Number Of Compatibility Conditions Research Articles

Compatibility conditions for discrete planar structures

Compatibility conditions are investigated for planar network structures consisting of nodes and connecting bars. These conditions restrict the elongations of bars and are analogous to the compatibility conditions of deformation in continuum mechanics. Two problems are considered: the discrete problem for structures with prescribed lengths and its linearization, the discrete problem of prescribed elongations. These problems approximate two continuum problems, the nonlinear continuum problem of given Cauchy Green tensor and its linearization, the continuum problem of prescribed strain. The requirement that the deformations remain planar imposes solvability conditions of all four problems. For triangulated structures, compatibility conditions for the nonlinear problem are expressed as a polynomial equation in the lengths of edges of the star domain surrounding each interior node. In the linearized discrete problem, the compatibility conditions become linear relations for the elongations of edges on the same domains. The continuum limits of the compatibility conditions for both discrete problems are proved to be the compatibility conditions for the continuum problems.The compatibility equations may be summed along a closed curve to give conditions supported on a strip along the curve. Similarly, for continuous materials, the compatibility equation for the prescribed strain problem may be integrated along a closed curve to provide an integral condition, analogous to how the prescribed Green tensor problem may be integrated to give the Gauss-Bonnet integral formula.Compatibility conditions are investigated for general trusses such as plates connected by girders or triangulated domains with holes or missing edges. Compatibility conditions on general trusses may be non-local. There may be rigid trusses without compatibility conditions in contrast to continuous materials. The number of compatibility conditions is the number of bars that may be removed from a structure and still keep it rigid. This number measures the resilience of the structure. The compatibility equations around a hole involve the edges in the neighborhood surrounding the hole. An asymptotic density of compatibility conditions for periodically damaged triangular structures is found to be sensitive to the number and location of the removed edges.

The Linear KdV Equation with an Interface

The interface problem for the linear Korteweg-de Vries (KdV) equation in one-dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain. The location of the interface is known and a number of compatibility conditions at the boundary are imposed. We provide an explicit characterization of sufficient interface conditions for the construction of a solution using Fokas's Unified Transform Method. The problem and the method considered here extend that of earlier papers to problems with more than two spatial derivatives.

An approximate solution for spherical and cylindrical piston problem

A new theory of shock dynamics (NTSD) has been derived in the form of a finite number of compatibility conditions along shock rays. It has been used to study the growth and decay of shock strengths for spherical and cylindrical pistons starting from a non-zero velocity. Further a weak shock theory has been derived using a simple perturbation method which admits an exact solution and also agrees with the classical decay laws for weak spherical and cylindrical shocks.

An approximate solution of one-dimensional piston problem

A new theory of shock dynamics has been developed in the form of a finite number of compatibility conditions along shock rays. It has been used to study the growth or decay of shock strength for accelerating or decelerating piston starting with a nonzero piston velocity. The results show good agreement with those obtained by Harten's high resolution TVD scheme.

Isomorphismen von rechtseitr�umen

Spaces called rectangular spaces were introduced in [5] as incidence spaces (P,G) whose set of linesG is equipped with an equivalence relation ∥ and whose set of point pairs P2 is equipped with a congruence relation ≡, such that a number of compatibility conditions are satisfied. In this paper we consider isomorphisms, automorphisms, and motions on the rectangular spaces treated in [5]. By an isomorphism ϕ of two rectangular spaces (P,G, ∥, ≡) and (P′,G′, ∥′, ≡′) we mean a bijection of the point setP onto P′ which maps parallel lines onto parallel lines and congruent points onto congruent points. In the following, we consider only rectangular spaces of characteristic ≠ 2 or of dimension two. According to [5] these spaces can be embedded into euclidean spaces. In case (P,G, ∥, ≡) is a finite dimensional rectangular space, then every congruence preserving bijection ofP onto P′ is in fact an isomorphism from (P,G, ∥, ≡) onto (P′,G′, ∥, ≡′) (see (2.4)). We then concern ourselves with the extension of isomorphisms. Our most important result is the theorem which states that any isomorphism of two rectangular spaces can be uniquely extended to an isomorphism of the associated euclidean spaces (see (3.2)). As a consequence the automorphisms of a rectangular space (P,G, ∥, ≡) are precisely the restrictions (onP) of the automorphisms of the associated euclidean space which fixP as a whole (see (3.3)). Finally we consider the motions of a rectangular space (P,G, ∥, ≡). By a motion of(P. G,∥, ≡) we mean a bijection ϕ ofP which maps lines onto lines, preserves parallelism and satisfies the condition(ϕ(x), ϕ(y)) ≡ (x,y) for allx, y ∈ P. We show that every motion of a rectangular space can be extended to a motion of the associated euclidean space (see (4.2)). Thus the motions of a rectangular space (P,G, ∥, ≡) are seen to be the restrictions of the motions of the associated euclidean space which mapP into itself (see (4.3)). This yields an explicit representation of the motions of any rectangular plane (see (4.4)).

The vortex rope in the draft tube of Francis turbines operating at partial load: a proposal for a mathematical model

The phenomenon under consideration is represented as the superposition of two potential-flow motion fields, endowed with singularities ofhelicoidal shape. In the first field the singularity is a helicoidal vortex (i.e. the circulation around it is finite); in the second one the singularity is a uniform distribution line source, sinusoidally pulsating in time. If the draft tube is rectilinear, the solution pertaining to the first field adequately represents the "rotating perturbation field", devoid of any synchronous component: it is not necessary in this case to assign a finite amplitude to the second flow field. If the draft-tube rectilinear stretch is followed by an elbow, qualitative considerations show that a synchronous component must needs arise. If, moreover, the vortex rope is cavitated, the linear combination (with suitable amplitudes) of the two flow fields allow the total perturbation field to be represented (rotating+ synchronous components). Obviously, a certain number of compatibility conditions have to be satisfied. Those concern on one hand the flow coming out of the runner (and this allows the parameters of the first flow field to be defined), on the other hand the other parts of the hydraulic system (and this leads to determination of the parameters of the second flow field, i.e. of the dynamic amplification of the synchronous excitation tied to the elbow). The main characters of the proposed mathematical model are in qualitative agreement with what is known from tests on reduced scale model or from measurements performed on actual installations. The proposed model, after proper quantitative validation, could provide a useful tool in order to advance comprehension of the phenomena. Moreover, it could be used in order to carry out a correct, reliable transposition from reduced-scale model tests to full-size prototype (it is a well-known fact that such a transposition is nowadays fraught with uncertainties).

Synthesis of Transfer Function Matrices in Control

A procedure is presented for the synthesis of some transfer function matrices that arise in control. This procedure is algebraic, and realization is accomplished with feedback control. A number of compatibility conditions on design specifications can be checked by elementary calculations, and specifications on required compensation filters are determined from these compatibility conditions. A design problem is included for illustrative purposes.