The control of a permanent rotational motion and equilibrium position of a spacecraft

This article has adopted an analytical method to obtain a non-linear control law to reach the exponential asymptotic stablity of the permanent rotational motion of a spacecraft. The control moments achieving this rotational motion are obtained. The control moments to establish exponential asymptotic stablity of the mentioned motion are obtained as non-linear functions of the phase coordinates of the spacecraft. The general solution of the equations of perturbed motion is derived. Furthermore, analysis and numerical simulation study of this solution are presented. For numerical examples the time needed for control is calculated. An equilibrium position of the spacecraft is proved to be exponentially asymptotically stable as a special case of the above-studied problem.     

Control of the rotational motion of the rigid body with the help of internal rotors using Rodrigues-Caley parameters

The purpose of this paper is to study the control of the rotational motion of the rigid body with the help of three rotors attached to the principal axes of the body. In such study the asymptotic stability of this motion is proved by using the Lyapunov technique. As a particular case of our problem, the equilibrium position of the rigid body, which occurs when the principal axes of inertia of the body coincide with the inertial axes, is proved to be asymptotically stable. The control moments that impose the stabilization of the rotational motion and equilibrium position are obtained.     

On the stability of relative programmed motion of satellite- gyrostat

This paper is devoted to study the asymptotic stability of the relative programmed motion of a satellite-gyrostat with the help of the three rotors attached to the principal axes of inertia of the satellite. The programmed control moments are obtained. The control moments on the rotors using the condition which impose the asymptotic stabilization of the programmed motion are obtained.     

On the control of programmed motion of satellite-gyrostat containing fluid

This article is devoted to study the control of the relative programmed motion of satellite-gyrostat containing fluid with the help of internal moving bodies. Using the Lyapunov function the asymptotic stability of the relative programmed motion is proved. The control moments corresponding to the programmed motion are obtained. The control moments ensuring the asymptotic stability of this motion are obtained. The existence of the fluid is shown to give more freedom to the control. This study is characterized by the fact that the control forces are exactly as non-linear functions of phase coordinates.     

On the stability of an equilibrium position and rotational motion of a gyrostat

This article is devoted to study the compulsory stability of equilibrium position and rotational motion of a rigid body containing fluid with the help of three rotors carried on the body. The control moments on the rotors using that condition which impose the stabilization of equilibrium position of the rigid body and rotational motion are obtained.     

Rotor oscillation and stability in complex motion

Precession vibration of a rigid disk with unequal axial moments of inertia is considered when the axis of rotation turns; the disk is located asymmetrically on a flexible axle. Periodic solutions of the equations of motion and the amplitude-frequency relations are obtained for various values of the angular velocity of the axis of rotation. The critical rotational velocities of disks with various moments of inertia are defined in terms of the gyroscopic forces. The stability of motion is analyzed for various angular velocities of the rotating axis. State Technical University of Building and Architecture, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 7, pp. 104–107, July, 1999.     

Numerical simulation of the hydrodynamic interaction between a sedimenting particle and a neutrally buoyant particle

A new method for the simulation of the translational and rotational motions of a system containing a sedimenting particle interacting with a neutrally buoyant particle has been developed. The method is based on coupling the quasi-static Stokes equations for the fluid with the rigid body equations of motion for the particles. The Stokes equations are solved at each time step with the boundary element method. The stresses are then integrated over the surface of each particle to determine the resultant forces and moments. These forces and moments are inserted into the rigid body equations of motion to determine the translational and rotational motions of the particles. Unlike many other simulation techniques, no restrictions are placed on the shape of the particles. Superparametric boundary elements are employed to achieve accurate geometric representations of the particles. The simulation method is able to predict the local fluid velocity, resolve the forces and moments exerted on the particles, and track the particle trajectories and orientations.     

“Pointwise Asymptotic Stability of Steady Fluid Motions”

We study pointwise asymptotic stability of steady incompressible viscous fluids. The region of the motion is bounded. Our results of stability are based on the maximum modulus theorem that we prove for solutions of the Navier–Stokes equations. The asymptotic stability is based on a variational formulation. Since the region of the motion is bounded, the time decay is of exponential type. Of course suitable assumptions are made about the smallness of the size of the uniform norm of the perturbations at the initial data. With no restrictions, we are able only to prove an existence theorem of the perturbation local in time.     

On the evolution of rigid-body rotations

The perturbed rotational motion of a rigid body with a nearly Lagrangian mass distribution is studied. It is assumed that the angular velocity of the body is sufficiently high, its direction is close to the axis of dynamic symmetry of the body, and the perturbing moments are small in comparison with the gravity moment. A small parameter is introduced in a special manner and the acceleration method is used. Averaged systems of motion equations are obtained in first and second approximations. The evolution of the precession angle is determined in the second approximation. Odessa Academy of Cold, Ukraine. Translated from Priknadnaya Mekhanika, Vol. 35, No. 1, pp. 98–103, January, 1999.     

Optimal control of a rotational motion of a gyrostat on circular orbit

A control scheme is proposed to guarantee an optimal stabilization of a given rotational motion of a symmetric gyrostat on circular orbit. The gyrostat controlled by the control action generated by rotating internal rotors. In such study the asymptotic stability of this motion is proved using Barbachen and Krasovskii theorem's and the optimal control law is deduced from the conditions that ensure the optimal asymptotic stability of the desired motion. As a particular case, the equilibrium position of the gyrostat, which occurs when the principal axes of inertia coincide with the orbital axes, is proved to be asymptotically stable. The present method is shown to more general than previous ones.     

Resonances of an in-extensional asymmetrical spinning shaft with speed fluctuations

In this study, main and parametric resonances of an asymmetrical spinning shaft with in-extensional nonlinearity and large amplitude are simultaneously investigated. The main resonance is due to inhomogeneous part of the equations of motion, which is due to dynamic imbalances of shaft whereas the parametric resonances are due to parametric excitations due to speed fluctuations and a shaft asymmetry. The shaft is simply supported with unequal mass moments of inertia and flexural rigidities in the direction of principal axes. The equations of motion are derived by the extended Hamilton principle. The stability and bifurcations are obtained by multiple scales method, which is applied to both partial and ordinary differential equations of motion. The influences of asymmetry of shaft, speed fluctuations, inequality between two eccentricities corresponding to the principal axes and external damping on the stability and bifurcation are studied. To investigate the effect of speed fluctuations on the bifurcations and stability the loci of bifurcation points are plotted as function of damping coefficient. The numerical solutions are used to verify the results of multiple scales method. The results of multiple scales method show a good agreement with those of numerical solutions.     

Robust stability of uncertain neutral linear stochastic differential delay system

The LaSalle-type theorem for the neutral stochastic differential equations with delay is established for the first time and then applied to propose algebraic criteria of the stochastically asymptotic stability and almost exponential stability for the uncertain neutral stochastic differential systems with delay. An example is given to verify the effectiveness of obtained results.     

Nonlinear stability and bifurcations of symmetric heavy gyroscope

The complete solutions of the upright and oblique permanent rotations of a symmetric heavy gyroscope with perfect dissipation are given. The asymptotic stability criteria and unstability criteria for these rotations in the sense of Liapunov and the sense of Movchan are also given on the basis of exact nonlinear motion equations respectively. The related oblique rotations are non-isolated. The main subdomains of the regions of asymptotic stability are obtained. The related bifurcation phenomena are discussed in detail. The project is supported by the National Natural Science Foundation of China.     

On the rotational motion of a gyrostat about a fixed point with mass distribution

The perturbed rotational motion of a gyrostat about a fixed point with mass distribution near to Lagrange’s case is investigated. The gyrostat is subjected under the influence of a variable restoring moment vector, a perturbing moment vector, and a third component of a gyrostatic moment vector. It is assumed that the angular velocity of the gyrostat is sufficiently large, its direction is close to the axis of dynamic symmetry, and the perturbing moments are small as compared to the restoring ones. These assumptions permit us to introduce a small parameter. Averaged systems of the equations of motion in the first and second approximations are obtained. Also, the evolution of the precession angle up to the second approximation is determined. The graphical representations of the nutation and precession angles are presented to describe the motion at any time.     

External stability of resonances in the motion of an asymmetric rigid body with a strong magnet in the geomagnetic field

We consider a precession motion, close to the classical Lagrange case, of an asymmetric rigid body with a strong magnet in an orbit in the geomagnetic field. For the principal moment we take the restoring torque due to the interaction between the planet magnetic fields and the rigid body. The perturbing actions are due to small moments of the rigid body mass-inertial asymmetry and small constant moments. We show that these perturbations result in the realization of secondary resonance effects in the rotational motion of the rigid body caused by the influence of resonance denominators in higher-order approximations of the averaging method. These effects were discovered in the study of rotational motion of a satellite with a magnetic damper in the nearly Euler case. In the present paper, we analyze both the secondary resonance effects themselves and the external stability of resonances. We obtain conditions ensuring a decrease in the angular velocity of the rigid body rotation about its center of mass. We also discover several new laws of influence of resonances on the nonresonance evolution of slow variables, which is related to the appearance of stable resonances.     

On the chaotic rotation of a liquid-filled gyrostat via the Melnikov-Holmes-Marsden integral

The non-linear motions of a gyrostat with an axisymmetrical, fluid-filled cavity are investigated. The cavity is considered to be completely filled with an ideal incompressible liquid performing uniform rotational motion. Helmholtz theorem, Euler's angular momentum theorem and Poisson equations are used to develop the disturbed Hamiltonian equations of the motions of the liquid-filled gyrostat subjected to small perturbing moments. The equations are established in terms of a set of canonical variables comprised of Euler angles and the conjugate angular momenta in order to facilitate the application of the Melnikov-Holmes-Marsden (MHM) method to investigate homoclinic/heteroclinic transversal intersections. In such a way, a criterion for the onset of chaotic oscillations is formulated for liquid-filled gyrostats with ellipsoidal and torus-shaped cavities and the results are confirmed via numerical simulations.     

Stability analysis of the whirl motion of a breathing cracked rotor with asymmetric rotational damping

The stability of the whirl motion of a breathing cracked rotor with the distinction of stationary damping and the asymmetric rotational damping is studied. By Lagrange’s principal, the motion equations are formed in rotational frame such that the multi-asymmetric system, i.e., asymmetric rotational damping and asymmetric time-periodic varying stiffness, is simplified to be a system with anisotropic damping and anisotropic time-periodical varying stiffness in rotational operation. Based on the multiple scales solution of the simplified whirling equation in moving frame, root locus method for stability analysis is proposed. Different from the former stability estimation method, the corresponding Campbell diagram, decay rate plot, and root locus plot of the fifth-order approach are derived to prove the effects of both crack depth and damping effects. The numerical results of the instabilizing effects of the crack depth are well agreeing with the previous studies. In addition, the destabilizing influence of the rotational damping on the breathing cracked rotor is presented for the first time.     

On two-dimensional large-scale primitive equations in oceanic dynamics （Ⅱ）

The initial boundary value problem for the two-dimensional primitive equations of largescale oceanic motion in geophysics is considered sequetially. Here the depth of the ocean is positive but not always a constant. By Faedo-Galerkin method and anisotropic inequalities, the existence and uniqueness of the global weakly strong solution and global strong solution for the problem are obtained. Moreover, by studying the asymptotic behavior of solutions for the above problem, the energy is exponential decay with time is proved.     

On two-dimensional large-scale primitive equations in oceanic dynamics(Ⅱ)

The initial boundary value problem for the two-dimensional primitive equations of largescale oceanic motion in geophysics is considered sequetially.Here the depth of the ocean is positive but not always a constant.By Faedo-Galerkin method and anisotropic inequalities,the existence and uniqueness of the global weakly strong solution and global strong solution for the problem are obtained.Moreover,by studying the asymptotic behavior of solutions for the above problem,the energy is exponential decay with time is proved.     

Biquaternion solution of the kinematic control problem for the motion of a rigid body and its application to the solution of inverse problems of robot-manipulator kinematics

The problem of reducing the body-attached coordinate system to the reference (programmed) coordinate system moving relative to the fixed coordinate system with a given instantaneous velocity screw along a given trajectory is considered in the kinematic statement. The biquaternion kinematic equations of motion of a rigid body in normalized and unnormalized finite displacement biquaternions are used as the mathematical model of motion, and the dual orthogonal projections of the instantaneous velocity screw of the body motion onto the body coordinate axes are used as the control. Various types of correction (stabilization), which are biquaternion analogs of position and integral corrections, are proposed. It is shown that the linear (obtained without linearization) and stationary biquaternion error equations that are invariant under any chosen programmed motion of the reference coordinate system can be obtained for the proposed types of correction and the use of unnormalized finite displacement biquaternions and four-dimensional dual controls allows one to construct globally regular control laws. The general solution of the error equation is constructed, and conditions for asymptotic stability of the programmed motion are obtained. The constructed theory of kinematic control of motion is used to solve inverse problems of robot-manipulator kinematics. The control problem under study is a generalization of the kinematic problem [1, 2] of reducing the body-attached coordinate system to the reference coordinate system rotating at a given (programmed) absolute angular velocity, and the presentedmethod for solving inverse problems of robotmanipulator kinematics is a development of the method proposed in [3–5].