Showing papers in "Quarterly of Applied Mathematics in 1944"

A method for the solution of certain non – linear problems in least squares

Abstract: The standard method for solving least squares problems which lead to non-linear normal equations depends upon a reduction of the residuals to linear form by first order Taylor approximations taken about an initial or trial solution for the parameters.2 If the usual least squares procedure, performed with these linear approximations, yields new values for the parameters which are not sufficiently close to the initial values, the neglect of second and higher order terms may invalidate the process, and may actually give rise to a larger value of the sum of the squares of the residuals than that corresponding to the initial solution. This failure of the standard method to improve the initial solution has received some notice in statistical applications of least squares3 and has been encountered rather frequently in connection with certain engineering applications involving the approximate representation of one function by another. The purpose of this article is to show how the problem may be solved by an extension of the standard method which insures improvement of the initial solution.4 The process can also be used for solving non-linear simultaneous equations, in which case it may be considered an extension of Newton's method. Let the function to be approximated be h{x, y, z, • • • ), and let the approximating function be H{oc, y, z, • • ■ ; a, j3, y, ■ • ■ ), where a, /3, 7, • ■ ■ are the unknown parameters. Then the residuals at the points, yit zit • • • ), i = 1, 2, ■ • • , n, are

On the deflection of a cantilever beam

Abstract: and this is satisfied when the factors a, b, c, ■ • ■ are all equal. Without loss of generality, they may be taken equal to unity. For this weighting system, the formation of the damped normal equations (10) may be thought of as being accomplished simply by the addition of a positive constant, 1 /w, to the coefficients of the principal diagonal of the standard normal equations (5). Another weighting system which has been used successfully is, a = \\aa ], b=m, ■ ■ • ; in this case the damped normal equations are formed by multiplying the principal diagonal coefficients of the standard normal equations by a constant greater than unity, 1 + 1 /w. The nature of the damping which we have imposed upon the parameter variables can be given a simple geometric interpretation. For instance, if the unity weighting system is considered, the \"overshooting\" of the solution is prevented by damping the distance (k dimensional) from the initial solution point, since Q is then the square of this distance. By this restriction of k dimensional distance (which would appear to be a natural way to prevent overshooting), we are not obliged to decide on an arbitrary preassigned procedure restricting the variables individually, as is done, for example, by the method of Cauchy (I.e.). The greater freedom given the individual variables by the method of damped least squares may account for the fact that it has solved, with a comparatively rapid rate of convergence, types of problems which are of much greater complexity than those to which the principle of least squares is ordinarilv applied.

A variational principle for a state of combined plastic stress

Abstract: In a recent paper1 M. A. Sadowsky has stated a heuristic principle of maximum plastic resistance which he has applied to several states of combined plastic stress. The principle states that \"among all statically possible stress distributions (satisfying all three equations of equilibrium, the condition of plasticity, and boundary conditions), the actual stress distribution in plastic flow requires a maximum value of the external effort necessary to maintain the flow.\" W. Prager, in a contribution to the discussion of this paper2, has shown that the principle can be so interpreted as to lead to the correct differential equation for a beam under combined torsion and tension. This note is concerned with an accurate statement of the principle together with a proof of its validity for the case of a beam in a perfectly plastic state under combined torsion and bending by couples, the cross-section of the beam having an axis of symmetry. Specifically, we shall prove the following variational principle for such a system. Among all statically possible stress distributions in a beam under a given torque (satisfying the equations of equilibrium, the condition of plasticity, and boundary conditions), the actual stress distribution when plastic flow occurs is the one for which the bending moment is stationary. Let us choose the coordinate axes in the following fashion, y lies along the axis of symmetry of the cross-section, z passes through the center of gravity of the crosssection and is parallel to the generators of the cylindrical beam, and x is perpendicular to y and z. We assume that the strain velocities, vx, vy, v„ are given by the same expressions as in the case of an incompressible elastic material; i.e.,

The numerical solution of Laplace’s and Poisson’s equations

Abstract: Briefly, the method of procedure, commonly called the Liebmann procedure,1 is to cover the region R with a rectangular network of lines at distances h apart, and to assume values at the interior lattice points of this network. Using these assumed values and the known boundary values, we traverse the region R moving in some definite geometrical pattern from lattice point to lattice point, replacing the assumed values of u at each lattice point by the arithmetic average of the values of u at the four neighboring lattice points. We then repeat the traverse moving in the same pattern to obtain a second improved value of u at each lattice point; and so on until a convergent stage is reached when the values of u are no longer changed materially by continued traversing. The purpose of this paper is to present a process which yields precisely the convergent values of u obtained by infinitely many traverses of the region. In more precise language, if m* is the £th approximation of the value of u after k traverses, our process yields the value u =limi_„ uk. 2. Notation and set up of the problem. Equation (1.2) can be transformed to

Electromagnetic waves in a bent pipe of rectangular cross section

Abstract: The analysis of electromagnetic wave propagation in a bent pipe of rectangular cross section, (x = R, x = R+a, y = 0, y = b), is based on the Maxwell field equations, expressed in cylindrical coordinates (r, 6, y) (Fig. 1). As in the case of the straight pipe,1 the time variation is given by the exponential ewhere co is the angular frequency. The angular variation is given by e~xe, where 2 is the propagation constant for the bent portion. The equations may be written

A geometrical interpretation of the relaxation method

Abstract: The solution is easily expressed as a set of quotients of determinants. However, as n increases, the task of calculating the determinants becomes excessively burdensome. The relaxation method1 provides a set of easy steps by which the solution of (1) is approached. The method has been compactly described by Temple.2 The purpose of the present note is to give a geometrical description of the relaxation method. For the trivial case n = 2 the geometrical description may be displayed accurately in a diagram. For w = 3 a model may be visualized. For n>3 we pass beyond the region of simple concrete geometrical representation, but in many ways geometry in an n-space is closely analogous to geometry in 2-space or 3-space, and the geometrical description continues to serve as a general guide to procedure. Let us regard x, as rectangular Cartesian coordinates in a Euclidean n-space. Let us define

A strain energy derivation of the torsional-flexural buckling loads of straight columns of thin-walled open sections

Abstract: In the thin-walled open section columns of modern aluminum alloy aircraft torsional buckling and combinations of torsional and flexural buckling are of considerable importance. The critical loads corresponding to these types of instability have been calculated by Wagner,1 Kappus,2 Lundquist and Fligg,3 and Goodier4 through integrating the differential equations of the problem. In the present paper the torsional-flexural buckling loads are determined with the aid of the Rayleigh-RitzTimoshenko method. This procedure obviates the derivation and integration of the differential equations as well as the geometric considerations connected with what Goodier termed \"Wagner's hypothesis.\" The equilibrium of a straight bar of a length L and a cross-sectional area A, loaded axially with a compressive force of a magnitude aA distributed uniformly over the end section, can be investigated by assuming that each section of the bar undergoes a virtual displacement. The end sections of the bar are assumed to be restrained in a manner which precludes translations as well as rotations about any axis perpendicular to the end section, but which permits rotations about axes in the plane of the end section and warping of the end section. Barring displacements that would change the shape of the cross section (such displacements lead to plateor shell-buckling), the most general virtual displacement pattern of the bar can be represented by the following infinite series: 00

On waves in bent pipes

Abstract: In a recent issue of this Quarterly,1 Karlem Riess obtained expressions for the fields of electromagnetic waves in bent pipes of rectangular cross section by the perturbation method. While it is true that in a bent pipe the waves cannot be classified into transverse electric and transverse magnetic types because in general both E and H have components in the direction of wave propagation, a different classification into two types is possible. This permits another method which yields the general solution in terms of Bessel functions. In the one wave type, the plane of the electric ellipse is normal to the axis of bending (the F-axis in Figure 1, p. 329 of Riess' paper); these waves have been called electrically oriented (EOm,n wave type) and the fields of these waves are obtainable from Hy which may be expressed as the product of Bessel and sine (or cosine) functions.

Kron’s method of subspaces

Abstract: Introduction. Gabriel Kron has introduced new and powerful methods of applying tensor analysis to complicated engineering problems, presenting his major contributions in the field of electrical engineering. The manner of presentation, and the rarity of a simultaneous knowledge of the hitherto almost unrelated subjects of electrical engineering and tensor analysis, have unfortunately served to limit his audience. Recent experimental confirmation of some of his investigations dealing with equivalent circuits, however, has attracted the serious attention of a wider engineering following. In view of the growing importance of the whole subject, and of the controversy which has surrounded it, it has seemed desirable to present some particular aspect of Kron's work in a form which may appeal to a less highly specialized audience. To avoid complications as far as possible, the present paper must ignore such important topics as electrical networks, electrical machines, and equivalent circuits. It confines itself to purely dynamical problems, and to that particular idea of Kron's which may be called the method of subs paces.] Much discussion has arisen over Kron's claim that he uses tensor analysis. It is the considered opinion of the present writer that Kron does indeed make a full and proper use of tensor analysis. Possibly the belief that Kron employs only matrices may have arisen from the fact that, in order to present his actual mathematical procedure in a form that may be understood and used by those not familiar with the intricacies of the tensor calculus, he often presents this procedure in matrix form. However, he is always careful to point out that the underlying concepts are wholly tensorial in character. In the present paper the method of subspaces will first be presented in terms of a simple example, and in purely matrix form, merely as a set of rules of procedure, the essentially tensorial significance of the procedure being discussed only after the actual procedure has been brought before the reader. The theoretical discussion will then be followed by three simple, related examples illustrative of various aspects of the method. Scope of the method of subspaces. Let us consider a system, which may be dynamical, electrodynamical, or otherwise, containing several standard parts, such as a fly-wheel, a governor, a pair of synchronous machines, a system of levers, etc. The equations of performance of the individual parts are usually well known, but the equations of performance of the complex whole will depend on the manner in which they are interconnected. Usually it is extremely difficult to trace out the full influence of each interconnection in setting up the equations of performance. The method

Rigorous solutions for the spanwise lift distribution of a certain class of airfoils

Abstract: In this equation y is the span coordinate, and —\\b^y^^b\\ V is the velocity of the air at infinity, its direction being parallel to the x-axis; c(y) is the chord function determining the shape or planform of the airfoil; m is a numerical constant, which the theory of wings of infinite span fixes at 2tt but which seems to have an experimental value in the vicinity of 5.5; a is the geometric angle of attack, which for flat wings is a constant but for twisted wings is a given function of y. For a derivation, the reader is referred to the papers mentioned in footnote 1; we only wish to note here that the downwash velocity w is essentially the first term of (1) and has the form

Studies in optics. I. General coordinates for optical systems with central or axial symmetry

Abstract: In previous papers1,2 the author has proposed an approach to geometrical optics different from that developed by Hamilton and his successors. The purpose of the present paper is to generalize the formulas in these papers, and to find the most general treatment of systems with central (point) symmetry and with axial symmetry. By leaving the coordinates general, subject only to the symmetry conditions of the problem, we retain the symmetry in the formulas up to the point where we desire to draw conclusions for a special problem. We can then introduce special coordinates adapted to the problem in question, and find the particular answers. The fundamental invariants of geometrical optics show no preference for either object or image side, nor for pbint or angle coordinates as variables. The different approaches suggested by Hamilton, as well as the direct approach just mentioned, are special cases of the methods developed here corresponding to special choices of coordinates. Several different choices of coordinates will be given as examples. The fundamental formulas (A, B, B', C below) are based only on symmetry conditions and on the validity of the Lagrange invariant (A). They are therefore not restricted to optical problems,3 but are also valid for problems in mechanics, hydrodynamics, and electron optics. 1. Ray tracing formulas, the Lagrangian invariant. Let us assume a ray traversing a number of optical media with refractive indexes n, «i2, n23, •••,«'. Let a(x, y, z), &'{x', y', z') be a vector from an arbitrary origin to a point on the object and image rays, respectively. Let ak(xk, y*, zk) be the vector from the same origin to the intersection point of the ray with the &th surface. Let sktk+\\($ik,k+i, Vk.k+i, £k,k+i) be a vector along the ray in the medium between &th and (& + l)th surface, a vector of length equal to the refractive index nk,k+i of the medium. Let Ok be a vector perpendicular to the fcth surface at the intersection point. Its length may remain arbitrary, for the moment. The refraction law then reads:

On the bending of a clamped plate

Abstract: The present paper contains an application of a recently developed variational method1 to the boundary value problem of the bending of a clamped plate of arbitrary shape. It will be shown that this problem can be linked to the simpler problem of the equilibrium of a membrane by a chain of intermediate problems, which can be solved explicitly and in finite form in terms of the membrane problem. In the intermediate problems, the deflection converges uniformly in the domain of the plate (including the boundary) to the deflection of the clamped plate, and the derivatives of all orders of the deflection converge uniformly in every domain completely interior to the plate. (In the Ritz method, not even the convergence of the slopes can be guaranteed.2) The method yields numerical results for plates of all shapes for which the membrane problem (which we shall call the base problem) admits an explicit solution. As an example we shall consider a clamped square plate under a uniform load. This problem has been the object of numerous investigations,3 some of which are theoretical, while others are purely numerical, use infinite simple and double series, and operate with an infinite number of linear equations and an infinite number of unknowns.4 An inspection of the general formulae derived in the present paper, formulae which become simple in numerical applications, would show how some of the numerical methods might be rendered rigorous.5 The convergence of higher derivatives is of great practical interest for the approximate computation of the stresses. (Cf. Handbuch der Physik, Springer, Berlin, Vol. VI, 1928, pp. 220-221.) Let us denote the domain of a clamped plate by 5 and its boundary by C. The deflection w(x, y) corresponding to a load q(x, y) and to a flexural rigidity D satisfies the differential equation AAw = q/D (1) with the boundary conditions