scholarly journals Solving the problem of moving a solid body around a fixed point in a resisting medium

2020 ◽  
Vol 32 ◽  
pp. 196-200
Author(s):  
G. U. Mamatova ◽  
B. Hamzina ◽  
A. О. Kabdoldina ◽  
А. K. Sugirbekova ◽  
R. A. Berkutbaeva
Keyword(s):  
Fixed Point ◽  
Solid Body ◽  
Resisting Medium

scholarly journals ON STABILIZATION OF UNSTABLE ROTATION IN THE RESISTING MEDIUM OF THE LAGRANGE GYROSCOPE USING THE SECOND GYROSCOPE AND ELASTIC HINGES

2021 ◽  
Vol 3 (2) ◽  
pp. 103-116
Author(s):  
Ya. Sviatenko ◽  
Keyword(s):  
Fixed Point ◽  
Angular Momentum ◽  
Angular Velocity ◽  
Elastic Recovery ◽  
Center Of Mass ◽  
Uniform Rotation ◽  
Resisting Medium ◽  
The Common ◽  
Elastic Coefficients ◽  
Kinetic Moment

The possibility of stabilizing an unstable uniform rotation in a resisting medium of a "sleeping" Lagrange gyroscope using a rotating second gyroscope and elastic spherical hinges is considered. The "sleeping" gyroscope rotates around a fixed point with an elastic recovery spherical hinge, and the second gyroscope is located above it. The gyroscopes are also connected by an elastic spherical restorative hinge and their rotation is supported by constant moments directed along their axes of rotation. It is shown that stabilization will be impossible in the absence of elasticity in the common joint and the coincidence of the center of mass of the second gyroscope with its center. With the help of the kinetic moment of the second gyroscope and the elasticity coefficients of the hinges, on the basis of an alternative approach, the stabilization conditions obtained in the form of a system of three inequalities and the conditions found on the elasticity coefficients at which the leading coefficients of these inequalities are positive. It is shown that stabilization will always be possible at a sufficiently large angular velocity of rotation of the second gyroscope under the assumption that the center of mass of the second gyroscope and the mechanical system are below the fixed point. The possibility of stabilizing the unstable uniform rotation of the "sleeping" Lagrange gyroscope using the second gyroscope and elastic spherical joints in the absence of dissipation is also considered. The "sleeping" gyroscope rotates at an angular velocity that does not meet the Mayevsky criterion. It is shown that stabilization will be impossible in the absence of elasticity in the common joint and the coincidence of the center of mass of the second gyroscope with its center. On the basis of the innovation approach, stabilization conditions were obtained in the form of a system of three irregularities using the kinetic moment of the second gyroscope and the elastic coefficients of the hinges. The condition for the angular momentum of the first gyroscope and the elastic coefficients at which the leading coefficients of these inequalities are positive are found. It is shown that if the condition for the angular momentum of the first gyroscope is fulfilled, stabilization will always be possible at a sufficiently large angular velocity of rotation of the second gyroscope, and in this case the center of mass of the second gyroscope can be located above the fixed point.

On the subject of influence of dissipative and constant of moments on the stability of uniform rotations of non-free two elastically connected gyroscopes of Lagrange

2021 ◽  
Vol 34 ◽  
pp. 50-61
Author(s):  
Yurii Kononov ◽  
Yaroslav Sviatenko
Keyword(s):  
Fixed Point ◽  
Asymptotic Stability ◽  
Stability Conditions ◽  
Inertial Coordinate System ◽  
Resisting Medium ◽  
Angular Velocities ◽  
The Subject ◽  
The Stability ◽  
Spherical Hinge ◽  
Asymptotic Stability Conditions

The conditions for asymptotic stability of uniform rotations in a resisting medium of two heavy Lagrange gyroscopes connected by an elastic spherical hinge are obtained in the form of a system of three inequalities. The bottom gyroscope has a fixed point. The rotation of the gyroscopes is maintained by constant moments in the inertial coordinate system. The influence of the elasticity of the hinge on the stability conditions is estimated. It is shown that for a sufficiently high rigidity of the hinge, the asymptotic stability conditions are determined by only one inequality, which coincides with the inequality obtained for the case of a cylindrical hinge. When the angular velocities of the gyroscopes' own rotations coincide, this inequality coincides with the well--known condition for one gyroscope. Cases of degeneration of an elastic spherical hinge into a spherical inelastic, cylindrical and universal elastic hinge (Hooke's hinge) are considered. For the Hooke hinge, it is shown that there is no asymptotic stability at a sufficiently high angular velocity of gyroscopes rotation.

Kinematics and mass geometry for a solid body with a fixed point in ℝn

2001 ◽  
Vol 46 (9) ◽  
pp. 663-666
Author(s):  
D. V. Georgievskii ◽  
M. V. Shamolin
Keyword(s):  
Fixed Point ◽  
Solid Body

On the equations of rotation of a solid body about a fixed point

1864 ◽  
Vol 13 ◽  
pp. 52-64
Keyword(s):  
Fixed Point ◽  
Solid Body ◽  
The Body ◽  
Principal Axes ◽  
Usual Notation

In treating the equations of rotation of a solid body about a fixed point, it is usual to employ the principal axes of the body as the moving system of coordinates. Cases, however, occur in which it is advisable to employ other systems; and the object of the present paper is to develope the fundamental formulæ of transformation and integration for any system. Adopting the usual notation in all respects, excepting a change of sign in the quantities F, G, H, which will facilitate transformations hereafter to be made, let A = Σ m ( y 2 + z 2 ), B = Σ m ( z 2 + x 2 ), C= Σ m ( x 2 + y 2 ), -F = Σ myz , -G = Σ mzx , -H = Σ mxy ;

Rapid rotation of a heavy gyrostat about a fixed point in a resisting medium

1982 ◽  
Vol 18 (7) ◽  
pp. 660-665 ◽  
Author(s):  
L. D. Akulenko ◽  
D. D. Leshchenko
Keyword(s):  
Fixed Point ◽  
Rapid Rotation ◽  
Resisting Medium ◽  
Heavy Gyrostat

scholarly journals II. On the equations of rotation of a solid body about a fixed point

1863 ◽  
Vol 12 ◽  
pp. 523-524
Keyword(s):  
Fixed Point ◽  
Solid Body ◽  
The Body ◽  
Principal Axes

In treating the equations of rotation of a solid body about a fixed point, it is usual to employ the principal axes of the body as the moving system of coordinates. Cases, however, occur in which it is advisable to employ other systems; and the object of the present paper is to develope the fundamental formulæ of transformation and integration for any system.

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