Liapunov's Direct Method Research Articles

Stability of a spinning liquid-filled spacecraft

The stability of a spinning liquid-filled spacecraft has been investigated in the present paper. Using Galerkin's method, the attitude dynamic equations have been given. The Liapunov direct method was employed to obtain a sufficient condition for stability. Three kinds of characteristic modals were investigated: free motion of inviscid fluid, slosh motion and non-slosh motion. All characteristic problems can be solved numerically by the Finite Element Method or the Boundary Element Method. It has been demonstrated that the viscosity of the fluid has a dissipative effect at large Reynolds number, while the slosh motion plays a destabilizing role. The non-slosh model of fluid does not affect the stability criterion.

Boundedness of Discrete Volterra Equations

Investigation of the boundedness of the solutions of Volterra equations with discrete time by means of the direct Liapunov method and comparison theorems is performed. Boundedness conditions of solutions of some particular classes of equations are derived. Using such conditions, stability of Direct Quadrature methods for second kind Volterra equations is analyzed.

The Stability of the Rotation of Steklov Gyro

In both uniform gravitational field and Newtonian central gravitational field, the conditions of stability and instability of the permanent rotation of the Steklov gyro with linear damping or nonlinear damping torques are studied by the Liapunov direct method and the first approximation method. Additionally, some new theorems are used in finding the stability and asymptotic stability of the gyro motion. In the Newtonian central gravitational field for the sake of high accuracy, the second order term of 1/R in the expression of potential energy has been considered, where R is the distance between the mass center of the gyro and the center of the earth. Numerical results are presented to show the domains of attraction in two cases by using a cell-to-cell mapping, the effects of types of damping torque and the system parameters on the gyro motion are also discussed in these cases.

Improved Stability of a Dual-Spin Satellite in Circular Orbit

Sufficient conditions for the global stability of equilibrium positions of a dual-spin satellite in a circular orbital system in a Newtonian central force field are obtained by Liapunov's direct method. The new results are compared with those already known. The problem is then studied with respect to all variables, including the angular velocity of the rotor. Further, sufficient conditions for the global stability of rotation with arbitrary angular velocity for an axisymmetric dual-spin satellite are presented by Liapunov's direct method, which improve on Kane's result of infinitesimal stability obtained by linearized equations of motion.

Stability of motion for a constrained Birkhoff's system in terms of independent variables

This paper proposes firstly the Pfaff-Birkhoff-D'Alembert's principle and obtains the equations of motion in terms of the independent variables for a constrained Birkhoff's system from the principle. Secondly, it establishes the equations of perturbation of the system. Finally, it obtains the stability criteria by using the Liapunov direct method and the firstly approximate method.

On the stability of dissipative mechanical systems with circulatory forces

The paper investigates the stability of a nonlinear dissipative mechanical system subjected to potential forces, gyroscopic forces, Rayleigh damping and circulatory forces, using direct Liapunov method. Assuming the gyroscopic matrix depends on a parameter, two asymptotically stable theorems are obtained, and the lower bounds of the parameter are evaluated.

Thermally Induced Instability of Laminated Beams and Plates

A theoretical investigation of dynamic stability for linear elastic structures due to non-uniform, time and space-dependent stochastic temperature fields is presented. The study is based on the reformulation of stochastic stability problems as a stability of Itoˆ type equations in some appropriate Hilbert space and is adopted for stability problems of structures with time and space-dependent stochastic coefficients. Uniform stochastic stability criteria of the structure equilibrium are derived using the Liapunov direct method. The energy-like functional and the generalized ltoˆ lemma are used to derive the sufficient stability conditions of the equilibrium state. A symmetrically laminated crossply plate subjected to the wide-band Gaussian temperature distribution and a laminated beam subjected to local short-time heatings are analysed in detail.

THE REGULAR AND CHAOTIC MOTIONS OF A SYMMETRIC HEAVY GYROSCOPE WITH HARMONIC EXCITATION

The non-linear motion of a symmetric gyro mounted on a vibrating base is investigated, with particular emphasis on its long-term dynamic behavior for a wide range of parameters. A single modal equation is used to analyze the qualitative behaviour of the system. External disturbance appears as vertical harmonic motion of the support point and linear damping is assumed. The complete equation of motion is a non-linear non-autonomous one. The stability of the system has been studied by damped Mathieu equation theory and the Liapunov direct method. As the system is subjected to external disturbance, the Melnikov method is used to show the existence of chaotic motion. Finally, the bifurcation of the parameter dependent system is studied numerically. The time evolutions of the non-linear dynamical system responses are described in phase portraits via the Poincaré map technique. The occurrence and the nature of chaotic attractors are verified by evaluating Liapunov exponents and average power spectra. The effect of the gyroscope's spinning speed is also studied, and it is shown that the gyroscope's spin velocity ω zhas a significant effect on the dynamic behaviour of the motion.

Perturbed steepest-descent technique in multiextremal problems

The steepest-descent technique dealing with the perturbed values of the objective function and its gradients and with nonexact line searches is considered. Attention is given to the case where the perturbations do not decrease on the algorithm trajectories; the aim is to investigate how perturbations at every iteration of the algorithm perturb its original attractor set. Based on the Liapunov direct method for attraction analysis of discrete-time processes, a sharp estimation of the attractor set generated by a perturbed steepest-descent technique with respect to the perturbation magnitudes is obtained. Some global optimization properties of finite-difference analogues of the gradient method are discovered. These properties are not inherent in methods which use exact gradients.

On quadratic Liapunov functionals for linear difference equations

This paper studies some stability properties of vector linear difference equations of the form x( t) = ∑ k j=1 A j x( t − r j ), 0 < r 1 < r 2 < … < r k < ∞. This is done through the direct method of Liapunov, by means of a quadratic functional defined in an appropriate L 2 space. The derivative of this functional along the equation, nevertheless, is defined in a finite dimensional euclidean space. The positive definiteness of certain matrices involved, together with the bounded real lemma of system theory, furnishes useful criteria to determine stability properties of the given equation. In particular, a criterion is given to determine stability with respect to the delays r i .

Necessary and Sufficient Conditions for Boundedness and Oscillation in the Retarded Liénard Equation

Consider the retarded Liénard equationx″+f(x)x′+g(x(t−h))=0, wheref,g:R→Rare continuous andh≥0. Using Liapunov's direct method, we give necessary and sufficient conditions to ensure boundedness and oscillation of all solutions and their derivatives.

NON-LINEAR DYNAMIC BUCKLING OF A SIMPLE MODEL VIA THE LIAPUNOV DIRECT METHOD

The non-linear dynamic response of a simple one-degree-of-freedom dissipative/non dissipative model under the more general case of partial follower loading is considered. The study is confined to imperfection sensitive systems which lose their static stability through a limit point. The analysis proceeds first by employing the inflection point criterion for dynamic buckling which is subsequently confirmed via the Liapunov direct method for global stability (instability). Attention is focused on determining the level of the dynamic buckling load above which the associated equilibria on the fundamental equilibrium path are globally unstable, although they are locally asymptotically stable.

Stability results for hyperbolic and parabolic equations

Abstract Some stability results for hyperbolic and parabolic P. D.Es. are presented and recent results are reviewed. The Liapunov direct method is used.

Control of seismically excited building structures using variable damper systems

This paper aims to study the effectiveness of using an active variable damper system to control seismically excited building structures. The Liapunov direct method was used to design the controller and the weighting matrix Q can be obtained by making the derivative of the Liapunov function as negative as possible. The control system, using an active variable damper, provides time-varying damping to the structure at every instant of time and the error estimate of system matrix A will not significantly effect the control result. A comparison of the efficiency of the control between classical optimal control and the present method is made. A combination of the active variable damper system with a passive base-isolation system proved to be the most effective form of control of the building response when subjected to seismic excitation even under different site conditions

Stability of elastic and viscoelastic systems under a random perturbations of their parameters

The present paper is concerned with the investigation of the almost sure stability of elastic and viscoelastic systems, when their parameters assume a random wide-band stationary process. The parameters are parametric loads, characteristics of external damping and material viscosity. With the help of Liapunov's direct method, the sufficient condition of the almost sure asymptotic stability for distributed parameter systems with respect to perturbations of initial conditions of an arbitrary form is obtained. It is shown that, in some cases, this condition coincides with a similar condition derived from the assumption that the form of sure and required perturbations coincides with the first eigenfunction of system oscillations. However, an example is given for the stability of a viscoelastic rod, when the perturbations of initial conditions are more dangerous, if their form differs from the first eigenfunction.

STABILITY OF EULER, LAGRANGE AND KOVALEVSKAYA GYROS IN A NEWTONIAN CENTRAL FORCE FIELD WITH SECOND ORDER TERMS

The stability of the permanent rotation of the Euler, Lagrange and Kovalevskaya gyros about a fixed point in the Newtonian central force field is investigated by Liapunov direct method, which improve Leimanis’ and Rumiantsev’s result of stability conditions. For the sake of high accuracy, the first and second order terms of R-1 expressed by potential energy of the Newtonian central force field have been considered, where the R is the distance from the fixed point to the center of attraction.

The Stability of the Rotation of a Magnetized Gyro with a Fixed Point

In conventional studies about the gyro motion the gyroscopic forces which are produced by the Lorentz action of the magnetic field on the electrical charges moving in space have not been considered. Euler gyro, Lagrange gyro and Kovalveskaya gyro are three representative gyros. In this paper, the stability and instability conditions of the rotation of the three representative gyros under the action of the Lorentz forces both in a uniform force field and in a central force field are obtained by using Liapunov's direct method.

Global Stability of a Shielded Dual-Spin Satellite

Sufficient conditions for the global stabilities of equilibrium positions of a dual-spin satellite equipped with a cylindrical electrostatic screen normal to the circular equatorial orbital plane system under the action of moments induced by the gravitational and Lorentz forces are obtained by Liapunov's direct method. The problem is then studied with respect to all variables, including the angular velocity of the rotor. The instabilities of the problem are solved by the first approximation theorem. Furthermore, sufficient conditions for the global stability of rotation with arbitrary angular velocity for an axisymmetric dual-spin satellite equipped with an electrostatic screen are presented by applying Liapunov's direct method.

STABILITY OF RATE GYRO IN A VEHICLE WITH UNCERTAIN ANGULAR VELOCITY AND ACCELERATION

Using the Liapunov direct method, we study the stability of motion of a single-axis rate gyro mounted on a space vehicle which is simultaneously accelerating (ω)x(t)) with respect to the output axis and rotating (ωz(t)) about the spin axis of gyro. The analytical relations, which must be satisfied by the gyro parameters, are obtained for the asymptotical stability of the gimbal motion. In addition, by using a fourth-order Runge-Kutta numerical method, the trajectories of motions of a rate gyro are obtained to verify its stability.

Collision avoidance in a two-point system via Liapunov's second method

The geometric problem of finding a path for a moving solid among other solid obstacles is well known in robotics in the area of real-time obstacle avoidance for manipulators and mobile robots. In this paper, a solution to the problem, also known as the findpath problem, is provided via the second or direct method of Liapunov. The method is used to construct control functions for the collision avoidance between two point masses which are required to move to designated areas or targets located in the horizontal plane. Two new results are presented. The first result opens up the possibility of analysing in a more effective manner the dynamics of more than two point masses. The second new result addresses, via generalized control functions, important collision avoidance issues which are (1) improving collision avoidance between objects, (2) obtaining low control inputs for collision avoidance and convergence to targets, and (3) having the best time to reach a target safely.