End conditions for cubic spline interpolation

End conditions are shown to be generated from the conditions on the derivatives of a class of piecewise cubic interpolatory polynomial approximations to the given function near the two end points. The new approach is illustrated by obtaining a number of end conditions, which include those discussed with special interest by Behforooz and Papamichael.     

Another approach to the quintic spline

A set of end conditions for interpolatory quintic spline is derived by use of integration. These end conditions are only in terms of function values at the knots (data), and rise to O(h⁶) spline approximation.     

Mathematical Statement of Boundary Conditions for Problems of Three-Dimensional Deformation of Plates

We consider the problem of mathematical statement of boundary conditions on the end surfaces of elastic plates for a model that describes three-dimensional deformation of these bodies. We show that there exist five conditions that have a certain physical meaning and characterize the conditions of fixation of a plate or loading on its end surfaces. We illustrate the results obtained for a circular plate rigidly fixed on its ends and loaded over the lateral surfaces.     

On Torsion of Prismatic Bodies

The method of decaying residual solution is applied to obtain an approximate interior solution for the torsion of slender prismatic elastic bodies under different end conditions. The approximate solution is generally accurate up to terms that are exponentially small in the length-to-cross-sectional-width ratio. For stress end conditions, the result is identical to the classical Saint-Venant torsion solution. Similar types of simple solutions, not known previously, are obtained for different types of mixed end conditions. For displacement conditions at both ends, the corresponding Saint-Venant type result requires an accurate solution of a canonical problem for a semi-infinite prismatic body that is to be obtained once and for all. The solution of the canonical problem is elementary for a circular cross section. The approximate interior solution in that case is identical to the known exact interior solution.     

Dynamic frictionless contact of a nonlinear beam with two stops

In this work, we consider mathematical and numerical approaches to a dynamic contact problem with a highly nonlinear beam, the so-called Gao beam. Its left end is rigidly attached to a supporting device, whereas the other end is constrained to move between two perfectly rigid stops. Thus, the Signorini contact conditions are imposed to its right end and are interpreted as a pair of complementarity conditions. We formulate a time discretization based on a truncated variational formulation. We prove the convergence of numerical trajectories and also derive a new form of energy balance. A fully discrete numerical scheme is implemented to present numerical results.     

Vibrational frequency distribution for nonconservative model of double-walled carbon nanotube

The paper presents the mathematical results for a double-walled carbon nanotube model with nonconservative boundary conditions, given in the form of two Timoshenko beams coupled through a distributed Van Der Waals force. The system is clamped at the left end and subject to a four-parameter family of dynamical boundary conditions at the right end. We also present the four-branch vibrational spectrum of the system. All the results are purely analytical; they will be complemented by numerical simulations in a forthcoming work.     

Effect of the end conditions on the critical loads for filament-wound glass-reinforced plastic columns

The behavior of glass-reinforced plastic columns with various end conditions has been experimentally investigated. The cross-wound columns had a length to mean diameter ratio of 18–20. Four types of support with different degrees of restraint on end rotation ranging from almost free ends to fixed ends were employed. The possibilities of increasing the critical loads by manipulating the boundary conditions are estimated. Numerical estimates are obtained for the various methods of support. Random imperfections and the number of loadings were not found to have any effect on the critical loads. It is shown that the use of Southwell's method makes it possible to estimate the critical loads for columns with different end conditions from the subcritical loading data.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Moscow. Translated from Mekhanika Polimerov, No. 1, pp. 54–62, January–February, 1976.     

具有复杂边界条件的杆的振动分析

本文研究一端带有集中质量并支以弹簧另一端作支承运动的杆的纵向振动.由于这个问题的边界条件比较复杂,且要考虑阻尼,因此本文只求稳态周期解.首先分析线性系统;然后考虑材料非线性,用摄动法求具有非线性边界条件的非线性方程的近似解析解.     

Low frequency resonances in the dynamic interaction of an elastic solid with a semi-bounded medium

The vibration of an elastic rod with a large point mass at one end to which a given harmonic load is applied is investigated. The other end of the rod is connected to a rigid punch in continuous contact with a semi-bounded elastic medium. Some lemmas are proved on systems with simplified boundary conditions with properties which majorize the characteristics of the initial system. The effect of the system parameters on the conditions under which an unbounded resonances occur is thereby determined. Analytic relations for determining the number of resonance frequences are obtained.     

Necessary extremum conditions for differential inclusion with aftereffect

Necessary extremum conditions for optimization problem on the minimum of a functional specified at the right-hand end of the trajectory of differential inclusion with aftereffect are obtained. The constraints, specified at the left-hand end of the trajectory, imply that the initial function belongs to some rnultivalued mapping. The problem is considered at the fixed interval of time     

Inverse problem for a vibrating beam

The problem of inferring the flexural rigidity and density of a beam from its eigenfrequencies is considered for the particular case in which one end is clamped. It is shown that three spectra associated with three sets of boundary conditions at the other end are required in order to insure a unique solution of the inverse problem. Furthermore, it is shown that this data set is equivalent to the information contained in the time history of the displacement and slope of the free end of the beam set in motion by a concentrated impulse.     

悬链线弛垂度的计算方法

分两种情况讨论了悬链线弛垂度的计算方法 .     

On the zeros of solutions of beam equations

Summary We consider extensible beam equations, Timoshenko beam equations and the system of coupled beam equations. We show that, under suitable conditions, there are bounded domains in which every solution satisfying certain end conditions has a zero. End conditions to be considered are hinged ends and hinged-sliding ends. The results are based on the conditions for the nonexistence of positive solutions of ordinary differential inequalities.     

End conditions for interpolatory cubic splines with unequally spaced knots

A class of end conditions is derived for cubic spline interpolation at unequally spaced knots. These conditions are in terms of function values at the knots and lead to 0 (h⁴) convergence uniformly on the interval of interpolation.     

Phragmén-Lindelöf and continuous dependence type results in generalized heat conduction

Using an energy method we investigate the decay of end effects for a generalized heat conduction problem defined on a semi-infinite cylindrical region. With homogeneous Dirichlet conditions on the lateral surface of the cylinder it is shown that solutions either grow exponentially or decay exponentially in the distance from the finite end of the cylinder. The effect of perturbing the equation parameters is also investigated.     

Existence and iteration of monotone positive solutions for an elastic beam equation with a corner

This paper is concerned with the existence of monotone positive solutions for an elastic beam equation with a corner. The boundary conditions mean that the beam is embedded at one end and fastened with a sliding clamp at the other end. By applying monotone iterative techniques, we not only obtain the existence of positive solutions, but also establish iterative schemes for approximating the solutions.     

Asymptotics of eigenfrequencies and eigenmodes of non‐homogeneous inextensible filament with an end load

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. We derive the spectral asymptotics for the aforementioned two‐parameter family of non‐selfadjoint operators. In the forthcoming papers, based on the asymptotical results of the present paper, we will prove the Riesz basis property of the eigenfunctions. The spectral results obtained in the aforementioned papers will allow us to solve boundary and/or distributed controllability problems for the filament using the spectral decomposition method. Copyright © 2001 John Wiley & Sons, Ltd.     

Second-order necessary conditions for differential–difference inclusion problems

We study an optimal control problem given by differential–difference inclusions with end point constraints. An approach concerning second-order optimality conditions is proposed.     

Regular singular Sturm-Liouville operators and their zeta-determinants

We consider Sturm-Liouville operators on the line segment [0,1] with general regular singular potentials and separated boundary conditions. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski-determinant of a fundamental system of solutions adapted to the boundary conditions. This generalizes the earlier work of the first author, treating general regular singular potentials but only the Dirichlet boundary conditions at the singular end, and the recent results by Kirsten-Loya-Park for general separated boundary conditions but only special regular singular potentials.     

G~1 Continuity Conditions of B-spline Surfaces

According to the B-spline theory and Boehm algorithm, this paper presents several necessary and sufficient G1 continuity conditions between two adjacent B-spline surfaces. In order to meet the need of application, a kind of sufficient conditions of G1 continuity are developed, and a kind of sufficient conditions of G1 continuity among N(N > 2) patch B-spline surfaces meeting at a common corner are given at the end.