Numerical simulation of the sedimentation of a tripole-like body in an incompressible viscous fluid

In this note, we discuss the application of a methodology combining distributed Lagrange multiplier based fictitious domain techniques, finite-element approximations and operator splitting, to the numerical simulation of the motion of a tripole-like rigid body falling in a Newtonian incompressible viscous fluid. The motion of the body is driven by the hydrodynamical forces and gravity. The numerical simulation shows that the distribution of mass of this rigid body and added moment of inertia compared to a simple cylinder (circular or elliptic) plays a significant role on the particle-fluid interaction. Apparently, for the parameters examined, the action of the moving rigid body on the fluid is stronger than the hydrodynamic forces acting on the rigid body.     

Oscillations of a rigid body containing an elastic element with distributed parameters

The motions of a hybrid (discrete-continual) system, consisting of a carrier rigid body and an elastic element with distributed parameters fastened to it are investigated. Two types of fastening are considered: (1) both ends are clamped, and (2) one of the ends is clamped while the other is free. A closed system of integro-differential equations is obtained which describes the state of the system under arbitrary initial conditions and forces applied to the rigid body. The perturbed motion of the rigid body in the case of a quasi-linear restoring force is investigated using asymptotic methods. The motions are studied both when there is internal resonance between the oscillations of the rigid body and the natural oscillations of the element, and when there are no such resonances. Qualitative effects are found.     

The motion of a body with a plane of symmetry over a three-dimensional sphere under the action of a spherical analogue of Newtonian gravitation

The problem of the motion of a rigid body possessing a plane of symmetry over the surface of a three-dimensional sphere under the action of a spherical analogue of Newtonian gravitation forces is considered. Approaches to introducing spherical analogues of the concepts of centre of mass and centre of gravity are discussed. The spherical analogue of “satellite approach” in the problem of the motion of a rigid body in a central field, which arises on the assumption that the dimensions of the body are small compared with the distance to the gravitating centre, is studied. Within the framework of satellite approach, assuming plane motion of the body, the question of the existence and stability of steady motions is investigated. A spherical analogue of the equation of the plane oscillations of a body in an elliptic orbit is derived.     

Locomotion and Control of a Self-Propelled Shape-Changing Body in a Fluid

In this paper we study the locomotion of a shape-changing body swimming in a two-dimensional perfect fluid of infinite extent. The shape changes are prescribed as functions of time satisfying constraints ensuring that they result from the work of internal forces only: conditions necessary for the locomotion to be termed self-propelled. The net rigid motion of the body results from the exchange of momentum between these shape changes and the surrounding fluid.     

An asymptotic form of the energy functional for an elastic body with a crack and a rigid inclusion. The plane problem

The plane problem in the linear theory of elasticity for a body with a rigid inclusion located within it is investigated. It is assumed that there is a crack on part of the boundary joining the inclusion and the matrix and complete bonding on the remaining part of the boundary. Zero displacements are specified on the outer boundary of the body. The crack surface is free from forces and the stress state in the body is determined by the bulk forces acting on it. The variation in the energy functional in the case of a variation in the rigid inclusion and the crack is investigated. The deviation of the solution of the perturbed problem from the solution of the initial problem is estimated. An expression is obtained for the derivative of the energy functional with respect to a zone perturbation parameter that depends on the solution of the initial problem and the form of the vector function defining the perturbation. Examples of the application of the results obtained are studied.     

Attitude stabilization of a rigid body in conditions of decreasing dissipation

The paper presents the problem of triaxial stabilization of the angular position of a rigid body. The possibility of implementing a control system in which dissipative torque tends to zero over time and the restoring torque is the only remaining control torque is considered. The case of vanishing damping considered in this study is known as the most complicated one in the problem of stability analysis of mechanical systems with a nonstationary parameter at the vector of dissipative forces. The lemma of the estimate from below for the norm of the restoring torque in the neighborhood of the stabilized motion of a rigid body and two theorems on asymptotic stability of the stabilized motion of a body are proven. It is shown that the sufficient conditions of asymptotic stability found in the theorems are close to the necessary ones. The results of numerical simulation illustrating the conclusions obtained in this study are presented.     

The motion of a body supported at three points on a rough plane

The problem of the motion of a heavy rigid body, supported on a rough horizontal plane at three of its points, is considered. The contacts at the support points are assumed to be unilateral and subject to the law of dry (Coulomb) friction. The dynamics of possible motions of such a body under the action of gravity forces and dry friction is investigated. In the case of a plane body, it is possible to obtain particular integrals of the equations of motion.     

The contact problem with the bulk application of intermolecular interaction forces (a refined formulation)

A refined formulation of the contact problem when there are intermolecular interaction forces between the contacting bodies is considered. Unlike the traditional formulation, it is assumed that these forces are applied to points within the body, rather than to the surface of the deformable body as a contact pressure, and that the body surface is load-free. Solutions of the contact problems for a thin elastic layer attached to an absolutely rigid substrate and for an elastic half-space are analysed. The refined and traditional formulations of the problem when there is intermolecular interaction are compared. ©2013     

A family of analytic extremals in problems of the optimal control of the rotation of a body

The problem of the optimal control of the rotation of an absolutely rigid body about the centre of mass is investigated. The main purpose of the control is to vary the angular velocity vector from its initial value to the required terminal value in a finite time so that the manoeuvre would require the smallest power consumption, which is characterized by an integral quadratic functional. The principal torque produced by the external forces applied to the body serves as the control. The change in orientation is not taken into account, i.e., the problem of the overspeed–braking control of the body, is studied. A new class of analytic extremals based on the use of space-time deformations of the solutions of the dynamical Euler equations for the free rotation of a rigid body is described. Sufficient conditions for the existence of such extremals for all types of symmetries are presented.     

A dynamically consistent model of the contact stresses in the plane motion of a rigid body

The problem of determining dry friction forces in the case of the motion of a rigid body with a plane base over a rough surface is discussed. In view of the dependence of the friction forces on the normal load, the solution of this problem involves constructing a model of the contact stresses. The contact conditions impose three independent constraints on the kinematic characteristics, and the model must therefore include three free parameters, which are determined from these conditions at each instant. When the body is supported at three points, these parameters (for which the normal stresses can be taken) completely determine the model, while indeterminacy arises in the case of a larger number of contact points and, in order to remove this, certain physical hypotheses have to be accepted. It is shown that contact models consistent with the dynamics possess certain new qualitative properties compared with the traditional quasi-static models in which the type of motion of the body is not taken into account. In particular, a dependence of the principal vector and principal moment of the friction forces on the direction of sliding or pivoting of the body, as well as on the magnitude of the angular velocity, is possible.     

On the linearized theory op flow around bodies. Method of force sources

The method of force sources is proposed for solving linear problems related to the interaction between rigid bodies, and fluids, or gases. Method is based on the introduction of perturbation force sources into equation of motion of fluid media. Boundary conditions at the rigid body surface make it possible to reduce the problem of hydrodynamic reactions to an integral equation defining the function of force sources. Method is illustrated by the solution of three simple problems in the field of acoustics, and of viscous, and compressible media flow around bodies.

In the linearized theory of flow around rigid bodies, as well as in acoustics, an important part of the sound wave generation analysis concerns the determination of hydrodynamic reactions of the medium on moving, pulsating, or oscillating bodies. Such reactions make themselves felt as constant, or variable mechanical forces, such as drag and lift, or in the case of sound wave emitters, as the wave resistance. Various methods had been proposed for the computation of such forces, as for example, in the monographs [1 to 6].

Here, a different approach to the problem of determination of surface forces exerted by liquids and gases on the rigid body is proposed. By resorting to the formalism of the generalized functions it is possible to introduce into the equations of motion of fluid media a perturbation source in the form volume density of forces exercised by the body on the gas. The distribution of surface tension entering into the expression of this force is selected in such a manner as to satisfy boundary conditions at the body surface. It becomes possible with the use of this device to reduce the problem of determination of forces acting on the body surface to the solution of certain Integral equations. The proposed method is in all respects completely analogous to the well-known method of sources and sinks [1 to 1]. Both methods reduce the problem of interaction between body and gas to the solution of Integral equations. The method of sources and sinks, however, leads to an integral equation which describes the distribution of fictitious sources and sinks in the volume of the body having the density of the medium, while the method of force sources yields an integral equation which directly defines the distribution of mechanical forces over the surface of the body (*).

We may note that the method of force sources had to a certain extent been already used in papers [6 and 7] for the determination of sound radiation by means of point-force sources.     

A contact problem for an elastic plate with a thin rigid inclusion

Under study is an equilibrium problem for a plate under the influence of external forces. The plate is assumed to have a thin rigid inclusion that reaches the boundary at the zero angle and partially contacts a rigid body. On the inclusion face, there is a delamination. We consider the complete Kirchhoff–Love model, where the unknown functions are the vertical and horizontal displacements of the middle surface points of the plate. We present differential and variational formulations of the problem and prove the existence and uniqueness of a solution.     

The stability of the steady motions of a rigid body in a central field

The problem of the translational-rotational motion of a rigid body with a triaxial ellipsoid of inertia in a central gravitational field is considered. The body is modelled by a weightless sphere, at the ends of the three mutually perpendicular diameters of which there are point masses. It is shown that, unlike the cases when the approximate expression for the potential of the gravity forces is used, there are not only “trivial” steady motions of the body, for which the main central axes of inertia of the body coincide with the axes of the orbital system of coordinates, but also other classes of steady motions. In addition, the stability of these “trivial” steady motions is investigated, and the possibility of secular stability of the motions, unstable in the satellite approximation, is pointed out.     

The motion in a perfect fluid of a body containing a moving point mass

The motion of a system (a rigid body, symmetrical about three mutually perpendicular planes, plus a point mass situated inside the body) in an unbounded volume of a perfect fluid, which executes vortex-free motion and is at rest at infinity, is considered. The motion of the body occurs due to displacement of the point mass with respect to the body. Two cases are investigated: (a) there are no external forces, and (b) the system moves in a uniform gravity field. An analytical investigation of the dynamic equations under conditions when the point performs a specified plane periodic motion inside the body showed that in case (a) the system can be displaced as far as desired from the initial position. In case (b) it is proved that, due to the permanent addition of energy of the corresponding relative motion of the point, the body may float upwards. On the other hand, if the velocity of relative motion of the point is limited, the body will sink. The results of numerical calculations, when the point mass performs random walks along the sides of a plane square grid rigidly connected with the body, are presented.     

Regularization of 2D frictional contacts for rigid body dynamics

Classic rigid body mechanics does not provide frictional forces acting in a 2D contact interface between two bodies during sticking. This is due to the statical undeterminacy related with this problem. Many technical systems, e.g. disk clutches, have such surface-to-surface contacts and it is sometimes desirable to treat them as rigid body systems despite the 2D contact. Alternatively it is possible to model the systems using elastic instead of rigid bodies, but this might lead to certain drawbacks. Here a new regularization model of such 2D contacts between rigid bodies is proposed. It is derived from a material model for elasto-plasticity in continuum mechanics. Only dry friction is taken into account. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Motion of a rigid body in a viscous fluid

We introduce a concept of weak solution for a boundary value problem modelling the motion of a rigid body immersed in a viscous fluid. The time variation of the fluid's domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem. Our main theorem asserts the existence of at least one weak solution for this problem. The result is global in time provided that the rigid body does not touch the boundary.     

The Dynamic Plane Deformation of a Rigid, Perfectly-Plastic Slab

A perturbation method is used to investigate the problem ofthe dynamic plane deformation of a rigid, perfectly-plasticslab by a pair of rough, rigid parallel plates. Perturbationsolutions for the stress and velocity fields are given whichare correct to first-order quantities in the non-dimensionalparameter pc²/2k. These solutions are used to examine a situationwhich occurs in forging practice where a slab is deformed bythe sudden application of constant forces in excess of the initialquasi-static yield load.     

A family of trigonometric extremals in the problem of reorienting a spherically symmetrical body with minimum energy consumption

The problem of the optimal control of the reorientation of an absolutely rigid, spherically symmetric body is investigated. An integral quadratic functional, which characterizes the total energy consumption, is chosen as the criterion of the efficiency of the manoeuvre. The resultant torque of the applied external forces serves as the control. Application of the formalism of the Pontryagin's maximum principle leads to an analysis of a third-order non-linear vector differential equation, whose general solution is still unknown at the present time. It is shown that this equation has a particular solution described by trigonometric functions of time, which can be used to completely reconstruct the explicit solution for the corresponding extremal rotation. An analogy with the free rotation of a certain axisymmetric body is proposed.     

Matrix formulation of the motion equations of a rigid body and identification of the ten inertia characteristics

The motion equations of a rigid body involve ten inertial characteristics: the mass, the mass center position and the inertia matrix. In order to identify these ten inertia characteristics, we propose an approach unifying them in a 4 × 4 positive definite symmetric matrix. The translation vector and the rotation matrix of the rigid body are also gathered in a 4 × 4 matrix. Therefore the motion equations are formulated as an equality between 4 × 4 skew–symmetric matrices: one representing the sum of external forces and torques, the second representing the dynamic force and torque. The identification is performed by a projected conjugate gradient algorithm developped in the 10–dimensional linear space of 4 × 4 symmetric matrices. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)