Optimization of axisymmetric membrane shells

Problems of the joint optimization of the shape and distribution along the meridian of the thickness of membrane shells of revolution under the action of axisymmetric loads are considered, taking account of the constraints concerning the strength of the shell and the volume of its cavity. General formulations of problems of the optimal design of shells of revolution are given and the optimal shape of a shell and the corresponding thickness distribution are investigated. Results of the exact solution of problems of the optimal design of closed shells of revolution when there is an internal pressure are presented. The simultaneous introduction of two control functions, describing the shape of the shell and the distribution of its thickness, not only ensures a substantial reduction in the mass of a shell but also leads to significant mathematical simplifications, which enable the solution of the optimization problem being considered to be obtained in an analytical form.     

转点频段旋转薄壳体的强迫振动和特殊效应

利用转点频段的一致有效解研究了截顶旋转薄壳受边缘薄膜力和弯矩驱动的强迫振动．薄壳强迫振动呈现出薄膜解和弯曲解的耦合特点:无论由薄膜力还是由弯矩激发,振动均俱含薄膜和弯曲运动．揭示了转点频段薄壳强迫振动的3种特殊效应:无弯曲效应、内部静止效应和内部薄膜运动外部弯曲运动效应．这些效应具有潜在的应用价值．     

Wrinkle-free solutions in the theory of annular elastic membranes

Rotationally symmetric deformations of a flat annular elastic membrane under a gravitational force are studied, with prescribed radial stresses or horizontal displacements at the edges. The small-finitedeflection theory of Föppl-Hencky as well as a simplified version of Reissner's static first approximation theory of thin shells of revolution are applied which lead to consider a single, second-order, ordinary differential equation for the derivation of the principal stresses in the membrane. Using analytical methods, the range of those boundary data is determined for which the solutions of the differential equation are wrinkle free in the sense that both the radial and the circumferential stress components are nonnegative everywhere.     

Mathematical study of the small oscillations of an incompressible and inviscid liquid under zero gravity in a container of revolution with elastic bottom and anchored edge

In this paper, the Authors study the small oscillations of an inviscid liquid under zero gravity in a container of revolution with elastic bottom, considered as a membrane or a plate, and anchored edge. They obtain the equations of the displacements of the free surface and the bottom and they reduce these equations to a functional equation in a suitable Hilbert space. By using the methods of Functional Analysis the Authors prove the existence of the small oscillations. Research partially supported by GNFM, MURST (Italy).     

Rationalization of the shape of wound membrane shells of revolution composed of high-modulus material

A method of designing composite membrane shells of revolution under axisymmetric loading is described. The properties of the shell material are analyzed. It is shown that for shells of high-modulus material in the presence of tensile membrane stresses the fibers fail in the matrix. A fiber arrangement and shell geometry ensuring isotensoid properties are proposed for this case. A technological and weight analysis is presented.S. Ordzhonikidze Moscow Aviation Institute. Translated from Mekhanika Polimerov, No. 5, pp. 822–828, September–October, 1975.     

Wrinkle-free solutions of circular membrane problems

Axisymmetric deformations of a plane circular membrane subjected to an axial surface load are studied, with prescribed radial stresses or radial displacements at the edge. Considering the small-finite-deflection theory of Föppl-Hencky as well as a simplified version of Reissner's nonlinear theory of thin shells of revolution, the determination of the principal stresses in the membrane is shown to reduce to the solution of a nonlinear second-order ODE. In the Föppl case, the basic equation becomes singular when the membrane edge is free of traction. Nevertheless, existence and uniqueness of nonnegative solutions for zero edge traction is proved in both Föppl and Reissner models. Thereby, a limit curve for the radial displacement at the boundary is induced in the Reissner case, which subdivides the actual parameter range into complementary domains of existence and nonexistence of tensile solutions. The limit curve is studied, analytically and numerically. Finally, a maximum principle is established in order to determine the more restricted subdomain of those parameters which admit wrinkle-free solutions, i.e. solutions governed by a nonnegative radial and circumferential stress component.     

The Washer Method

<正>You've learned the disk method to use integral finding the volume of solid revolution.You'll learn how to use other method for finding the volume of a special type of solids of revolution.Solids of revolution are used commonly in industry.Sometimes,slicing a solid of revolution results in disks with holes in the middle.It's called washer method.The washer is formed by revolving a rectangle about an axis,as shown in Figure 1&2.     

Some classification of surfaces of revolution in Minkowski 3-space

We study the surfaces of revolution with the non-degenerate second fundamental form in Minkowski 3-space. In particular, we investigate the surfaces of revolution satisfying an equation in terms of the position vector field and the 2nd-Laplacian in Minkowski 3-space. As a result, we give some new examples of the surfaces of revolution with light-like axis in Minkowski 3-space.     

关于SOL流形具有常平均曲率旋转曲面的注记

研究非欧流形SOL 空间上共形平均曲率方程的可解性,通过研究轮廓曲线对具有平均曲率的旋转曲面进行分类。当这些旋转曲面的平均曲率为给定函数时,计算出相应轮廓曲线的微分方程。通过求解这些微分方程,给出旋转函数是其上共形平均曲率的充分条件。     

坐标变换和旋转曲面的方程

利用坐标变换给出了一种求旋转曲面方程的方法,并举例说明用该方法可以直接从方程判断出它所表示的曲面类型。     

Hamiltonian approach to studying thin shells of revolution made of composite materials

A method for constructing defining relations of the linear theory of shells of revolution in complex Hamiltonian form has been proposed. Based on the Lagrange variational principle, we have constructed a mathematical model of a multilayer orthotropic shell of revolution. We have obtained explicit expressions for the coefficients and right-hand sides of the Hamiltonian complex system of equations describing the statics of shells of revolution in terms of their rigid characteristics and acting loads. The Hamiltonian resolving system of linear differential equations, formulated in the axially symmetric case, has some specific properties facilitating both analytical studies and numerical procedures of their solution.     

How to find the volume by using the definite integral

<正>This time you will learn how to find the volume of a solid of revolution continuously.And learn how to find volumes of other kind of solids are not the solid of revolution.For example:Find the volume of a sphere with the radius r.     

Large deformations of bodies of revolution made of elastic homogeneous and fiber-reinforced materials 1. Torsion of toroidal bodies

Equations of a mathematical model for bodies of revolution made of elastic homogeneous and fiber-reinforced materials and subjected to large deformations are presented. The volume content of reinforcing fibers is assumed low, and their interaction through the matrix is neglected. The axial lines of the fibers can lie both on surfaces of revolution whose symmetry axes coincide with the axis of the body of revolution and along trajectories directed outside the surfaces. The equations are obtained for the macroscopically axisymmetric problem statement where the parameters of macroscopic deformation of the body vary in its meridional planes, but are constant in the circumferential directions orthogonal to them. The equations also describe the torsion of bodies of revolution and their deformation behavior under the action of inertia forces in rotation around the symmetry axis. The results of a numerical investigation into the large deformations of toroidal bodies made of elastic homogeneous and unidirectionally reinforced materials under torsion caused by a relative rotation of their butt-end sections around the symmetry axis are presented.     

旋转曲面面积的曲线积分表示

利用旋转曲面方程,以及曲面积分和曲线积分的计算方法,可将旋转曲面的面积通过第一型曲线积分表示出来并进行计算.     

平面图形的形心在旋转体体积计算中的应用

本文讨论平面图形绕平面内的直线旋转的旋转体体积与被旋转的平面图形的形心的关系.当被旋转的平面图形的内部与旋转轴没有交点时,得到了用被旋转的平面图形的面积以及被旋转的平面图形的形心到旋转轴的距离表示的旋转体体积公式.     

The volume of a solid of revolution in calculus

<正>On the last time,you have known what a right circular cylinder is in calculus.Right circular cylinders are just a part of solids of revolution.Today you will learn solids of revolution arent right circular cylinders.In general way,a solid is bounded by the region under the curve y=f(x)by rotating a-     

从Beltrami-Schaefer应力函数导出几个特殊问题的应力函数

本文从弹性力学中的Beltrami-Schaefer应力函数出发,导出了扭转问题、平面问题、轴对称问题和回转体扭转问题的应力函数.     

Constant Mean Curvature Surfaces Foliated by Circles in Lorentz–Minkowski Space

We prove that a spacelike surface in L3 with nonzero constant mean curvature and foliated by pieces of circles in spacelike planes is a surface of revolution. When the planes containing the circles are timelike or null, examples of nonrotational constant mean curvature surfaces constructed by circles are presented. Finally, we prove that a nonzero constant mean curvature spacelike surface foliated by pieces of circles in parallel planes is a surface of revolution.     

Some special surfaces in the pseudo-Galilean space

We describe some special surfaces in pseudo-Galilean spaces such as helical surfaces, ruled screw surfaces, surfaces of revolution and in particular tori of revolution. We define special surfaces and find their main properties.      

Das Gleichgewicht eines Wasserringes mit freier Oberfläche in einem rotierenden Hohlkörper

Summary The problem of the critical speed of revolution of a hollow rotor which has a water ring revolving inside and with it, is investigated and it is found to be identical with the critical speed of revolution of a rotor completely filled with water. As the calculation of the oscillations of the water ring itself shows, dangerous resonances with the rotor oscillations need not be expected.