On a Replacement Problem of a Cumulative Damage Model

Items are assumed to fail only by degradation. An appropriate stochastic model of such items is a cumulative process in which an item can fail only when the total amount of damage exceeds a prespecified failure level. This paper introduces a replacement policy in which an item is replaced at a certain level of damage before failure or at failure, whichever occurs first. The optimum replacement level of damage which will minimize the total expected cost per unit of time for an infinite time span is obtained. A numerical example is also presented. The total expected cost for a finite time span is also discussed.     

Analysis of a Diffusion-Convection System Modeling a Contamination Problem

A system of coupled diffusion-convection equations which model a contamination problem are analyzed. The equations are reformu-lated as an abstract problem which is used to obtain existence, uniqueness and posit ivity results for the solutions. A minimum principle is also proved and a special class of solutions which have bearing on the model are derived     

An inverted pendulum on a fixed and a moving base

The plane motions of a controlled single-link pendulum with a fixed suspension point and a pendulum with its suspension point located at the centre of a wheel which rolls without sliding along a flat horizontal surface are considered. The control torque, applied to the pendulum at the suspension point, is bounded in absolute magnitude. A controllability domain is constructed in the linear approximation for the one and the other pendulum, from all points of which the pendulum can be brought into the upper unstable equilibrium position without oscillations about the lower equilibrium. It is shown that the domain of controllability is greater for a pendulum mounted on a wheel, as a result it is more easily stabilizable. Control laws are constructed, under which the domain of attraction is identical to the controllability domain and is thereby the largest possible domain.     

Fast computation of a rational point of a variety over a finite field

We exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence. Its time-space complexity is roughly quadratic in the logarithm of the cardinality of the field and a geometric invariant of the input system. This invariant, called the degree, is bounded by the Bézout number of the system. Our algorithm works for fields of any characteristic, but requires the cardinality of the field to be greater than a quantity which is roughly the fourth power of the degree of the input variety.

     

Fuzzy sets as a basis for a theory of possibility

The theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable. More specifically, if F is a fuzzy subset of a universe of discourse U={u} which is characterized by its membership function μ_F, then a proposition of the form “X is F,” where X is a variable taking values in U, induces a possibility distribution ∏X which equates the possibility of X taking the value u to μF(u)—the compatibility of u with F. In this way, X becomes a fuzzy variable which is associated with the possibility distribution ∏_x in much the same way as a random variable is associated with a probability distribution. In general, a variable may be associated both with a possibility distribution and a probability distribution, with the weak connection between the two expressed as the possibility/probability consistency principle.A thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is, in the main, possibilistic rather than probabilistic in nature. Thus, by employing the concept of a possibility distribution, a proposition, p, in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p. Several types of conditional translation rules are discussed and, in particular, a translation rule for propositions of the form “X is F is α-possible,” where α is a number in the interval [0, 1], is formulated and illustrated by examples.     

On a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam

This paper is concerned with a model which describes the interaction of sound and elastic waves in a structural acoustic chamber in which one “wall” is flexible and flat. The model is new in the sense that the composite dynamics of the three-dimensional structure is described by the linearized equations for a gas defined on the interior of the chamber and the Reissner-Mindlin plate equations on the two-dimensional flat wall of the chamber, while, if a two-dimensional acoustic chamber is considered, the Timoshenko beam equations describe the deflections of the one-dimensional “wall.” With a view to achieving uniform stabilization of the structure linear feedback boundary damping is incorporated in the model, viz. in the wave equation for the gas and in the system of equations for the vibrations of the elastic medium. We present the uniform stability result for the case of a two-dimensional chamber and outline the method for the three-dimensional model which shows strong resemblance with the system of dynamic plane elasticity.     

Bifurcation in a time-optimal problem for a second-order non-linear system

A time-optimal problem for a second-order non-linear system with one degree of freedom is considered. The system describes the dynamics of an inertial object under the action of a control force of limited modulus which appears linearly and a perturbing force which is periodic in coordinate. The terminal set represents points on the abscissa of the phase plane, and the distance between two neighbouring points is equal to the period of the perturbing force. An estimate is obtained for the amplitude of the control for which the control has the simplest structure: the number of switchings is not greater than one.     

The problem of the reinforcement of a plate with a cutout by a two-dimensional patch

The problem of the reinforcement of an elastic plate with a cutout by means of a two-dimensional patch, which completely covers the cutout and is rigidly fixed to the plate along its boundary, is considered. The cutout and the patch can be of arbitrary shape. The problem is reduced to a system of three singular integral equations using of special integral representations which describe the stress state in the plate and in the patch. The unique solvability of the system is proved. Examples are presented.     

Scheduling a repairman in a finite source system

The scheduling of a single server in a finite source model is considered. TheN customers in the system have different failure and repair rates. Also the costs depend on the customers which are broken down. We give a condition under which the average costs are minimized by a simple list policy, and with a counterexample we show that in the general case no optimal list policy may exist. This motivates us to derive policies which are optimal under low and high traffic conditions. They are again list policies, which behave well numerically.     

The steady motions of a gyrostat suspended on a rod in a central gravitational field

The problem of the existence, stability and bifurcation of the steady motions of two bodies in an orbital tethered system, when one of the bodies is a symmetrical satellite with a rotor on the axis of symmetry, is considered. One-parameter families of steady motions are indicated, and their stability and bifurcations are investigated. The conditions which relate the parameters of the system for which stabilization of the families obtained is possible using a rotating rotor are obtained.     

Conditions for a two-dimensional surface in E 5 to be contained in a hypersphere or a hyperplane

Two theorems on conditions under which a two-dimensional surface in Euclidean 5-space is contained in a hypersphere and one theorem on conditions under which such a surface is contained in a hyperplane are proved. The notion of hyperbolic and elliptic domains on a surface are introduced. The conditions in the theorems are expressed in terms of the behavior of the plane of the normal curvature ellipse of the surface and certain boundary conditions. An example which shows that the boundary conditions are essential is constructed.     

Analysis of a single-server queue interacting with a fluid reservoir

We consider a single-server queueing system with Poisson arrivals in which the speed of the server depends on whether an associated fluid reservoir is empty or not. Conversely, the rate of change of the content of the reservoir is determined by the state of the queueing system, since the reservoir fills during idle periods and depletes during busy periods of the server. Our interest focuses on the stationary joint distribution of the number of customers in the system and the content of the fluid reservoir, from which various performance measures such as the steady-state sojourn time distribution of a customer may be obtained. We study two variants of the system. For the first, in which the fluid reservoir is infinitely large, we present an exact analysis. The variant in which the fluid reservoir is finite is analysed approximatively through a discretization technique. The system may serve as a mathematical model for a traffic regulation mechanism - a two-level traffic shaper - at the edge of an ATM network, regulating a very bursty source. We present some numerical results showing the effect of the mechanism. This revised version was published online in June 2006 with corrections to the Cover Date.     

Aerodynamic Sound Due to a Source Near a Half-Plane

The effect of applying a Kutta-Joukowski condition at the edgeof a semi-infinite plane which is generating noise in a turbulentfluid at low Mach numbers is examined. It is found that in somecircumstances the noise is increased and the intensity of thedistant sound field may depend upon the third power of a typicalfluid velocity. When the sound field is convected the orders of magnitude ofthe acoustic far field are the same whether or not the Kutta-Joukowskicondition is applied, provided that the point of observationis not near the wake. Near the wake there is an acoustic "surface"wave which is much stronger than the distant field elsewhere.     

Nonlinear resonances of a semi-infinite cable on a nonlinear elastic foundation

The Multiple Time Scale (MTS) method is applied to the study of nonlinear resonances of a semi-infinite cable resting on a nonlinear elastic foundation, subject to a constant uniformly distributed load and to a linear viscous damping force. The zero order solution provides the static displacement, which is governed by a nonlinear equation which has been solved in closed form. The first order solution provides the linear resonances, which are seen to be functions of the nonlinearity parameter and of the static displacement at the finite boundary only. Although the first-order governing equation is linear, it has non constant coefficients and cannot be solved in closed form, so that a numerical solution is considered; the eigenfrequencies obtained in this way are also compared with the approximate eigenvalues obtained by the WKB method. At the second order of the MTS expansion, we see that the solution is independent of the intermediate time scale; some additional terms are present, including a time-independent shift of the average position of the oscillations. Finally, the nonlinear frequency–amplitude response curves, which are investigated in detail and which represent the main result of this work, are obtained from the solvability condition at the third order.     

Optimization of the motion in a resistant medium of a body with a movable internal mass

The motion, in a resistant medium, of a system consisting of a rigid body and movable internal mass is considered. The external medium acts on the body by a force that piecewise linearly depends on its speed. The class of periodic motions of the internal mass for which the speed of this mass relative to the body is piecewise constant is studied. It is shown that, under certain conditions, the forward movement of the whole system in the medium is possible. The average speed of this movement over a period is determined. Optimal parameters of the motion of the internal mass for which the average speed of the system movement is maximal are found.     

A qualitative investigation of the oscillations of a pendulum with a periodically varying length and a mathematical model of a swing

The behaviour of the amplitude-frequency characteristics of families of periodic solutions, produced from the equilibrium position of a system, is established by a qualitative investigation of the equation of the oscillations of a pendulum, the length of which is an arbitrary periodic function of time. The non-local conditions for their stability and instability, expressed in terms of the amplitude and frequency of the oscillations, are obtained. The results are used when discussing the parametric and self-excited oscillatory model of a swing. In the parametric model the length of a swing is a specified periodic function of time, and in the self-excited oscillatory model it is a function of the phase coordinates of the system. For an appropriate choice of these functions, both systems have a common periodic solution. It is shown that the parametric model leads to an erroneous conclusion regarding the instability of the periodic mode, which is in fact realized in the oscillations of a swing, whereas the self-excited oscillatory model indicates its stability.     

On a nonsingular singular surface of a differential game

A differential game in which nonterminating play is assigned a fixed value is examined. The solution exhibits a new kind of singular surface which an optimal path takes an infinite time to cross. By playing nonoptimally in a sufficiently small neighborhood of this surface, the player who prefers the optimal (terminating) value to the value of nonterminating play can force the state across this surface and obtain a value arbitrarily close to the optimal.This work was carried out with the support of a CSIRO postgraduate studenship.     

Joint deformation of a circular inclusion and a matrix

An elastic infinite plane containing a circular inclusion with given jumps of tractions and displacements along the interface and nonzero conditions at infinity is considered. Explicit expressions are derived for the Goursat-Kolosov complex potentials of this problem. The solution constructed can be used to examine various circular interfacial defects, including interfacial cracks and rigid parts of the interface. The problem under consideration is fundamental for the superposition method, which solves many problems in which a circular region is an element of a polyphase elastic medium. In such cases, the well-posedness of the problem, which depends on the interrelation between the jumps of tractions and displacements, follows from the very superposition method. The application techniques of this method are demonstrated for singular problems on the action of a point force and an edge dislocation located inside an inclusion or in the matrix. Computational results for the tractions arising at the interface under the action of a point force concentrated in the inclusion are given.     

A subdirectly irreducible band accepting a = a

Subdirectly irreducible bands which generate all proper subvarieties of bands (idempotent semigroups) were exhibited in [7]. In this paper a very general construction is presented which can be used to produce subdirectly irreducible bands which generate the variety of bands.     

The contact problem for a piecewise-homogeneous plane with a finite inclusion

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite inclusion, which meets the interface at a right angle and is loaded with shear forces, is considered. The contact stresses along the contact line are determined, and the behaviour of the contact stresses in the neighbourhood of singular points is established. By using the methods of the theory of analytic functions, the problem is reduced to a singular integro-differential equation in a finite section. Using an integral transformation, a Riemann problem is obtained, the solution of which is presented in explicit form.