Search

We consider the motion of a mechanical system with several degrees of freedom near zero of the phase space of states under conditions of constantly acting small perturbations. The generalized forces are represented in the dynamic equations by homogeneous forms of the first and the third degree with respect to the phase coordinates and by small time functions characterizing the constantly acting perturbations. It is assumed that there are no multiple eigenvalues of the matrix of the system linear part. For a definitely positive quadratic Lyapunov function, we define a differential inequality with Riccati differential comparison equation, together with an exponential differential inequality that is integrable in quadratures. When solving Riccati quadratic differential inequality, we assume one particular solution of Riccati equation to be known. As a result of integrating in quadratures of the exponential differential inequalities, the estimate of transient processes in the finite domain of the phase coordinates is obtained.

We consider a system of differential equations in the Cauchy form, containing homogeneous linear and cubic forms with respect to phase variables and small perturbing functions. We consider the system as a polynomial dynamical system under constant disturbances. We propose a method for constructing a nonlinear exponential differential inequality for a quadratic Lyapunov function and a square inequality. By integrating the exponential differential inequality, we obtain an estimate of the Lyapunov function from a function of time and initial conditions and an estimate of the phase variables.

The paper proposes a method for identification of axial moment of inertia of the mechanical system called reaction wheel pendulum with a viscous friction in the bearings of the suspension. The method is based on the reversible symmetric motions. Pendulum system motion includes a free measured motion and reverse symmetrical motion at the same angular interval. The pendulum includes a rod with a low-power DC motor with a flywheel attached to the end of the rod. The angle of rotation and velocity of the rod and the flywheel are measured by encoders. The paper introduces a new method,presents a design formula,a mathematical model of the pendulum system and a robust motor control law for it. The method is based on energy algorithm and control residing in electric motor operational changes by means of a flywheel. The mechanical system moves symmetrically that is provided by nonuniform controlled flywheel rotation. As a result, the influence of dissipative factors on identification results is eliminated. Dynamic modeling is carried out for the pendulum system and proves high accuracy of the method. The research results can be used for identification of complex mechanical systems under the action of resistance, dissipative and other forces.

In this paper we consider an autonomous mechanical system in a finite neighborhood of the zero of the phase space of states. The system is given as a matrix differential equation in the Cauchy form with the right-hand side of the polynomial structure. We propose a method for constructing a sequence of linear inhomogeneous differential inequalities for Lyapunov functions. As a result, we obtain estimates of transient processes in the form of functional inequalities.

We consider an equation of an autonomous dynamical system with one degree of freedom. It contains the linear and cubic forms relative to phase variables. In order to simplify the mathematical model we use the modified asymptotic method of Poincare-Dulac conversion. Using Chebyshev approximations of high-degree polynomials with polynomials of smaller degrees we reduce the residual error. We consider non-oscillatory mechanical systems in the case of absence of internal Poincare resonances that leads to a linear form.

A knowledge of the inertia tensors and centre of gravity locations of rigid bodies and rigid mechanical systems is required in precise control of motion and in many other applications whenever the dynamics of them is significantly determined by these inertia parameters. This paper presents a new method for identifying the inertia tensor and coordinates of the centre of mass of a rigid body or of a rigid mechanical system. The proposed method is based on the work-energy principle and provides high accuracy of identification even on devices with essential dissipation. The tensor and centre of gravity location are defined on a single spherical motion — symmetric precession. The paper gives a survey of existing inertia parameter identification methods, the theoretical fundamentals of the proposed identification method, the hybrid identification device and data processing procedure with error estimation on dissipative systems using proposed and classical methods.

This paper presents a new approach for matrix root-clustering in sophisticated and multiply-connected regions of the complex plane. The parametric sweeping method and a concept of the closed forbidden region covered by a set of modified three-parametrical Cassini regions are used. A conformal mapping approach was applied to formulate the main results of the paper. An application of the developed method to the problem of matrix root-clustering in a multiply connected region is shown for illustration.

A dynamic equation of a mechanical system with one degree of freedom with functionally odd nonlinear recover forces and small dissipative forces is considered. An improved method of transformation and integration of the equation, based on a method of normalization of Poincare-Dulac is developed. The improvement of the method is in application of the Chebyshev's economization to high-order nonlinear terms.