The capability of a balancing of a rotated body by rigid bodies with a fixed point on rotation axis is investigated. In problems of the first type by bodies, which are counterbalanced there can be rotors, which speed rotated, spindles of machine tools etc. In problems of the second type by bodies, which are counterbalanced there can be artificial satellites of the Earth, spaceships, which position in space is stabilized by rotation.

Further:

To define conditions, which are imposed on geometry of masses, at which realization of CM with fixed points on rotation axis of a rotor will exhibit an autobalancing property. To create new designs of passive autobalancers. To create a classification of passive autobalancers. To investigate theoretically and experimentally dynamics of classical and nonclassical autobalancers. To define conditions imposed on parameters of a system, at which realization the auto balancing will occur.

It is offered to create the simplified models of a body, which is counterbalanced also of autobalancers. In models of the first type the plane-parallel driving of a body is taken into account only, and attitudes round cross-sectional axes, concerning rotation axis is neglected. The models are aplicable for rotors on symmetrical support, rotors on the cantilever shaft, provided that the dimensions of a rotor in five and more times are less than length of the shaft, for artificial satellites of ground, which have the rather flat form and rotation axis of perpendicular their plane, and under other conditions. In models of the second type it are taken into account only of driving of a body round a fixed point (center of masses of a material system). The models are aplicable for a research of dynamics of gyroscopes, rotors on the cantilever shaft, provided that the dimensions of a rotor are close to length of the shaft, for satellites or spacecraft provided that under the form they are close to a full-sphere.

At a construction of mathematical models it is expedient to use a mobile frame, which synchronously rotates together with a rotor. Then the system of autonomous differential second-kind equations is received. At compiling of the equations of dynamics it is expedient to use the common theorems of dynamics, since they enable by a short way to receive the differential equations of driving of a system. The fixed solutions of an obtained system of the differential equations are systems, steady-state driving. They are formed by so-called basic motions, in which the rotor is counterbalanced, and spurious motions, in which the rotor is unbalanced. The research of dynamics of a system is reduced to a research of a stability of various steady-state motions of a system. For functionability of autobalancers, it is necessary, that on a working interval of angular velocities of rotation of a rotor steady were basic steady-state of driving, and spurious - were unstablis. The stability of driving is investigated under the theory of a stability of fixed motions of nonlinear autonomous systems of ordinary differential second-kind equations.

There are three such as classical passive autobalancers: ring-type, pendulum-type, ball-type. Their dynamics analytically is investigated in case, when mass of an autobalancer is much less than mass of a rotor. Others are not investigated practically important cases - fast rotated of a rotor, small and large forces of damping, and other. The basic researches are devoted to a flat model of a rotor and autobalancers. As far as we know, the problem of the stabilization by autobalancers of rotation of artificial satellites or spacecraft was not investigated.

Is established, that CM of a definite form, exhibits autobalancing property, if it to establish, with a possibility of motion round a fixed point on rotation axis of a rotor. If CM can be realized of spherical driving round a point of suspend, with it capable to damp small vibrations of a rotor, both longitudinal, and cross-sectional. It is offered for using in autobalancer - dampers. If CM install on an axes, perpendicular rotation axes, they rotate together with a rotor synchronously. It eliminates sensitivity of autobalancers to a change of an angular velocity of rotation of a rotor. If CM can turn round two mutually of perpendicular axes, which pass through a point of suspend, it can balance a rotor in a plane of a correction, which perpendicular axes the rotation and passes through a point of suspend. It considerably simplifies a design of autobalancers. If CM can turn round rotation axis of a rotor, it is so-called classical autobalancers and they make one of subsets of new autobalancers as bodies with a fixed point on rotation axis of a rotor. At superposition on motions of CM of defined connections it is possible to achieve defined properties of autobalancers, for example: to eliminate sensitivity to gravity: to reduce sensitivity to a change of an angular velocity of rotation of a rotor, to do a boost or braking of a rotor quieter and other.

At a theoretical research of dynamics of autobalancers the mobile frame is used which synchronously rotates together with a body, which is counterbalanced. The system of autonomous differential second-kind equations is received. The stability of various steady-state motions of a system is investigated. The conditions are become clear, at which the basic systems, steady-state motion are stable, and spurious - are unstable. The stability of motions is investigated under the theory of a stability of fixed motions of nonlinear autonomous systems of ordinary differential second-kind equations. It allows to investigate a stability not only in case, when mass of CM is much less than mass of a rotor, but also in others, important from a point of view of practice cases. In particular: at small forces of a resistance; for fast speed of rotation of a rotor; in case of large forces of interior or exterior damping, etc.

At an experimental research of dynamics of autobalancers are used: the method of stroboscopic illumination - for study of driving of CM concerning a rotor; the method of a laser ray - for observation of the motion of a centerline of a rotor.