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Proposed work is the seventh work of the cycle on differential and topological diagnostics. Diagnostics ofmalfunctions in the system of indirect control of an object whose motion is described by nonlinear ordinary differential equations of the third order (the problem of B.V. Bulgakov) is considered. In the diagnostic algorithm used, the scope of control is selected, when a certain constant value is assigned to each faulty system, and according to certain rules, the numbers obtained in the process of integrating equations and characterizing the functional state of the system are compared with it. Troubleshooting is also considered in one system of direct control of the movement of the aircraft, which can be described by nonlinear differential equations of the second order. At the same time, a certain diagnostic algorithm is built and used in accordance with the previously developed methodology.

Proposed work is the sixth work of the cycle on differential and topological diagnostics. It is shown that the diagnostics in the case of trajectorial measurements corrupted by noise, which is a stochastic process of the normal white noise type with zero mean value and bounded spectrum, can be performed by using the diagnostic algorithms obtained in [5], i.e., the results of this section remain valid even in this rather general case; moreover, the diagnostic functional, which was introduced in the theorem of [5] a priori, is now obtained a posteriori.

We systematize some results on the study of the equations of spatial motion of dynamically symmetric fixed rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of a spatial motion of a free rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint, or the center of mass of the body moves rectilinearly and uniformly; this means that there exists a nonconservative couple of forces in the system

Proposed activity presents next stage of the study of the problem of the plane-parallel motion of a rigid body interacting with a resistant medium through the frontal plane part of its external surface. Under constructing of the force acting of medium, we use the information on the properties of medium streamline fl w around in quasistationarity conditions (for instance, on the homogeneous circular cylinder input into the water). The medium motion is not studied, and we consider such problem in which the characteristic time of the body motion with respect to its center of masses is comparable with the characteristic time of motion of the center of masses itself.

The so-called pendulum-like systems arise in dynamics of a rigid body in a non-conservative field, in the theory of oscillations, and in theoretical physics. In this article, the methods of analysis are described which allow us to generalize the previous results. Herewith, we deal with some qualitative questions of the theory of ordinary differential equations whose solution facilitates studying some dynamical systems. In result of investigating more general classes of systems, we show that these general systems possess the already known family of nonequivalent phase portraits. We also deal with the aspect of integrability.

In this review, we discuss new cases of integrable systems on the tangent bundles of finite-dimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in nonconservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

In the proposed cycle of work, we begin the study of the motion of an aircraft which is described bynonlinear ordinary differential equations. Based on these equations, the probable malfunctions in the motioncontrol system are classified, the concepts of reference malfunctions and their neighborhoods are introduced,the mathematical modeling of these malfunctions and their neighborhoods is carried out, the concept ofdiagnostic space is introduced, and the mathematical structure of this space is defined. Proposed work isthe first in the cycle, therefore, the classification of malfunctions is given. This activity is also a preparatorypart of the diagnostic problem, which can be represented in the form of two successively solved problems,i.e., control problem, that is the problem of determining the presence of a malfunction in the system, anddiagnostic problem, that is the recognition problem of malfunction specification. This activity is just anillustration of the proposed approach.

Under study is the problem of spatial free deceleration of a rigid body in a resisting medium. It is assumed that a axisymmetric homogeneous body interacts with the medium only through the front part of its outer surface that has the shape of flat circular disk. Under the simplest assumptions about the impact forces from the medium, it is demonstrated that any oscillations of bounded amplitude are impossible. In this case, any precise analytical description of the force-momentum characteristics of the medium impact on the disk is missing. For this reason, we use the method of “embedding” the problem into a wider class of problems. This allows us to obtain some relatively complete qualitative description of the body motion. For the dynamical systems under consideration, it is possible to obtain particular solutions as well as the families of phase portraits in the three-dimensional space of quasi-velocities which consist of a countable set of the phase portraits that are trajectory-nonequivalent and have different nonlinear qualitative properties.

Some qualitative analysis is carried out of the rectilinear and spatial problems concerning the motion of a rigid body in a resisting medium.Anonlinearmodel is constructed of impact of the mediumon the rigid body, which takes into account the dependence of the arm of force on the reduced angular velocity of the body. Moreover, the moment of this force itself is also a function of the angle of attack. As was shown by the processing the experimental data on the motion of homogeneous circular cylinders in water, these circumstances should be taken into account in the simulation. The analysis of the plane and spatial models of the interaction of a rigid body with a medium reveal the sufficient conditions of stability of the key regime of motion, i.e., the translational rectilinear deceleration. It is also shown that, under certain conditions, both stable or unstable auto-oscillating regimes can be presented in the system.

A mathematical model of the influence of a medium on a rigid body with some part of its external surface being flat is considered with due allowance for an additional dependence of the moment of the medium action force on the angular velocity of the body. A full system of equations of motion is given under quasi-steady conditions; the dynamic part of this system forms an independent third-order system, and an independent second-order subsystem is split from the full system. A new family of phase portraits on a phase cylinder of quasi-velocities is obtained. It is demonstrated that the results obtained allow one to design hollow circular cylinders (“shell cases”), which can ensure necessary stability in conducting additional full-scale experiments.

A mathematical model of the effect of a medium on a homogeneous rigid body whose outer surface includes a circumferential cone is considered. The complete system of equations of motion under quasistationarity conditions is given. In the dynamic part forming an independent third-order system, an independent second-order subsystem is distinguished. A new two-parameter family of phase portraits on the phase cylinder of quasivelocities is obtained.

A complete analysis of the phase trajectories representing the motion of a rigid body in a resisting medium in quasistationary conditions is carried out. More general systems characterized by hidden nontrivial symmetries are studied

A certain single-phase problem with free border is studied. The local solvability of such problem is proved. The more general method investigated earlier is used in concrete event in this work.

The paper studies the spatial motion of a rigid body in a resisting medium under the action of a follower force that causes the center of mass to move rectilinearly and uniformly. The body, which is axisymmetric and homogeneous, interacts with the medium by its frontal area that has the form of a flat circular disk. Since there is no exact analytic description of the forces and torques exerted by the medium on the disk, the problem is “immersed” in a wider class of problems. Partial solutions and phase portraits in the three-dimensional space of quasivelocities are obtained for the dynamic systems under consideration. The transcendental first integrals of the dynamic part of the equations of motion are listed

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

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

The paper presents a complete qualitative analysis of the model plane-parallel motion of a body (plate) through a resisting medium with jet or detached flow. A simplified system is analyzed on the phase plane. A geometrical interpretation is given to the motion of the plate

A dynamic model of the interaction of a rigid body with a jet flow of a resistant medium is considered. This model allows us to obtain three-dimensional analogs of plane dynamic solutions for a solid interacting with the medium and to reveal new cases where the equations are Jacobi integrable. In such cases, the integrals are expressed in terms of elementary functions. The classical problems of a spherical pendulum in a flow and three-dimensional motion of a body with a servoconstraint are shown to be integrable. Mechanical and topological analogs of these problems are found

The results of this work appeared in the process of studying a certain problem on the rigid body motion in a medium with resistance [Chaplygin, 1933], [Chaplygin, 1976], [Shamolin-1, 2007], [Shamolin-2, 2007], where we needed to deal with first integrals having nonstandard properties. Precisely, they are not analytic, not smooth, and on certain sets, they can be even discontinuous. Moreover, they are expressed through a finite combination of elementary functions. However, the latter circumstances allowed us to carry out a complete analysis of all phase trajectories and show those their properties which have a ”roughness” and are preserved for systems of a more general form having certain symmetries of latent type. Therefore, it is interesting to study sufficiently wide classes of dynamical systems that have analogous properties and moreover, coming from the dynamics of a rigid body interacting with a medium.

Institute of Mechanics, Lomonosov Moscow State University, Michurinskii Ave., 1, 119899 Moscow, Russian FederationThe results of the presented work are due to the study of the applied problem of the rigid body motion in a resisting medium.Moreearlierthecomplete listsoftranscendentalfirstintegralsexpressed througha finitecombinationofelementary functionswere obtained. This circumstance allowed the author to perform a complete analysis of all phase trajectories and highlightthose properties of them which exhibit the roughness and preserve for systems of a more general form.

The purpose of the activity is to elaborate the qualitative methods for studying the dynamics of rigid bodies interacting with a resisting medium under quasistationarity conditions. This material refers equally to the qualitative theory of ordinary differential equations and the dynamics of rigid bodies. We use the properties of body’s motion in a medium under conditions of the jet flow past this body. We study the plane model problems of the motion of a body with the cone form of its shape in a resisting medium. The new families of phase portraits of variable dissipation systems are obtained, their absolute or relative roughness is demonstrated. The integrable cases of equations of motion of rigid bodies are found. c � 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

A vast number of papers are devoted to studying the complete integrability of equations of four-dimensional rigid-body motion. Although in studying low-dimensional equations of motion of quite concrete (two- and three-dimensional) rigid bodies in a nonconservative force field, the author arrived at the idea of generalizing the equations to the case of a four-dimensional rigid body in an analogous nonconservative force field. As a result of such a generalization, the author obtained the variety of cases of integrability in the problem of body motion in a resisting medium that fills the four-dimensional space in the presence of a certain tracing force that allows one to reduce the order of the general system of dynamical equations of motion in a methodical way.