Mathematical modeling of ambiguity domains in the aerodynamic pendulum model

This paper presents a mathematical model of the self-oscillations of an aerodynamic pendulum in a fluid flow. The external forces acting on the body from the fluid are generated using the assumption of quasi-stationary flow past a plate. A mathematical model of the body's motion is constructed. It includes differential equations describing the body's motion and algebraic transcendental kinematic constraint equations. Experimental aerodynamic functions for lift and drag are also included in the right-hand side of the equations of motion. After introducing new dimensionless variables, equations are derived that describe the mathematical model of the body in dimensionless form. The simplest steady-state regimes are found, and stability regions are depicted in the space of geometric and mechanical parameters. In the numerical solution of the problem, the angle of attack is first determined from the constraint equations, and then the fundamental equations are integrated. Using simple operations, additional algebraic transcendental relations are transformed. The geometric meaning of the coefficients included in the ambiguity equation and its graphical interpretation are demonstrated. An analysis of the number of solutions to the ambiguity equation was conducted, and in some cases, violations of the uniqueness of the angle of attack were demonstrated. Using the MATLAB computer mathematics system, ambiguity regions were found in the subspace of initial conditions. Numerical studies of the behavior of trajectories in these regions are being conducted.