THE FIRST ORDER APPROXIMATION OF NON-KIRCHHOFF-LOVE THEORY FOR ELASTIC CIRCULAR PLATE WITH FIXED BOUNDARY UNDER UNIFORM SURFACE LOADING (Ⅰ)

Based on the approximation theory adopting non-kirchhoff-Love assumption for three dimensional elastic plates with arbitrary shapes^[1],[2], the author derives a functional of generalized variation for three dimensional elastic circular plates, thereby obtains a set of differential equations and the relate boundary conditions to establish a first order approximation theory for elastic circular plate with fixed boundary and under uniform loading on one of its surface. The analytical solution of this problem will present in another paper.     

Determining the limiting solutions of nonstationary axisymmetric Hele-Shaw problems

Numerical solution of the Hele-Shaw problem reduces to solution of three boundary-value problems of determining analytic functions of a complex variable in each time step: conformal mapping of the range of the parametric variable to the physical plane, the Dirichlet problems for determining the electric-field strength, and the Riemann-Hilbert problem for calculating partial time derivatives of the coordinates of points of the interelectrode space (the images of the points on the boundary of the parametric plane are fixed). Unlike in the two-dimensional problem, the electric-field strength is determined using integral transformations of an analytic function. Approximation by spline function is performed, and more accurate and steady (than the well-known ones) general solution algorithms for the nonstationary axisymmetric problems are described. Results of a numerical study of the formation of stationary and self-similar configurations are presented. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 87–99, July–August, 2009.     

THE EXISTENCE OF PERIODIC SOLUTION OF THE FOURTH ORDINARY NONLINEAR DIFFERENTIAL EQUATION CAUSED BY FLOW－IN

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Bulgakov problem of the accumulation of perturbations

The Bulgakov problem of maximizing the Euclidean norm of the solution to a linear time-invariant system at a fixed instant of time on the set of piecewise continuous scalar controls bounded in absolute value is considered. A control law as a function of time and of the state of the system is proposed. Some conditions ensuring that the proposed control is a solution to the problem under study are formulated.     

New solutions of classical problems in rigid body dynamics

The equations of motion of a rigid body acted upon by general conservative potential and gyroscopic forces were reduced by Yehia to a single second-order differential equation. The reduced equation was used successfully in the study of stability of certain simple motions of the body. In the present work we use the reduced equation to construct a new particular solution of the dynamics of a rigid body about a fixed point in the approximate field of a far Newtonian centre of attraction. Using a transformation to a rotating frame we also construct a new solution of the problem of motion of a multiconnected rigid body in an ideal incompressible fluid. It turns out that the solutions obtained generalize a known solution of the simplest problem of motion of a heavy rigid body about a fixed point due to Dokshevich.     

三维粘弹性层状半空间埋置集中荷载动力格林函数求解—修正刚度矩阵法

针对三维粘弹性层状半空间埋置集中荷载作用下动力响应问题,在柱面坐标下,结合径向Hankel积分变换,提出了一种新的求解方法—修正刚度矩阵法。方法基于位势函数理论,将三维问题分解为平面内反应(P-SV波型)和平面外反应（SH波型）两个二维问题的叠加;借鉴结构力学中超静定结构的位移法原理,首先固定荷载所在层的上下界面,通过对波动方程的特解和齐次解叠加得到“固端”反力。进而放松两“固端约束”,利用直接刚度法求得各层面位移,荷载作用层内反应另需叠加上该“固定层”内解,并将特解部分积分（直达波）由全空间解析解代替,解决了当接收点和源点作用水平面接近时的积分收敛问题。算例分析表明,对于低频（可退化为静力状态）和高频问题,本文方法均具有很高的计算效率和精度。     

Exact solution and stability of postbuckling configurations of beams

We present an exact solution for the postbuckling configurations of beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions. We take into account the geometric nonlinearity arising from midplane stretching, and as a result, the governing equation exhibits a cubic nonlinearity. We solve the nonlinear buckling problem and obtain a closed-form solution for the postbuckling configurations in terms of the applied axial load. The critical buckling loads and their associated mode shapes, which are the only outcome of solving the linear buckling problem, are obtained as a byproduct. We investigate the dynamic stability of the obtained postbuckling configurations and find out that the first buckled shape is a stable equilibrium position for all boundary conditions. However, we find out that buckled configurations beyond the first buckling mode are unstable equilibrium positions. We present the natural frequencies of the lowest vibration modes around each of the first three buckled configurations. The results show that many internal resonances might be activated among the vibration modes around the same as well as different buckled configurations. We present preliminary results of the dynamic response of a fixed–fixed beam in the case of a one-to-one internal resonance between the first vibration mode around the first buckled configuration and the first vibration mode around the second buckled configuration.     

Enhancement of diffusive transfer by periodic pulsation between dirichlet and neumann type boundary conditions

A solution to the equation of heat conduction is computed for the case where the boundary condition alternates periodically between fixed temperature and fixed heat flow. This problem is equivalent to a highly non-linear parametric oscillation. Applications to energy conservation and non-destructive heating are discussed.     

Lyapunov-stable equilibrium distribution of charged particles in an ionized-gas cloud

The solution of several problems requires establishing the equilibrium spatial distribution of the charged particles in an ionized-gas cloud having fixed shape and given dimensions.In the present study we find the Lyapunov-stable equilibrium spatial distribution of the charged particles in an ionized-gas cloud enclosed in a region of space with fixed form and given dimensions. An expression is obtained for the limiting energy stored by the ionized gas for given degree of ionization of neutral gas, and expressions are found for the pressure and chemical potential of the ionized gas in the cloud. We use the general method for the solution of a particular type of extremal problem presented in [1, 2], which reduces the optimization problem to the solution of the corresponding inverse problem of potential theory.In conclusion I would like to thank N. N. Bogolyubov, B. B. Kadomtsev, and Yu. N. Barabanenkov for their discussions of the present study and helpful comments.     

Three coplanar Griffith cracks in an infinite elastic strip

We consider the problem of determining the stress intensity factors and crack energy in an infinitely long isotropic, homogeneous elastic strip containing three coplanar Griffith cracks. The three coplanar Griffith cracks are situated symmetrically on a line perpendicular to the edges of the strip. We assume that the cracks are opened by an internal pressure and the edges of the strip are fixed. By using the theory of Fourier series we reduce the problem to solving a set of quadruple trigonometrical series equations with a cosine kernel. Closed form solution is obtained for the quadruple series equations. Closed form analytical expressions are derived for the stress intensity factors, the shape of the deformed cracks and the crack energy. Solutions to some particular problems are derived as limiting cases.     

Hybrid projection method for generalized mixed equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces

A new hybrid projection iterative scheme is introduced to approximate a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of fixed points of a relatively weak nonexpansive mapping in the Banach spaces. The obtained results generalize and improve the recent results announced by many other authors.     

Hybrid projection method for generalized mixed equilibrium problems,variational inequality problems,and fixed point problems in Banach spaces

A new hybrid projection iterative scheme is introduced to approximate a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of fixed points of a relatively weak nonexpansive mapping in the Banach spaces. The obtained results generalize and improve the recent results announced by many other authors.     

Motion of a vertical wall fixed on springs under the action of surface waves

A two-dimensional unsteady hydroelastic problem of interaction between surface waves and a moving vertical wall fixed on springs is studied. An analytical solution of the problem is constructed using a linear approximation, and a numerical solution within the framework of a nonlinear model of a potential fluid flow is found by a complex boundary element method. By means of analysis of the linear and nonlinear solutions, it is found that the linear solution can be used to predict some important characteristics of the wall motion and the fluid flow in the case of moderate wave amplitudes.     

Diffusional growth of cloud particles: existence and uniqueness of solutions

Diffusional growth of cloud particles is commonly described by a coupled system of parabolic equations and ordinary differential equations. The Dirichlet boundary condition for the parabolic equation is obtained from the solution of the ordinary differential equations, but this solution itself depends on the solution of the parabolic equations. We first present the governing equations describing diffusional growth of cloud particles. In a second step, we consider a simplified model problem, motivated by the diffusional growth equations. The main difference between the simplified model problem and the diffusional growth equations consists in neglecting the dependence of the domain for the parabolic equations on the solution. For the model problem, we show unique solvability using a fixed point method. Finally, we discuss application of the main result for the model problem to the diffusional growth equations and illustrate these equations with the help of a numerical solution.     

Existence and stability of solutions to inverse variational inequality problems

In this paper, two new existence theorems of solutions to inverse variational and quasi-variational inequality problems are proved using the Fan-Knaster-KuratowskiMazurkiewicz(KKM) theorem and the Kakutani-Fan-Glicksberg fixed point theorem.Upper semicontinuity and lower semicontinuity of the solution mapping and the approximate solution mapping to the parametric inverse variational inequality problem are also discussed under some suitable conditions. An application to a road pricing problem is given.     

Dual boundary element analysis of wave scattering from singularities

The dual boundary element method is used to obtain an efficient solution of the Helmholtz equation in the presence of geometric singularities. In particular, time-harmonic waves in a membrane which contains one or more fixed edge stringers (or cracks) are investigated. The hypersingular integral equation is used in the procedure to ensure a unique solution for the problem with a degenerate boundary. The method yields a solution for the entire membrane as well as the dynamic stress intensity factor. Numerical results are presented for a circular membrane containing a single edge stringer, two edge stringers and an internal stringer. Also, the first three critical wave numbers of the membrane with the homogeneous boundary condition are determined, and the dynamic stress intensity factors are found for problems with the nonhomogeneous boundary condition. Good agreement is found after comparing the results with exact solutions, and with results obtained using DtN-FEM and ABAQUS.     

A second order stationary solution of the three dimensional wave diffraction potential

A theory on the second order wave diffraction by a three dimensional body fixed in a regular sea has been developed in the present paper. By regarding the sinusoidal disturb potential as a stationary solution of an initial value problem, and using Laplace transformation method and Tauberian theorem, the boundary value problems of stationary solution of the first and second order diffraction potential have been derived in this paper. Furthermore, the explicit solution of the second order stationary diffraction potential has been obtained with the method of the double Fourier transformation. It is found that the asymptotic behaviour of the second order stationary solution at far field is dependent on two wave systems, the first is “free wave”, travelling independently of the first order wave system, the other is “phase locked waves”, which accompany the first order waves. At the same time, the radiation conditions of the second order diffraction problems are derived. We also find that one can still pursue a steady state formulation with the inclusion of Rayleigh damping. Finally, as an example, the second order wave forces upon a fixed vertical circular cylinder have been calculated, and the numerical results agree well with the experimental data.     

Construction of the Exact Solution to a Torsion Problem for an Elastic Shaft of Variable Cross Section

The exact solution is constructed to a torsion problem for a circular elastic shaft in a medium referred to a spherical coordinate system. One end of the shaft is rigidly fixed and the other is subjected to either tangential forces or a torque. New integral transforms are obtained to solve the problem.