Comparison of different notation for equations of motion of a body in a medium flow

In 1–6], a model of a nonstationary action of a medium flow on a body moving in this flow was constructed in the form of an associated dynamical system of second order. In the literature, the representation of the aerodynamic force in integral form with a Duhamel type integral is often used (e.g., see 7, 8]). In the present paper, we pay attention to the fact that a system of ODE is equivalent not to a single integro-differential equation but to a family of such equations. Therefore, it is necessary to discuss the problem of the correspondence between their solutions. The integro-differential representation of the aerodynamic force is reduced to a form convenient to realize the procedure of separation of motions. In this case, we single out the first two approximations with respect to a small parameter. It turns out that in the case of actual airfoils one can speak of “detached” rather than “attached” mass. In the problem on the forced drag of an airfoil in a flow, it is shown that for a sufficiently large acceleration the aerodynamic force can change its direction and turn from a drag force into an “accelerating” force for some time. At the same time, in the case of free drag of a sufficiently light plate, the “acceleration” effect is not observed, but in the course of deceleration the plate moves from it original position in the direction opposite to the initial direction of motion.