Two Saint-Venant problems for anisotropic circular cylinder

In the paper the method of homogeneous solutions is used to reduce the torsion and tension Saint-Venant problems for a circular cylinder with rhombohedral cylindrical anisotropy to construction of solutions of two boundary value problems for a system of ordinary differential equations. By integrating it is found that if the twisting moment is nonzero, then a longitudinal strain, which is proportional to the twisting moment, is brought about in the cylinder, provided that the boundary conditions on at least one of the ends of the cylinder do not impede longitudinal displacements. Otherwise, an axial force arise in the cylinder under torsion.     

A Saint-Venant principle for a class of second-order elliptic boundary value problems

Zusammenfassung Ein Saint-Venantsches Prinzip für eine gewisse Art von Randwertproblemen elliptischer Differentialgleichungen zweiter Ordnung wird bewiesen. Der Beweis erfordert die Abschätzung verschiedener energie-artiger Funktionale, die zur Herleitung von lokalen Schranken für die Lösung erforderlich sind. Zur Veranschaulichung wird das Prinzip auf die Berechnung der Kanalströmung einer zähigkeitslosen Flüssigkeit angewendet.

---

---

This research was supported in part by the United States Office of Naval Research under Contract Nonr 220(56). Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Departure from linear mechanical behaviour of a helical spring

We model small deviations from linear mechanical behaviour of a loaded vertically suspended helical spring. A Taylor expansion of the elasticity equations governing the axial extension of the spring is used to determine the relative magnitudes of linear and (quadratic and cubic) nonlinear terms in the force–extension relationship. This relationship is the basis for the derivation of a model for the static extension of a loaded spring, and a wave equation that models small amplitude oscillation. The models account for the natural decline in pitch angle down a suspended spring, and provide accurate fits to measurements of static extension and periods of oscillation that are not adequately represented by equations based on Hooke’s law. The static and dynamic data yield consistent estimates of the spring rate.     

Homogeneous solutions in three-dimensional problems of the theory of elasticity for transtropic plates with restrained ends

Homogeneous solutions of equations of the theory of elasticity for transtropic plates, on the flat faces of which the components of the displacement vector are zero, are obtained. The solution consists of the turbulent and potential states of the plate.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 10–13, 1988.     

Blow-up solutions and global solutions for nonlinear parabolic problems

This paper deals with the blow-up of positive solutions for a nonlinear parabolic equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in a finite time, by a new approach. Moreover, upper estimates of the “blow-up time”, blow-up rate and global solutions are obtained also.     

Blow-up solutions and global solutions for nonlinear parabolic problems

This paper deals with the blow-up of positive solutions for a nonlinear parabolic equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in a finite time, by a new approach. Moreover, upper estimates of the “blow-up time”, blow-up rate and global solutions are obtained also.     

Viscosity solutions for elliptic-parabolic problems

We study an elliptic-parabolic problem appearing in the theory of partially saturated flows in the framework of viscosity solutions. This is part of current investigation to understand the theory of viscosity solutions for PDE problems involving free boundaries. We prove that the problem is well posed in the viscosity setting and compare the results with the weak theory. Dirichlet or Neumann boundary conditions are considered.     

Positive solutions for a system of generalized Lidstone problems

In this paper, we study the existence and multiplicity of positive solutions for the system of the generalized Lidstone problems We use fixed point index theory to establish our main results based on a priori estimates achieved by utilizing some properties of concave functions, so that the nonlinearities f and g are allowed to grow in distinct manners, with one of them growing superlinearly and the other growing sublinearly.     

Multibump solutions for a class of nonlinear elliptic problems

The paper is concerned with a class of semilinear elliptic Dirichlet problems approximating degenerate equations. By using variational methods, it is proved that, if the degeneration set consists of connected components, then there exist at least multibump positive solutions. Received March 25, 1996 / Accepted May 26, 1997     

Homogeneous solutions of the dirichlet problem for an anisotropic layer

A complete system of homogeneous solutions of the Dirichlet problem for an anisotropic layer is constructed. These solutions represent series containing metaharmonic functions of a complex argument which depends on all three coordinates. The solution obtained can be used when considering boundary-value problems of potential theory for a piecewise-homogeneous layer.     

Multiple equilibria,periodic solutions and a priori bounds for solutions in superlinear parabolic problems

Consider the Dirichlet problem for the parabolic equation in , where $\Omega$ is a bounded domain in and f has superlinear subcritical growth in u. If f is independent of t and satisfies some additional conditions then using a dynamical method we find multiple (three, six or infinitely many) nontrivial stationary solutions. If f has the form where m is periodic, positive and m,g satisfy some technical conditions then we prove the existence of a positive periodic solution and we provide a locally uniform bound for all global solutions.     

Numerical modelling of twin helical spring under tensile loading

This paper presents a 3D geometric modelling of a twin helical spring and its finite element analysis to study the spring mechanical behaviour under tensile axial loading. The spiralled shape graphic design is achieved through the use of Computer Aided Design (CAD) tools, of which a finite element model is generated. Thus, a 3D 18-dof pentaedric elements are employed to discretise the complex “wired-shape” of the spring, allowing the analysis of the mechanical response of the twin spiralled helical spring under an axial load. The study provides a clear match between the evolution of the theoretical and the numerical tensile and compression normal stresses, being of sinusoidal behaviour. The overall equivalent stress isovalues increases radially from 0° to 180°, being maximal on the internal radial zone at the section 180°. On the other hand, the minimum stress level is located in the centre of the filament cross section.     

Homogeneous solutions to fully nonlinear elliptic equations

We classify homogeneous degree solutions to fully nonlinear elliptic equations.

     

An existence principle for solutions to a singular boundary-value problems

The singular boundary-value problem

$ \left\{ {\begin{array}{*{20}{c}} {{u^{\prime\prime}} + g\left( {t,u,{u^{\prime}}} \right) = 0\quad {\text{for}}\quad t \in \left( {0,1} \right),} \hfill \\ {u(0) = u(1) = 0} \hfill \\ \end{array} } \right. $

is studied. The singularity may appear at u?=?0, and the function g may change sign. An existence theorem for solutions to the above boundary-value problem is proposed, and it is proved via the method of upper and lower solutions.

     

Closed-form solutions for a class of optimal quadratic tracking problems

Closed-from solutions are derived for a class of tracking problems including a linear optimal regulator and a prefilter for a time-invariant plant. The solutions for the prefilter equation and state trajectory, coupled by the Riccati equation, are exponentially related to the stability matrix of the plant. A computational procedure is presented in recursive form when the desired output state dynamics is assumed linear and time-invariant. Several examples are given for illustration.     

Existence of solutions for a class of nonlinear control problems

We consider problems of control and problems of optimal control, monitored by an abstract equation of the formEx=N _u x in a finite interval [0,T]; here,x is the state variable with values in a reflexive Banach space;u is the control variable with values in a metric space;E is linear and monotone; andN _u is nonlinear of the Nemitsky type. Thus, by well-known devices, the results apply also to parabolic partial differential equations in a cylinder [0,T]×G,G ⁿ , with Cauchy data fort=0 and Dirichlet or Neumann conditions on the lateral surface of the cylinder. We prove existence theorems for solutions and existence theorems for optimal solutions, by reduction to a theorem of Kemochi for reflexive Banach spaces.