Showing papers in "Quarterly of Applied Mathematics in 1951"

The principle of minimized iterations in the solution of the matrix eigenvalue problem

Abstract: An interpretation of Dr. Cornelius Lanczos' iteration method, which he has named \"minimized iterations\", is discussed in this article, expounding the method as applied to the solution of the characteristic matrix equations both in homogeneous and nonhomogeneous form. This interpretation leads to a variation of the Lanczos procedure which may frequently be advantageous by virtue of reducing the volume of numerical work in practical applications. Both methods employ essentially the same algorithm, requiring the generation of a series of orthogonal functions through which a simple matrix equation of reduced order is established. The reduced matrix equation may be solved directly in terms of certain polynomial functions obtained in conjunction with the generated orthogonal functions, and the convergence of the solution may be observed as the order of the reduced matrix is successively increased with the order of the original matrix as a limit. The method of minimized iterations is recommended as a rapid means for determining a small number of the larger eigenvalues and modal columns of a large matrix and as a desirable alternative for various series expansions of the Fredholm problem. 1. The conventional iterative procedures. It is frequently required that real latent roots, or eigenvalues, and modal columns be determined for a real numerical matrix, u, of order, n, in the characteristic homogeneous equation,*

On a quasi-linear parabolic equation occurring in aerodynamics

Abstract: where u — u(x, t) in some domain and v is a parameter. The occurrence of the first derivative in t and the second in x clearly indicates the equation is parabolic, similar to the heat equation, while the interesting additional feature is the occurrence of the non-linear term u du/dx. The equation thus shows a structure roughly similar to that of the Navier-Stokes equations and has actually appeared in two separate problems in aerodynamics. An equation simply related to (1) appears in the approximate theory of a weak non-stationary shock wave in a real fluid. This is discussed in Ref. 1 (pp. 146-154) where a general solution of (1) is given. The equation is also given in J. Burgers' theory of a model of turbulence (Ref. 2) where he notes the relationship between the model theory and the shock wave. Historically, the equation (1) first appears in a paper by H. Bateman (Ref. 3) in 1915 when he mentioned it as worthy of study and gave a special solution. Eq. (1) is of some mathematical interest in itself and may have applications in the theory of stochastic processes. The aim of this paper is to study the general properties of (1) and relate the various applications. I wish to thank Professor P. A. Lagerstrom and F. K. Chuang for helpful collaboration. 2. Relationship of (1) to Shock Wave Theory. The solutions to Eq. (1) can approximately describe the flow through a shock wave in a viscous fluid. They can be related to the shock wave in several ways. In Ref. 1 an approximation based on the NavierStokes equations for one-dimensional non-stationary flow of a compressible viscous fluid gives

A note on the existence of a solution to a problem of Stefan

Abstract: When certain metals are heated slowly, the temperature rises until it reaches a critical temperature at which the structure of the metal changes from one crystalline form to another. As for example, iron changes from a to (3 crystals at 1643°F. Accompanying this change of crystalline form is a latent heat of recry&tallization. In order to study the process we investigate the associated mathematical problem, which requires the solution of a partial differential equation in a region with an undetermined boundary. Our analysis establishes the existence and uniqueness of the solution. In a previous paper2 this problem is treated from the point of view of computing the solution. Suppose a metal slab having two infinite parallel faces is brought uniformly to the critical temperature and then heated by a uniform source covering the front face while an insulator covers the back face. Under these conditions, the new crystals are first formed at the front face, and the interface between the new and old crystals travels from the front face to the back face. Mathematically the problem can be stated as follows, where u = 0 is taken as the critical temperature: Find the temperature, u = u(x, t), and the curve, x = x(t), which satisfy the following conditions

Note on the stresses produced by nuclei of thermo-elastic strain in a semi-infinite elastic solid

Abstract: where v is Poisson's ratio, a, the coefficient of linear expansion and r, the distance of the point from C. The singularity at C is the simple 'centre of dilatation' which is supposed to form a nucleus of thermo-elastic strain at the point. The problem solved in this paper is that of determining displacements and stresses in an isotropic semi-infinite elastic solid which has a nucleus of thermo-elastic strain as defined above at any point C below the plane boundary. 2. Statement of the problem. We suppose that the solid is bounded by the plane Z = 0, the axis of Z being drawn into the body. The coordinates of C are taken as (0, 0, c). Putting

The oscillating rectangular airfoil at supersonic speeds

Abstract: The pressure distribution on a quarter infinite, zero thickness airfoil having a prescribed distribution of down wash (on the wing only), which exhibits a harmonic time dependence, is determined by a Fourier transform solution of the linearized, potential equation for supersonic flow. The solution is effected with the aid of the Wiener-Hopf technique and leads to a Green's function, which may be expressed either as a finite, definite integral or as an expansion in powers of a dimensionless frequency parameter. It is shown that the results are applicable to the calculation of the forces and moments on rectangular airfoils of effective aspect ratio (A cot 6, where 6 is the Mach angle) greater than unity. It appears that the force and moment coefficients of practical interest may be expressed in terms of known functions, including certain integrals which have been calculated for the two-dimensional, oscillating airfoil. The extension of the two-dimensional results to rectangular wings for which the prescribed down wash is constant along the span is particularly simple. The extension of the results for harmonic time dependence to the step function (Heaviside) case is indicated.

Minimal problems in airplane performance

Abstract: We develop here the theory of operating an airplane so as to minimize an arbitrary function of the end-values of the generalized coordinates. A propeller-driven airplane is treated as a particle in equilibrium, subject to the forces of drag, lift, thrust, and gravity. We assume that the specific fuel consumption is a function of the power only, and that the available power is independent of the altitude. The problem is shown to be of the Bolza type in the Calculus of Variations, with the complications arising from the presence of inequalities, discontinuities, and variables whose derivatives do not enter the problem explicitly. The Euler-Lagrange equations are derived and discussed. Notation. A subscript will sometimes denote an index, at other times the argument of partial differentiation. A superscript dot will indicate differentiation with respect to the parameter t. The Summation Convention will be observed. In referring to equations decimals may be used; e.g. (59.4) is the fourth equation of the set (59). 8a is the Kronecker delta.

The torsion and stretching of spiral rods. II

Abstract: In this paper the torsion or the stretching problem for a spiral rod is treated theoretically. The equations of equilibrium expressed in terms of displacements are reduced to forms which are independent of one co-ordinate. They are readily integrated for the particular case where the helix angle is small, and the corresponding displacements and stresses can be expressed in forms which contain three arbitrary plane harmonic functions, determination of which is dependent upon the shape of the section. As an application of the general solution, the problem for an elliptic section is solved explicitly. Two-dimensional problems in elasticity have been studied extensively from early times on account of their simplicity in stress analysis and their useful applications in many engineering problems. For a similar reason, various problems of axially symmetrical stress distribution have been investigated by many writers. In this paper we shall treat the torsion or the stretching problem for a spiral rod. The stress distribution for this case is neither two-dimensional nor axially symmetrical, and each stress does not vanish in general and consequently the analysis becomes somewhat complicated. But the problem is not a three dimensional one without any restriction, since if we rotate the co-ordinate axes about the axis of the helix so as to coincide with the fixed directions with respect to a section which is perpendicular to the axis of helix, then the stress distribution referred to the rotating axes is the same in any section. Starting from the equations of equilibrium expressed in terms of displacements, we shall introduce equations which are independent of the position of the section. The differential equations of displacements are readily integrated for the particular case where the helix angle is small. The corresponding displacements and stresses are expressed in forms which contain three arbitrary plane harmonic functions, determination of which is dependent upon the shape of the section, and thus we can considerably simplify the problem. We shall take the axis of the helix as the z-axis, and shall denote the displacements in the x, y, z directions by u, v and w, respectively. Then the equations of equilibrium can be expressed in the forms1

Quasi-stationary airfoil theory in subsonic compressible flow

Abstract: A solution of the integral equation for an oscillating, two-dimensional, thin airfoil in a compressible flow (subsonic and inviscid) is obtained by retaining only first order terms in frequency. The results are applied to the calculation of the damping derivative of a tail in rotary motion about a forward center, and it is shown that the damping is considerably less than that calculated on the basis of stationary airfoil theory. A brief investigation of induction effects shows this reduction to be considerably less for a wing of finite aspect ratio.

The method of characteristics applied to problems of steady motion in plane plastic stress

Abstract: 5. Conclusion. The methods presented above have been applied to more complicated frames and readily give the required solutions for collapse design under constant or varying loads. For shakedown design, the iterative numerical method converges fairly rapidly. It may be found easier for highly redundant frames to obtain a new elastic solution at each stage using the numerical values obtained from the previous analysis, since an analysis with numerically unspecified flexural rigidities is extremely tedious. For all three types of design, the introduction at an early stage of the numerical values of the loads simplifies the work greatly, since it will be found that a large proportion of the inequalities generated become redundant and can be ignored. Only examples of concentrated loads on straight members of uniform cross-section between joints have been examined, making it possible to pick by inspection the critical cross-sections. However, the basic ideas are not altered by the introduction of other variables; the analysis will be more complicated, but aids to calculation may be introduced which leave the basic problem unchanged.

Relation between Bergman’s and Chaplygin’s methods of solving the hodograph equation

Abstract: When a perfect gas is in steady irrotational isentropic motion in two dimensions, the stream function 1p satisfies a linear differential equation in which the independent variables are components of velocity. For this 'hodograph equation', general forms of solution have been given by Chaplygin1 and Bergman2. The purpose of this note is to show how Bergman's form of solution can be converted into Chaplygin's. Hereby we obtain the specification of the same solution by means of two quite different series, and are in the position to check the extensive computations which are required (in general) to evaluate either of the series. The results of §1 are due to Chaplygin1, Lighthill3 and Cherry4; for proofs of the keyformulae (4), (6), (12) reference may be made to [3] or [4]. For Bergman's form of solution the most convenient reference is v. Mises and Schiffer.5 The different authors use different notations, and the present paper uses a blend of them. 1. Let the rectangular velocity-components be r1/2 cos 6, r1/2 sin 0, with the unit of speed so chosen that the limiting speed, at which the pressure vanishes, corresponds to r = 1. Then the hodograph equation is

A third order boundary value problem arising in aeroelastic wing theory

Abstract: where X is a parameter. The boundary conditions are z(0) = z'il) = (EI(x)z'(x))'x= 0. Since this represents a non self-adjoint boundary value problem, the ordinary RayleighRitz variational method for approximating its characteristic values is not applicable. Formal extensions of this variational approach have been suggested by Flax [1] and used by Cheng [2] but no mathematical evidence for their validity is at present available. In this connection, two questions have been raised [1], (A) Under what conditions on EI(x) and c(x) are all the characteristic values of (la) real? (B) Under what conditions on EI(x), c(x) and fix) can the solution zx(x) of

Source-superposition method of solution of a periodically oscillating wing at supersonic speeds

Abstract: Introduction and summary. In a recent paper Evvard (Ref. 1) discussed the linearized theory of the non-steady motion of three dimensional wings by methods which he had previously developed for the treatment of the corresponding steady flow problems (Refs. 2 and 3). Evvard represented the wing by a distribution of sources, and the important result of his steady state theory concerned the determination of the flow in a region influenced by a subsonic leading edge or wing tip. He showed that the influence of the flow around this subsonic edge of a flat lifting wing on the velocity potential at

A note on the principal frequency of a triangular membrane

Abstract: The principal frequency of a membrane of triangular shape is exactly known in two simple cases: for the 45°, 45°, 90° and the 60°, 60°, 60° triangles.2 As will be shown in this note, an exact solution of comparable simplicity exists also for the 30°, 60°, 90° triangle-a result which, to the author's knowledge, has not been observed before. The three lines the equations of which in rectangular coordinates x, y are

On torsion of prisms with longitudinal holes

Abstract: This paper presents a method of solution, called the method of images, for the torsion of prisms having one or more longitudinal holes. The method is applicable to prisms of the following four, and only four, sections: a rectangle, an equilateral triangle, an isosceles triangle and a 30°-60°-90° triangle. These four sections form a group by themselves. The solution is obtained by adding to the known solution of a corresponding solid prism without holes a system of harmonic functions which vanish on the entire external boundary of the given section, and besides possess a singularity at the centre of each hole. Such a system of functions may be constructed from Weierstrass' Sigma function and its allied functions. The solution is illustrated by applying it in detail to a rectangular prism having a central longitudinal hole. Numerical results are shown for the special case of a square prism. Introduction. The torsion of a circular cylinder having longitudinal circular holes, with or without a central hole, has been investigated by Kondo1 and by the present writer.2 In the present paper, the investigation will be extended to a prism, which is also pierced by such longitudinal holes. Both problems in fact belong to the same general class of torsion problems dealing with cylinders of multi-connected sections. Analytic solutions of such problems are generally difficult except in some simple cases, and indeed very few solutions have ever been found. It appears, however, that certain prisms of this nature can be solved by adapting to them the method of images. There are altogether four such prisms, the cross sections of which are as follows: (1) a rectangle, including square as a special case, (2) an equilateral triangle, (3) an isosceles right triangle, (4) a 30°-60°-90° triangle. It is not difficult to show that by reflection about the edges each of the above four sections forms a doubly infinite set of images. Furthermore, it can be shown conclusively by theory of groups3 that these four sections are the only ones which form such images. The images formed in each case are shown in Figs. 1-4 respectively. In each figure the shaded area represents the fundamental region, or the given section of the prism: *Received August 9, 1950. XM. Kondo, The stresses in twisted circular cylinder having circular holes, Phil. Mag. (7) 22, 1089-1108 (1936). 2C. B. Ling, Torsion of a circular tube with longitudinal circular holes, Q. Appl. Math., 5, 168-181 .(1947). SW. Burnside, Theory of groups, 2nd ed., Cambridge University Press, 1911, 410-418. 248 CHIH-BING LING [Vol. IX, No. 3 The regions marked by positive signs represent the images which are formed by an even number of reflections, while the regions marked by negative signs represent those which are formed by an odd number of reflections. Note that any two adjacent regions must have alternate positive and negative signs. The set of points in each figure represents the images due to a given point in the fundamental region.

On the method of inversion in the two-dimensional theory of elasticity

Abstract: 1. Chaplygin, Gas jets, Moscow 1902, see also NACA TM 1063. 2. Goldstein, Lighthill and Craggs, On the hodrograph transformation for highspeed flow, Q. J. of Mech. Appl. Math. 1, 344-357 (1948). 3. Garrick and Kaplan, On the flow of a compressible fluid by the hodograph method, NACA Report No. 790 (1944). 4. J. Horn, Veber eine lineare Differentialgleichung zweiter Ordnung mit einem willkuerlichen Parameter, Math. Ann. 52, 271-292 (1899).