scholarly journals The Stability Conditions for a Heavy Solid Motion

2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
A. I. Ismail
Keyword(s):  
Equations Of Motion ◽  
Center Of Mass ◽  
Sufficient Conditions ◽  
The Body ◽  
Stability Conditions ◽  
Moments Of Inertia ◽  
Principal Moments Of Inertia ◽  
The Stability ◽  
Nonlinear Equations Of Motion ◽  
Necessary And Sufficient

In this paper, the stability conditions for the rotary motion of a heavy solid about its fixed point are considered. The center of mass of the body is assumed to lie on the moving z-axis which is assumed to be the minor axis of the ellipsoid of inertia. The nonlinear equations of motion and their three first integrals are obtained when the principal moments of inertia are distributed as I 1 < I 2 < I 3 . We construct a Lyapunov function L to investigate the stability conditions for this motion. We give a numerical example to illustrate the necessary and sufficient conditions for the stability of the body at certain moments of inertia. This problem has many important applications in different sciences.

On the Stability of Linear Stochastic Differential Equations

1973 ◽  
Vol 40 (1) ◽  
pp. 87-92 ◽  
Author(s):  
F. Kozin ◽  
C.-M. Wu
Keyword(s):  
White Noise ◽  
Stability Region ◽  
Sufficient Conditions ◽  
Stability Conditions ◽  
Density Functions ◽  
Close Approximation ◽  
Sample Stability ◽  
Gaussian Coefficient ◽  
The Stability ◽  
Necessary And Sufficient

In this paper we present a study of the almost-sure sample stability properties of second-order linear systems with stochastic coefficients. Using knowledge of the first density functions of the coefficient processes, stability conditions are obtained. Based upon recent necessary and sufficient conditions for white-noise coefficient systems, the conditions obtained may yield a close approximation of the exact stability region for the Gaussian coefficient case.

scholarly journals Dynamics of constrained many body problems in constant curvature two-dimensional manifolds

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170425 ◽  
Author(s):  
Andrzej J. Maciejewski ◽  
Maria Przybylska
Keyword(s):  
Constant Curvature ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Rigid Bodies ◽  
The Body ◽  
Two Dimensional ◽  
Many Body ◽  
Finite Dimensional ◽  
Necessary And Sufficient

In this paper, we investigate systems of several point masses moving in constant curvature two-dimensional manifolds and subjected to certain holonomic constraints. We show that in certain cases these systems can be considered as rigid bodies in Euclidean and pseudo-Euclidean three-dimensional spaces with points which can move along a curve fixed in the body. We derive the equations of motion which are Hamiltonian with respect to a certain degenerated Poisson bracket. Moreover, we have found several integrable cases of described models. For one of them, we give the necessary and sufficient conditions for the integrability. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals New stability conditions for positive continuous-discrete 2D linear systems

2011 ◽  
Vol 21 (3) ◽  
pp. 521-524 ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Asymptotic Stability ◽  
Linear Systems ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stability Conditions ◽  
Stability Tests ◽  
Numerical Examples ◽  
The Stability ◽  
Necessary And Sufficient

New stability conditions for positive continuous-discrete 2D linear systemsNew necessary and sufficient conditions for asymptotic stability of positive continuous-discrete 2D linear systems are established. Necessary conditions for the stability are also given. The stability tests are demonstrated on numerical examples.

scholarly journals The necessary and sufficient condition for the stability of a rigid body

2017 ◽  
Vol 13 (4) ◽  
pp. 4999-5003 ◽  
Author(s):  
W. S. Amer
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
First Integrals ◽  
Body Motion ◽  
Sufficient Condition ◽  
Stability Criteria ◽  
Necessary And Sufficient Condition ◽  
The Stability ◽  
Necessary And Sufficient

In this paper, the stability of the unperturbed rigid body motion close to conditions, related with the center of mass, is investigated. The three first integrals for the equations of motion are obtained. These integrals are used to achieve a Lyapunov function and to obtain the necessary and sufficient condition satisfies the stability criteria.

Stability and Regularization of Difference Schemes in Banach and Hilbert Spaces

2013 ◽  
Vol 13 (2) ◽  
pp. 139-160
Author(s):  
Ivan P. Gavrilyuk ◽  
Volodymyr L. Makarov
Keyword(s):  
Banach Space ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Iteration Scheme ◽  
Difference Schemes ◽  
Stability Conditions ◽  
Initial Operator ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Stability Results

Abstract. The necessary and sufficient conditions for stability of abstract difference schemes in Hilbert and Banach spaces are formulated. Contrary to known stability results we give stability conditions for schemes with non-self-adjoint operator coefficients in a Hilbert space and with strongly positive operator coefficients in a Banach space. It is shown that the parameters of the sectorial spectral domain play the crucial role. As an application we consider the Richardson iteration scheme for an operator equation in a Banach space, in particulary the Richardson iteration with precondition for a finite element scheme for a non-selfadjoint operator. The theoretical results are also the basis when using the regularization principle to construct stable difference schemes. For this aim we start from some simple scheme (even unstable) and derive stable schemes by perturbing the initial operator coefficients and by taking into account the stability conditions. Our approach is also valid for schemes with unbounded operator coefficients.

On the Necessary and Sufficient Conditions for the Stability of Linear Generalized Ordinary Differential, Linear Impulsive and Linear Difference Systems

2009 ◽  
Vol 16 (4) ◽  
pp. 597-616
Author(s):  
Shota Akhalaia ◽  
Malkhaz Ashordia ◽  
Nestan Kekelia
Keyword(s):  
Linear System ◽  
Difference Equations ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Difference Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Generalized Ordinary Differential Equations ◽  
Lyapunov Sense

Abstract Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

Quasi-Periodic Motion and Hopf Bifurcation of a Two-Dimensional Aeroelastic Airfoil System in Supersonic Flow

2021 ◽  
Vol 31 (02) ◽  
pp. 2150018
Author(s):  
Wentao Huang ◽  
Chengcheng Cao ◽  
Dongping He
Keyword(s):  
Hopf Bifurcation ◽  
Sufficient Conditions ◽  
Theoretical Method ◽  
Simulation Method ◽  
Runge Kutta Method ◽  
Complex Roots ◽  
The Stability ◽  
Necessary And Sufficient ◽  
2D And 3D ◽  
Imaginary Roots

In this article, the complex dynamic behavior of a nonlinear aeroelastic airfoil model with cubic nonlinear pitching stiffness is investigated by applying a theoretical method and numerical simulation method. First, through calculating the Jacobian of the nonlinear system at equilibrium, we obtain necessary and sufficient conditions when this system has two classes of degenerated equilibria. They are described as: (1) one pair of purely imaginary roots and one pair of conjugate complex roots with negative real parts; (2) two pairs of purely imaginary roots under nonresonant conditions. Then, with the aid of center manifold and normal form theories, we not only derive the stability conditions of the initial and nonzero equilibria, but also get the explicit expressions of the critical bifurcation lines resulting in static bifurcation and Hopf bifurcation. Specifically, quasi-periodic motions on 2D and 3D tori are found in the neighborhoods of the initial and nonzero equilibria under certain parameter conditions. Finally, the numerical simulations performed by the fourth-order Runge–Kutta method provide a good agreement with the results of theoretical analysis.

Coupled Stability of Multiport Systems—Theory and Experiments

1994 ◽  
Vol 116 (3) ◽  
pp. 419-428 ◽  
Author(s):  
J. E. Colgate
Keyword(s):  
Stability Property ◽  
Force Feedback ◽  
Linear Time ◽  
Experimental Studies ◽  
Sufficient Conditions ◽  
Direct Drive ◽  
Property A ◽  
Linear Time Invariant ◽  
The Stability ◽  
Necessary And Sufficient

This paper presents both theoretical and experimental studies of the stability of dynamic interaction between a feedback controlled manipulator and a passive environment. Necessary and sufficient conditions for “coupled stability”—the stability of a linear, time-invariant n-port (e.g., a robot, linearized about an operating point) coupled to a passive, but otherwise arbitrary, environment—are presented. The problem of assessing coupled stability for a physical system (continuous time) with a discrete time controller is then addressed. It is demonstrated that such a system may exhibit the coupled stability property; however, analytical, or even inexpensive numerical conditions are difficult to obtain. Therefore, an approximate condition, based on easily computed multivariable Nyquist plots, is developed. This condition is used to analyze two controllers implemented on a two-link, direct drive robot. An impedance controller demonstrates that a feedback controlled manipulator may satisfy the coupled stability property. A LQG/LTR controller illustrates specific consequences of failure to meet the coupled stability criterion; it also illustrates how coupled instability may arise in the absence of force feedback. Two experimental procedures—measurement of endpoint admittance and interaction with springs and masses—are introduced and used to evaluate the above controllers. Theoretical and experimental results are compared.

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