Control variational method approach to bending and contact problems for Gao beam

This paper deals with a nonlinear beam model which was published by D.Y.Gao in 1996. It is considered either pure bending or a unilateral contact with elastic foundation, where the normal compliance condition is employed. Under additional assumptions on data, higher regularity of solution is proved. It enables us to transform the problem into a control variational problem. For basic types of boundary conditions, suitable transformations of the problem are derived. The control variational problem contains a simple linear state problem and it is solved by the conditioned gradient method. Illustrative numerical examples are introduced in order to compare the Gao beam with the classical Euler-Bernoulli beam.     

Variant of an Analytical Shear Model for the Stress-Strain State of Heterogeneous Composite Beams

A nonclassical analytical model for the stress-strain state of composite beams with account of shear strains is suggested. It is assumed that the beam is piecewise heterogeneous across its height. Normal and tangential loads operate on its upper and lower surfaces and on interfaces. The model describes the distribution of tangential displacements across the thickness of plies by a third-degree polynomial. The corresponding system of differential equations is obtained by the variational method and contains two equations. The first one is an analog of the equation of classical theory of beams for deflections, and the second one is an analog of the equation of the theory for the bending moment from the generalized load. The solutions to test problems are compared with three-dimensional solutions and with experimental results for simply supported and clamped beams of different composite structure. An applied engineering problem is solved for a multispan statically indeterminate beam.     

The linear theory of dislocations and disclinations in elastic shells

A stress state of a thin linearly elastic shell containing both isolated as well as continuously distributed dislocations and disclinations is considered using the classical Kirchhoff–Love model. A variational formulation of the problem of the equilibrium of both a multiply connected shell with Volterra dislocations as well as shells containing dislocations and disclinations distributed with a known density is given. The mathematical equivalence between the boundary-value problem of the stress state of a shell caused by distributed dislocations and disclinations and the boundary-value problem of the equilibrium of a shell under the action of specified distributed loads is established. A number of problems on dislocations and disclinations in a closed spherical shell is solved. The problem of infinitesimally deformations of a surface when there are distributed dislocations is formulated.     

A new Bernoulli–Euler beam model incorporating microstructure and surface energy effects

A new Bernoulli–Euler beam model is developed using a modified couple stress theory and a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equation and complete boundary conditions for a Bernoulli–Euler beam. The new model contains a material length scale parameter accounting for the microstructure effect in the bulk of the beam and three surface elasticity constants describing the mechanical behavior of the beam surface layer. The inclusion of these additional material constants enables the new model to capture the microstructure- and surface energy-dependent size effect. In addition, Poisson’s effect is incorporated in the current model, unlike existing beam models. The new beam model includes the models considering only the microstructure dependence or the surface energy effect as special cases. The current model reduces to the classical Bernoulli–Euler beam model when the microstructure dependence, surface energy, and Poisson’s effect are all suppressed. To demonstrate the new model, a cantilever beam problem is solved by directly applying the general formulas derived. Numerical results reveal that the beam deflection predicted by the new model is smaller than that by the classical beam model. Also, it is found that the difference between the deflections predicted by the two models is very significant when the beam thickness is small but is diminishing with the increase of the beam thickness.     

Buckling analysis of a ring under the action of internal pressure in a cylindrical shell

Buckling analysis of a thin cylindrical shell stiffened by rings with T-shaped cross section under the action of uniform internal pressure in the shell is performed. An annular plate stiffened over the outer edge by a circular beam is used as the ring model. The classical ring model, which is a beam with a T-shaped cross section, is inappropriate in this problem, since in the case of the loss of stability, buckling deformations are localized on the ring surface. The beam model does not allow one to find the critical pressure that corresponds to such a loss of stability. In the first approximation, the problem of the loss of stability of the annular plate connected with the shell is reduced to solving the boundary value problem for finding eigenvalues of the annular plate bending equation. Approximate formulas for determining critical pressure are obtained under the assumption that the plate width is much smaller than its inner radius. The results found using the Rayleigh method and the shooting method differ slightly from each other. It has been demonstrated that the critical pressure for rings with rectangular cross section is higher than that for rings with a T-shaped cross section.     

Structural identifiability of a model of dialysis

In this paper, structural identifiability from input–output data is considered. A mathematical model to describe a dialysis process based on linear dynamic systems is used and the identifiability of this model is tested. The problem of estimating the parameters and obtaining conditions to assure the uniqueness of the parameters is solved. Some conditions to obtain attractive points for the considered model are given, and finally, the stability problem is analyzed.     

纤维增强聚合物加固木柱的非线性稳定性分析

考虑加固层中纤维增强聚合物布（FRP布）拉伸与压缩时的不同弹性模量,基于梁大挠度变形假定,首先建立了FRP加固细长木梁大挠度弯曲的一般数学模型,给出了考虑梁弯曲二阶效应的非线性控制方程．其次,研究了FRP布加固细长简支木柱的非线性稳定性问题,得到了FRP加固简支木柱的临界载荷公式．理论证明了其过屈曲解的存在性,并利用摄动法,得到了临界载荷附近过屈曲状态的渐近解析解．进行了参数分析,结果表明:FRP加固层对临界载荷有显著的影响,而对其无量纲过屈曲状态影响较小．     

Mathematical modeling of marine sounding by the field of a high-power distant source in the presence of a complex coastline

The article considers a method for three-dimensional mathematical modeling of the effect of a bending coastline on the electromagnetic field. A sea with a peninsula and with a bay is examined. The effect of the peninsula and the bay on the coast skin effect is investigated. The problem is solved by the integral equation method.     

Goal programming using multiple objective tabu search

Goal programming (GP) is one of the most commonly used mathematical programming tools to model multiple objective optimisation (MOO) problems. There are numerous MOO problems of various complexity modelled using GP in the literature. One of the main difficulties in the GP is to solve their mathematical formulations optimally. Due to difficulties imposed by the classical solution techniques there is a trend in the literature to solve mathematical programming formulations including goal programmes, using the modern heuristics optimisation techniques, namely genetic algorithms (GA), tabu search (TS) and simulated annealing (SA). This paper uses the multiple objective tabu search (MOTS) algorithm, which was proposed previously by the author to solve GP models. In the proposed approach, GP models are first converted to their classical MOO equivalent by using some simple conversion procedures. Then the problem is solved using the MOTS algorithm. The results obtained from the computational experiment show that MOTS can be considered as a promising candidate tool for solving GP models.     

Some mathematical models related to the Zhukovskii problem of the flow around a sheet pile

The seepage under a Zhukovskii sheet pile through a layer of soil underlain by a highly permeable pressurized horizon is considered. The left semi-infinite part of the roof of this horizon is simulated by an impermeable foundation. The flow when the velocity on the edges of the sheet pile is equal to infinity and, on the two water permeable parts of the boundary of the domain of motion, the flow rate takes extremal values, is investigated. The limiting cases, associated with the absence of both a backwater and an impermeable inclusion, are mentioned. The problem of seepage from a foundation pit formed by two Zhukovskii sheet piles is solved within the limits of a flow with a highly permeable pressurized stratum lying below. In the case when there is no infiltration onto the free surface, a solution of the well-known Vedernikov problem is obtained. A contact scheme, arising when there are no such indicated critical points, is considered; it is described outside the scope of the constraints imposed on the unknown conforming mapping parameters ensuring the realization of the basic mathematical model. Solutions are given for two schemes of motion in a semi-inverse formulation. The classical Zhukovskii problem is the limiting case of one of them. The special features of such models are mentioned. The Polubarinova-Kochina method is used to study all the above-mentioned flows. This method enables exact analytical representations of the elements of the motion to be obtained. The results of numerical calculations and an analysis of the effect of all the physical factors on the seepage characteristics are presented.     

Solution of boundary-value problems of plates by the Vekua method for approximations <Emphasis Type="Italic">N</Emphasis> = 1 and <Emphasis Type="Italic">N</Emphasis> = 2

In this paper, the problem of bending for an isotropic plate with constant thickness 2h is considered. Problems of bending for infinite plates with a circular hole in which a rigid body is placed in the cases of approximations N = 1 and N = 2 of the Vekua theory are solved. We consider the case where the body is soldered. Problems of bending of circular rings are also solved. The results obtained are compared to the corresponding results obtained by plane classical bending theory. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.     

Spectral scale of self-adjoint operators and trace inequalities

Two mathematical models are presented which yield some mechanical aspects of this elastoplastic plate bending. The frist model proves insufficient if the classical Sobolev space framework is kept up. With the second model, an existence result for the transverse displacement problem formulation is obtained when the load does not exceed a specific critical value. The study of the stability problem leads to differentiation of a projector on a closed convex set, which is a difficult question; nevertheless, a hypothesis of regularity of the solution of some plasticity problem is introduced, and the existence of a critical load under which there is stability in Some sense is shown.     

Particle simulation of large carbon dioxide bubbles in water

A method that can be applied to simulate the motion of fluid drops within fluids is described through a detailed study of a prototype problem, the motion of carbon dioxide bubbles in water. The mathematical formulation uses classical molecular dynamics type formulas and results in an n-body problem that is solved numerically. The rise of the bubbles is described, as is the motion of the water near the bubbles. For variety, both H₂O water and D₂O heavy water are considered. Only workstation computer capabilities are required.     

Techniques for approximating a spatially varying Euler-Bernoulli model with a constant coefficient model

Developing accurate models to describe the behaviour of a physical system often results in differential equations with spatially varying coefficients. A notable example of this that appears in many applications is the Euler-Bernoulli beam equation for transverse vibrations. This equation with spatially varying coefficients, such as when the bending stiffness or mass per unit length varies along the length of the beam, is of interest in the current research. Methods for approximating the Euler-Bernoulli equation with periodically varying coefficients have been proposed yet there is still a need for methods that approximate the more general, non-periodically varying, cases. The goal of this research is to obtain a constant coefficient Euler-Bernoulli equation that accurately approximates the original spatially varying equation using an inverse problem approach. Obtaining such an approximation has advantages in control applications where a constant coefficient model is strongly preferred for computational efficiency. The motivation for this research stems from previous work by the authors on modelling cable-harnessed structures. The spatially varying equation is solved using the Lindstedt-Poincaré perturbation method and these results are used to determine the approximate model. Multiple inverse problem methods for determining the coefficients in the approximate model are considered including metric minimization, the modal participation factor (MPF), and the proper orthogonal decomposition (POD). Continuous version of POD and MPF methods are obtained. Several wrapping patterns and boundary conditions are considered for comparison and the results are in good agreement with analytical and finite element analysis (FEA) results.     

Mathematical justification of a one-dimensional model for general elastic shallow arches

We present a bending model for a shallow arch, namely the type of curved rod where the curvature is of the order of the diameter of the cross section. The model is deduced in a rigorous mathematical way from classical tridimensional linear elasticity theory via asymptotic techniques, by taking the limit on a suitable re-scaled formulation of that problem as the diameter of the cross section tends to zero. This model is valid for general cases of applied forces and material, and it allows us to calculate displacements, axial stresses, bending moments and shear forces. The equations present a more general form than in the classical Bernoulli–Navier bending theory for straight slender rods, so that flexures and extensions are proved to be coupled in the most general case. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Algorithm of Exact Methods Obtaining of the Solution and Dynamical Flexibilities of Subsystem of the Mechatronic system

In this paper the application of analysis of transverse vibrating subsystem of mechatronic systems by means of the exact were the main purposes of work to solve the task of assignment of frequency-modal analysis and characteristics of mechatronic system. At first the problem of analysis in the form of the one differential equation of motion of mechanical subsystem or of the set of state equations of considered mechatronic model of object has been formulated and solved. Classic method to solve this problem have been used. The considered transverse vibrating mechanical subsystems of mechatronic system are a continuous beams with constant cross-section, with free ends and/or clamped on one end. A ring transducer, which is the integral part of mechatronic system, extorted by harmonic force or voltage excitation, is assumed to be perfectly bonded to the beam surface. Parameters of the transducer have important influence on values of natural frequencies and on form of characteristics of considered mechatronic system. The poles of dynamical characteristic calculated by mathematical exact method and the Galerkin's method have approximately different or the same values; it is depended on the combination of boundary conditions of elementary beam. The results of the calculations were not only presented in mathematical form but also as a transients of examined dynamical characteristic which are function of frequency of assumed excitation. Practical implications of this work is to present the introduction to synthesis of considered class of mechanical and/or mechatronic beam-systems with a constant changeable cross-section. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Method for Constructing Piecewise Quadratic Value Functions in a Control Problem for a Switched System

The problem of constructing internal approximations to solvability sets and the control synthesis problem for a piecewise linear system with control parameters and disturbances (uncertainties) are solved. The solution is based on the comparison principle and piecewise quadratic value functions of a special form. Relations defining such functions and, in particular, “continuous binding conditions” for the functions and their first derivatives are obtained. The results are used to construct numerical methods for solving the control synthesis problem for the class of switched systems under study. An example of approximate solution of the control synthesis problem in a target control problem for a nonlinear mathematical model of a pendulum with a flywheel is considered.     

Reissner板弯曲的辛求解体系

基于Reissner板弯曲问题的Hellinger-Reissner变分原理,通过引入对偶变量,导出Reissner板弯曲的Hamilton对偶方程组．从而将该问题导入到哈密顿体系,实现从欧几里德空间向辛几何空间,拉格朗日体系向哈密顿体系的过渡．于是在由原变量及其对偶变量组成的辛几何空间内,许多有效的数学物理方法如分离变量法和本征函数向量展开法等均可直接应用于Reissner板弯曲问题的求解．这里详细求解出Hamilton算子矩阵零本征值的所有本征解及其约当型本征解,给出其具体的物理意义．形成了零本征值本征向量之间的共轭辛正交关系．可以看到,这些零本征值的本征解是Saint-Venant问题所有的基本解,这些解可以张成一个完备的零本征值辛子空间．而非零本征值的本征解是圣维南原理所覆盖的部分．新方法突破了传统半逆解法的限制,有广阔的应用前景．     

Modeling and analysis of nonlinear rotordynamics due to higher order deformations in bending

A mathematical model incorporating the higher order deformations in bending is developed and analyzed to investigate the nonlinear dynamics of rotors. The rotor system considered for the present work consists of a flexible shaft and a rigid disk. The shaft is modeled as a beam with a circular cross section and the Euler Bernoulli beam theory is applied with added effects such as rotary inertia, gyroscopic effect, higher order large deformations, rotor mass unbalance and dynamic axial force. The kinetic and strain (deformation) energies of the rotor system are derived and the Rayleigh–Ritz method is used to discretize these energy expressions. Hamilton’s principle is then applied to obtain the mathematical model consisting of second order coupled nonlinear differential equations of motion. In order to solve these equations and hence obtain the nonlinear dynamic response of the rotor system, the method of multiple scales is applied. Furthermore, this response is examined for different possible resonant conditions and resonant curves are plotted and discussed. It is concluded that nonlinearity due to higher order deformations significantly affects the dynamic behavior of the rotor system leading to resonant hard spring type curves. It is also observed that variations in the values of different parameters like mass unbalance and shaft diameter greatly influence dynamic response. These influences are also presented graphically and discussed.     

Bending of a two-layer beam with non-rigid contact between the layers

The bending, under plane stress state conditions, of a two-layer beam-strip with identical isotropic linearly elastic layers with non-rigid contact between them is considered. The effect of the contact interaction between the layers, simulated by an elastic or elastoplastic gasket of negligibly small thickness with a finite shear stiffness, on the deflection of the beam is studied. Absolute slippage and rigid contact between the layers are the two limiting values of the shear stiffness. The values of the flexural stiffness of the beam differ by a factor of four in these limiting situations. The problem is reduced to a one- dimensional problem in the case of harmonic external load and an asymptotic solution is constructed for it. In the case of a load of general form, the Kirchhoff - Love hypotheses are used to construct an approximate solution and the problem is reduced to a one-dimensional problem. The difficulties which arise in simulating of the interaction forces between the layers using Coulombic dry friction forces are discussed.