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Two Rossler’s problems (three-dimensional and four-dimensional) and a problem with a set of singular points (a model of a rigid body with a linear feedback loop) are considered. The equation of the set of singular points is derived. A new interpretation of the symmetry principle in three-dimensional systems is given. The mechanisms of generation of regular attractors and their degeneration into strange ones are established.

The error of the Wentzel–Kramers–Brillouin solution of the equations describing the angular motion of the axis of symmetry of rotation of a rigid body (projectile) is estimated. It is established that order of this estimate does not depend on whether the low-frequency oscillations of the axis of symmetry are damped or not

Studying the influence of a number of design parameters of solid-propellant rockets on the longitudinal and lateral dispersion is an important applied problem. A mathematical model of a rigid body of variable mass moving in a disturbed medium exerting both wave drag and friction is considered. The model makes it possible to determine the coefficients of aerodynamic forces and moments, which affect the motion of vehicles, and to assess the effect of design parameters on their accuracy

The developed approaches to and results of studying the impact of rigid and deformable blunt bodies on an elastic half-space or a cavity in an elastic medium are stated using a plane problem of linear elasticity

A refined mathematical dynamic model of a rigid body with a gravity–friction seismic damper moving inside a movable hemispherical rough depression is set up. This model describe the frequency and decrement properties of a rigid body undergoing translational vibrations on a gravity–friction seismic-isolation mechanism. The mechanism consists of two platforms. The lower (supporting) platform has spherical supports that contact with the upper (supported) platform within its hemispherical depressions. The equations of motion incorporate the nonideality of the unilateral constraints and the shock interaction of a material point with the sphere. It is shown how to join the models describing the free motion and the sliding motion of the point over a spherical surface

A nonlinear mathematical model of a gravitational vibratory system with a controlled electromagnetic seismic damper is developed. The dependence of the force of attraction of ferromagnetic bodies by the solenoid of the frictional device on the solenoid current is established for a specific solenoid design. The analytic expression for this force is derived by the least-squares method using a system of continuous piecewise-linear functions. It is used to describe the pressure of the solenoid on the friction surface. For multifrequency inertial excitation, the acceleration amplification factor is evaluated depending on the time constant and gain for two cases of control. The possibility of damping vibrations by controlling the absolute velocity and acceleration is established

The similarity transform and the properties of quaternion matrices are used to derive invariant expressions for the inertia matrix and kinetic energy of an asymmetric rigid body undergoing spatial motion around a fixed point in body-fixed and inertial frames of reference. The elements of the quaternion matrixes are Rodrigues–Hamilton parameters and quasi-velocities

The paper discusses a nonlinear model that describes the interaction of a rigid body with a medium and takes into account (based on experimental data on the motion of circular cylinders in water) the dependence of the arm of the force on the normalized angular velocity of the body and the dependence of the moment of the force on the angle of attack. An analysis of plane and spatial models (in the presence or absence of an additional follower force) leads to sufficient stability conditions for translational motion, as one of the key types of motions. Either stable or unstable self-oscillation can be observed under certain conditions

A set of four quaternion matrices is used to represent the equations of finite rotation theory and to describe the kinematics and nonlinear dynamics of an asymmetric rigid body in space. The results obtained are tested in setting up direction-cosine matrices, calculating three-index symbols, establishing the relationship between the components of angular velocity in body-fixed and space-fixed frames of reference, and using a set of three independent rotations. Euler–Lagrange equations and a set of four quaternion matrices are used to construct a block-matrix model describing the nonlinear dynamics of a free asymmetric rigid body in three-dimensional space. The model gives the matrix Euler’s equations of motion and other special cases. Algorithms adapted to use in a numerical experiment are developed

The nonstationary indentation of a blunt rigid body into an elastic layer is studied. The general formulation of the problem includes different boundary conditions in the contact area and on the free surface of the layer. The simplified nonmixed problem that arises at the early stage of interaction and allows obtaining approximate results for later times is solved exactly. The solution obtained is compared with that for the plane case

The shock interaction of a spherical rigid body with a spherical cavity is studied. This nonstationary mixed boundary-value problem with an unknown boundary is reduced to an infinite system of linear Volterra equations of the second kind and the differential equation of motion of the body. The hydrodynamic and kinematic characteristics of the process are obtained

The paper proposes computer algebra system (CAS) algorithms for computer-assisted derivation of the equations of motion for systems of rigid bodies with holonomic and nonholonomic constraints that are linear with respect to the generalized velocities. The main advantages of using the D’Alembert-Lagrange principle for the CSA-based derivation of the equations of motion for nonholonomic systems of rigid bodies are demonstrated. Among them are universality, algorithmizability, computational efficiency, and simplicity of deriving equations for holonomic and nonholonomic systems in terms of generalized coordinates or pseudo-velocities

The paper addresses a problem of plane elasticity for a doubly connected body with outer and inner boundaries in the form of regular polygons with a common center and parallel sides. The neighborhoods of the vertices of the inner boundary are unknown equal full-strength smooth arcs symmetric about the rays coming from the vertices to the center. It is assumed that this elastic body is inserted into a hole of a rigid body, with the hole boundary coinciding with the outer boundary of the elastic body. Absolutely smooth rigid punches with rectilinear bases are pressed into all the rectilinear sections of the inner polygonal boundary of the elastic body. There is no friction between the elastic and rigid bodies. The unknown arcs are free from external stresses. Complex variable theory is used to determine the unknown arcs and the stress state of the elastic body

The problem of stabilizing the axis of a solid by coupled perfectly rigid bodies (PRBs) is solved. The solid executes a plane-parallel motion. The PRBs can rotate as a single rigid body about the centroidal axis of the solid and counterrotate about its transverse axes through equal angles. There is a particle inside the solid which causes its imbalance. It is established that the principal state (if any) of the system—rotation about the centroidal axis—is stable, whereas the rest (unwanted) states are unstable

The theory behind analytical algorithms for computer-algebra systems is discussed. A computer system is described which is capable of optimizing input data, constructing equations of equilibrium and motion, formulating and solving the basic mechanics problems for a broad class of holonomic systems with elastic and dissipative constraints on the basis of the Lagrange-D'Alembert principle

A technique is proposed to identify the principal central axis of inertia in a joint-suspended inhomogeneous rigid body from its vertical revolution. The problem is reduced to synchronizing the revolution of the body with the revolution of a fictitious mathematical pendulum

Delamination of a symmetric circular plate made of viscoelastic composite is studied. It is assumed that the plate has a penny-shaped crack whose edges have a minor axisymmetric imperfection. The lateral surface of the plate is clamped and is compressed by uniform radial normal forces through a rigid body. The studies are made using the exact geometrically nonlinear equations of the theory of viscoelasticity. The delamination criterion is assumed unrestrained growth of the initial imperfection. The Laplace transform and FEM are employed. In particular cases, the results are compared with those for elastic composites

A problem is solved on the quadratic boundedness of motions of a quasilinear uncertain system. The sufficient boundedness conditions are established. The result is illustrated by an example of studying the rotary motions of a rigid body in a medium with uncertain resistance.

A dynamic model of the interaction of a rigid body with a jet flow of a resistant medium is considered. This model allows us to obtain three-dimensional analogs of plane dynamic solutions for a solid interacting with the medium and to reveal new cases where the equations are Jacobi integrable. In such cases, the integrals are expressed in terms of elementary functions. The classical problems of a spherical pendulum in a flow and three-dimensional motion of a body with a servoconstraint are shown to be integrable. Mechanical and topological analogs of these problems are found

This article is the second part of a review of the dynamics of rigid and elastic bodies in a compressible viscous liquid in a linearized formulation. The following processes are investigated: the forced harmonic vibrations of rigid bodies in moving and resting compressible viscous liquids, the nonstationary motion of rigid bodies in a compressible viscous liquid at rest, the movement of rigid bodies in a resting compressible viscous liquid under the action of radiation forces that are due to the interaction with propagating acoustic harmonic waves, the propagation of harmonic waves in thin-walled cylindrical elastic shells containing a compressible viscous liquid, and the propagation of harmonic waves in hydroelastic systems consisting of a resting compressible viscous liquid and elastic compressible or incompressible bodies with initial stresses. Publications concerning the above problems are analyzed.

The article is the first part of a survey of problems in compressible, viscous fluid dynamics as related to the dynamics of rigid and elastic bodies in a compressible, viscous fluid in the linearized formulation. The formulation of basic problems is discussed, along with a method of solution based on general solutions of the Navier-Stokes equations in vector and scalar form in dynamical problems. Forced harmonic vibrations of rigid bodies in rest and moving compressible, viscous fluids are discussed. Publications relevant to the stated problems are analyzed.

The problem of interaction of a rigid body and a boundary of a viscous fluid under acoustic wave propagation is investigated. An example of the motion of a cylinder in the vicinity of a flat rigid wall caused by acoustic field forces is examined.

The plane asymmetric problem of vertical impact of a rigid body against the surface of a compressible fluid is considered. It is assumed that the penetrating circular cylinder has a shifted center of mass. Solution of the boundary-value problem is reduced to solution of an infinite system of linear integral Volterra equations of the second kind. We present an analysis of the results as a function of initial values of the angle of asymmetry.

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.

Controllability with respect to certain variables. We will examine dynamic systems that can be described by ordinary differential equations. Where x {element_of} D {improper_subset} R{sup n} is the phase vector; u {element_of} U {improper_subset} R{sup m} is the control vector. The latter is a bounded measureable function of time t, t {element_of} T = [0, {infinity}]. We assume that the regions D and U are convex and contain the coordinate origin. We also assume the function f to be a function of its arguments that is continuously differentiable a sufficient number of times. We divide the phase vector into two subvectors x{sup T} = (x{sub {alpha}}{sup T}, x{sub {beta}}{sup T}) (x{sub {alpha}} {element_of} D{sub {alpha}} {improper_subset} R{sup {alpha}}, x{sub {beta}} {element_of} D {improper_subset} R{sup {beta}}) and we introduce the following definition for system (1.1).

scopic instruments [I], rotors on magnetic or gas bearings [2], and also ball bearings in the presence of gaps, there arises the problem of the impact-oscillatory motion of rigid bodies in a space bounded by two rectilinear rough surfaces converging at a sharp angle. In the present article this problem is examined in its "plane formulation," with the cases of a rigid body with two translational degrees of freedom (of a material point) and a round body possessing an additional rotational mobility being considered separately. It is assumed that in the intervals between collisions with the stops the body moves rectilinearly through inertia. Collision of the body with the stops is described within the framework of stereomechanical impact theory [3, 4]. The necessary and sufficient conditions of realization of quasiplastic collision [4] an infinite-impact process of finite duration at the end of which the body arrives at the apex of the angle - are determined. The Case of a Material Point. If a rigid body undergoes translational, uniform, and rectilinear movement in the intervals between collisions, then, without loss of generality, it may be taken to be a material point. We will assume that the stops are identical to one another in their elastic and dissipative properties, such that the impact interaction with them of a material point is characterized by identical values of the velocity restoration coefficient R and the impact friction coefficient f. Let us assume that an arbitrary collision of the body with a stop has a nonsliding character and that therefore the vector of its restored velocity is perpendicular to the stop. It is clear that the subsequent impact-oscillatory movement of the body will be characterized by continuous withdrawal from the apex of the angle. Consequently, the above-mentioned quasiplastic impact in this case should be characterized by a sequence of damping sliding collisions [4]. Let us enumerate all collisions of a point with the stops from the beginning of the movement. Then an arbitrary K-th sliding impact (K = 0, I, 2, ...) will be characterized by the relations vh = RVksin ~h, uh = Vh(cos~h--×sinek). (i) Here V k is the modulus of drop velocity; the angle of its incline to the stop %k for a quasiplastic impact may be considered as within the interval (0, 0/2), K = f(l + R), while

We sum up the results regarding the stabilization of the investigated motion. The system (1.2) is stabilizable in the linear approximation if the conditions (2.2) on the parameter are not satisfied, and also in the case 4 if a2α2α3ω0>0. The system (1.2) is stabilizable in the nonlinear approximation for ω0=0, α1≠0. The system (1.2) is nonstabilizable in the case 4 if a2α2α3ω0