广义静态梁方程非负解的存在性

运用Leray—Schauder拓扑理论,证明了广义静态梁方程和静态梁方程非负解的存在性,仅要求非线性项f在原点的某个邻域满足一定的符号条件,突破了以往对非线性项f的增长性限制．所获结果对工程设计具有重要的理论意义和实用价值．     

Two-end observability of elastic vibrations in distributed and lumped parameter systems

The boundary observation problems (initial state reconstruction) of vibrations in objects with distributed and lumped parameters are solved. The vibrations in an object with distributed parameters are described by boundary value problems with boundary conditions of various types. An object with lumped parameters is described by a second-order ordinary differential equation.     

Existence and uniqueness results for the bending elastic beam equations

This paper deals with the solvability of the fourth-order boundary value problem $\left\{\begin{array}{c}{u}^{\left(4\right)}=f(x,u,u^{\prime \prime }),0\le x\le 1,\hfill \\ u\left(0\right)=u\left(1\right)=u^{\prime \prime }\left(0\right)=u^{\prime \prime }\left(1\right)=0,\hfill \end{array}\right.$ which models a statically bending elastic beam whose two ends are simply supported, where $f:[0,1]\times {\mathbb{R}}^2\to \mathbb{R}$ is continuous. Inequality conditions on $f$ guaranteeing the existence and uniqueness of solution are presented. The inequality conditions allow that $f(x,u,v)$ may be superlinear growth on $u$ and $v$ as $\left|(u,v)\right|\to \infty$.     

Existence and multiplicity of positive solutions for an elastic beam equation

This paper investigates the boundary value problem for elastic beam equation of the form

Stability of a matrix finite-difference scheme for solving the problem of elastic vibrations

The stability of a matrix finite-difference scheme is analyzed. The scheme is based on central differences and is designed for the differential equation governing the vibrations of an elastic system.     

Controllability of vibrations of a net of coupled objects with distributed and lumped parameters

The problem of damping vibration of a net consisting of m distributed parameter objects is solved. A lumped parameter controlled object affects the distributed parameter objects through boundary conditions.     

Existence of positive solutions of fractional‐order elastic beam equation with a non‐Carathéodory nonlinearity

In this article, the existence of positive solutions of a boundary value problem for nonlinear singular fractional‐order elastic beam equation is established. Here, f depends on t,x, and x′; f may be singular at t = 0 and t = 1; and f is a non‐Carathéodory function. The results obtained are based upon fixed‐point theorems in a cone in Banach space. An example is included to illustrate the main results. Copyright © 2015 John Wiley & Sons, Ltd.     

Vibrations of elastic string with nonhomogeneous material

In this paper we analyze from the mathematical point of view a model for small vertical vibrations of an elastic string with fixed ends and the density of the material being not constant. We employ techniques of functional analysis, mainly a theorem of compactness for the analysis of the approximation of Faedo-Galerkin method. We obtain strong global solutions with restrictions on the initial data u₀ and u₁, uniqueness of solutions and a rate decay estimate for the energy.     

非齐次边值条件下一类静态梁方程非负解的存在性

运用Schauder不动点定理，在非齐次边值条件，讨论带导数项的一端简单支撑另一端滑动的静态梁方程y^(4)(x)=f(x,y(x),y′(x),y″，y^(3)（x）），x∈[0,1] y(0)=a,y′（1）＝b,y″（0)=c,y^(3)（1）＝d非负解的存在性，其中a≥0，b≥0，c≤0,d≤0。假定f在零点次线性增长，在无穷远点超线性增长，则上述非齐次边值问题当max{a,b,-c,-d}充分小时有非负解存在，当max{a,b,-c,-d}充分大时无非负解存在。     

Controllability of vibrations of a system of objects with distributed and lumped parameters

The problem of vibration damping in a system described by the set of a wave equation and a second-order ordinary differential equation is considered. The state functions of the system are related through the boundary conditions for the wave equation.     

A monotone iterative technique for solving the bending elastic beam equations

This paper discusses the solvability of the fourth-order boundary value problem

     

Existence of maximal and minimal solutions to a boundary value problem for the equation of the bending of an elastic beam

By introducing a partial order and using the Mönch fixed point theorem, we establish the existence of maximal and minimal solutions in Banach spaces to a boundary value problem for the equation of the bending of an elastic beam.     

Mathematical analogy of a beam on elastic supports as a beam on elastic foundation

This paper presents the mathematical hypothesis that a beam on equidistant elastic supports (BOES) can be considered as a beam on an elastic foundation (BOEF) in static and free vibration problems. This modeling of BOES as BOEF is presumed to be applicable to a limited range of support stiffness, spacing and flexural rigidity of the beam. The authors investigate the applicability of the modeling of BOEF from the property of characteristic solutions obtained from governing equations of both BOES and BOEF. In this study, the formulation of BOES leads to governing difference equations, and the motions of BOEF are expressed by differential equations. This is because exact solutions must be employed in order to verify their analogy accurately. The characteristic solutions obtained from these two governing equations are compared to each other in order to investigate the relationship between them.     

Strong convergence of difference approximations in the problem of transverse vibrations of thin elastic plates

The problem of transverse vibrations of a thin elastic plate is considered. It is proved that the differential operators of the boundary value problem are regularly elliptic, and weak solutions are estimated. For a previously developed difference method, the solution to the difference problem is proved to converge strongly to a weak solution of the original differential problem and the rate of convergence is estimated.     

An existence theorem of positive solutions for elastic beam equation with both fixed end-points

Abstract. By using the degree theory on cone an existence theorem of positive solution for aclass of fourth-order two-point BVP‘s is obtained. This class of BVP‘s usually describes the de-formation of the elastic beam with both fixed end-points.     

一类含隅角和弯矩的奇异梁方程三个正解的存在性

利用格林函数方法和Avery-Peterson不动点定理研究了一类非线性四阶两点边值问题u(4)(t) =f(t,u(t),u′(t),u″(t)), 0 ＜ t ＜ 1,u(0) =u′(1) =u″(0) =u′″(1) =0多个正解的存在性,其中允许非线性项f(t,u,v,w)在t=0,t=1,u=0,v=0,w=0处奇异.在力学上该问题模拟了左端简单支撑右端被滑动夹子夹住的弹性梁的挠曲.由于非线性项同时涉及隅角和弯矩,因此主要结论对于梁的稳定性分析是有益的.最后我们给出了一个例子,进一步证实本文理论的严密性和可行性.     

一类弹性梁的能量衰减估计

研究一类弹性梁的耗散性.首先应用非线性算子半群理论证明了系统的适定性;进而运用能量方法结合乘子技巧得到弹性梁系统的能量衰减估计.     

Boundary Controllability and Observability of a One-Dimensional Nonuniform SCOLE System

A hybrid system, composed of an elastic beam governed by an Euler-Bernoulli beam equation with variable coefficients and a linked rigid body governed by an ordinary differential equation, is considered. Various controllability/observability properties of the system under bounday control/observation are studied. It is shown that an open-loop smooth/singular controller of either torque control or force control is sufficient to make the system exactly controllable in arbitrarily short time duration. The work was carried out with the support of the National Natural Science Foundation of China and the Russian Foundation for Fundamental Researches, Grant 02-01-00554.     

Functional-differential equations in a class of analytic functions and its application to elastic composites

According to Muskhelishvili’s approach, two-dimensional elastic problems for media with non-overlapping inclusions are reduced to boundary value problems for analytic functions in multiply connected domains. Using a method of functional equations developed by Mityushev, we reduce such a problem for a circular multiply connected domain to functional-differential equations. It is proved that the operator corresponding to the functional-differential equations is compact in the Hardy–Sobolev space. Moreover, these equations can be solved by the method of successive approximation under some natural conditions.