Reuven Segev

A simple velocity-free controller for attitude regulation of a spacecraft with delayed feedback

In this note, we consider the application of a velocity-free controller for attitude regulation of a rigid spacecraft with gas jet actuators when the effects of time-delays in the feedback loop are taken into consideration. Simple sufficient conditions for exponential stability are presented. Some structural properties of the resulting closed-loop system are studied, and relevant design tools are demonstrated.

Forces and the existence of stresses in invariant continuum mechanics

In an invariant formulation of pth‐grade continuum mechanics, forces are defined as elements of the cotangent bundle of the Banach manifold of C p embeddings of the body in space. It is shown that forces can be represented by measures which generalize the stresses of continuum mechanics. The mathematical representation procedure makes the restriction of forces to subbodies possible. The local properties of the stress measures are examined. For the case where stresses are given in terms of smooth densities, it is shown that the structure of forces agrees with the form of forces one assumes in the traditional formulation, and the equilibrium differential equations are obtained.

Cauchy's Theorem on Manifolds

A generalization of the Cauchy theory of forces and stresses to the geometry of differentiable manifolds is presented using the language of differential forms. Body forces and surface forces are defined in terms of the power densities they produce when acting on generalized velocity fields. The normal to the boundary is replaced by the tangent space equipped with the outer orientation induced by outward pointing vectors. Assuming that the dimension of the material manifold is m, stresses are modelled as m − 1 covector valued forms. Cauchys formula is replaced by the restriction of the stress form to the tangent space of the boundary while the outer orientation of the tangent space is taken into account. The special cases of volume manifolds and Riemannian manifolds are discussed.

A GEOMETRICAL FRAMEWORK FOR THE STATICS OF MATERIALS WITH MICROSTRUCTURE

A geometrical framework for the formulations of the invariant mechanical theories of materials with microstructure is presented. The suggested framework is based on the construction of an infinite-dimensional manifold containing the configurations of the body. Each configuration includes both the macro- and the micro-states of the body. The notions of forces, stresses, balance laws and latent microstructure are discussed. Liquid crystals are used as an example.

On the consistency conditions for force systems

Abstract In analogy with the classical Cauchy conditions, this work presents conditions so that a force system, the assignment of a force to each subbody of a given body, can be represented by a stress. The setting in which the theory is formulated is more general than that of classical continuum mechanics as stresses can be as irregular as measures, equilibrium is not assumed, it applies to continuum mechanics of order higher than one and it may be extended to the case where the body and space are modelled by general differentiable manifolds. The consistency conditions presented are that of additivity of the force system on pairs of disjoint subbodies, continuity and boundedness.

METRIC-INDEPENDENT ANALYSIS OF THE STRESS-ENERGY TENSOR

The stress-energy tensor of field theory is defined and analyzed in a geometric setting where a metric is not available. The stress is a linear mapping that transforms the three-form representing the flux of any given property, e.g., charge-current density, to the three-form representing the flux of energy. The example of the electromagnetic stress-energy tensor is given with the additional structure of a volume element.

Cauchy’s Flux Theorem in Light of Geometric Integration Theory

This work presents a formulation of Cauchys flux theory of continuum mechanics in the framework of geometric integration theory as formulated by H. Whitney and extended recently by J. Harrison. Starting with convex polygons, one constructs a formal vector space of polyhedral chains. A Banach space of chains is obtained by a completion process of this vector space with respect to a norm. Then, integration operators, cochains, are defined as elements of the dual space to the space of chains. Thus, the approach links the analytical properties of cochains with the corresponding properties of the domains in an optimal way. The basic representation theorem shows that cochains may be represented by forms. The form representing a cochain is a geometric analog of a flux field in continuum mechanics.

Geometric aspects of singular dislocations

The theory of singular dislocations is placed within the framework of the theory of continuous dislocations using de Rham currents. For a general n-dimensional manifold, an (n − 1)-current describes a local layering structure and its boundary in the sense of currents represents the structure of the dislocations. Frank’s rules for dislocations follow naturally from the nilpotency of the boundary operator.

GENERALIZED STRESS CONCENTRATION FACTORS

The classical stress concentration factor is regarded as the ratio between the maximal value of the stress in a body and the maximal value of the applied force for a given distribution of material properties. An optimal stress concentration factor is defined as the lowest stress concentration factor if we allow any stress field that is in equilibrium with the given load. The generalized stress concentration factor, a purely geometric property of a body, is the maximal optimal stress concentration factor for any applied force. We show that the generalized stress concentration factor is equal to the norm of an extension mapping of Sobolev functions.

Numerical solution of field problems by nonconforming Taylor discretization

Abstract An algorithm for the numerical solution of field problems is presented. The method is based on expanding the unknown function in a Taylor series about some nodal points so that the coefficients of the series for the various nodes are considered as unknowns. Discrepancy between the values resulting from Taylor expansions about distinct nodes is allowed if it is on the same order of magnitude as the estimated error resulting from the discretization. This enables considerable savings in computation effort in addition to the advantage of specifying the accuracy of the obtained solution. Geometrical flexibility, which enables handling complex boundaries for a large variety of field problems, is another advantage of the proposed scheme. The algorithm is applied to nonlinear steady-state heat-conduction. A test case is treated numerically, and for the evaluation of the scheme the results are compared with the analytical solution.