The equations of the approximate theory of the motion of a rigid body with a vibrating suspension point

The motion of a rigid body in a uniform gravity field is investigated. One of the points of the body (the suspension point) performs specified small-amplitude high-frequency periodic or conditionally periodic oscillations (vibrations). The geometry of the body mass is arbitrary. An approximate system of differential equations is obtained, which does not contain the time explicitly and describes the rotational motion of the rigid body with respect to a system of coordinates moving translationally together with the suspension point. The error with which the solutions of the approximate system approximate to the solution of the exact system of equations of motion is indicated. The problem of the stability with respect to the equilibrium of the rigid body, when the suspension point performs vibrations along the vertical, is considered as an application.     

Approximate equations of rotational motion of a rigid body carrying a movable point mass

The dynamics of a compound system, consisting of a rigid body and a point mass, which moves in a specified way along a curve, rigidly attached to the body is investigated. The system performs free motion in a uniform gravity field. Differential equations are derived which describe the rotation of the body about its centre of mass. In two special cases, which allow of the introduction of a small parameter, an approximate system of equations of motion is obtained using asymptotic methods. The accuracy with which the solutions of the approximate system approach the solutions of the exact equations of motion is indicated. In one case, it is assumed that the point mass has a mass that is small compared with the mass of the body, and performs rapid motion with respect to the rigid body. It is shown that in this case the approximate system is integrable. A number of special motions of the body, described by the approximate system, are indicated, and their stability is investigated. In the second case, no limitations are imposed on the mass of the point mass, but it is assumed that the relative motion of the point is rapid and occurs near a specified point of the body. It is shown that, in the approximate system, the motion of the rigid body about its centre of mass is Euler–Poinsot motion.     

Geometric Aspects of Degenerate Modulation Equations

Geometric aspects of degenerate modulation equations associated with spatially reversible systems are considered. Our primary observation is that stationary solutions of such equations always have a Poisson structure that is reminiscent of the equations governing the rigid body in mechanics. The Poisson structure is used to study the geometry of “spatial” phase space: A nontrivial Casimir of the Poisson structure provides a foliation of the phase space, spatially periodic states are given by critical points on level sets of the Casimir and stability type is given by the rate of change of the Casimir function. The bifurcation of spatially periodic states is then studied using singularity theory. The case where branches intersect transversely is treated in detail.     

The stability of a non-autonomous functional-differential equation relative to part of the variables

The asymptotic stability and instability of the trivial solution of a functional-differential equation of delay type relative to part of the variables are investigated using limit equations and a Lyapunov function whose derivative is sign-definite. The theorems thus obtained are used to solve the problem of stabilizing mechanical control systems with delayed feedback. As examples, solutions of problems of the uniaxial and triaxial stabilization of the rotational motion of a rigid body with a delay in the control system are presented.     

Numerical analysis method for studying local forms of stability loss of bearing layers of three-layered shells using mixed forms

A revised formulation of linearized stability problems of three-layered shells with a sofi filler has been presented. The form of stability loss of the rigid layers is mixed in the shells when the moment precritical stress-strain state (SSS) is reached and is localized in the principal moment SSS zones. If the filler thickness is much greater than the thickness of the rigid layers, the size of the bulges and thickness of the filler have the same order of magnitude. Thus, a very fine grid must be used for a numerical solution of the stability loss equations, which poses considerable computational difficulties. A numerical analysis method is proposed for the local forms of mixed mode stability loss of the rigid layers of a three-layered shell. Using this method, the solution of equations for the precritical SSS by the finite element scheme is found but an analytical solution of reduced stability loss equations is presented for estimating the critical load. This solution is an asymptotic approximation for local modes of stability loss implemented into design.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 1, pp. 88–100, January–February, 1995.     

Dynamic behavior of an AMB supported rotor subject to harmonic excitation

This paper presents a study of the non-linear response of a simple rigid disk-rotor, supported by active magnetic bearings (AMB), without gyroscopic effects. The case of primary resonance is examined under multi-excitation forces. The rotating shaft is described by a coupled second order non-linear ordinary differential equations. Approximate solutions are sought applying the method of multiple scales. Numerical simulations are carried out to illustrate the steady-state response and the stability of the solutions for various parameters using the frequency response function method. It is shown that the system parameters have different effects on the non-linear response of the rotor. For steady-state response, however, multiple-valued solutions and jump phenomenon occur. Results are compared to previously published work.     

Some exact results basic to the linearized Boltzmann equations for a binary mixture of rigid spheres

Six exact solutions (related to the conservation of number, energy and momentum) of the linearized Boltzmann equations for a binary mixture of rigid spheres, for the case of isotropic scattering in the center-of-mass system, are reported. The verification of the reported exact solutions (collisional invariants) is based on a recently reported explicit formulation of the linearized Boltzmann equation for a binary mixture of rigid spheres. Elementary analysis is used also to establish a basic flow condition.     

Some exact results basic to the linearized Boltzmann equations for a binary mixture of rigid spheres

Six exact solutions (related to the conservation of number, energy and momentum) of the linearized Boltzmann equations for a binary mixture of rigid spheres, for the case of isotropic scattering in the center-of-mass system, are reported. The verification of the reported exact solutions (collisional invariants) is based on a recently reported explicit formulation of the linearized Boltzmann equation for a binary mixture of rigid spheres. Elementary analysis is used also to establish a basic flow condition.     

EVANS FUNCTIONS AND INSTABILITY OF A STANDING PULSE SOLUTION OF A NONLINEAR SYSTEM OF REACTION DIFFUSION EQUATIONS

In this paper, we consider a nonlinear system of reaction diffusion equations arising from mathematical neuroscience and two nonlinear scalar reaction diffusion equations under some assumptions on their coefficients. The main purpose is to couple together linearized stability criterion (the equivalence of the nonlinear stability, the linear stability and the spectral stability of the standing pulse solutions) and Evans functions to accomplish the existence and instability of standing pulse solutions of the nonlinear system of reaction diffusion equations and the nonlinear scalar reaction diffusion equations. The Evans functions for the standing pulse solutions are constructed explicitly.     

Evolution of the characteristics of the Prandtl solution

We transform the well-known Prandtl solution using point symmetries. This yields a new class of exact solutions to the plasticity equations. We dwell on those solutions that can be used to describe the flat flows arising as a plastic layer is compressed by the rigid plates.     

Integral Representations of Solutions of Differential Equations Free from Accessory Parameters

We show that every accessory parameter free system of differential equations of Okubo normal form has integral representation of solutions. The proof is constructive; we study the change of solutions under the operations—the extension and the restriction, which have been introduced by Yokoyama [Construction of systems of differential equations of Okubo normal form with rigid monodromy, preprint] in order to construct every such system of differential equations. Several examples are given.     

Analysis of the Euler–Poisson equations by methods of power geometry and normal form

New approaches and methods for studying non-linear problems are applied to the classical problem of the motion of a heavy rigid body about a fixed point, i.e., to the system of Euler–Poisson equations. All the asymptotic expansions of the solutions of the Kowalewski equations, to which the Euler–Poisson equations reduce when certain constraints are imposed on the parameters, are found using power geometry. They form 24 families. Then all the exact solutions of the Kowalewski equations of a specific class (which includes almost all the known exact solutions) are found on the basis of these expansions. Five new families of such solutions are found. Instead of the conventional technique of studying the global integrability of the Euler–Poisson equations, studying their local integrability near stationary and periodic solutions is proposed. Normal forms are used for this purpose. Sets of real stationary solutions, in the vicinity of which these equations are locally integrable, are discovered using them. Other real stationary solutions, in the vicinity of which the Euler–Poisson equations are locally non-integrable, are also found. This is established using the theory of resonant normal forms developed and computer calculations of the coefficients of a normal form.     

Linearized stability and instability of nonconstant periodic solutions of Lagrangian equations

This paper is motivated by the stability problem of nonconstant periodic solutions of time‐periodic Lagrangian equations, like the swing and the elliptic Sitnikov problem. As a beginning step, we will study the linearized stability and instability of nonconstant periodic solutions that are bifurcated from those of autonomous Lagrangian equations. Applying the theory for Hill equations, we will establish a criterion for linearized stability. The criterion shows that the linearized stability depends on the temporal frequencies of the perturbed systems in a delicate way.     

On approximate methods of analysing certain singularly-perturbed systems

A certainclass of sigularly-perturbed systems which have a variety of m-dimensional stationary positions is considered. When a small parameter disappears, the system also has an m-dimensional manifold of stationary positions and, therefore, the corresponding characteristic equation has m zero roots. The conditions under which the solution of a stability problem reduces to the same problem for a degenerate system are defined. As an application in practice gyroscopic stabilizing systems (the critical case corresponds to such systems) with elastic elements of high stiffness are discussed. The conditions under which the solution of the problem of the stability of steady motion follows from the solution of this problem for an ideal system (with absolutely rigid elements) are obtained. The problem of the closeness of the corresponding solutions of the complete and a simplified system of differential equations over an infinite time interval is discussed.     

Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach spaces

A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified theoretical foundation for the stability analysis of solutions to nonlinear stiff problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and VFDEs of other type which appear in practice.     

Mathematical modelling and stabilization of large flexible spacecraft subject to orbital perturbation

In this paper the problem of modelling of large flexible spacecraft and their stabilization under the influence of orbital (radial) perturbation is considered. A complete dynamics of the spacecraft consisting of a rigid bus and a flexible beam is derived using Hamilton's principle. The equations of motion consist of a coupled system of partial differential equations governing the vibration of the flexible beam and ordinary differential equations describing the translational and rotational motions of the rigid bus. The asymptotic stability of the system is proved using Lyapunov's approach. Simple feedback controls are suggested for the stabilization of the system. For illustration, numerical simulations are carried out, giving interesting results.     

Integrable variational equations of non-integrable systems

Paper is devoted to the solvability analysis of variational equations obtained by linearization of the Euler-Poisson equations for the symmetric rigid body with a fixed point on the equatorial plain. In this case Euler-Poisson equations have two pendulum like particular solutions. Symmetric heavy top is integrable only in four famous cases. In this paper is shown that a family of cases can be distinguished such that Euler-Poisson equations are not integrable but variational equations along particular solutions are solvable. The connection of this result with analysis made in XIX century by R. Liouville is also discussed.     

An algorithm for normalizing Hamiltonian systems in the problem of the orbital stability of periodic motions

A constructive procedure is proposed for constructing equations of perturbed motion convenient for investigating the orbital stability of periodic motion in an autonomous Hamiltonian system with two degrees of freedom. An algorithm for normalizing these equations is described, and formulae for evaluating the coefficients of the normal form are presented. The results are used to investigate the stability of motion in certain special cases of the regular Grioli precession of a heavy rigid body with one fixed point.     

Stability analysis of the servohydraulic equations with linear state feedback and harmonic reference signal

The servohydraulic equations describe the dynamic behaviour of a hydrostatic drive featuring a servo‐valve and a hydraulic cylinder. Under the assumptions of constant supply pressures, a rigid support of the cylinder, and a rigid mass attached to the piston, the evolution of cylinder pressures as well as piston position and velocity is governed by a 4th order non‐linear ordinary differential equation. The servo valve is assumed to be much faster than the dynamics of the mass‐cylinder system. Thus, the valve dynamics is neglected. Adding a feedback control law makes up the servohydraulic equations. These equations are partially singularly perturbed. The perturbation parameter is associated with fluid compressibility and the reduced system corresponds to the case of an incompressible fluid. This paper deals with the stability analysis of periodic orbits arising in the case of harmonic reference position trajectories. Geometric singular perturbation theory is used to generate stability charts. Numerical computations verify the analytical results. For a given set of hydraulic system parameters, the stability of periodic orbits depends on controller gains as well as on the amplitude and frequency of the reference signal.     

Absolute stability of the solutions of linear differential equations with time lags in Banach space

The connection between the exponential stability of the solutions of linear differential equations in space of multidimensional bounded vector sequences and the absolute asymptotic stability of the solutions of differential equations with several time lags is investigated.