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The dynamic response of an infinite, elastic plate to a steadily moving transverse force is studied under the so-called improved plate theory. Analytic solutions for the subsonic and the intersonic cases are obtained using the method of integral transforms. The usual difficulty of the nonuniqueness of the steady-state solutions is circumvented by formulating the problems as initial value problems and then seeking the large time limits of the solutions of such initial value problems. The subsonic and intersonic solutions obtained here are similar to that obtained previously for the supersonic case. However, in contrast with the supersonic solution, which contains two families of Mach lines, the subsonic solution contains no Mach lines and the intersonic solution contains only one family of Mach lines.

Two lower bounds of Sperb for the first eigenvalue of an elastically attached membrane are proved in an elementary way.

The so-called ‘Gauss-Jordan’ procedure for inverting matrices has been described in many papers and is well known. The present paper gives an unusual approach to this method insofar as the inverse is obtained by repeated application of a certain non-similarity transformation to the original matrix. Of course this transformation is related to the Gauss elimination process, but formally has rather the character of an exchange of components in two vector spaces.

This is the first of a series of papers dealing with the theory of oscillation type viscometers with particular emphasis on those in which a finite right circular cylinder (cup or disk) oscillates in contact with a fluid. The purpose is to obtain formulae which relate the density and viscosity of the fluid to the frequency and logarithmic decrement of the oscillation and the various physical properties of the suspension system. The study is to include an accurate analysis of the fluid motion in the vicinity of the edges since this is the main source of error in most existing theories. The first paper gives a general formulation of the problem and an exact solution (including transients) for the oscillating cup viscometer with the fluid inside the cup.

This paper is concerned with the problem of axisymmetric torsion by terminal loads of elastic shells of revolution. Such shells are considered here to be three-dimensional bodies occupying a region of space which includes all points whose distances from a given surface - the midsurface - do not exceed h/2, where h is the shell thickness. The analysis is based on the classical linear theory of elasticity for homogeneous, isotropic materials, and it may be regarded as an extension of the work described in [1] and [2]. The purpose of the present paper is to assess the quality of an approximate solution of the thin shell problem. The case of axisymmetric torsion of thin shells of revolution is perhaps tile simplest one for this purpose, since simple approximate solutions - constructed from two-dimensional shell theories or otherwise - are known [3]. Our results provide estimates, based on the three-dimensional theory of elasticity, for the error involved in a stress analysis in which the 'exact' solution is replaced by an approximate one.

We are dealing with the problem of calculating the numberN of zeros of a polynomialw(z) in certain regions Γ of the complex plane. If Γ is a halfplane or a disc, there exist many well-known solutions of the problem. If Γ is of a more general type, there is only a paper ofSherman4) for the case, that Γ is the intersection of two halfplanes. In this paper we give the solution for the case that the region Γ is simply connected and the boundary of Γ is composed by a finite number of pieces of rational curves. Our method generalizes the method ofWielandt3) for the halfplane and the method ofSherman4). The core of our paper is the way of computing the variation of argw, ifw runs along a piece of a rational curve. This computation is based on a lemma about generalized Sturm sequences.

For the solutions of a system of ordinary differential equations restricted Liapunov stability and the notion of Liapunov stable sets are discussed, in particular for separable and almost separable Hamiltonian systems. We consider stabilization procedures, i.e. modifications of the original differential equations such that the solutions of the modified are Liapunov stable and thus the new system is better suited for numerical integration. In this paper some theoretical aspects rather than numerical methods are pointed out.

In numerous places in the literature of eigenvalue problems of mathematical physics one finds curves which approach each other and suddenly veer away. The author postulates that this ugly behavior may be the result of approximation in the representation of physical reality. In the present paper such behavior is demonstrated to arise from the application of the well-known Ritz-Galerkin method to the classical eigenvalue problem of the free vibration of a rectangular membrane.

In this paper a study is made of finite Grashof number effects upon the flow, in an unbounded domain, in the buoyant plume which forms over a horizontal line source of heat. The consequences of introducing a plane boundary into the domain are also assessed for two particular configurations.

This paper examines the temperature distribution in a low Reynolds number flow past a circular cylinder following a sudden change in the temperature of the cylinder. Solutions describing the initial development of the temperature field and the final approach to the eventual steady state are obtained. Expressions for the Nusselt number for small and large values of time are determined.

In this paper we consider a dynamical system whose position is defined by a certain number of the Poincare-Cetaev variables and which moves with nonlinear nonholonomic constraints of Cetaev's type. The canonical equations of motion of the system are derived and, for integrating them, a generalisation of the Hamilton-Jacobi theorem is presented.

General inequalities of type (5) for eigen values were established as early as 1912 byH. Weyl [11] through consideration of integral equations; he used them for studying the asymptotic behaviour of λn whenn→∞. These inequalities seem to have been forgotten, as some of them were independently re-discovered by several authors (see [4], p. 314 and [10], pp. 483–484). —Such inequalities are here derived and applied in various ways to special problems (vibrating strings, rods, membranes and plates). Explicit lower bounds for λ1 are obtained in a somewhat similar manner in terms of theGreen's function. ForSchrodinger's equation (15′), the very simple inequalities (17′) are found. Analogous inequalities are established for a class of equilibrium problems. Many of the inequalities thus obtained express convexity properties (cf. thePost-scriptum above and the paper byPolya andSchiffer quoted there).

The paper deals with an application of Galerkin's method to a nonconservative problem of the elastic stability of bars. It is shown that the relation between the critical loads of the two-parametric eigenvalue problem is nearly linear.

For the numerical solution of an ordinary differential equation of then-th order (1) with the initial conditions (2)Zurmuhl has developed general Runge-Kutta formulas which integrate this equation directly without separating it into a system ofn equations of the first order. In the present paper some estimations of the errors of this Runge-Kutta method are given, which are valid for any value ofn. It is assumed that the functionf(x, y, y′,...) on the right side of the differential equation and its partial derivatives up to the fourth order are one-valued and continuous in a certain neighbourhood of the initial values and that bounds of these functions are known.

In this paper, we consider the problem of stability of one-dimensional structures and give a general formulation by means of the energy method. We propose then a numerical method of deriving the critical rate requiring eigenvalues of a sequence of small matrices rather than of a big one.

The paper gives general formula (17) for coefficients of any real power ζ of a generalized power series (4). Four first coefficients are given by (19). An example of application to Bessel functions is described.

This paper points out some disadvantages ofM. Levinson's extension of the Galerkin and Ritz methods. In particular M. Levinson's method, as shown by his treatment of Beck's problem, gives an equivalent variational problem only by prescribing an arbitrary restriction. The numerical results obtained byM. Levinson do not convincingly demonstrate the usefulness of his method, because the underlying algebraic equations happen to coincide with those which may be obtained by a different well-founded method.

Deflation methods which are used to eliminate dominant eigenvalues of a matrix in order to compute also non-dominant ones, in general destroy special forms of a matrix, such as the property $$a_{ik} = {\mathbf{ }}0{\mathbf{ }}for{\mathbf{ }}\left| {i - k} \right| > m$$ . (In some cases even the symmetry of the matrix is destroyed by deflation.) In the present paper, the author describes a deflation method which maintains the symmetry as well as the property (1).

The starting point of the present paper is a simple method ofPayne-Weinberger to obtain lower bounds for the first eigenvalue of a membrane through comparison with the Rayleigh principle for homogeneous strings. A generalization of this idea to auxiliary non-homogeneous strings leads to a maximum principle (7) (of which an inequality ofBarta-Polya is a special case) or (7′); in its form (8), it was essentially known (although with some unnecessary restrictive hypotheses) toM. H. Protter in 1958. — This principle is closely related to theThomson principle of boundary value problems; in particular, the allowed discontinuities of the concurrent vector fields (in the formulation (7′)) are the same as inThomson's principle.—Someapplications of the principle to the calculation of rather sharp lower bounds are indicated, as well as an intuitivephysical interpretation (in terms of variation of masses and constraints). It is valid inthree dimensions as well; we indicate its extension toSchrodinger's equation.

An initial value investigation into the development of two-dimensional surface waves generated by a harmonically oscillating pressure acting on the free surface of a fluid of limited, unlimited and very shallow depth is made in this paper in considerable detail. A rigorous mathematical treatment of the problem is given with the aid of integral transform technique in conjunction with asymptotic methods. The limiting behaviour of the asymptotic solution as timet tends to infinity has been given due attention. It is shown that the ultimate steady state is attained in the case of physically realisable pressure distributions. Finally, a discussion of the wave motions including the main predictions of this theory is presented in some detail. A careful consideration has also been given to examine the validity of Lighthill's prediction regarding the steady state.

A set of three integral equations for the components of the relative displacement vector are used to describe the flutter oscillations of an aircraft. Two methods for the approximate solution are presented, one making use of the eigenvectorfunctions and eigenvalues of the problem without airforces, the other beingRauscher's method of stationfunctions in a slightly generalised form. Then numerical methods for finding the zeros of the flutterdeterminants are given. The paper is concluded by a short description of actual numerical computations on automatic computing machines.

The paper investigates the non-stationary flows of a compressible frictionless medium, which are caused by the accelerated motion of rigid ducts with variable cross-sections. The method of characteristics for non-steady one-dimensional flows is adapted to the problem in the form of a strip method. Thereby a smoothly varying ‘duct function’ is replaced by a sequence of discrete impulsive accelerations of discontinuous changes of the cross-section. The discrete waves produced by the acceleration can be calculated either exactly by numerical methods, or in first and second approximation by means of given diagrams. Examples demonstrate that the intensity of the waves varies strongly with the Mach number in the relative system; the different behaviour of the waves going upstream and downstream is discussed.

The paper studies the steady plane flow of a non-viscous incompressible and perfectly conductive fluid limited by an insulatings wall. The flow, analysed under the hypothesis of the linearized theory, takes place in presence of a magnetic field arbitrarilys oriented with respect to the direction of the velocity of the fluid at infinity. The case of the aligned fields is developed taking into account the propagation of the perturbations at infinity upon the direction of the magnetic field. The mathematical problem is that of determining a piecewise holomorphic function, two of the actual connecting relations on the boundary being known.

The paper deals with the application ofGalerkin's method to non-selfadjoint problems of the elastic stability of bars. The author shows that for three special problems which have been treated formerly [1], the method of Galerkin converges, if one uses competing functions of the form\(y = \sum\limits_i {a_i y_i } \). Herea i are constants and they1;i=1,2... are the eigenfunctions of that selfadjoint problem, which one gets from the original problems by omitting the terms which cause the non-selfadjoint character.

The paper is based on the principle of maximal specific rate of dissipation work, which yields the theory of the plastic potential including the convexity of the yield surface. The author shows that in isotropic bodies the stress and the strain rate always have a common system of principal axes. He further demonstrates that, for soils with a sufficiently slender conical yield surface, the principal stress corresponding to a zero principal extension rate is always the intermediate one. It follows that a large class of yield surfaces are equivalent in plane flow.

This paper deals with the solution of two-dimensional variational problems of first order by the finite element method. The expressions for the basic functions in a triangular or rectangular element are given in a general form which is especially suitable for computation. These finite elements are used for plane elasticity problems, for which the basic functions ensure the continuity of the displacement field or of the displacement and stress fields. It is shown how the change of variables allows to carry out numerical integrations once for all; thus, the computing time depends very little on the choice of the basic functions. The different approximated solutions are numerically compared.

J. C. F. Francis proposes for the determination of eigenvalues of an arbitrary matrix theQR-transformation which is a unitary analogue of theLR-transformation. In this process the current approximation matrix is always factorized into a unitary and a triangular matrix and the 2 factors are multiplied together in reverse order; this yields the next approximant. The present paper shows how this process may be extended to a process which transforms a given matrix with quadratic convergence orthogonally toWintner-Murnaghan's normal form. The process is formulated in the real domain.

The paper is concerned with the dynamics of a thin circular plate made of a plastic-rigid material that obeys the yield condition ofTresca and the associated flow rule. The basic equations are established in the case of rotational symmetry, and their complete solution is given for a simply supported plate subjected to a uniformly distributed load which is brought on suddenly and, after a certain time interval, removed suddenly.

Numerical treatment of the integral in Cauchy's integral formula produces approximations for the derivatives of an analytic functionf; this fact has already been utilized byLyness andMoler [3, 4]. In the present paper this idea is investigated especially in view of the accuracy of these formulas regarded as quadrature formulas. Since the integration can be reduced to the integration of a periodic analytic function, it is possible to continue the considerations ofDavis [2] in order to find bounds for the error of the differentiation rules. For the application of these bounds one essentially needs estimations of the maximum off on a circle inside of its region of analyticity. Examples show the practical use of the bounds.

The initial boundary value problem of a thin profile in a compressible subsonic flow is investigated, being important for the gust problem. Existence and uniqueness of the solution are shown. By a generalizedWiener-Hopf-equation, the solution is reduced to the solution of a system of integro-differential equations for the downwashes in the profile plane. This system can be solved exactly by an iterative calculation of integrals for any value of timet>0. Finally, for smallt>0, the indicial-admittance function for vertical translation is found and is compared with a result given byLomax [6]. The asymptotic case for larget>0 will be investigated in a later paper.

The well-known regularization of Kepler motion given by Levi-Civita is very well suited for computing perturbations because the transformed equations are linear. RecentlyP. Kustaanheimo andE. Stiefel have generalized the 2-dimensional Levi-Civita-transformation to 3-dimensional motion and adapted for perturbation-purposes. The paper is an account on corresponding numerical experiments. It turns out that the regularized computation is in many cases (and in particular for orbits of high excentricity) superior to classical methods.

A method consisting of three conformal transformations is used for the numerical solution of a class of harmonic mixed boundary value problems containing boundary singularities. The method, although less flexible than the well-known finite-difference and finite element techniques, is very well suited to problems of this class. In the present paper the transformations are described in detail and the merits and shortcomings of the method are discussed objectively. © 1972 Birkhäuser-Verlag. 23 4 655 664 Cited By :42

The stability of a viscous flow between two co-axial circular cylinders, produced by the simultaneous effect of rotation of the interior cylinder and by a constant pressure gradient in peripherical direction, has been examined. (The two single problems had been studied first byTaylor [1] resp.Dean [2].) The perturbations investigated are vortices of the Taylor-Gortler-type and are assumed to be small. It therefore becomes possible to operate on the basis of the linear differential equations for perturbations. The present stability problem had been treated under the same assumptions byDiPrima [5] for small gap-distances by using the series method ofTaylor. In this paper, however, arbitrary gap-distances are admitted, and the integral equation method ofGortler [3] andHammerlin [4] has been applied. The calculated dependences of the critical Reynolds number upon the pressure gradient are in good qualitative agreement with the results ofDiPrima. However, our computations performed by more exact methods yielded somewhat different results with regard to the dependence of the critical vortex number upon the pressure gradient.

The paper describes a method for numerically solving an elliptic type differential equation connected with a diffusion problem with superimposed convection. The method consists in replacing the differential equation by a set of simultaneous difference equations which are solved exactly, thus avoiding difficulties inherent to the relaxation method. An example is treated in detail and the truncation error is discussed.

Various aspects of the diffraction of plane shock waves by moving antisymmetric thin bodies are examined. A method is presented for removing the inhomogeneity in the shock condition. This is done without knowledge of the explicit form of the solution, and is applicable to both symmetric and antisymmetric problems. With this reduction, explicit solutions have been previously obtained for symmetric bodies. For the antisymmetric case, the problem has been previously reduced to a complicated set of integrodifferential equations. In the cases where the flow field exhibits conical properties or when the body is slender, solutions may be obtained without recourse to solving the governing integral equations. In this paper it is shown that two-dimensional conical solutions are obtainable for moving antisymmetric wings only in the restricted case of the wing Mach numbers relative to the stream ahead of and behind the shock being the same. An analytical solution is presented for the above problem. In the case of slender bodies, it is shown that the classical steady slender body theory may be extended to shock diffraction problems and again analytical solutions are presented. It is shown that for general slender bodies, the lift and moment coefficients, and therefore the center of lift is influenced by the shock wave only through the Mach number change across the shock. Results are presented for the pressure distribution on the body, the lift and moment coefficients, and the center of lift for various slender body configurations at angle of attack.

The quotient-difference (=QD) algorithm developed by the author may be considered as an extension ofBernoulli's method for solving algebraic equations. WhereasBernoulli's method gives the dominant root as the limit of a sequence of quotientsq 1 () =s 1 (+1) /s 1 () formed from a certain numerical sequences 1 () , the QD-algorithm gives (under certain conditions) all the rootsλ σ as the limits of similiar quotient sequencesq σ () =s σ (+1) /s σ () . Close relationship exists between this method and the theory of continued fractions. In fact the QD-algorithm permits developing a function given in the form of a power series into a continued fraction in a remarkably simple manner. In this paper only the theoretical aspects of the method are discussed. Practical applications will be discussed later.

The paper provides a short survey of the actual state of the theoretical exterior ballistics. In particular the fundammental equations defining the motion of a rotating projectile are given in a form suitable for numerical calculation. Then, using the argument that a projectile possesses full rotational and also mirror summetry, the explicit development of the force and moment components is given in terms of the linear and angular velocities. Finally this development is compared with the experimentally known aerodynamic coefficients and stability derivatives.

In this paper we develop a multi-step method of order nine for obtaining an approximate solution of the initial value problemy'=f(x,y),y((x0)=y0. The present method makes use of the second derivatives, namely,\(y'' = \frac{{\partial f}}{{\partial x}} + f(x,y)\frac{{\partial f}}{{\partial y}}\) at the grid points. A sufficient criterion for the convergence of the iteration procedure is established. Analysis of the discretization error is performed. Various numerical examples are presented to demonstrate the practical usefulness of our integration method.

The purpose of the paper is to consider the stability for wavelike disturbances in the steady, twodimensional, laminar boundary layer of a magnetic field, which is applied uniformly normal to the flat plate. The results show that the critical Reynolds number (Rc*) increases remarkably with the characteristic parameter (δ). The increase of the critical Reynolds number depends not only on the shape parameter of the velocity distribution in the boundary layer but also on the peculiarity of the velocity profile. It is also found that the boundary layer holds itself laminar all over the flat plate, when the magnetic parameterN is greater than 1.25×10−7, then a reduction of the skin-frictin drag might be expeced.

The heating of cylinder, sphere and cube is computed, the starting and boundary conditions being specified for cases of technical importance. The solution of nonstationary heating problems (considering heat sources, and starting and boundary conditions of very general nature) can always be synthetized by superposition of a finite number of partial solutions, which can, in principle, be found in the same manner as those worked out in this paper.

The paper deals with a new method for transforming digital data tables into continuous shaft rotations. The method is characterized by a special type of parabolic interpolation incorporated into the analog part of the machine and conceived in a way to allow for perfect stability and smoothness of output Formulae are given for predetermining the errors of the system for interpolating any desired function.

In the present paper we wish to describe a Saint-Venant principle for a special class of boundary value problems for second order elliptic equations in two independent variables. While the context thus presented is much simpler than that encountered in most problems arising in elasticity, the methods developed here may be of some relevance in more involved situations. Moreover, our estimates are directly applicable to certain problems of physical interest, as we shall point out below; they may therefore be of some interest in themselves.

The author has recently shown thatv. Mises' theory of the plastic potential, if valid for the elements of a rigid-plastic body, also holds for the representation in generalized strains and stresses. This paper supplies the analogous proof for an elastic-plastic body. Here, however, the yield surface deforms, even in the case of perfect plasticity. Besides, it is impossible to determine the shape of the yield surface without considering the real state of stress. Thus, the simplification resulting from the use of generalized coordinates is less than in the case of a rigid-plastic body.

The paper deals with the convergence ofR. Grammel's method for the solution of boundary-eigenvalue-problems. It is further shown thatGrammel's method can be extended to the case where it is difficult to find the exactGreen function. In this case one may use a different, ‘related’Green function if one adds certain supplemental terms to the properGrammel equations.

The paper presents two torsion solutions, based on method of images, for an equilateral triangular prism: (i) when the prism contains a central longitudinal hole, and (ii) when the prism contains three equal longitudinal holes situated symmetrically in two different ways. The two solutions differ from each other in their formulation. Numerical examples are given in each solution.

Stability problems of rotating shafts are usually solved by linearization without investigating the influence of nonlinearity, and in most cases persistant disturbances are not taken into account. The present paper makes a contribution to these questions. For this purpose a shaft with one disk under different boundary conditions and periodic loading is considered. If the mass of the shaft is neglected a system with two degrees of freedom has to be treated. Therefore application of Liapunov's ideas and its generalizations is possible. Stability can be determined easily by linear approximation after introducing damping forces.

The solution of the multigroupP3-equations by Bessel-functions leads to difficulties as the number of particular integrals is gradually increasing. This paper treats the central zone of a cylindrically symmetric problem inP3-approximation using Taylor-series. The recursive relations are developed in order to calculate numerically the coefficients of the series.

The line integral obtained by the use of the Inversion Theorem of the Laplace Transformation is usually evaluated by applying the calculus of residues. This method can also be applied to find the behavior function of a regulation circuit from its Laplace Transform, if neither the regulator nor the process has a dead time. If there is a dead time in the regulation circuit, the Laplace Transform of the behavior function has an infinite number of poles. Therefore it is very difficult to evaluate exactly the inversion integral, and it is only possible if the arising infinite sum is convergent. In the present paper it is shown that in this case we nevertheless can apply the calculus of residues, because we thus obtain a good approximation applying it only to a finite number of poles. Thus the sum of the residues of a finite number of poles is proved to be a function approximating asymptoticly the exact function for large values of the time. But it must be assumed that always all the poles which are on the right side of a parallel to the imaginary axis are used.

The spatial decay of solutions of parabolic partial differential equations has been the subject of two recent papers by Edelstein [1] and Sigillito [2]. The results obtained by these authors are in some respects analogous to estimates derived in connection with Saint-Venant's principle in the linear theory of elasticity [3], [4].

In [3], paragraph 6, the author showed how to find the zeros of a polynomial with the aid of the QD-Algorithm. The method required a certain rational function to be developed into a continued fraction of Stieltjes type to give the starting values for the QD-Algorithm. In the present paper the author shows how starting values can be obtainedwithout computing a continued fraction.