally shorten the measuring time by acquiring less data than needed according to the sampling theorem. In order to obtain a proper image from such undersampled data, the reconstruction is commonly defined as the solution of an inverse problem, which is regularized by a priori assumptions about the object. While practical realizations have hitherto been surprisingly successful, strong assumptions about the continuity of image features may affect the temporal fidelity of the estimated images. Here we propose a novel approach for the reconstruction of serial real-time MRI data which integrates the deformations between nearby frames into the data consistency term. The method is not required to be affine or rigid and does not need additional measurements. Moreover, it handles multi-channel MRI data by simultaneously determining the image and its coil sensitivity profiles in a nonlinear formulation which also adapts to non-Cartesian (e.g., radial) sampling schemes. Experimental results of a motion phantom with controlled speed and in vivo measurements of rapid tongue movements demonstrate image improvements in preserving temporal fidelity and removing residual artifacts.Comment: This is a preliminary technical report. A polished version is published by Magnetic Resonance in Medicine. Magnetic Resonance in Medicine 201

Multi-frame scene-flow estimation using a patch model and smooth motion prior

This paper addresses the problem of estimating the dense 3D motion of a scene over several frames using a set of calibrated cameras. Most current 3D motion estimation techniques are limited to estimating the motion over a single frame, unless a strong prior model of the scene (such as a skeleton) is introduced. Estimating the 3D motion of a general scene is difficult due to untextured surfaces, complex movements and occlusions. In this paper, we show that it is possible to track the surfaces of a scene over several frames, by introducing an effective prior on the scene motion. Experimental results show that the proposed method estimates the dense scene-flow over multiple frames, without the need for multiple-view reconstructions at every frame. Furthermore, the accuracy of the proposed method is demonstrated by comparing the estimated motion against a ground truth

Observer design for piecewise smooth and switched systems via contraction theory

The aim of this paper is to present the application of an approach to study contraction theory recently developed for piecewise smooth and switched systems. The approach that can be used to analyze incremental stability properties of so-called Filippov systems (or variable structure systems) is based on the use of regularization, a procedure to make the vector field of interest differentiable before analyzing its properties. We show that by using this extension of contraction theory to nondifferentiable vector fields, it is possible to design observers for a large class of piecewise smooth systems using not only Euclidean norms, as also done in previous literature, but also non-Euclidean norms. This allows greater flexibility in the design and encompasses the case of both piecewise-linear and piecewise-smooth (nonlinear) systems. The theoretical methodology is illustrated via a set of representative examples.Comment: Preprint accepted to IFAC World Congress 201

Long wave expansions for water waves over random topography

In this paper, we study the motion of the free surface of a body of fluid over a variable bottom, in a long wave asymptotic regime. We assume that the bottom of the fluid region can be described by a stationary random process $\beta(x, \omega)$ whose variations take place on short length scales and which are decorrelated on the length scale of the long waves. This is a question of homogenization theory in the scaling regime for the Boussinesq and KdV equations. The analysis is performed from the point of view of perturbation theory for Hamiltonian PDEs with a small parameter, in the context of which we perform a careful analysis of the distributional convergence of stationary mixing random processes. We show in particular that the problem does not fully homogenize, and that the random effects are as important as dispersive and nonlinear phenomena in the scaling regime that is studied. Our principal result is the derivation of effective equations for surface water waves in the long wave small amplitude regime, and a consistency analysis of these equations, which are not necessarily Hamiltonian PDEs. In this analysis we compute the effects of random modulation of solutions, and give an explicit expression for the scattered component of the solution due to waves interacting with the random bottom. We show that the resulting influence of the random topography is expressed in terms of a canonical process, which is equivalent to a white noise through Donsker's invariance principle, with one free parameter being the variance of the random process $\beta$. This work is a reappraisal of the paper by Rosales & Papanicolaou \cite{RP83} and its extension to general stationary mixing processes