Application of the integral manifold method to the analysis of the spatial motion of a rigid body fixed to a cable

We analyze the spatial motion of a rigid body fixed to a cable about its center of mass when the orbital cable system is unrolling. The analysis is based on the integral manifold method, which permits separating the rigid body motion into the slow and fast components. The motion of the rigid body is studied in the case of slow variations in the cable tension force and under the action of various disturbances.We estimate the influence of the static and dynamic asymmetry of the rigid body on its spatial motion about the cable fixation point. An example of the analysis of the rigid body motion when the orbital cable system is unrolling is given for a special program of variations in the cable tension force. The conditions of applicability of the integral manifold method are analyzed.     

Construction of optimal laws of variation of the angular momentum vector of a rigid body

We consider the problem of constructing optimal preset laws of variation of the angular momentum vector of a rigid body taking the body from an arbitrary initial angular position to the required terminal angular position in a given time. We minimize an integral quadratic performance functional whose integrand is a weighted sum of squared projections of the angular momentum vector of the rigid body. We use the Pontryagin maximum principle to derive necessary optimality conditions. In the case of a spherically symmetric rigid body, the problem has a well-known analytic solution. In the case where the body has a dynamic symmetry axis, the obtained boundary value optimization problem is reduced to a system of two nonlinear algebraic equations. For a rigid body with an arbitrarymass distribution, optimal control laws are obtained in the form of elliptic functions. We discuss the laws of controlled motion and applications of the constructed preset laws in systems of attitude control by external control torques or rotating flywheels.     

Strong Solutions for the Interaction of a Rigid Body and a Viscoelastic Fluid

We study a coupled system of equations describing the movement of a rigid body which is immersed in a viscoelastic fluid. It is shown that under natural assumptions on the data and for general geometries of the rigid body, excluding contact scenarios, a unique local-in-time strong solution exists.     

Taking into account the influence of elastic elements of a free structure on the accuracy of its stabilization under a bang-bang control

We consider the problem of stabilization with respect to a prescribed position for the translational motion of a rigid body with interior material points connected with each other and with the exterior body by linear viscoelastic constraints. The motion occurs under the action of a constant exterior perturbation and a bang-bang control force that are directed along the line of motion. We assume that the bang-bang force control channel has a fixed delay, so that arbitrarily frequent switchings are impossible. We suggest a positional control ensuring the solution of this problem. We estimate the amplitude of the rigid body vibrations about the center of mass of the entire structure and the accuracy of stabilization of the prescribed position of the rigid body depending on the mechanical characteristics of the system and the control force magnitude. We also consider the problem of maximizing the stabilization accuracy depending on the control parameters. By way of example, we consider the controlled motion of a two-mass oscillatory system. This work is closely related to [1–3] and continues the studies of the guaranteed optimal bang-bang controllers with delay in the control channel [4–9]. The dynamics of a rigid body with elastic and dissipative elements was studied in [10] under the assumption that the period of natural vibrations and their decay time are small compared with the characteristic time of motion.     

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.     

Global Existence of Weak Solutions for Viscous Incompressible Flows around a Moving Rigid Body in Three Dimensions

We study the motion of a rigid body of arbitrary shape immersed in a viscous incompressible fluid in a bounded, three-dimensional domain. The motion of the rigid body is caused by the action of given forces exerted on the fluid and on the rigid body. For this problem, we prove the global existence of weak solutions.     

Analysis of a rigid body obliquely impacting granular matter

The model of the oblique rigid body impact with a granular matter is studied. The force acting on the body is a linear superposition of a static (velocity-independent) friction force and a dynamic (velocity-dependent) resistance force. The impact of a sphere, a mathematical and a compound pendulum are modeled and simulated using different initial impact velocity conditions and different impact angles. We analyze how rapidly the rigid body impacting a granular media slows upon collision. For most of the analyzed cases the rigid body under high-force impact (higher initial velocity) comes to rest faster in a granular matter than the same body under low-force impacts (lower initial velocity). Researchers were able to explain this interesting phenomena, not shared by solids or liquids, for the vertical impact of spheres. The simulations for some configurations with small initial impact angles show that as the speed at which the rigid body impacts the media increases, the later it will come to rest.

     

Construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric rigid body

We consider the problem of construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric rigid body so as to ensure the transition of the rigid body from an arbitrary initial angular position to the required final angular position. For the functionals to be minimized, we use combined performance functionals, one of which characterizes the expenditure of time and of the squared modulus of the angular momentum vector in a given proportion, while the other characterizes the expenditure of time and momentum of the modulus of the angular momentum vector necessary to change the rigid body orientation. The control (the vector of the rigid body angular momentum) is assumed to be bounded in the modulus. The problem is solved by using Pontryagin’s maximum principle and the quaternion differential equation [1, 2] relating the vector of the dynamically symmetric rigid body angular momentum to the quaternion of orientation of the coordinate system rotating with respect to the rigid body about its dynamical symmetry axis at an angular velocity proportional to the angular momentum vector projection on the axis. The use of such a model of rotational motion leads to the problem of optimal control with the moving right end of the trajectory and significantly simplifies the analytic study of the problem of construction of optimal laws of variation in the angular momentum vector, because this model explicitly exploits the body angular momentum quaternion (control) instead of the rigid body absolute angular velocity quaternion. We construct general analytic solutions of the differential equations for the boundary-value problems which form systems of nine nonlinear differential equations. It is shown that the process of solving the differential boundary-value problems is reduced to solving two scalar algebraic transcendental equations.     

Existence of Weak Solutions for a Two-dimensional Fluid-rigid Body System

We consider the problem of a rigid body immersed in an inviscid incompressible fluid in two dimensional space. The motion of the fluid is described by the incompressible Euler equations and the motion of the rigid body is governed by the balance of linear and angular momentum. A global weak solution is obtained, without any assumption on the weighted norm of the initial vorticity.     

On the dynamical symmetry points and the orientations of the principal axes of inertia of a rigid body

For an arbitrary rigid body, all dynamical symmetry points are found, and the directions of the axes of dynamical symmetry are determined for these points. We obtain conditions on the principal central moments of inertia under which the Lagrange and Kovalevskaya cases can be realized for the rigid body. We also analyze the set of orientations of the bases formed by the principal axes of inertia for various points of the rigid body.     

Averaging method in the motion stability problem for a rotating body suspended on a long rigid string

We use the averaging method to study the stability of the vertical rotation of a rigid body suspended on a long rigid string.     

Synthesis of a controller for stabilizing the motion of a rigid body about a fixed point

A method for the approximate design of an optimal controller for stabilizing the motion of a rigid body about a fixed point is considered. It is assumed that rigid body motion is nearly the motion in the classical Lagrange case. The method is based on the common use of the Bellman dynamic programming principle and the averagingmethod. The latter is used to solve theHamilton–Jacobi–Bellman equation approximately, which permits synthesizing the controller. The proposed method for controller design can be used in many problems close to the problem of motion of the Lagrange top (the motion of a rigid body in the atmosphere, the motion of a rigid body fastened to a cable in deployment of the orbital cable system, etc.).     

Steady-state rotational motions of a rigid body with a strong magnet in an alternating magnetic field in the presence of dissipation

We consider steady-state rotational motions of a satellite, i.e., a rigid body with a passive magnetic attitude control system consisting of a strong constant magnet and a set of magnetic hysteresis rods. We use asymptotic methods to show that in the absence of dissipation there exists a one-parameter family of steady-state rotations of the rigid body with the strong magnet and that this one-parameter family passes into an isolated solution if a model dissipation is introduced. The motion thus obtained was discovered when processing the telemetry data from the first Russian nano-satellite TNS-0 launched in 2005.     

Solution of the optimal turn problem for a spherically symmetric rigid body with arbitrary boundary conditions in the class of generalized conical motions

We consider the problem of time- and energy consumption-optimal turn of a rigid body with spherical mass distribution under arbitrary boundary conditions on the angular position and angular velocity of the rigid body. The optimal turn problem is modified in the class of generalized conical motions, which allows one to obtain closed-form solutions for equations of motion with arbitrary constants. Thus, solving the optimal control boundary value problem is reduced to solving a system of nonlinear algebraic equations for the constants. Numerical examples are considered to illustrate the proximity between the solutions of the traditional and modified problems of optimal turn of a rigid body.     

Dynamic model and optimal setup of a vibroprotective system

We consider low-frequency vibrations of a vibroprotective system of rigid bodies consisting of a roller vibration suppressor and a movable carrying body under the action of an external harmonic excitation. We write out the dynamic equations of the common motion of the damper working body along the hinged roller and of the carrying body. We propose a graphical method for determining the optimal adjustment parameters of the roller damper, which is part of the vibroprotective system.     

Quasi-Euler-Poinsot motion of a rigid body

The phase-plane method of nonlinear oscillation is used to discuss the influence of the small dissipation upon the Euler-Poinsot motion of a rigid body about a fixed point. The equations of phase coordinates are applied instead of Eulerian equations, and the global characteristics of the motion of rigid body are analysed according to the distribution and the type of the singular points. A Chaplygin's sphere on a rough plane, a rigid body in viscous medium and one with a cavity filled with viscous fluid are discussed as examples. It is shown that the motions of rigid bodies dissipated by various physical factors have a common qualitative character. The rigid body tends to make a permanent rotation about the principal axis of the largest moment of inertia. The transitive process can change from oscillatory to aperiodic with the decrease in dissipation.     

Planar dynamics of a dimer on a wave

Nonlinear Dynamics - We study the 2D dynamics of a rigid dimer, a dumbbell-shaped extended body, on an elastic surface carrying a harmonic traveling wave. The impact of the dimer with the surface...     

On contact interaction between a moving massive body and a linear elastic half space

We study a class of problems involving the motion of a linear elastic body in frictional contact with a linear elastic half space. The dynamic effects considered are the inertial properties of the body regarded as rigid. We study only those regimes of contact interaction for which the slip velocity with the body taken as absolutely rigid and the time rate of change of the elastic displacements of points of the body and the half space that are on the contact surface are of the same order of magnitude. This work generalizes previous work on similar problems in that we simultaneously consider inertia forces of the body and the convective term in the slip-velocity due to the rigid-body velocity of the slider/indentor. Thus regimes of contact interaction investigated include rolling/sliding and shift-torsion type. We propose a variational formulation of the following two problems: (a) finite contact area and shift-torsion type of contact kinematics, (b) local contact area and general kinematics at the contact surface. Results for an elastic cylinder contacting an elastic half-plane are also given.     

New solvable problems in the dynamics of a rigid body about a fixed point in a potential field

We determine the general form of the potential of the problem of motion of a rigid body about a fixed point, which allows the angular velocity to remain permanently in a principal plane of inertia of the body. Explicit solution of the problem of motion is reduced to inversion of a single integral. A several-parameter generalization of the classical case due to Bobylev and Steklov is found. Special cases solvable in elliptic and ultraelliptic functions of time are discussed.     

On the non-persistence of irrotational motion in a viscous heat-conducting fluid

We consider the possibility of irrotational flow in a fluid exterior to a moving rigid obstacle, or interior to a moving rigid shell. Observations show that when a rigid body is impulsively set into motion an irrotational flow may exist initially but does not persist. The breakup of this irrotational flow and the associated phenomenon of generation of vorticity at the wall are generally attributed to the condition of adherence at the fluid-solid interface. Since this condition itself is derived from observation, one can ask whether there is another explanation for the phenomenon. The purpose of this paper is to show that a persistent irrotational flow is incompatible with the second law of thermodynamics.