Quasi-optimal deceleration of rotational motion of a dynamically symmetric rigid body in a resisting medium

We study the problem of quasi-optimal (with respect to the response time) deceleration of rotational motion of a free rigid body which experiences a small retarding torque generated by a linearly resisting medium. We assume that the undeformed body is dynamically symmetric and its mass is concentrated on the symmetry axis. A system of nonlinear differential equations describing the evolution of rotation of the rigid body is obtained and studied.     

A model problem of deceleration of a body in a resisting medium with a jet flow around the body

A mathematical model is constructed describing the deceleration of a solid body moving in a medium with a jet flow around the body. The regime of translational deceleration is shown to be normally unstable. This has made it possible to develop a relatively simple technique for determining model parameters experimentally. An example of the application of this technique to a cylindrical body is presented.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 23–27, May–June, 1995.The work was financially supported by the Russian Foundation for Fundamental Research (project No. 93-013-17637).     

Stability of a rigid body translating in a resisting medium

The paper discusses a nonlinear model that describes the interaction of a rigid body with a medium and takes into account (based on experimental data on the motion of circular cylinders in water) the dependence of the arm of the force on the normalized angular velocity of the body and the dependence of the moment of the force on the angle of attack. An analysis of plane and spatial models (in the presence or absence of an additional follower force) leads to sufficient stability conditions for translational motion, as one of the key types of motions. Either stable or unstable self-oscillation can be observed under certain conditions     

论刚体的平衡

就刚体在一般情况下的平衡状态, 运动特征,及刚体平衡的实验方法作一讨论.     

Stability problem for pendulum-type motions of a rigid body in the Goryachev-Chaplygin case

We consider the motion of a rigid body with a single fixed point in a homogeneous gravity field. The body mass geometry and the initial conditions for its motion correspond to the case of Goryachev—Chaplygin integrability. We study the orbital stability problem for periodic motions corresponding to vibrations and rotations of the rigid body rotating about the equatorial axis of the inertia ellipsoid.In [1], it was proved that these periodic motions are orbitally unstable in the linear approximation. It was also shown that, to solve the stability problem in the nonlinear setting, it does not suffice to analyze terms up to the fourth order in the expansion of the Hamiltonian function in the canonical variables.The present paper shows that in this problem one deals with a special case where standard methods for stability analysis based on the coefficients in the normal form of the Hamiltonian of the perturbed equations of motion do not apply. We use Chetaev’s theorem to prove the orbital instability of these periodic motions in the rigorous nonlinear statement of the problem. The proof uses the additional first integral of the Goryachev—Chaplygin problem in an essential way.     

The supersonic motion of a rigid body in an elastic medium with friction

The plane problem of supersonic steady motion of a body in an elastic medium is solved. Two possible cases of body motion are considered depending on its velocity. In the first case, the body moves at a velocity greater than the velocity of transverse waves but smaller than the velocity of longitudinal waves. In the second case, the body moves at a velocity greater than the velocity of longitudinal waves. An analytic solution of the problem under study is obtained and analyzed. It is shown that friction substantially influences the penetration process.     

A research on the problem of the motion of a rigid body with quasi gyroscopic structure and variable mass,free in space

Summary The problem of the motion of a rigid body, with variable mass, free in space is considered. The problem is studied starting from the Eulerian differential equations, which are modified in relation to the variability of the mass and of the principal moments of inertia.The case of the quasi-gyroscopic structure of the vehicle is considered and the problem of motion using Poincarè's variational equations of the Eulerian dynamical equations is studied. The external moments depend on time only. The variational equations are solved and the stability of the movements considered is demonstrated.

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Sommario Si prende in considerazione il problema del moto di un corpo rigido di massa variabile libero nello spazio. Ciò partendo dalle equazioni differenziali euleriane corrette per effetto della variazione della massa e dei momenti principali d'inerzia. Nella aprossimazione in cui è valida una tale impostazione, si suppone esista una piccola differenza fra i due primi momenti principali d'inerzia del corpo, cioè che il corpo stesso sia a struttura quasi giroscopica. I momenti sollecitanti vengono supposti dipendenti dal solo tempo. Si risolvono le equazioni alle variazioni e si fa vedere che le variazioni si mantengono sempre piccole nel tempo, consentendo così di concludere per una affermazione di stabilità dei moti nel caso a struttura giroscopica.

     

Some model problems of dynamics for a rigid body interacting with a medium

The paper addresses the stability of solutions of ordinary differential equations of particular type for different statements and assumptions. The equations are interpreted as models of motion of a rigid body under the action of the ambient medium __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 49–67, October 2007.     

A multiparameter family of phase portraits in the dynamics of a rigid body interacting with a medium

The problem of plane-parallel motion of a uniform symmetric rigid body interacting with a medium only through a flat region of its outer surface is studied. The force field is constructed on the basis of information on the properties of jet flow under quasistationarity conditions. The motion of the medium is not studied. The problem of rigid body dynamics is considered for the case when the characteristic time of motion of the body relative to its center of mass is comparable with the characteristic time of motion of this mass center.     

Kinematic problem of optimal nonlinear stabilization of angular motion of a rigid body

The problem of optimal transfer of a rigid body to a prescribed trajectory of preset angular motion is considered in the nonlinear statement. (The control is the vector of absolute angular velocity of the rigid body.) The functional to be minimized is a mixed integral quadratic performance criterion characterizing the general energy expenditure on the control and deviations in the state coordinates.Pontryagin’s maximum principle is used to construct the general analytic solution of the problem in question which satisfies the necessary optimality condition and ensures the asymptotically stable transfer of the rigid body to any chosen trajectory of preset angular motion. It is shown that the obtained solution also satisfies Krasovskii’s optimal stabilization theorem.     

One approach to solving the problem of the contact of a cylindrical shell with a rigid body

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Sverdlovsk. Translated from Prikladnaya Mekhanika, Vol. 26, No. 5, pp. 36–42, May, 1990.