Optimal shape of a vertical rotating column

We determine the shape of the lightest rotating column that is stable against buckling, positioned in a constant gravity field, oriented along the column axis. In deriving the optimality conditions, the Pontryagin's principle was used. Optimal cross-sectional area is obtained from the solution of a non-linear boundary value problem. For this problem a variational principle and a first integral are formulated. Also a priori estimates of the cross-sectional area at the lower end are presented. The procedure is illustrated by three concrete examples. The problem treated here may be considered as a step in the dynamic optimization procedure of a heavy rotating column.     

On a method for determining the fixing conditions for a rectangular plate from the natural frequencies

We prove the duality of solutions for the problem of determining the boundary conditions on two opposite sides of a rectangular plate from the frequency spectrum of its bending vibrations. A method for determining the boundary conditions on two opposite sides of a rectangular plate from nine natural frequencies is obtained. The results of numerical experiments justifying the theoretical conclusions of the paper are presented. Rectangular plates are widely used in various technical fields. They serve as printed circuit boards and header plates, bridging plates, aircraft and ship skin, and parts of various mechanical structures [1–4]. If the plate fixing cannot be inspected visually, then one can use the natural bending vibration frequencies to find faults in the plate fixing. For circular and annular plates, methods for testing the plate fixing were found in [5–7], where it was shown that the type of fixing of a circular or annular plate can be determined uniquely from the natural bending vibration frequencies. The following question arises: Is it possible to determine the type of fixing of a rectangular plate on two opposite sides of the plate from the natural bending vibration frequencies if the other two sides are simply supported? Since the opposite sides of the plate are equivalent to each other, a plate with “rigid restraint—free edge” fixing will sound exactly the same as a plate with “free edge—rigid restraint” fixing. Hence we cannot say that the type of fixing of a rectangular plate on two opposite sides can be uniquely determined from its natural bending vibration frequencies. But it turns out that we can speak of duality in the solution of this problem. Here we observe an analogy with the problem of determining the rigidity coefficients of springs for elastic fixing of a string [8]: the rigidity coefficients of the springs are determined by the natural frequencies uniquely up to permutations of the springs.     

On a Rayleigh Wave Equation with Boundary Damping

In this paper an initial-boundary value problem for a weakly nonlinear string (or wave) equation with non-classical boundary conditions is considered. One end of the string is assumed to be fixed and the other end of the string is attached to a dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a simple model describing oscillations of flexible structures such as overhead transmission lines in a windfield. An asymptotic theory for a class ofinitial-boundary value problems for nonlinear wave equations is presented. Itwill be shown that the problems considered are well-posed for all time t. A multiple time-scales perturbation method incombination with the method of characteristics will be used to construct asymptotic approximations of the solution. It will also be shown that all solutions tend to zero for a sufficiently large value of the damping parameter. For smaller values of the damping parameter it will be shown how the string-system eventually will oscillate. Some numerical results are alsopresented in this paper.     

On Boundary Damping for a Weakly Nonlinear Wave Equation

In this paper an initial-boundary value problem for a weakly nonlinear string(or wave) equation with non-classical boundary conditions is considered. Oneend of the string is assumed to be fixed and the other end of the string isattached to a spring-mass-dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a rather simple model describing oscillationsof flexible structures such as suspension bridges or overhead transmission lines in a windfield. A multiple-timescales perturbation method will be usedto construct formal asymptotic approximations of the solution. It will also beshown that all solutions tend to zero for a sufficiently large value of thedamping parameter. For smaller values of the damping parameter it will be shownhow the string-system eventually will oscillate.     

Variational approach to non-linear boundary value problems for elasto-plastic incompressible bending plate

Non-linear boundary value problems for inelastic isotropic homogeneous incompressible bending plate, within the range of J₂-deformation theory, are considered. An existence of the weak solution of the non-linear problem with clamped boundary condition is obtained in H²(Ω) by using monotone operator theory and Browder-Minty theorem. For linearization of the non-linear problem a monotone iteration scheme is constructed. It is shown that the sequence of potentials obtained from the sequence of approximate solutions (i.e. iterations), is a monotone decreasing one. Convergence of the iteration process in H²-norm is proved by using the convexity argument. Numerical solutions, based on finite-difference scheme, are given for linear bending problems with rigid clamped as well as simply supported boundary conditions. Further numerical examples are presented to illustrate the convergence of approximate solutions and monotonicity of the potentials as applied to the non-linear problems.     

Dynamics and Stability of a Liquid Triaxial Ellipsoid

The problem of the dynamics of a liquid triaxial ellipsoid confined inside a uniformly rotating rigid shell is formulated and solved within the framework of the Poincare-Joukovski problem. The system of three non-linear first-order differential equations has kinetic energy and circulation integrals representing two ellipsoids with displaced centers in parameter space. All the steady-state and time-dependent solutions are studied; time-dependent solutions exist in four zones, where they are represented by elliptic Jacobi functions, and on three zone boundary interval (represented in terms of elementary functions). On the three other boundary intervals there exist liquid steady-state ellipsoids for which the vortex vector either coincides with one of the principal axes of the ellipsoid or lies in one of its principal planes of symmetry.     

The boundary value problem for the free oscillations of an ideal rotating liquid

We consider the problem, in its linear formulation, of the free oscillations of an ideal incompressible liquid, partially or completely filling a rotating cavity in a field of body forces. The unperturbed motion is the rotation of the liquid-cavity system at constant angular velocity as a solid body. The problem of determining the frequency spectrum is reduced to the solution of a boundary value problem for the eigenvalues, which can be solved analytically or numerically. Particular cavity shapes and types of motion are discussed for which exact and approximate solutions of the boundary value problem are obtained. The results of numerical computations are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 81–88, July–August, 1973.In conclusion, the author wishes to thank G. S. Narimanov, B. I. Rabinovich, and V. M. Rogovoi for valuable advice, and also I. A. Artemov who gave great help with the calculations.     

Unsteady Couette flow in a rotating system

The unsteady Couette flow between two infinite horizontal plates induced by the non-torsional oscillations of one of the plates in a rotating system under the boundary layer approximations is investigated. An exact solution of the governing equations has been obtained by using Laplace transform technique. It is shown that when the oscillating plate situated at an infinite distance from stationary plate then the problem reduces to the unsteady boundary layer problem in a rotating system with non-torsional oscillations of the free-stream velocity.     

A regularization method for a boundary variational inequality of the second kind associated to a friction problem

A scalar contact problem with friction is formulated as a boundary variational inequality of the second kind. The presence of the non-differentiable friction functional causes difficulties when approximating it. We present two approaches to overcome these difficulties: A regularization procedure leading to a non-linear boundary variational equation, for which we propose an iterative process and the second one is a boundary mixed variational formulation involving Lagrange multiplier. We reformulate our problem in terms of a saddle point problem for the corresponding boundary Lagrangian and describe Uzawa's algorithm to compute it.     

Post-buckling of cantilever columns having variable cross-section under a combined load

Post-buckling of a cantilever column is examined under a combined load consisting of a tip-concentrated load and a distributed axial load, through dynamic formulation. The formulation of the problem is based on the moment–curvature relationship. The two-point boundary value problem described by the governing equations is dependent on the frequency parameter and the two load parameters. The buckling loads are those loads at which the eigencurve, namely, the load versus frequency curve of the column meets the load axis. A simple and reliable iterative procedure to convert the two-point boundary value problem into an initial value problem is followed and solved the non-linear differential equations utilizing a fourth-order Runge–Kutta integration scheme. To demonstrate the potentiality of the adopted numerical scheme, linear vibration frequencies of truncated, tapered cantilever wedges and cones are determined and compared with the published analytical and test results. Buckling and post-buckling loads of a simply supported stepped column are obtained and compared with the published test results. The loads and deflections of non-uniform cantilever columns are obtained for various slopes at the tip. The interaction of load parameters for a free–free truncated conical column has also been examined. The numerical results indicate that the path represented by the two load parameters turns out to be nearly a straight line.     

A PIECEWISE–LINEARIZED METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS: TWO–POINT BOUNDARY–VALUE PROBLEMS

Piecewise-linearized methods for the solution of two-point boundary value problems in ordinary differential equations are presented. These problems are approximated by piecewise linear ones which have analytical solutions and reduced to finding the slope of the solution at the left boundary so that the boundary conditions at the right end of the interval are satisfied. This results in a rather complex system of non-linear algebraic equations which may be reduced to a single non-linear equation whose unknown is the slope of the solution at the left boundary of the interval and whose solution may be obtained by means of the Newton–Raphson method. This is equivalent to solving the boundary value problem as an initial value one using the piecewise-linearized technique and a shooting method. It is shown that for problems characterized by a linear operator a technique based on the superposition principle and the piecewise-linearized method may be employed. For these problems the accuracy of piecewise-linearized methods is of second order. It is also shown that for linear problems the accuracy of the piecewise-linearized method is superior to that of fourth-order-accurate techniques. For the linear singular perturbation problems considered in this paper the accuracy of global piecewise linearizat ion is higher than that of finite difference and finite element methods. For non-linear problems the accuracy of piecewise-linearized methods is in most cases lower than that of fourth-order methods but comparable with that of second-order techniques owing to the linearization of the non-linear terms.     

Non-linear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement and finite element discretization

This paper deals with the non-linear viscoelastodynamics of three-dimensional rotating structure undergoing finite displacement. In addition, the non-linear dynamics is studied with respect to geometrical and mechanical perturbations. On part of the boundary of the structure, a rigid body displacement field is applied which moves the structure in a rotation motion. A time-dependent Dirichlet condition is applied to another part of the boundary. For instance, this corresponds to the cycle step of a helicopter rotor blade. A surface force field is applied to the third part of the boundary and depends on the time history of the structural displacement field. For example, this might corresponds to general unsteady aerodynamics forces applied to the structure. The objective of this paper is to model the non-linear dynamic behavior of such a rotating viscoelastic structure undergoing finite displacements, and to allow small geometrical and mechanical (mass, constitutive equations) perturbations analysis to be performed. The model is constructed by the introduction of a reference configuration which is deduced from the non-linear steady boundary value problem. A constitutive equation deduced from the Coleman and Noll theory concerning the viscoelasticity in finite displacement is used. Thereafter, the weak formulation of the boundary value problem is constructed and discretized using the finite element method. In order to simplify the mathematical study of the equations, multilinear forms are introduced in the algebraic calculation and their mathematical properties are presented.     

On the unsteady flow of a compressible liquid through a porous medium

Summary The unsteady flow of a compressible liquid in a porous medium can be described in terms of a non-linear partial differential equation for the liquid pressure or a linear differential equation for the density if gravitational effects are negligible. In gravitational flow fields the formulation yields non-linear equations for both density and pressure. A transformation is given which shows that in the absence of gravitational effects, the solution of the non-linear boundary value problem in terms of the pressure involves no more labour than the solution of the linear problem in terms of the density, contrary to a misconception in the petroleum literature. Furthermore this transformation offers in addition the solution to a heretofore unsolved problem in gravity flow.This research was supported in part by the Office of Naval Research under Contract Nonr-222(04).     

An iterative method for solution of a boundary value problem in non-newtonian fluid flow

The boundary value problem

arises in boundary layer equations for the steady flow of a power-law fluid over an impermeable, semi-infinite flat plane. The parameter μ is equal to $\text{1}\text{n}$ where n is the exponent of the strain rate in the expression for the shear stress. We develop and prove the convergence of an iterative method for the solution of the given boundary value broblem for dilatant fluids (0 < μ <1). The iterative method can be easily implemented computationally. An added feature of our technique is that it accurately yields y(0), an important parameter which is related to the drag at the plate. The iterative method works well computationally not only for 0 < μ < 1 but for the range 1 < μ < 4 (pseudoplastic fluids with 1 > n > $\text{1}\text{4}$), as well.     

Lane-Emden equations of second kind modelling thermal explosion in infinite cylinder and sphere

We study a modified version of the Lane-Emden equation of the second kind modelling a thermal explosion in an infinite cylinder and a sphere. We first show that the solution to the relevant boundary value problem is bounded and that the solutions are monotone decreasing. The upper bound, the value of the solution at zero, can be approximated analytically in terms of the physical parameters. We obtain solutions to the boundary value problem, using both the Taylor series （which work well for weak nonlinearity） and the b-expansion method （valid for strong nonlinearity）. From here, we are able to deduce the qualitative behavior of the solution profiles with a change in any one of the physical parameters.     

Effect of Low Rotation Rate on Steady Convection During the Solidification of a Ternary Alloy

We consider the problem of steady convective flow during the directional solidification of a horizontal ternary alloy system rotating at a constant and low rate about a vertical axis. Under the limit of large far-field temperature, the flow region is modeled to be composed of two horizontal mushy layers, which are referred to here as a primary layer over a secondary layer. We first determine the basic state solution and then carry out linear stability analysis to calculate the neutral stability boundary and the critical conditions at the onset of motion. We find, in particular, that there are two flow solutions and each solution exhibits two neutral stability boundaries, and the flow can be multi-modal in the low rotating rate case with local minima on each neutral boundary. The critical Rayleigh number and the wave number as well as the vertical volume flux increase with the rotation rate, but the flow is found to be less stabilizing as compared to the binary alloy counterpart flow. The effects of low rotation rate increase the solid fraction and the liquid fraction at certain vertically oriented fluid lines, and the highest value of such increase is at a horizontal level close to the interface between the two mushy layers.     

A semi-analytical approach for the non-linear large deflection analysis of laminated rectangular plates under general out-of-plane loading

A semi-analytical approach for the geometrically non-linear analysis of rectangular laminated plates with general inplane and out-of-plane boundary conditions under a general distribution of out-of-plane loads is developed. The analysis is based on the elastic thin plate theory with geometrically non-linear von Kármán strains. The solution of the non-linear partial differential equations is reduced to an iterative sequential solution of non-linear ordinary differential equations using the multi-term extended Kantorovich method. The efficiency, accuracy, and convergence of the proposed method are examined through a comparison with other semi-analytical methods and with finite element analyses. The capabilities of the approach and its applicability to the non-linear large deflection analysis of plate structures are demonstrated through various numerical examples. Emphasis is placed on combinations of lamination, boundary, and loading conditions that cannot be analyzed using alternative semi-analytical methods.     

优化迭代步长的两种改进增量谐波平衡法

增量谐波平衡法(IHB法)是一个半解析半数值的方法, 其最大优点是适合于强非线性系统振动的高精度求解. 然而, IHB法与其他数值方法一样, 也存在如何选择初值的问题, 如初值选择不当, 会存在不收敛的情况. 针对这一问题, 本文提出了两种基于优化算法的IHB法: 一是结合回溯线搜索优化算法(BLS)的改进IHB法(GIHB1), 用来调节IHB法的迭代步长, 使得步长逐渐减小满足收敛条件; 二是引入狗腿算法的思想并结合BLS算法的改进IHB法(GIHB2), 在牛顿-拉弗森(Newton-Raphson)迭代中引入负梯度方向, 并在狗腿算法中引入2个参数来调节BSL搜索方式用于调节迭代的方式, 使迭代方向沿着较快的下降方向, 从而减少迭代的步数, 提升收敛的速度. 最后, 给出的两个算例表明两种改进IHB法在解决初值问题上的有效性.      

Problems of Hydrodynamics for a Triaxial Ellipsoid

Problems of motion of a triaxial ellipsoid in an ideal liquid and in a viscous liquid in the Stokes approximation and also equilibrium shapes of the rotating gravitating liquid mass are considered. Solutions of these problems expressed via four quadratures depending on four parameters are significantly simplified because they are expressed via the only function of two arguments. The efficiency of the proposed approach is demonstrated by means of analyzing the velocity and pressure fields in an ideal liquid, calculating the added mass of the ellipsoid, determining the viscous friction, and studying the equilibrium shapes and stability of the rotating gravitating capillary liquid. The pressure on the triaxial ellipsoid surface is expressed via the projection of the normal to the impinging flow velocity. The shape of an ellipsoid that ensures the minimum viscous drag at a constant volume is determined analytically. A simple equation in elementary functions is derived for determining the boundary of the domains of the secular stability of the Maclaurin ellipsoids. An approximate solution of the problem of equilibrium and stability of a rotating droplet is presented in elementary functions. A bifurcation point with non-axisymmetric equilibrium shapes branching from this point is found.