Stability Boundary for Haptic Rendering: Influence of Damping and Delay

2009 ◽  
Vol 9 (1) ◽  
Author(s):  
Jorge Juan Gil ◽  
Emilio Sánchez ◽  
Thomas Hulin ◽  
Carsten Preusche ◽  
Gerd Hirzinger
Keyword(s):  
Stability Boundary ◽  
Sampling Frequency ◽  
Haptic Interface ◽  
Viscous Damping ◽  
Light Weight ◽  
Stability Boundaries ◽  
Linear Condition ◽  
Virtual Stiffness ◽  
The Stability ◽  
Theoretical Results

The influence of viscous damping and delay on the stability of haptic systems is studied in this paper. The stability boundaries have been found by means of different approaches. Although the shape of these stability boundaries is quite complex, a new linear condition, which summarizes the relation between virtual stiffness, viscous damping, and delay, is proposed under certain assumptions. These assumptions include a linear system, short delays, fast sampling frequency, and relatively low physical and virtual damping. The theoretical results presented in this paper are supported by simulations and experimental data using the DLR light-weight robot and the large haptic interface for aeronautic maintainability (LHIfAM).

On the spin-up and spin-down of a rotating fluid. Part 2. Measurements and stability

1976 ◽  
Vol 77 (4) ◽  
pp. 709-735 ◽  
Author(s):  
Patrick D. Weidman
Keyword(s):  
Stability Boundary ◽  
Fluid Motion ◽  
Azimuthal Velocity ◽  
Laser Doppler Velocimeter ◽  
Rotating Fluid ◽  
Constant Acceleration ◽  
Centrifugal Instability ◽  
The Stability ◽  
Theoretical Results

Measurements of the azimuthal velocity inside a cylinder which spins up or spins down at constant acceleration were obtained with a laser-Doppler velocimeter and compared with the theoretical results presented in part 1. Velocity profiles near the wave front in spin-up indicate that the velocity discontinuity given by the inviscid Wedemeyer model is smoothed out in a shear layer whose thickness varies with radius and time but scales with hE1/4Ω. The spin-down profiles are always in excellent agreement with theory when the flow is stable. Visualization studies with aluminium tracers have made possible the determination of the stability boundary for Ekman spiral waves (principally type II waves) observed on the cylinder end walls during spin-up. For spin-down to rest the flow always experienced a centrifugal instability which ultimately disrupted the interior fluid motion.

The Stability of Self-Acting Gas Journal Bearings With Noncircular Members and Additional Elements of Flexibility

1969 ◽  
Vol 91 (1) ◽  
pp. 113-119 ◽  
Author(s):  
H. Marsh
Keyword(s):  
Stability Boundary ◽  
Journal Bearings ◽  
Satisfactory Explanation ◽  
Unusual Behavior ◽  
Stability Boundaries ◽  
Linearized Theory ◽  
New Type ◽  
The Stability ◽  
Theory And Experiments ◽  
Bearing System

The linearized theory for the stability of self-acting gas bearings is extended to include bearing systems with noncircular members or additional elements of flexibility and damping. The theory offers a satisfactory explanation for the unusual behavior of a bearing system with a three-lobed rotor, including the whirl at low speeds and the whirl cessation. A comparison between the theory and experiments for a flexibly mounted bearing system shows that the theory can be applied to predict the stability boundaries of bearing systems with additional elements of flexibility. A new type of bearing apparatus is proposed in which it would be possible to obtain information about bearing stability without operating at the stability boundary.

Phase-Locked Mode Stability for Coupled van der Pol Oscillators

1999 ◽  
Vol 122 (3) ◽  
pp. 318-323 ◽  
Author(s):  
Duane W. Storti ◽  
Per G. Reinhall
Keyword(s):  
Variational Equation ◽  
Stability Boundary ◽  
Series Expansions ◽  
Van Der Pol ◽  
Stability Boundaries ◽  
Power Series Expansions ◽  
Mode Stability ◽  
Van Der Pol Oscillators ◽  
The Stability ◽  
The Hill

The critical variational equation governing the stability of phase-locked modes for a pair of diffusively coupled van der Pol oscillators is presented in the form of a linear oscillator with a periodic damping coefficient that involves the van der Pol limit cycle. The variational equation is transformed into a Hill’s equation, and stability boundaries are obtained by analytical and numerical methods. We identify a countable set of resonances and obtain expressions for the associated stability boundaries as power series expansions of the associated Hill determinants. We establish an additional “zero mean damping” condition and express it as a Pade´ approximant describing a surface that combines with the Hill determinant surfaces to complete the stability boundary. The expansions obtained are evaluated to visualize the first three resonant surfaces which are compared with numerically determined slices through the stability boundaries computed over the range 0.4<ε<5. [S0739-3717(00)00502-X]

Global Static Instability of Risers

1984 ◽  
Vol 28 (04) ◽  
pp. 261-271
Author(s):  
Michael M. Bernitsas ◽  
Theodore Kokkinis
Keyword(s):  
Boundary Conditions ◽  
Internal Pressure ◽  
Stability Boundary ◽  
Critical Length ◽  
Bending Rigidity ◽  
Buckling Mode ◽  
Stability Boundaries ◽  
Various Boundary Conditions ◽  
Static Instability ◽  
The Stability

Global instability of risers depends on riser weight, internal and external fluid static pressure forces, tension exerted at the top of the riser, and boundary conditions. The purpose of this work is to study the effects of these factors on the stability boundaries of risers and specifically.(i) compare buckling loads for various boundary conditions; (ii) find the long-riser instability behavior from the asymptotics of the stability boundaries; (iii) find the short-riser instability behavior; (iv) analyze the relative effects of boundary conditions, weight, internal pressure, and bending rigidity on stability; (v) show the variation of the stability boundary shape with the order of the buckling mode; and (vi) compare the critical length at which risers in tension over their entire length may buckle due to internal pressure, for various boundary conditions.

Numerical study of nonlinear stability boundaries for orifice-compensated hole-entry hybrid journal bearings

2019 ◽  
Vol 234 (12) ◽  
pp. 1867-1880 ◽  
Author(s):  
Jian Li ◽  
Runchang Chen ◽  
Haiyin Cao ◽  
Zhuxin Tian
Keyword(s):  
Finite Length ◽  
Nonlinear Stability ◽  
Critical Speed ◽  
Stability Boundary ◽  
Initial Point ◽  
Journal Bearings ◽  
Stability Boundaries ◽  
Rotating Speed ◽  
The Stability ◽  
Bearing System

A high-performance and finite-length bearing system requires that the shaft can be stabilized even under a strong perturbation. The linear stability theory neglects the effects of nonlinear forces and the initial point of the shaft. Therefore, the stability of the bearing system is largely determined by the rotating speed of the shaft. In the present numerical investigation, the nonlinear forces and initial point of the shaft are accounted for to obtain the nonlinear stability boundary. The objective of this study is extended to orifice-compensated and hole-entry hybrid journal bearings with finite length. The critical rotating speed and the shaft center trajectory are acquired by solving Reynolds equation using the finite element method. By identifying the states of the orbits (stable or unstable), the nonlinear stability boundaries can be obtained. Results show that for the hybrid bearing system under the nonlinear conditions, the critical speed is a determinant factor while the initial location is another key factor. The shaft can be unstable if the initial point is outside of the stability boundary, although the speed is lower than the critical speed. There exists an obvious transitional region between the stable and unstable condition when the speed approaches the critical speed.

STABILITY BOUNDARY AND DURATION TIME OF SYNCHRONIZATION FOR COUPLED CHAOTIC MATHIEU–DUFFING OSCILLATORS

2008 ◽  
Vol 22 (27) ◽  
pp. 4817-4831
Author(s):  
JIANHE SHEN ◽  
JIANPING CAI ◽  
SHUHUI CHEN ◽  
KECHANG LIN
Keyword(s):  
Coupling Strength ◽  
Stability Boundary ◽  
Hill Equation ◽  
Critical Values ◽  
Stability Boundaries ◽  
Synchronization Time ◽  
Duration Time ◽  
Error System ◽  
The Stability ◽  
The Hill

The stability boundaries and behaviors of the duration time of synchronization for chaotic Mathieu–Duffing oscillators are investigated. Based on the unidirectional or bidirectional linear state error feedback coupled scheme, the error system is derived. After replacing the chaotic orbit by a regular orbit containing multi-harmonics, we analyze the asymptotic stability of the error system, which leads to a Hill equation. According to Floquet theory and the properties of the Hill equation, the evolution of the discriminant of the Hill equation with respect to the coupling strength is traced to determine the stability boundaries between the synchronization and desynchronization domains. Thus, the critical values of coupling strength are obtained. These critical values are in good agreement with those from numerical simulations. The behaviors of the synchronization time are numerically investigated in the synchronization domain. It is found that the synchronization time reaches an asymptotic minimal value when the oscillators are unidirectionally or bidirectionally coupled, and the two asymptotic minimal values are almost the same. It is also noted that the slowing down behavior of the synchronization time can occur inside the synchronization domain when the coupling is bidirectional.

Double-diffusive instabilities of autocatalytic chemical fronts

2007 ◽  
Vol 576 ◽  
pp. 445-456 ◽  
Author(s):  
J. D'HERNONCOURT ◽  
A. DE WIT ◽  
A. ZEBIB
Keyword(s):  
Stability Boundary ◽  
Lewis Number ◽  
Double Diffusion ◽  
Stable Stratification ◽  
Additional Contribution ◽  
Stability Boundaries ◽  
Density Perturbations ◽  
The Stability ◽  
Global And Local ◽  
Double Diffusive

Convective instabilities of an autocatalytic propagating chemical front in a porous medium are studied. The front creates temperature and concentration gradients which then generate a density gradient. If the front propagates in the direction of the gravity field, adverse density stratification can lead to Rayleigh–Taylor or Rayleigh–Bénard instabilities. Differential diffusivity of mass and heat can also destabilize the front because of the double-diffusive phenomena. We compare the stability boundaries for the classical hydrodynamic case of a bounded layer without reaction and for the chemical front in the parameter space spanned by the thermal and solutal Rayleigh numbers. We show that chemical reactions profoundly affect the stability boundaries compared to the non-reactive situation because of a delicate coupling between the double-diffusive and Rayleigh–Taylor mechanisms with localized density perturbations driven by the reaction. In the reactive case, a linear stability analysis identifies three distinct stationary branches of the instability. They bound a region of stability that shrinks with increasing Lewis number, in marked contrast to the classical double-diffusive layer. In particular a region of global and local stable stratification is susceptible to a counter-intuitive mechanism of convective instability driven by chemistry and double-diffusion. The other two regions display an additional contribution of localized Rayleigh–Taylor instabilities. Displaced-particle arguments are employed in support of and to elucidate the entire stability boundary.

Lyapunov’s Exponents for Non-Smooth Dynamical Systems: Behavior Identification Across Stability Boundaries With Bifurcations

2005 ◽  
Author(s):  
Alessio Ageno ◽  
Anna Sinopoli
Keyword(s):  
Closed Form ◽  
Stability Boundary ◽  
Period Doubling ◽  
Numerical Algorithms ◽  
Theoretical Method ◽  
Smooth Functions ◽  
Stability Boundaries ◽  
Strong Discontinuities ◽  
Starting Point ◽  
The Stability

In this paper some recently developed techniques to evaluate both analytically and numerically stability features of a non-smooth dynamical system, are used to investigate in detail the stability boundaries of regions corresponding to given stable periodic responses. The problem analyzed is that of a rocking block simply supported on a harmonically moving rigid ground; in this case, if the block is assumed to be a rigid body, the strong discontinuities characterizing the dynamic evolution are due to the impacts occurring each time the block crosses the initial equilibrium configuration. Therefore some special tools, specific for non-smooth functions, must be introduced to perform the stability analysis. In the present study, the theory due to Mu¨ller [19] is used to handle the evaluation of Lyapunov’s exponents upon discontinuities, by introducing the treatment, both analytical and numerical, of “saltation matrices”. Such a general theoretical method on one hand has been adapted to the numerical algorithms needed for the solution of the complete, non-linearized, problem and on the other hand, it allowed the development of the closed-form analytical reference solutions, obtained by linearizing assumptions less restrictive than those used by Hogan [8, 9]. The approximated stability boundaries obtained by the linearized closed-form solutions have been the starting point to guide the choice of the system parameters values to locate the responses in regions where bifurcations can arise. Inside these ranges several examples can be presented to illustrate the trends exhibited by the numerically evaluated Lyapunov’s exponents when the values of the forcing amplitude increase over the stability boundary of the symmetric responses while the value of the forcing frequency is fixed. Among these, investigations on sequences of responses composing period doubling cascades toward chaos, can provide a good and interesting test to appreciate the indications offered by the numerically derived Lyapunov’s exponents.

Symmetrizable Difference Dchemes

2005 ◽  
Vol 5 (1) ◽  
pp. 3-50 ◽  
Author(s):  
Alexei A. Gulin
Keyword(s):  
Initial Data ◽  
Difference Scheme ◽  
Difference Schemes ◽  
Time Dependent ◽  
Stability Boundaries ◽  
Typical Representative ◽  
Variable Weight ◽  
The Stability ◽  
Conditions Of Stability ◽  
The Right

AbstractA review of the stability theory of symmetrizable time-dependent difference schemes is represented. The notion of the operator-difference scheme is introduced and general ideas about stability in the sense of the initial data and in the sense of the right hand side are formulated. Further, the so-called symmetrizable difference schemes are considered in detail for which we manage to formulate the unimprovable necessary and su±cient conditions of stability in the sense of the initial data. The schemes with variable weight multipliers are a typical representative of symmetrizable difference schemes. For such schemes a numerical algorithm is proposed and realized for constructing stability boundaries.

scholarly journals Dynamics of the rumor-spreading model with hesitation mechanism in heterogenous networks and bilingual environment

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuai Yang ◽  
Haijun Jiang ◽  
Cheng Hu ◽  
Juan Yu ◽  
Jiarong Li
Keyword(s):  
Lyapunov Method ◽  
Reproduction Number ◽  
Model Parameters ◽  
Rumor Spreading ◽  
Spreading Model ◽  
Reductio Ad Absurdum ◽  
The Stability ◽  
The Impact ◽  
Theoretical Results ◽  
The Basic Reproduction Number

Abstract In this paper, a novel rumor-spreading model is proposed under bilingual environment and heterogenous networks, which considers that exposures may be converted to spreaders or stiflers at a set rate. Firstly, the nonnegativity and boundedness of the solution for rumor-spreading model are proved by reductio ad absurdum. Secondly, both the basic reproduction number and the stability of the rumor-free equilibrium are systematically discussed. Whereafter, the global stability of rumor-prevailing equilibrium is explored by utilizing Lyapunov method and LaSalle’s invariance principle. Finally, the sensitivity analysis and the numerical simulation are respectively presented to analyze the impact of model parameters and illustrate the validity of theoretical results.

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