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Methods of Analysis of Dynamic Systems with the Various Dissipation in Dynamics of a Rigid Body

Maxim V. Shamolin
Due to its complexity, the problem of the motion of a rigid body in an unbounded medium requires the introduction of certain simplifying restrictions. The main aim in this connection is to introduce hypotheses that would make it possible to study the motion of the rigid body separately from the motion of the medium in which the body is embedded. On the one hand, a similar approach was realized in the classical Kirchhoff problem on the motion of a body in an unbounded ideal incompressible fluid that undergoes an irrotational motion and is at rest at infinity. On the other hand, it is obvious that the above-mentioned Kirchhoff problem does not exhaust the possibilities of this kind of simulation.
In this activity, we consider the possibility of transferring the results of the dynamics of the plane-parallel motion of a homogeneous axisymmetric rigid body interacting with a uniform flow of a resisting medium through its forward circular face to the case of three-dimensional motion. In contrast to the preceding works, the medium action on the rigid body is simulated with the inclusion of the effects of the so-called rotary derivatives of the moment of hydroaerodynamic forces with respect to the components of the angular velocity of the body itself.
On the basis of certain hypotheses, the main one of which is the quasi-stationarity hypothesis, a three-dimensional dynamic model of the medium action on the body was developed. In this connection, the possibility arises to formalize the model assumptions and derive a complete system of equations.
In what follows, we will address some typical "representatives" of the classes of the medium action functions under consideration, namely, the Chaplygin functions.
We will use Chaplygin's result as a reference point. Chaplygin calculated the medium action functions for an infinitely long plate in plane-parallel motion in the oncoming flow following the jet flow laws. In this case, the distance between the drag application point (center of pressure) and the plate center is proportional to the sine of the angle of attack, while the Newtonian drag coefficient is proportional to its cosine.
Moreover, integrable cases in the dynamics of the three-dimensional motion of a rigid body were also found for other model problems in the early works of the present author. For the Chaplygin medium action functions, the systems had a complete set of transcendental first integrals, which could be expressed in terms of a finite combination of elementary functions. In this case, transcendence is understood in the sense of the theory of functions of a complex variable (that is, their continuations to the complex plane have essentially singular points).
Dynamic systems investigated are the dynamic systems with variable dissipation with zero mean value (over the angle of attack, in our case). This means that the integral over a period of the angle of attack from the divergence of its right-hand side is equal to zero [this integral is responsible for the phase volume variation (after the corresponding reduction of the system)]. In this sense, the system is "semiconservative."
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