New invariant relations for the generalized two-field gyrostat

In the paper, we consider a completely integrable Hamiltonian system with three degrees of freedom found by V.V. Sokolov and A.V. Tsiganov. This system is known as the generalized two-field gyrostat. For the case of only gyroscopic forces present, we find new invariant four-dimensional submanifolds such that the induced dynamical systems are almost everywhere Hamiltonian with two degrees of freedom.     

New invariant relations for one critical subsystem of a generalized two-field gyrostat

In this paper new invariant relations for one critical subsystem of a completely integrable Hamiltonian system with three degrees of freedom found by V.V. Sokolov and A.V. Tsyganov, known as a generalized two-field gyrostat, are obtained. The dynamic system that is induced on the invariant four-dimensional submanifolds is almost everywhere Hamiltonian with two degrees of freedom. The type of system motions on this invariant manifold is determined.     

Stable periodic motions of an impact-pair

The asymptotically stable motion of two rigid masses separated by a clearance is formulated most generally when the masses collide periodically. Predictions are shown to agree satisfactorily with comprehensive experimental results. Difficulties encountered in the practical application of this and similar stability theories are discussed with the help of empirical data. A technique is proposed for determining a clearance's dimensions in situ which could provide a basis for detecting progressive wear in mechanisms like gear trains.     

Experiments on periodic and chaotic motions of a parametrically forced pendulum

An experimental study of periodic and chaotic type aperiodic motions of a parametrically harmonically excited pendulum is presented. It is shown that a characteristic route to chaos is the period-doubling cascade, which for the parametrically excited pendulum occurs with increasing driving amplitude and decreasing damping force, respectively. The coexistence of different periodic solutions as well as periodic and chaotic solutions is demonstrated and various transitions between them are studied. The pendulum is found to exhibit a transient chaotic behaviour in a wide range of driving force amplitudes. The transition from metastable chaos to sustained chaotic behaviour is investigated.     

Stable periodic motions in the problem on passage through a separatrix

A Hamiltonian system with one degree of freedom depending on a slowly periodically varying in time parameter is considered. For every fixed value of the parameter there are separatrices on the phase portrait of the system. When parameter is changing in time, these separatrices are pulsing slowly periodically, and phase points of the system cross them repeatedly. In numeric experiments region swept by pulsing separatrices looks like a region of chaotic motion. However, it is shown in the present paper that if the system possesses some additional symmetry (like a pendulum in a slowly varying gravitational field), then typically in the region in question there are many periodic solutions surrounded by stability islands; total measure of these islands does not vanish and does not tend to 0 as rate of changing of the parameter tends to 0.(c) 1997 American Institute of Physics.     

Analytical periodic motions in a parametrically excited, nonlinear rotating blade

The stability and bifurcation analyses of periodic motions in a rotating blade subject to a torsional excitation are investigated. For high speed rotations, cubic geometric nonlinearity and gyroscopic effects of the rotating blade are considered. From the Galerkin method, the partial differential equation of the nonlinear rotating blade is simplified to the ordinary differential equations, and periodic motions and stability of the rotating blade are studied by the generalized harmonic balance method. The analytical and numerical results of periodic solutions are compared. The rich dynamics and co-existing periodic solutions of the nonlinear rotating blades are investigated.     

The existence of periodic motions in rub-impact rotor systems

The existence of rub-impact periodic motions in an eccentric rotor system is considered. A criterion for the periodicity condition of n-periodic impacts is derived and other conditions for real rub-impacts are also discussed. A method consisting of analytical and numerical techniques is presented to solve the existence problem of rub-impact periodic motions. Some special results are obtained by theoretical analyses for rub-impact rotor systems, including the existence of grazing circle motions and that of single-impact periodic motions.     

Explicit solution for the optical properties of a uniaxial crystal in generalized ellipsometry

The complex ordinary (N_o) and extraordinary (N_e) refractive indices of an absorbing uniaxial crystal can be explicitly derived from the normalized diagonal (α) and off-diago-nal (β) elements of one oblique-incidence (at angle φ) reflection matrix measured by generalized ellipsometry on a crystal face that contains the optic axis at an oblique orientation (ω ≠ O or $\text{π}\text{2}$) with respect to the plane of incidence. At most four solution sets (N_o, N_e) are mathematically and physically consistent with one measurement set (α,β,φ,ω). In many cases, identification of the correct solution is feasible without additional measurements. If necessary repeated measurements at a different value of φ or ω will resolve the ambiguity; only one solution set remains invariant upon a change of φ or ω, the correct set. A single measurement of reflectance (e.g., for p-polarized light) may also be adequate. The analytic inversion method is used in an error analysis to determine optimum choices of angle of incidence (φ ? 50–70°) and crystal orientation (ω ? 45°) that lead to minimum percentage errors of N_o and N_e for several uniaxial crystals.     

Explicit solutions of a certain class of differential equations by means of fractional calculus

Judging by the remarkably large number of recent publications on Fractional Calculus and Its Applications in several widely diverse areas of mathematical, physical, and engineering sciences, the current popularity and importance of the subject of fractional calculus cannot be overemphasized. Motivated by some of these interesting developments, many authors have recently demonstrated the usefulness of fractional calculus in the derivation of explicit particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. The main object of the present paper is to show how several recent contributions on this subject, involving a certain class of ordinary differential equations, can be obtained (in a unified manner) by suitably applying some general theorems on explicit particular solutions of a family of linear ordinary fractional differintegral equations.     

Explicit Runge-Kutta-like schemes to solve certain quantum operator equations of motion

We propose a kind of explicit sympletic Runge-Kutta-like scheme to solve operator equations of motion in quantum mechanics and in a quantum scalar field theory.     

Existence and analytical predictions of periodic motions in a periodically forced,nonlinear friction oscillator

The grazing bifurcation, stick phenomena and periodic motions in a periodically forced, nonlinear friction oscillator are investigated. The nonlinear friction force is approximated by a piecewise linear, kinetic friction model with the static force. The total forces for the input and output flows to the separation boundary are introduced, and the force criteria for the onset and vanishing of stick motions are developed through such input and output flow forces. The periodic motions of such an oscillator are predicted analytically through the corresponding mapping structure. Illustrations of the periodic motions in such a piecewise friction model are given for a better understanding of the stick motion with the static friction. The force responses are presented, which agreed very well with the force criteria. If the fully nonlinear friction force is modeled by several portions of piecewise linear functions, the periodically forced, nonlinear friction oscillator can be predicted more accurately. However, for the fully nonlinear friction force model, only the numerical investigation can be carried out.     

The local time of self-intersections of Brownian motions as generalized Brownian functionals

In this Letter, we show that the local time of self-intersections ofd-dimensional Brownian motions is a generalized Brownian functional in the sense of Hida.     

A generalized modelling technique for linearized motions of mechanisms with flexible parts

Analysis and control of vibrations of agricultural machines to improve machine performance and vibration comfort of the operator is a major concern of manufacturers these days. In this paper, an analytical method to build the linearized equations of motion of an elastic tree structured mechanism, is presented. The method is based on the principle of virtual work resulting in a set of parameterized linear equations that are functions of the mechanical parameters and the geometry and the interconnection structure of different bodies in the mechanism. The rigid-body motions of the mechanical system are represented by Lagrangian generalized co-ordinates while elastic deformations are described by nodal co-ordinates from a finite element formulation. Explicit expressions for external distributed and concentrated forces and internal concentrated forces acting on the mechanism are given.