COLLINEAR CONTROL OF OSCILLATION MODES OF SPATIAL DOUBLE PENDULUM WITH VARIABLE GAIN

This article is devoted to the study of controlled movements of spatial double pendulum with non-parallel cylindrical joints axes. The collinear control is used to swinging of the system by feedback. The most important property of collinear control is the ability of increasing system oscillations only on one oscillation mode. A modification of the collinear control law with variable gain depending on the energy level is investigated. It allows to control the system motions more flexible than in the case of constant gain. As a result, it is possible to observe a smooth transition from a linear oscillation mode to a nonlinear one with a gradual output to a steady oscillation motion with a given energy level. The obtained results are clearly illustrated by graph dependencies that demonstrate the swinging of the system on one oscillation mode from small to finite amplitudes.

Formation of singularities in a stratified fluid in the presence of a critical level

The paper is concerned with formation of singularities in a density stratified fluid subject to a monochromatic point source of frequency σ. The frequency of the source is assumed to be such that the steady-oscillation equation is hyperbolic in the neighbourhood of the source and degenerates at a critical level. We obtain asymptotic formulae demonstrating how the solution diverges as t → ∞ on the characteristic surface emanating from the source. It is shown that, at points of the surface that belong to the critical level, the solution behaves as t⅔ exp {i(σt + π/2)} as t → ∞, whereas its large time behaviour at the other points of the surface is given by t½ exp {i(σt + π/2 ± π/4)}.

A solution of the heat conduction equation in the finite cylinder exposed to periodic boundary conditions: the case of steady oscillation and constant thermal properties

The problem of the temperature distribution in the finite cylinder with periodic boundary conditions over all the surfaces is treated analytically and numerically. The solution in the case of steady oscillation and constant thermal properties is presented in reduced coordinates and can be used to correct thermal diffusivity measurements by the Angstrom method or to study, for example, the influence of a destabilized controller which induces a periodic temperature of the sample during thermal treatment.

The Behaviour of Damped Linear Systems in Steady Oscillation

In this paper (published in The Aeronautical Quarterly, Vol. VII, Part 2, p. 156, May 1956), a set of equations is derived giving the responses at the generalised coordinates q1, q2, … qn which are caused by a harmonic generalised force Φseiωt corresponding to qs. They are equations (26), namelyThis note is concerned with a sentence which follows these equations which says (of them) that “ … there is no longer any assurance that the ratios between the constants mBrs in any column of partial fractions are the same for all co-ordinates qs at which excitation is applied, since the previous limiting condition cannot now be applied. Thus the shapes of the ‘ modes ’ which are associated with the various columns depend upon the nature of the applied harmonic excitation.”

The Behaviour of Damped Linear Systems in Steady Oscillation

SummaryThe classical theory of small harmonic vibrations of a linear damped system embodies the notion of “ viscous damping.” The equations of motion which result are somewhat complicated and, when there are more than two degrees of freedom, they are usually too unwieldy to be of much practical value. When the damping is small, however, approximating assumptions may be made which permit the treatment of systems which are near resonance as if they possess but one degree of freedom. But the effects of making these assumptions are by no means easily assessed, and even their justification is tedious.It is shown that these difficulties may be greatly diminished by postulating hysteretic damping instead of viscous damping; the concept of hysteretic damping has been dealt with in two previous papers. The equations then take a much simpler form and the justification for, and validity of, the foregoing approximations are more easily seen. Moreover, the effects of damping upon the principal modes which the system possesses in the absence of its damping may be elucidated in this way.