MATHEMATICAL MODEL OF SMALL BODY VIBRATIONS IN THE STREAM OF THE ENVIRONMENT
Abstract
This article explores the mathematical model of a body that performs self-oscillations in a flow of a quasistatic medium under the influence of aerodynamic forces. The equations of motion of the body under consideration and the kinematic relations connecting the phase coordinates with the angle of attack are derived. The equilibrium equations were solved and it was shown that the only equilibrium position is the state of rest. The equations of the first approximation are obtained and the stability of the state of rest is studied using the Hurwitz criterion. It is shown that as a result of interaction with the medium, the body in question can oscillate with increasing amplitude (flutter) in the medium flow. Areas of stability are constructed on the plane of geometric parameters: spring stiffness and rod length. In the mathematical package MATLAB, a set of programs has been proposed that allow for numerical studies that implement the numerical integration of the equations describing the vibrations of a plate with a fixed center of pressure. Such a model is possible provided that the length of the rod is much greater than the width of the plate. When the program is started, the stability region is constructed and the geometrical parameters are entered on it: the spring stiffness and the rod length. Next, a vector of initial conditions is introduced. When searching for a numerical solution, the ode45 procedure is used, which implements fourth and fifth order Runge – Kutta methods with a variable step. When searching for a numerical solution, experimental aerodynamic functions are interpolated by a cubic spline. The solution obtained by integrating is depicted on the graph as Lissajous figures. Thus, a mathematical model of plate oscillations was developed, a parametric stability analysis was carried out, with the help of a complex of programs based on a specialized computer mathematics system, it is possible to confirm the obtained analytical results.
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