Peter C. Varadi

On Some Peculiar Aspects of Axial Motions of Closed Loops of String in the Presence of a Singular Supply of Momentum

We consider the dynamics of a closed loop of inextensible string which is undergoing an axial motion. At each instant, one material point of the string is in contact with a singular supply of linear momentum (also known as an external constraint). Several peculiar features of this problem which have not been previously discussed are presented. These include the possible presence of an arbitrary number of kinks, the vanishing nature of the singular supply of momentum, and the critical nature of the tension in the string. When the linear momentum is supplied by a mass-spring-dashpot system, we are also able to establish an exact expression for the frequency of the resulting vibrations, prove that dissipation cannot be present, show that these vibrations only occur for discrete speeds of axial motion, and establish that Coulomb friction is absent.

Hoberman's Sphere, Euler Parameters and Lagrange's Equations

In this classroom note, we explore how the Euler parameters can be used to represent a particular homogeneous deformation of a continuum. One possible application is Hobermans sphere. With the assistance of the theory of a pseudo-rigid body, we show how the motion of the continuum can be determined. We also present a new derivation of Lagranges equations for the rotational dynamics of a rigid body where the rotation tensor is parameterized using Euler parameters.

Elastic equilibria of translating cables

SummaryThe equations of motion describing the non-linear behavior of a perfectly flexible travelling cable are derived from first principles. Influences due to changes in the cross-sectional area of the cable and mass conservation are included. A homogenous isotropic non-linearly elastic cable material is assumed and the qualitative nature of a class of its equilibria is analyzed. The dependence of this equilibrium on the constitutive equations and the translational speed is discussed. It is shown that, under gravitational loading, the stretch in this equilibrium is a monotonically increasing function of the translational speed. Furthermore, if this speed is unbounded, so too is the stretch. Related results are proven for the particular cases of a cable composed of a St. Venant-Kirchhoff and an inextensible material.

A TRAVELER'S WOES: SOME PERSPECTIVES FROM DYNAMICAL SYSTEMS

Some features of vehicular dynamics are explored in the context of a simple model for a two-wheeled suitcase. Despite the simplicity of the model, we find some peculiar phenomena. These include an uncountable infinity of heteroclinic orbits, bifurcations in the presence of reversible symmetries and complex dynamics.