A recurrent problem in materials science is the prediction of the percolation threshold of suspensions and composites containing complex-shaped constituents. We consider an idealized material built up from freely overlapping objects randomly placed in a matrix, and numerically compute the geometrical percolation threshold ${\mathit{p}}_{\mathit{c}}$ where the objects first form a continuous phase. Ellipsoids of revolution, ranging from the extreme oblate limit of platelike particles to the extreme prolate limit of needlelike particles, are used to study the influence of object shape on the value of ${\mathit{p}}_{\mathit{c}}$. The reciprocal threshold 1/${\mathit{p}}_{\mathit{c}}$ (${\mathit{p}}_{\mathit{c}}$ equals the critical volume fraction occupied by the overlapping ellipsoids) is found to scale linearly with the ratio of the larger ellipsoid dimension to the smaller dimension in both the needle and plate limits. Ratios of the estimates of ${\mathit{p}}_{\mathit{c}}$ are taken with other important functionals of object shape (surface area, mean radius of curvature, radius of gyration, electrostatic capacity, excluded volume, and intrinsic conductivity) in an attempt to obtain a universal description of ${\mathit{p}}_{\mathit{c}}$. Unfortunately, none of the possibilities considered proves to be invariant over the entire shape range, so that ${\mathit{p}}_{\mathit{c}}$ appears to be a rather unique functional of object shape. It is conjectured, based on the numerical evidence, that 1/${\mathit{p}}_{\mathit{c}}$ is minimal for a sphere of all objects having a finite volume.

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Geometrical percolation threshold of overlapping ellipsoids

An excursion into large rotations

An approach is presented for the estimation of object motion parameters based on a sequence of noisy images. The problem considered is that of a rigid body undergoing unknown rotational and translational motion. The measurement data consists of a sequence of noisy image coordinates of two or more object correspondence points. By modeling the object dynamics as a function of time, estimates of the model parameters (including motion parameters) can be extracted from the data using recursive and/or batch techniques. This permits a desired degree of smoothing to be achieved through the use of an arbitrarily large number of images. Some assumptions regarding object structure are presently made. Results are presented for a recursive estimation procedure: the case considered here is that of a sequence of one dimensional images of a two dimensional object. Thus, the object moves in one transverse dimension, and in depth, preserving the fundamental ambiguity of the central projection image model (loss of depth information). An iterated extended Kalman filter is used for the recursive solution. Noise levels of 5-10 percent of the object image size are used. Approximate Cramer-Rao lower bounds are derived for the model parameter estimates as a function of object trajectory and noise level. This approach may be of use in situations where it is difficult to resolve large numbers of object match points, but relatively long sequences of images (10 to 20 or more) are available.

Estimation of Object Motion Parameters from Noisy Images

Publisher's description: The use of geometric methods in classical mechanics has proven fruitful, with wide applications in physics and engineering. In this book, Professor Marsden concentrates on these geometric aspects, especially on symmetry techniques. The main points he covers are: the stability of relative equilibria, which is analyzed using the block diagonalization technique; geometric phases, studied using the reduction and reconstruction technique; and bifurcation of relative equilibria and chaos in mechanical systems. A unifying theme for these points is provided by reduction theory, the associated mechanical connection and techniques from dynamical systems. These methods can be applied to many control and stabilization situations, and this is illustrated using rigid bodies with internal rotors, and the use of geometric phases in mechanical systems. To illustrate the above ideas and the power of geometric arguments, the author studies a variety of specific systems, including the double spherical pendulum and the classical rotating water molecule.

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Lectures on Mechanics

Direct methods for the search of the second invariant

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE GENERAL THEORY OF ORBITS

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: NON-HOLONOMIC SYSTEMS. DISSIPATIVE SYSTEMS

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: PRINCIPLES AVAILABLE FOR THE INTEGRATION

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE REDUCTION OF THE PROBLEM OF THREE BODIES

The ideas of rest and motion. In the previous chapter we have frequently used the terms “fixed” and “moving” as applied to systems. So long as we are occupied with purely kinematical considerations, it is unnecessary to enter into the ultimate significance of these words; all that is meant is, that we consider the displacement of the “moving” systems, so far as it affects their configuration with respect to the systems which are called “fixed,” leaving on one side the question of what is meant by absolute “fixity.” When however we come to consider the motion of bodies as due to specific causes, this question can no longer be disregarded. In popular language the word “fixed” is generally used of terrestrial objects to denote invariable position relative to the surface of the earth at the place considered. But the earth is rotating on its axis, and at the same time revolving round the Sun, while the Sun in turn, accompanied by all the planets, is moving with a large velocity along some not very accurately known direction in space. It seems hopeless therefore to attempt to find anything which can be really considered to be “at rest.” In the nineteenth century it was supposed that the aether of space (the vehicle of light and of electric and magnetic actions) was (apart from small vibratory motions) stagnant, and so was capable of providing a basis for absolute fixity.

Sir William McCrae

Papers

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE GENERAL THEORY OF ORBITS

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: NON-HOLONOMIC SYSTEMS. DISSIPATIVE SYSTEMS

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: PRINCIPLES AVAILABLE FOR THE INTEGRATION

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE REDUCTION OF THE PROBLEM OF THREE BODIES

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE EQUATIONS OF MOTION