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.     

Slow nonstationary vertical motions of a die on the surface of an elastic half-space

We consider the dynamic contact problem on vertical motions of an absolutely rigid body on an elastic half-space. We assume that the contact region does not vary during the motion and there is no friction under the die bottom. We construct an approximate solution of the problem under the assumption that the variation in the contact pressure under the die bottom on the time interval in which the Rayleigh wave runs the distance equal to the contact area diameter is small. Computational formulas are obtained for the cases of circular and elliptic dies.     

To the problem of plane periodic rotations of a satellite in an elliptic orbit

We study the motion of a satellite (a rigid body) with respect to its center of mass in an elliptic orbit of small eccentricity. We analyze the nonlinear problem of the existence and stability of periodic (in the orbital coordinate system) rotations of the satellite with a period multiple of the period of revolution of its center of mass in the orbit. We study the direct and reverse rotations. In particular, we find and investigate the set of bifurcation values of the satellite dimensionless inertial parameter near which the branching of the periodic reverse rotations occurs. We consider three specific examples of application of the obtained general theoretical conclusions. In one of these examples, we prove the stability of the direct resonance rotations of Mercurial type. In the other two examples, we consider the branching problem for reverse rotations with a period whose ratio to the period of motion of the center of mass in the orbit is equal to 1 or 2.     

Theorems in linear elastodynamics for plane cracked bodies

The motion of a plane cracked body is studied by assuming that the elasticity tensor is either positive semi-definite or strongly elliptic. It is proved that the basic theorems of linear elastodynamics hold in the class of all motions having bounded velocities near the crack tips.     

Analytical solutions for dynamics of dual-spin spacecraft and gyrostat-satellites under magnetic attitude control in omega-regimes

The attitude dynamics of a dual-spin spacecraft (a gyrostat with one rotor) with magnetic actuators attitude control is considered in the constant external magnetic field at the presence of the spacecraft’s own magnetic dipole moment, which is created proportionally to the angular velocity components (this motion regime can be called as “the omega-regime” or “the omega-maneuver”). The research of the dual-spin spacecraft angular motion under the action of the magnetic restoring torque is fulfilled in the generalized formulation close to the classical mechanics’ task of the heavy body/gyrostat motion in the Lagrange top. Analytical exact solutions of differential equations of the motion are obtained for all parameters in terms of elliptic integrals and the Jacobi functions. New obtained analytical solutions can be classified as results developing the classical fundamental problem of the rigid body and gyrostat motion around the fixed point. The technical application of the omega-regime to the angular reorientation of the spacecraft longitudinal axis along the angular momentum vector is considered.     

Unsteady motion of an elliptic cylinder under a free surface

A problem of generation of nonlinear unsteady waves on the surface of an ideal liquid in an infinitely deep fluid due to the motion of a submerged elliptic cylinder is considered. The initial formulation of the problem is reduced to an integrodifferential system of equations for the function defining the free surface shape and for the normal and tangential components of velocity on the free surface. The small-time asymptotics of the solution is constructed for the case of cylinder motion with a constant acceleration from the state at rest.     

Arbitrary motion of an elliptic disc at a fluid interface

The problem of calculating the disturbance due to finite elliptic discs at the interface ($\text{x}{}_3\text{= 0}$) of two incompressible immiscible fluids of different coefficients of viscosity is solved, assuming that body and inertia forces are negligible. When the direction of motion is parallel to the interface, our solution, which is based on potential functions analogous to the Papkovitch-Neuber functions of linear elasticity, satisfies not only the interface conditions of continuity of fluid velocity and stresses but also that of zero normal velocity at the interface. It is also remarkable that this solution produces in each of the fluids a flow field that is totally independent of the properties of the other fluid. These results are not peculiar to elliptic discs, but also hold for finite discs of other shapes. The method of solution presented here can be readily applied to the more general cases where the two-phase fluid, in the absence of the disc, moves with an arbitrarily directed velocity which is a general polynomial function of the coordinates $\text{x}{}_1$ and $\text{x}{}_2$ at the interface. The procedure for carrying this out is demonstrated by treating the case of an elliptic disc in linear shear flow.     

The Stress State of a Ferromagnetic with an Elliptic Crack in a Homogeneous Magnetic Field

The problem on the stress–strain state of an infinite isotropic body made of a magnetically soft material and containing an elliptic crack is considered. It is assumed that the body is under an external magnetic field perpendicular to the crack plane. The basic characteristics of the stress–strain state and the magnetic field induced are determined and their singularities near the elliptic crack are studied. Formulas are given for the stress intensity factors for the force and magnetic fields near the crack tip     

Hydrodynamic impact of an elliptic paraboloid on cylindrical waves

The linearized Wagner theory is used to describe the initial stage of the penetration of an elliptic paraboloid on the crest of a regular wave. It is shown that the asymptotic solution for small wave steepness and large enough radii of curvature of the body is obtained from a slight modification of the standard impact problem without a wave. In practice the boundary value problem is formulated for a fictitious elliptic paraboloid: its radii of curvature are modified compared to the actual ones and its kinematics of penetration makes mainly a horizontal velocity appear due to the velocity of the propagating crest.To validate the present approach, an experimental campaign is carried out. The combined choice of the wave parameters and the geometric characteristics of the body leads to a circular expanding wetted surface. The experimental data confirm the theoretical results. Comparisons made for the pressure and the force show a satisfactory agreement.     

Chaotic attitude motion of a magnetic rigid spacecraft in an elliptic orbit and its control

This paper deals with the chaotic attitude motion of a magnetic rigid spacecraft with internal damping in an elliptic orbit. The dynamical model of the spacecraft is established. The Melnikov analysis is carried out to prove the existence of a complicated nonwandering Cantor set. The dynamical behaviors are numerically investigated by means of time history, Poincaré map, Lyapunov exponents and power spectrum. Numerical simulations demonstrate the chaotic motion of the system. The input-output feedback linearization method and its modified version are applied, respectively, to control the chaotic attitude motions to the given fixed point or periodic motion. The project supported by the National Natural Science Foundation of Chine (10082003)     

Stability of a planar resonance satellite motion under spatial perturbations

The satellite motion relative to the center of mass in a central Newtonian gravitational field on an elliptic orbit is considered. The satellite is a rigid body whose linear dimensions are small compared with the orbit dimensions. We study a special case of planar motion in which the satellite rotates in the orbit plane and performs three revolutions in absolute space per two revolutions of the center of mass in the orbit. Perturbations are assumed to be arbitrary (they can be planar as well as spatial). In the parameter space of the problem, we obtain Lyapunov instability domains and domains of stability in the first approximation. In the latter, we construct third- and fourth-order resonance curves and perform nonlinear stability analysis of the motion on these curves. Stability was studied analytically for small eccentricity values and numerically for arbitrary eccentricity values.     

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.     

Two-dimensional approach to the motion of a red blood cell in a plane couette flow of plasma

In relation to microrheology of blood, a theoretical approach to the motion of a red blood cell in a plane Couette flow between two parallel plates is made with emphasis on effects of wall. The red blood cell is assumed to be an elliptic cylindrical particle with a thin, inextensible membrane moving like a tank-tread along its perimeter and to contain a Newtonian fluid inside. Fluid motions are analysed numerically both inside and outside the particle on the basis of the Stokes equations, using the finite element method.A quasi-static equilibrium condition leads to the solution for the motion of the particle. It is shown that two types of motion exist (a stationary orientation motion and a flipping motion), depending on the viscosity ratio of inner to outer fluid, the axis ratio of the elliptic cylinder and the ratio of particle size to channel width. The results are applied to capillary blood flow.     

Formation of separation zones in an asymmetric motion of a body in an elastic medium

We obtain an analytical solution of the problem on the motion of a body with wedge-shaped nose in an elastic medium for the case in which a medium separation zone may occur near the nose owing to asymmetry. The character of the dependence of the separation region length on the body velocity, the nose opening angle, the motion asymmetry degree, and the friction coefficient is found. It is shown that if the body moves at a velocity greater than the transverse wave velocity, then there is a limit velocity at which the separation region near the nose of the body disappears.     

Action of an elliptic punch on a transversally isotropic half-space

The 3D contact problem on the action of a punch elliptic in horizontal projection on a transversally isotropic elastic half-space is considered for the case in which the isotropy planes are perpendicular to the boundary of the half-space. The elliptic contact region is assumed to be given (the punch has sharp edges). The integral equation of the contact problem is obtained. The elastic rigidity of the half-space boundary characterized by the normal displacement under the action of a given lumped force significantly depends on the chosen direction on this boundary. In this connection, the following two cases of location of the ellipse of contact are considered: it can be elongated along the first or the second axis of Cartesian coordinate system on the body boundary. Exact solutions are obtained for a punch with base shaped as an elliptic paraboloid, and these solutions are used to carry out the computations for various versions of the five elastic constants. The structure of the exact solution is found for a punch with polynomial base, and a method for determining the solution is proposed.     

Dynamics of a satellite carrying a point mass moving about it

We study the dynamics of a complex system consisting of a solid and a mass point moving according to a prescribed law along a curve rigidly fixed to the body. The motion occurs in a central Newtonian gravitational field. It is assumed that the orbit of the system center of mass is an ellipse of arbitrary eccentricity.We obtain equations that describe the motion of the carrier (satellite) about its center of mass. In the case of a circular orbit, we present conditions that should be imposed on the law of the relative motion of the mass point carried by the satellite so that the latter preserves a constant attitude with respect to the orbital coordinate system. In the case of a dynamically symmetric satellite, we consider the problem of existence of stationary and nearly stationary rotations for the case in which the carried point moves along the satellite symmetry axis.We consider several problems of dynamics of the satellite plane motion about its center of mass in an elliptic orbit of arbitrary eccentricity. In particular, we present the law of motion of the carried point in the case without eccentricity oscillations and study the stability of the satellite permanent attitude with respect to the orbital coordinate system.     

超晶格的位错模型与系统的动力学稳定性

在经典力学框架内和Seeger方程基础上,讨论了超晶格界面附近的位错动力学行为,指出了由于系统的分叉或混沌将导致位错的运动与堆积,造成了超晶格的分层或断裂;同时,也指出了,将生长过程中的超晶格置于适当的声场中将应力减至最小,或者适当调节系统参数就可最大限度的保证系统的动力学稳定性.首先,引入阻尼项,把描述一般位错运动的...     

Forced vibrations of three-layer elliptic cylindrical shells under distributed loads

The dynamic behavior of a three-layer elliptic cylindrical shell is studied and associated problems are formulated. The refined Timoshenko model is used. The numerical method used to solve the equations of motion is based on the integro-interpolation differencing method. The dynamic behavior of a three-layer elliptic cylindrical shell under a distributed nonstationary load is studied.     

Numerical simulation of the unsteady flow over an elliptic cylinder at different orientations

A numerical method is developed for investigating the two‐dimensional unsteady viscous flow over an inclined elliptic cylinder placed in a uniform stream of infinite extent. The direction of the free stream is normal to the cylinder axis and the flow field unsteadiness arises from two effects, the first is due to the flow field development following the start of the motion and the second is due to vortex shedding in the wake region. The time‐dependent flow is governed by the full conservation equations of mass and momentum with no boundary layer approximations. The parameters involved are the cylinder axis ratio, Reynolds number and the angle of attack. The investigation covers a Reynolds number range up to 5000. The minor–major axis ratio of the elliptic cylinder ranges between 0.5 and 0.6, and the angle of attack ranges between 0° and 90°. A series truncation method based on Fourier series is used to reduce the governing Navier–Stokes equations to two coupled infinite sets of second‐order differential equations. These equations are approximated by retaining only a finite number of terms and are then solved by approximating the derivatives using central differences. The results reveal an unusual phenomenon of negative lift occurring shortly after the start of motion. Various comparisons are made with previous theoretical and experimental results, including flow visualizations, to validate the solution methodology. Copyright © 2001 John Wiley & Sons, Ltd.     

Oblique impact of an elongated three-dimensional body on a thin liquid layer

The problem of the impact of an elongated solid body with a blant bottom on a thin layer of an ideal incompressible liquid is considered in the case where the horizontal component of the body velocity is much greater than its vertical component. The initial stage of the impact, during which the contact area between the body and the liquid is rapidly expanding, is studied. The loads on the body are determined by strip theory. The method of matched asymptotic expansions is used to determine the position and size of the contact area in each section. The considered problem is coupled: the liquid flow due to the motion of the body and the body motion itself are determined simultaneously. A system of integrodifferential equations was derived and used for both numerical investigation of the body motion under the action of hydrodynamic loads and for determination of the hydrodynamic pressure distribution over the contact area.