Method of an Asymptotic Analysis of the Nonlinear Monotonic Stability of the Oscillation at the Problem of Damping of the Angle of Attack of a Symmetric Spacecraft

Abstract

One of the current directions in the development of the modern theory of oscillations is the elaboration of effective methods for analyzing the stability of solutions of dynamical systems. The aim of the work is to develop a new asymptotic method for studying the nonlinear monotonic stability of the amplitude of plane oscillations in a dynamic system of equations with one fast phase. The method is based on the use of the method of variation of an arbitrary constant, the averaging method, and the classical method of mathematical research of the function of one independent variable. It is assumed that the resulting approximate analytical function is defined and twice continuously differentiable on the entire considered interval of change of the independent variable. It describes the nonlinear and monotonic evolution of the oscillation amplitude on the entire considered interval of change of the independent variable. In the paper, this method is applied to the problem of nonlinear monotonic aerodynamic damping of the amplitude of oscillations of the angle of attack during the descent of a symmetric spacecraft in the atmosphere of Mars. The method presented in this paper made it possible to find all characteristic cases of nonlinear monotonic stability and instability of the oscillation amplitude of the angle of attack. In addition, one should speak of a symmetrical quantity of different cases of stability and instability, located on different sides of the zero value of the first average derivative of the angle of attack.