Analysis of the Ambiguity of the Angle of Attack for the Model of an Aerodynamic Pendulum in the Flow of the Medium
Abstract
The work is devoted to the construction and study of a mathematical model of self-oscillations of an aerodynamic pendulum in a medium flow. The model of quasi-static flow of the plate by the medium was adopted as a model of the impact of the medium on the body. According to this hypothesis, the aerodynamic forces acting on the body are applied at the center of pressure. In the problem under consideration, the center of pressure is movable relative to the plate. A transition to new dimensionless variables is carried out. The violation of uniqueness in determining the angle of attack is shown. A parametric analysis of ambiguity domains is carried out. It is shown that in the most characteristic position of equilibrium, corresponding to the state of rest, there are no ambiguity regions. The stability of the equilibrium position corresponding to the state of rest was studied, in which the Hurwitz criterion was implemented, and the stability region was depicted. It is shown that aerodynamic forces for bodies with some forms can contribute to the development of self-oscillations, and for others, attenuation. In the MATLAB mathematical package, a set of programs is proposed allowing to carry out numerical research, realizing the numerical integration of equations describing the vibrations of a plate with a fixed pressure center. Such a model is possible, provided that the length of the rod is much greater than the width of the plate. When starting the program, a stability area is built and geometric parameters are entered on it: spring stiffness and rod length.
The initial conditions vector is then entered. To find a numerical solution, the ode45 procedure is used, which implements the Runge-Kutt methods of the fourth and fifth orders with a variable step. In the function-file, at each step of the Runge-Kutta method, a nonlinear equation is solved from which the angle of attack is determined. When searching for a numerical solution, the experimental aerodynamic functions are interpolated by a cubic spline. The solution obtained by integration is shown on the graph in the form of Lissajous figures. Thus, a mathematical model of plate oscillations has been developed, a parametric analysis of stability has been carried out, and with the help of a set of programs based on a specialized system of computer mathematics, it is possible to confirm the obtained analytical results.
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