On the Stability of the Vibrations of Two Coupled Particles in the Plane

1969 ◽  
Vol 36 (3) ◽  
pp. 417-419
Author(s):  
R. H. Rand ◽  
S. F. Tseng
Keyword(s):  
Initial Stress ◽  
Floquet Theory ◽  
Stability Parameter ◽  
Identical Particles ◽  
Variational Equations ◽  
The Stability

The stability of the vibrations of two identical particles constrained to a plane and restrained by three identical linear springs with initial stress is studied by uncoupling the first variational equations and applying Floquet theory and perturbations. A new stability parameter is introduced which indicates stability by virtue of its sign.

On the Stability of a Differential Equation With Application to the Vibrations of a Particle in the Plane

1969 ◽  
Vol 36 (2) ◽  
pp. 311-313 ◽  
Author(s):  
Richard H. Rand ◽  
Shoei-Fu Tseng
Keyword(s):  
Differential Equation ◽  
Fourier Analysis ◽  
Initial Stress ◽  
Floquet Theory ◽  
The Stability

The stability of the equation Z¨ + f(t)Z = 0, where f(t) = (δ − ε cos2t)/(1 − ε cos2t), is studied by using Floquet theory, Fourier analysis and perturbations. The results are used to study the stability of the vibrations of a particle constrained to a plane and restrained by two identical linear springs with initial stress.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

Influence of Interaction Between Torsion and Bending on Long-Term Nonlinear Rotor Dynamics

1999 ◽  
Author(s):  
Jiazhong Zhang ◽  
Bram de Kraker ◽  
Dick H. van Campen
Keyword(s):  
Critical Speed ◽  
Floquet Theory ◽  
Rotor Dynamics ◽  
Steady State Response ◽  
Bending Vibrations ◽  
Bending Vibration ◽  
State Response ◽  
Stability And Bifurcation ◽  
The Stability

Abstract An elementary system with gears and excited by unbalance mass has been constructed for analyzing the interaction between torsion and bending vibration in rotor dynamics. For this system, only the interaction caused primarily by unbalance mass has been investigated. The stability and bifurcation characteristics of the system have been studied by numerical computation based on Hopf bifurcation and Floquet theory. The results show that the interaction between torsion and bending vibrations can affect the stability and bifurcation of the unbalance response, in particular the onset speed of instability. In addition to the above, the interaction also affects the steady-state response. To investigate the influence of unbalance mass, the periodic solution and its stability have been studied near the first bending critical speed of the decoupled system. All the results show that the coupling of torsion and bending vibrations can have a significant influence on the nonlinear dynamics of the whole system.

Dynamics of Two Van Der Pol Oscillators Coupled via a Bath

2003 ◽  
Author(s):  
Erika Camacho ◽  
Richard Rand ◽  
Howard Howland
Keyword(s):  
Software Package ◽  
Bifurcation Theory ◽  
Floquet Theory ◽  
Van Der Pol Oscillator ◽  
Direct Connection ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Existence And Stability ◽  
Van Der Pol Oscillators ◽  
The Stability

In this work we study a system of two van der Pol oscillators, x and y, coupled via a “bath” z: x¨−ε(1−x2)x˙+x=k(z−x)y¨−ε(1−y2)y˙+y=k(z−y)z˙=k(x−z)+k(y−z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

scholarly journals Cylinder wakes in shallow oscillatory flow: the coastal island wake problem

2019 ◽  
Vol 874 ◽  
pp. 158-184 ◽  
Author(s):  
Paul M. Branson ◽  
Marco Ghisalberti ◽  
Gregory N. Ivey ◽  
Emil J. Hopfinger
Keyword(s):  
Boundary Layer ◽  
Oscillatory Flow ◽  
Bottom Friction ◽  
Stability Parameter ◽  
Bottom Boundary Layer ◽  
Ekman Pumping ◽  
Bottom Boundary ◽  
Island Wakes ◽  
Island Wake ◽  
The Stability

Topographic complexity on continental shelves is the catalyst that transforms the barotropic tide into the secondary and residual circulations that dominate vertical and cross-shelf mixing processes. Island wakes are one such example that are observed to significantly influence the transport and distribution of biological and physical scalars. Despite the importance of island wakes, to date, no sufficient, mechanistic description of the physical processes governing their development exists for the general case of unsteady tidal forcing. Controlled laboratory experiments are necessary for the understanding of this complex flow phenomenon. Here, three-dimensional velocity field measurements of cylinder wakes in shallow-water oscillatory flow are conducted across a parameter space that is typical of tidal flow around shallow islands. The wake form in steady flows is typically described in terms of the stability parameter $S=c_{f}D/h$ (where $D$ is the island diameter, $h$ is the water depth and $c_{f}$ is the bottom boundary friction coefficient); in tidal flows, there is an additional dependence on the Keulegan–Carpenter number $KC=U_{0}T/D$ (where $U_{0}$ is the tidal velocity amplitude and $T$ is the tidal period). In this study we demonstrate that when the influence of bottom friction is confined to a Stokes boundary layer the stability parameter is given by $S=\unicode[STIX]{x1D6FF}^{+}/KC$ where $\unicode[STIX]{x1D6FF}^{+}$ is the ratio of the wavelength of the Stokes bottom boundary layer to the depth. Three classes of wake form are observed with decreasing wake stability: (i) steady bubble for $S\gtrsim 0.1$; (ii) unsteady bubble for $0.06\lesssim S\lesssim 0.1$; and (iii) vortex shedding for $S\lesssim 0.06$. Transitions in wake form and wake stability are shown to depend on the magnitude and temporal evolution of the wake return flow. Scaling laws are developed to allow upscaling of the laboratory results to island wakes. Vertical and lateral transport depend on three parameters: (i) the flow aspect ratio $h/D$; (ii) the amplitude of tidal motion relative to the island size, given by $KC$; and (iii) the relative influence of bottom friction to the flow depth, given by $\unicode[STIX]{x1D6FF}^{+}$. A model of wake upwelling based on Ekman pumping from the bottom boundary layer demonstrates that upwelling in the near-wake region of an island scales with $U_{0}(h/D)KC^{1/6}$ and is independent of the wake form. Finally, we demonstrate an intrinsic link between the dynamical eddy scales, predicted by the Ekman pumping model, and the island wake form and stability.

A STUDY OF HIGHER-ORDER DISCONTINUOUS GALERKIN AND QUADRATIC LEAST-SQUARES STABILIZED FINITE ELEMENT COMPUTATIONS FOR ACOUSTICS

2006 ◽  
Vol 14 (01) ◽  
pp. 1-19 ◽  
Author(s):  
ISAAC HARARI ◽  
RADEK TEZAUR ◽  
CHARBEL FARHAT
Keyword(s):  
Least Squares ◽  
Discontinuous Galerkin ◽  
Large Scale ◽  
Dispersion Analysis ◽  
Stability Parameter ◽  
New Approach ◽  
One Dimensional ◽  
Stabilized Finite Element ◽  
Quadratic Interpolation ◽  
The Stability

One-dimensional analyses provide novel definitions of the Galerkin/least-squares stability parameter for quadratic interpolation. A new approach to the dispersion analysis of the Lagrange multiplier approximation in discontinuous Galerkin methods is presented. A series of computations comparing the performance of [Formula: see text] Galerkin and GLS methods with Q-8-2 DGM on large-scale problems shows superior DGM results on analogous meshes, both structured and unstructured. The degradation of the [Formula: see text] GLS stabilization on unstructured meshes may be a consequence of inadequate one-dimensional analysis used to derive the stability parameter.

Consensus stability in multi-agent systems with periodically switched communication topology using Floquet theory

2020 ◽  
pp. 014233122096905
Author(s):  
Mohammad Maadani ◽  
Eric A Butcher
Keyword(s):  
Feedback Linearization ◽  
Floquet Theory ◽  
Kuramoto Model ◽  
Multi Agent Systems ◽  
Agent Systems ◽  
Unstable Subsystems ◽  
Special Cases ◽  
Communication Topology ◽  
Multi Agent ◽  
The Stability

The stability of consensus in linear and nonlinear multi-agent systems with periodically switched communication topology is studied using Floquet theory. The proposed strategy is illustrated for the cases of consensus in linear single-integrator, higher-order integrator, and leader-follower consensus. In addition, the application of Floquet theory in analyzing special cases such as switched systems with joint connectivity, unstable subsystems, and nonlinear systems, including the use of feedback linearization and local linearization in the Kuramoto model, is also studied. By utilizing Floquet theory for multi-agent systems with periodically switched communication topologies, one can simultaneously evaluate the effects of each subsystem’s convergence rate and dwell time on overall behavior. Numerical simulation results are presented to demonstrate the efficacy of the proposed approach in stability analysis of all these cases.

Instability and Unbalance Response of Dissymmetric Rotor-Bearing Systems

1988 ◽  
Vol 110 (3) ◽  
pp. 288-294 ◽  
Author(s):  
P. M. Guilhen ◽  
P. Berthier ◽  
G. Ferraris ◽  
M. Lalanne
Keyword(s):  
Differential Equations ◽  
Coordinate System ◽  
Transfer Matrix ◽  
Differential System ◽  
Industrial Application ◽  
Floquet Theory ◽  
Unbalance Response ◽  
Stationary Coordinate System ◽  
The Matrix ◽  
The Stability

The study deals with the instability and unbalance response of dissymmetric rotors, when periodic differential equations are impossible to avoid. The method which yields motion instability is based on an extension of the well-known Floquet theory. A transfer matrix over one period of the motion is obtained, and the stability of the system can be tested with the eigenvalues of the matrix. To find the instability and the unbalance response, the Newmark formulation is used. Here, the dissymmetry comes either from the rotor or from the bearings in such a way that it is possible to solve a regular differential system without periodic coefficients, either in the stationary coordinate system or in the rotating one. Three examples are given, including an industrial application. The results show that the method proposed is satisfactory.

Stability Analysis of Complex Multibody Systems

2005 ◽  
Vol 1 (1) ◽  
pp. 71-80 ◽  
Author(s):  
Olivier A. Bauchau ◽  
Jielong Wang
Keyword(s):  
Stability Analysis ◽  
Multibody Systems ◽  
Floquet Theory ◽  
Numerical Models ◽  
Characteristic Exponent ◽  
Prony's Method ◽  
Linearized Stability ◽  
The Stability ◽  
Curve Fitting Method ◽  
Prony’S Method

The linearized stability analysis of dynamical systems modeled using finite element-based multibody formulations is addressed in this paper. The use of classical methods for stability analysis of these systems, such as the characteristic exponent method or Floquet theory, results in computationally prohibitive costs. Since comprehensive multibody models are “virtual prototypes” of actual systems, the applicability to numerical models of the stability analysis tools that are used in experimental settings is investigated in this work. Various experimental tools for stability analysis are reviewed. It is proved that Prony’s method, generally regarded as a curve-fitting method, is equivalent, and sometimes identical, to Floquet theory and to the partial Floquet method. This observation gives Prony’s method a sound theoretical footing, and considerably improves the robustness of its predictions when applied to comprehensive models of complex multibody systems. Numerical and experimental applications are presented to demonstrate the efficiency of the proposed procedure.

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