Problems of Qualitative Analysis in the Spatial Dynamics of Rigid Bodies Interacting with Media
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PROBLEMS OF QUALITATIVE ANALYSIS IN THE SPATIAL DYNAMICS OF RIGID BODIES INTERACTING WITH MEDIA M. V. Shamolin
UDC 517.933
Abstract. In this paper, we examine the problem on the spatial free deceleration of a rigid body in a resistive medium under the assumption that the interaction of the homogeneous axisymmetric body with the medium is concentrated on the frontal part of the surface, which has the shape of a flat circular disk. In earlier works of the author, under the simplest assumptions on interaction forces, the impossibility of oscillations with bounded amplitude was proved. Note that an exact analytic description of forces and moments of the body-medium interaction is unknown, so we use the method of “embedding” of the problem into a wider class of problems; this allows one to obtain a sufficiently complete qualitative description of the motion of the body. For dynamical systems considered, we obtain particular solutions and families of phase portraits of quasi-velocities in the three-dimensional space that consist of countable sets of nonequivalent portraits with different nonlinear qualitative properties. Keywords and phrases: rigid body, resistive medium, qualitative analysis, numerical analysis. AMS Subject Classification: 70G60
1. Model assumptions and equations of motion. Consider the problem on the spatial motion of a homogeneous, axisymmetric rigid body of mass m, whose surface contains a planar circular disk interacting with a medium by the laws of jet flows (see [4, 6, 13]). Assume that the other part of the surface of the body is located inside the domain bounded by a jet surface breaking from the boundary of the disk and is not affected by the medium. Similar conditions appear, for example, when a homogeneous circular cylinder is immersed in liquid (see [5, 8, 13]). Assume that tangent forces do not act on the disk. Then the force S acting on the body from the medium does not change its orientation with respect to the body (it is directed along the normal of the disk) and is quadratic with respect to the speed of its center (see Fig. 1). We also assume that the gravitational force acting on the body is negligible compared with the resistance of the medium. Under the conditions listed above, among motions of the body there exists a mode of rectilinear translational deceleration: the body moves along its symmetry axis, i.e., perpendicularly to the plane of the disk. Note that the application point N of the force S acting on the body from the medium coincides with the geometric center D of the disk. Under perturbations of the rectilinear translational deceleration, the velocity vector v of the point D, generally speaking, deviates from the axis of geometric symmetry by a certain angle α, called the attack angle. The application point N of the resistance force S shifts by a value of DN = R1 from the center of the disk; it lies in the plane containing the velocity vector v and the symmetry axis of the body. These arguments are confirmed by the fact that a plate located at an angle to the direction of the
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